$t0.50
STATISTICS
IN RESEARCH
Which statistical tool shoaLl I use?
How should I use if?
How can I interpret my results?
SUCH QUESTIONS plague every engineer,
scientist, research worker, student, and
teacher, and they will tmd this book an
indispensable reference which contains
the answers to these and many other
questions.
Modern techniques arc presented in
ways conducive to easy leaning and
application. The most difficult part of
statistical method is learning when and
where to apply a particular technique.
Statistics in Research carefully indicates
the assumptions underlying ruii tech
nique so that it will be applied properly
within its limitations,
Especially noteworthy features in
clude;
* emphasis on theory for a thorough
undorstaiuling of fundamentals
* a comprehensive coveuv^ of re
gression analysis
* useful check-lists on the various
aspects of experimental design
* presentation of rigorously de
veloped procedures in the form
best suited for computation
* inclusion of a number of non-para
metric techniques in recognition
of the growing importance of this
area in research
* a separate chapter on quality con.-
trol
* an excellent collection of impor
tant and useful tables with stand
ardized formats
* statistical inference (both estima
tion and testing hypotheses) well
organised, thoroughly discussed
(Continued on beck flap)
OCT 8 1975
NOY 4
jMAj; MAR 2 9 1976
JUN 1 1976
2 1976
7 - T976
SEP 1 1 1978
!MAI JUL 14 1992"
If III f j
28 0651
519-9 085s2
Ostle
Statistics in research
63-09055
519.9 085s2
Ostle $10.50
Statistics in research
63-09055
M|IJ..I.. , ,
STATISTICS IN RESEARCH
i .
-1"
r
SECOND EDITION
STATISTICS IN RESEARCH
BASIC CONCEPTS AND TECHNIQUES FOR RESEARCH WORKERS
BERNARD OSTLE
PROFESSOR OF ENGINEERING, ARIZONA STATE UNIVERSITY
THE IOWA STATE UNIVERSITY PRESS
_xVm#4, IOWA, U.S.A.
About the author . . .
BERNARD OSTLES, Professor of ^Engineering at Arizona State University, includes
operations research, quality control, reliability, the statistical design of experi
ments, and the statistical analysis of data in his principal areas of teaching
and research. He also serves as special consultant for several industrial firms-
EJarlier, Dr. Gstle was Supervisor of the Statistics Section in the Reliability
Department of Sandia Corporation, Albuquerque, New Mexico. From 1952 to
J957 he was Profossor of Mathematics, Agricultural Experiment Station Statisti
cian, and Director of the Statistical Laboratory at Montana State College.
Prior to that he taught statistics at Iowa State University and the University of
Minnesota. In addition, he served as Special Lecturer in Statistics at the Uni
versity of New Mexico, 1958-60, and as Special Lecturer in Statistics for the
National Science Foundation Summer Institute at Oklahoma State University
in 1959.
He received his B.A. degree from the University of British Columbia in 1945
with honors in mathematics, and the M.A. degree In economics from the same
institution in 194(5. He wan awarded the Ph.D. degree in statistics at Iowa State
University iix 1949. He IB a member of the American Society for Quality Control,
the American Statistical Association, the Operations Tlosearch Society of
America, Alpha Pi Mu, Phi Kappa Phi, Pi Mu Kpsilon, and Sigma Xi. Dr. Ostle
alno is the author of numerous artiolos in various scientific journals.
1963 The Iowa State University Press.
All rights rofiorvod.
Manufactured in U.S.A.
Library of Congress catalog card number: 03-7548.
To
RUTH JEAN
KANSAS cny cm.) puer re
I O , , O 630905
THE PAST FEW YEARS have seen many changes in the science of statis
tics. New techniques of analysis and inference have been developed
by mathematical statisticians while applied statisticians have been
busily engaged in applying both old and new techniques in novel
situations. These facts, plus the tendency toward an increased use of
mathematics in research, indicated that a revision of this text was
needed.
This revised edition has been prepared with the following goals in
view: (1) to provide a book giving those statistical methods that have
been found useful by workers in many areas of scientific research, (2)
to present these methods as integral parts of a complete discipline, and
(3) to provide a textbook that will facilitate teaching the science of
statistics.
In attempting to achieve the above goals, I have taken the position
that it is possible to write a book which will prove acceptable to stu
dents, teachers, and research workers in many fields of specialization.
Consequently, this book presents the techniques of modern statistics as
statistical methods per se* To demonstrate the universality of statistical
methods, many examples from varied fields of application ai*e included.
In this book considerable attention has been given to the assumptions
underlying the techniques presented. Withoxit a thorough understand
ing of the limitations of various techniques, one might apply them in
situations where they should not be used. The learning of methods is
easy; learning when and where to use them is not so easy. I have at
tempted to achieve a reasonable balance between these two ends.
This edition has been designed in such a manner that it should prove
xiseful for several purposes, namely: (1) as a text for a standard course
in statistical methods, (2) as a text for an integrated course in theory
and methods for students in engineering and the physical sciences, and
(3) as a reference book for research workers and other users of statistical
methods, whether they be affiliated with government, industry, re
search institutes, or universities.
Because of the multiple-purpose design of the book, some topics will
be of interest only to special groups. In addition, changes in order of
presentation by individual teachers will also occur. However, it is my
belief that a reasonable compromise among descriptive statistics,
mathematical statistics, statistical methods, and the design and analy
sis of experiments has been achieved, and that the book will prove
suitable for all the purposes for which it was planned.
I am indebted to Sir Ronald A. Fisher, to Dr. Frank Yates, and
to Oliver and Boyd Ltd., Edinburgh, for permission, to reprint Table III
from Statistical Tables for Biological, Agricultural and Medical Research.
I am also indebted to Dr. O. L, Davies and to Oliver and Boyd Ltd.,
CvIII
viii PREFACE
Edinburgh, for permission to reprint Table 6.G1 from, the second
edition of Statistical Methods in Research and Production, and Tables
7.7, 7.72, E, E.I, G, and H from the second edition of The Design and
Analysis of Industrial Experiments. Many other persons have also
graciously given permission for the reproduction of published material,
and acknowledgment has been made at the appropriate places in the
text. The author is deeply appreciative and wishes to express his
thanks for their cooperation.
Acknowledgment is due to Paul G. Homeyer, David V. Huntsberger,
Bmil H. Jebe, Oscar Kempthorne, and George W. Snedecor for their
encouragement during the preparation of the first edition. To my
former co-workers at Sandia Corporation, Albuquerque, New Mexico,
and in particular to John M. Wiesen, I also wish to express my appreci
ation. The suggestions which resulted from our many stimulating con
versations contributed greatly to the improvement of this new edition.
My greatest personal indebtedness is to my wife, Ruth Jean Ostle,
without whose help this revision would still be far from complete. In
particular, I wish to thank her publicly for her diligence in typing the
manuscript, for her editorial assistance, and for her tmfailing patience
and understanding during the entire project.
BERNABB OST&E
Tempe, Arizona
TABLE OF CONTENTS
1. THE ROLE OF STATISTICS IN RESEARCH
1.1 The Nature and Purpose of Research 1
1.2 Research and Scientific Method 2
1.3 What Is Statistics? 2
1.4 Statistics and Research 3
1.5 Further Remarks on Science, Scientific Method, and Statistics. . 3
1.6 Applications of Statistics in Research 5
1.7 Summary 12
Problems 14
References and Further Reading 14
2. MATHEMATICAL CONCEPTS
Set Theory 17
Notation 18
Permutations and Combinations 20
Some Useful Identities and Series 20
Some Important Functions 21
Matrices , 22
Linear Equations 24
Problems 25
References and Further Reading 28
3. A SUMMARY OF BASIC THEORY IN PROBABILITY AND STATISTICS
Probability 29
Mathematical Expectation 33
Probability Distributions 33
Expected Values 35
Other Descriptive Measures 36
Special Probability Distributions 37
Problems 37
References and Further Reading 43
4. ELEMENTS OF SAMPLING AND DESCRIPTIVE STATISTICS
4.1 The Population and the Sample 44
4.2 Types of Samples 45
4.3 Sampling From a Specified Population 46
4.4 Presentation of Data , 47
4.5 Calculation of Sample Statistics 52
4.6 The Arithmetic Mean 53
4.7 The Midrange 55
4.8 The Median 55
4.9 Percentile, Decile, and Quartile Limits 56
4.10 The Mode 58
4.11 The Hange 60
4.12 The Standard Deviation and Variance 60
4.13 The Coefficient of Variation 64
4.14 Summary 65
Problems 66
Uxl
x CONTENTS
5. SAMPLING DISTRIBUTIONS
5.1 Sample Moments 70
5.2 Variance of the Sample Moan 70
5.3 TchebyohefTPs Inequality 71
5.4 Law of Large Numbers 72
5.5 Central Limit Theorem 72
5.6 Random Sampling From a Specified Population 73
5.7 The Hypergeomotrie Distribution , 73
5.8 The Binomial Distribution 74
5.9 Binomial Approximation to the Hypergeo metric 74
5.10 Poisson Approximation to the Binomial 75
5.11 Normal Approximation to the Binomial 76
5.12 The Multinomial Distribution 78
5.13 The Negative Binomial Distribution and the Geometric Dis
tribution 79
5.14 Distribution, of a. Linear Combination of Normally Distributed
Variables 80
5.15 Distribution of the Sample Moan for Normal Populations 80
5,1(5 Distribution of the Difference of Two Sample Moan** 81
5.17 Chi-Square Distribution 81
5.18 Distribution of the Sum of Squares of Independent Standard Nor
mal Variates . 82
5.19 Distributions of the Sample Variance ami Standard Deviation for
Normal Population** , 82
5.20 Distribution of "StxulontV t 83
5.21 Distribution of F . . , . 83
5.22 Order Statistics 84
Problems 85
References and Further Rending 86
6. STATISTICAL INFERENCE; ESTIMATION
6,1 Some Preliminary Idean 87
l 0.2 Methods of Obtaining Point Kntimatorw 88
6.3 Maximum Likelihood Katimutorw 89
6.4 Confidence Intervals: Crtmenil DIHCUHHIOII , 89
(5.5 Confidence Interval for the Mean of a Normal Population , , . . . 00
6.6 Confidence Interval for the Moan of a Nonnormal Population . . 92
6.7 Confidence* Interval for the Variance of a Normal Population. , . 03
6.8 Confidence) Interval for p, the Parameter of a Binomial Population 94
6.0 Confidence Interval for the Difference Between the Means* of Two
Normal Populations , , , . , 95
6.10 Cofiderico Interval for the Ratio of th& Variances of Two Normal
Populations , 97
6.11 Tolerance Limitn: General DuicuaHum , ...,,., 98
6.12 Tolerance Limits (Two-Bided; One-Sided) for Normal Popula
tions ..,.,.....,. ..,,>,... 98
6.13 Distribution- Free Tolerance* LimitH . . 100
Prohlemn . . , 101
Reference** and Further Heading , . , 105
7. STATISTICAL INFERENCE: TESTING HYPOTHESES
7.1 < Jeneral OonHidorntumH ,,,,.,,,... 107
7.2 KHtnhliHhment of Tewt Procedure**. , , * 111
CONTENTS xi
7.3 Normal Population; H'.jm juiQ Versus A:/A^pt ................. 113
7.4 Normal Population; ^IM^/XO Versus A:A*>MOJ or H:JJL>JULO Versus
A : M <MO ................................................. 113
7.5 Normal Population; H:<r 2 =<rl Versus A:a- 2 ^o-Q ................ 114
7.6 Normal Population; H:<7 2 <o-o Versus A:<T*>CT*, or Hi<r z ><rl
Versus A :o- 2 <o-Q .......................................... 115
7.7 Binomial Population; H:p~p Q Versus A:p^p Q ............... 115
7.8 Binomial Population; H:p<.p Versus A:p>p Q , or ff-p>po
Versus A :p <p ........................................... 118
7.9 Two Normal Populations; jT : /-n = >U2 Versus A'.J^I^JJ.^ .......... 119
7.10 Two Normal Populations; T:/zi< M2 Versus A:/xi>M2, or H :jun >^t 2
Versus A :/xi <At 2 .......................................... 122
7.11 Two Normal Populations, H i&l =<T% Versus A '.a\^<r\ .......... 123
7.12 Two Normal Populations; H'.a\ <cr~ Versus A iv\ ><r%, or H:crl>cr^
Versus Ai<r\<<ri .......................................... 123
7.13 Multinomial Data ......................................... 124
7.14 Poisson Data ............................................. 124
7.15 Chi-Square Test of Goodness of Fit ......................... 126
7.16 Binomial Population; More Than One Sample ................ 128
7.17 Contingency Tables ....................................... 129
7.18 Special Approximate Methods for 2X2 Tables ................ 131
7.19 The Exact Method for 2 X 2 Tables .......................... 132
7.20 Several Normal Populations; H : MI = ju 2 = - =M fc ............ 133
7.21 Several Normal Populations; ff:crl~<r%~ - =cr ............ 136
7.22 Sample Size .............................................. 136
7.23 Sequential Tests .......................................... 140
Problems ................................................ 143
References and Further Reading ............................ 157
8. REGRESSION ANALYSIS
8.1 Functional Relations Among Variables ....................... 159
8.2 A Word of Caution About Fimctional Relations .............. 160
8.3 The Choice of a Functional Relation ........................ 160
8.4 Curve Fitting ............................................ 160
8.5 The Method of Least Squares ............................... 161
8.6 Graphical Interpretation of the Method of Least Squares ...... 162
8.7 Simple, Linear Regression ............ . ..................... 164
8.8 Partitioning the, Sum of Squares of tho Dependent Variable, .... 164
8.9 A Practical .Example ...................................... 167
8.10 AwHumptionH Necessary for JBJstimation and Testing Hypotheses in
Simple Linear Regression .................................. 168
8.11 K&timates of Krror Associated With Simple Linear degression
Analyses ..... . ........................................... 170
8.12 Confidence and Prediction Intervals In Simple Linear Regression 170
8.13 Tests of Hypotheses in Simple Linear Regression .............. 174
8.14 Inverse Prediction in Simple Linear Regression. , . . . . ......... 176
8.35 The Abbreviated Doolittle Method. ... ...................... 177
8.16 Some Additional Remarks With Regard to Generalized Regression
Analyses ................................................. 186
8.17 Teats for Lack of Fit ...................................... 188
8.18 Nonlinear Models ......................................... 190
8.19 Second Order Models ...................................... 191
8-20 Orthogonal Polynomials ................................... 192
xii CONTENTS
8.21 Simple Exponential Regression 194
8.22 The Special Case: 77 =.X" 196
8.23 Weighted Regressions 197
8.24 Sampling From a Bivariate Normal Popxilation 198
8.25 Adjusted F Values 199
8.26 The Problem of Several Samples or Groups 201
8.27 Some Uses of Regression Analysis 205
Problems 206
References and Further Reading 221
9. CORRELATION ANALYSIS
9.1 Measures of Association , 222
9.2 An Intuitive Approach to Correlation 222
9.3 The Correlation Index 223
9.4 Correlation in Simple Linear Regression 223
9.5 Sampling From a Bivariate Normal Population 225
9.6 Correlation in Multiple Linear Regression 227
9.7 The Correlation Ratio 229
9.8 Biaerial Correlation 231
9.9 Tetrachoric Correlation 232
9.10 Coefficient of Contingency , . .* , 232
9.11 Rank Correlation 233
9.12 IntraehusH Correlation 235
9.13 Correlations of Sums and Differences ,....,... 238
Problems , 239
References and Further Reading , 243
10. DESIGN OF BXPERIMBNTAL INVESTIGATIONS
10 1 Some General Remarks , 244
10.2 What IB Moant by "The Design of an Experiment"? , . 244
10.3 The Need for an Experimental Dcmgn, 244
10.4 The Purpose of an Kxporimental Design 245
10*5 Battle Principles of Experimental Demgn , . . , . 246
10.6 Replication 246
10.7 Experimental Krror and Experimental Units 247
10.8 Confounding , , 248
10*9 Randomisation *.*,*. ......,,,.,,. 249
10.10 Local Control . . . 250
10*11 Balancing, Blocking, and Grouping. I.,......,............*.. 251
10,12 Treatments and Treatment Combinations * . , , 4 . 252
10. IS Factors, Factor levels, and Factorials. t , . . 253
10,14 Effects and Interactions ..,.... 256
10*35 Treatment Comparisons. k * . , * 261
10.10 Htepa in Designing an Kxperimcmt. . * . . - * . 264
10.17 IlhitftrutioziH of the Statistician's Approach to !>emgn Problem**. , 266
10.18 Advantagen and Disadvantages of Statistically Designed Kxperi-
mont.H , . , * . , 27 1
10.19 Hummary , , 273
Problomn ,.,,., 273
eeH and Further Reading. , , . , . . , 275
CONTENTS xiii
11. COMPLETELY RANDOMIZED DESIGN
11.1 Definition of a Completely Randomized Design 278
11.2 Completely Randomized Design With One Observation per
Experimental Unit 279
11.3 The Relation Between a Completely Randomized Design and
"Student's" Z-Test of JET: AXIOMS Versus A: v\^m 288
11.4 Subsampling in a Completely Randomized Design 288
11.5 Expected Mean Squares, Components of Variance, Variances of
Treatment Means, and Relative Efficiency 298
11.6 Some Remarks Concerning F-Ratios That Are Less Than Unity . . 301
11.7 Satterthwaite's Approximate Test Procedure 302
11.8 Selected Treatment Comparisons: General Discussion 303
11.9 Selected Treatment Comparisons: Orthogonal and TSTonorthogonal
Contrasts 306
11.10 All Possible Comparisons Among Treatment Means 310
11.11 Response Curves: A Regression Analysis of Treatment Means
When the Various Treatments Are Different Levels of One
Quantitative Factor 312
11.12 Analysis of a Completely Randomized Design Involving Factorial
Treatment Combinations 316
11.13 Nonconformity to Assumed Statistical Models 338
11.14 The Relation Between Analysis of Variance and Regression
Analysis 340
11.15 Presentation of Results 341
Problems 344
References and Further Reading 360
12. RANDOMIZED COMPLETE BLOCK DESIGN
12.1 Definition of a Randomized Complete Block Design 363
12.2 Randomized Complete Block Design With One Observation per
Experimental Unit 364
12.3 The Relation Between a Randomized Complete Block Design
and "Student's" if-Test of H : && ** When Paired Observations Are
Available 368
12.4 Subaampling in a Randomized Complete Block Design 368
12.5 Preliminary Testa of Significance 371
12.6 legitimation, of Components of Variance and Relative Efficiency . 373
12.7 Efficiency of a Randomized Complete Block Design Relative to a
Completely Randomized Design 374
12.8 Selected Treatment Comparisons 376
12.9 Subdivision of the Experimental Error Sxim. of Squares When
Considering Selected Treatment Comparisons 376
12.10 All Possible Comparisons Among Treatment Means 380
12.11 Response Curves in a Randomized Complete Block Design. .... 380
12.12 Factorial Treatment Combinations in a Randomized Complete
Block Design 380
12.13 Missing Data in a Randomized Complete Block Design. . 390
Problems 394
References and Further Reading 408
13. OTHER DESIGNS
13,1 Latin arid Graeco-Latirj. Squares 410
xiv CONTENTS
13.2 Split Plots 415
13.3 Complete Factorials Without Replication, Fractional Factorials,
and Incomplete Blocks 417
13.4 Unequal but Proportionate Subclass Nximbers 421
13.5 Unequal arid Disproportionate Subclass Numbers 423
13.6 Response Surface Techniques 424
13.7 Random Balance 425
13.8 Other Designs and Techniques 425
Problems 426
References and Further Reading 434
14, ANALYSIS OF COVARIANCE
14.1 Uses of Oovnrianec Analysis. 437
14.2 Assumptions Underlying Analyses of Oovarianee 43S
14.3 Completely Randomized Design 439
14.4 Randomized Complete Block Design 444
14.5 Latin Square Design 449
14.6 Two-Factor Factorial in a Randomized Complete Block Design . . 452
14.7 Covariaixce When the X Variable Is AfTeeted by the Treatments . . 456
14.8 Multiple Co variance , , 457
Problems , 460
References and Further Rending , 465
15. DISTRIBtJTI ON-FREE METHODS
35.1 DiHtributitm-Froe Methodw Included in Previous Chapters 466
15.2 The Sign Tout - 466
15.3 The Signed Rank Tost 468
15.4 The Rxm Tewt , 470
15.5 The Kolmogorov-Smirnov Tent of Goodno.wfl of Fit 471
15.6 Median Testa 473
Problems , , 473
References and Further Reading 474
16. STATISTICAL QUALITY CONTROL
16.1 Control Charts 477
16.2 Acceptance Sampling Plans , , , . , 4<S5
Problems . . 49 1
Roferouees und Further Reading. ,...,,...,... 408
17, SOME OTHE& TECHNIQUES AND APPLICATIONS
17.1 Some Pseudo t Statistics ,.,..,.,,,..,,, 500
17.2 A PHoudo F Statistic . . , , , 501
J 7.3 Kvolutionary Operation. * .,.,.. 501
1 7.4 TolerancoH * 502
17.5 The Entimatum of 8ytem Reliability, ...,.,..., 506
Problems . , 508
Reference and Further Heading* . , , .....,, 510
APPENDIX
I . ( Jreek Alphabet . , , , , , . . . 511
iJ, Cumulative PoiHwon Dmtributicm 512
3. CUimulativ< Standard Normal Difttrihution. , . ...,..,, , . , , , 517
CONTENTS xv
4. Cumulative Chi-Square Distribution 523
5. Cumulative ^-Distribution 528
6. Cumulative F- Distribution 529
7. Random Numbers 544
S. Control Chart Constants 548
9. Number of Observations for -Test of Mean 550
10. Number of Observations for -Test of Difference Between Two
Means 552
11. Number of Observations Required for the Comparison of a Popula
tion Variance With a Standard Value Using the % 2 -Test 554
12. Number of Observations Required for the Comparison of Two
Population Variances Using the F-Test, 555
13. Critical Values of r for the Sign Test 556
14. Table of Critical Values of T in the Wilcoxon Signed Rank Test .... 557
15. Table of Critical Values of r in the Run Test 558
10. Table of Critical Values of D in the Kolmogorov-Smirnov Goodness
of Fit Test 560
17. Percentage Points of Pseudo t and F Statistics 561
INDEX 565
CH APTE R 1
THE ROLE OF STATISTICS
IN RESEARCH
EVERY DAY each of us engages in some observation in which statistics
is used. Such common, events as noting the weather forecast, weighing
oneself, checking the position of a favorite ball team in its league, or
testing a new food product are typical. The element of statistics creeps
in when you mentally evaluate your research. In weighing yourself, you
automatically compare your observation with your average weight
(deviation from the mean) and conclude the present weight is usual
(no significance to the difference) or unusual (a significant difference) ,
basing your judgment upon previous measurements of your weight and
your knowledge of the variation generally observed. These common
results are easily obtained, are of only local importance, and are soon
forgotten. However, the formal research which means so much to
improving man's lot is of infinitely greater importance and must be
conducted with much greater care. It is with the latter type of research
that this book is concerned.
1.1 THE NATURE AND PURPOSE OF RESEARCH
Research, according to Webster, is studious inquiry or examination
critical and exhaustive investigation or experimentation having for its
aim the discovery of new facts and their correct interpretation. It also
aims at revising accepted conclusions, theories, or laws in the light of
newly discovered facts or the practical applications of such new or
revised conclusions. Research, therefore, means continued search for
knowledge and understanding; scientific research is continued research
using scientific methods. Scientific research is essentially compounded
of two elements: observation, by which knowledge of certain facts is
obtained through sense-perception; and reasoning, by which the mean
ing of these facts, their interrelation, and their relation to the existing
body of knowledge are ascertained insofar as the present state of
knowledge and the investigator's ability permit.
In any discussion of research, two important facts should be noted.
They are : (1) there is an ever increasing trend towards extreme speciali
zation on the part of individual scientists, and (2) most research prob
lems are such that many disciplines and fields of specialization can
contribute in a significant manner to their solutions. Thus, it is evident
that more and more research will be handled on an interdisciplinary
team basis rather than by individual scientists working in "solitary
confinement/' (NOTE: This is not to say that very little individual
til
2 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
research will continue to be performed. Sucli research always should
and always will be performed. The statement was meant to imply
only that research is becoming predominantly a team or cooperative
effort.)
In summary, research is an inquiry into the nature of, the reasons for,
and the consequences of any particular set of circumstances- whether
these circumstances are experimentally cozitrolled or recorded just as
they occur. Further, research implies the researcher is interested in
more than particular results he is interested in the repeatability of
the results and in their extension to more complicated and general
situations.
1.2 RESEARCH AND SCIENTIFIC METHOD
Although the techniques of investigation may vary considerably
from one science to another, the philosophy common to all is generally
referred to as scientific method. There are, perhaps, as many definitions
of scientific method as there are workers in research. For our purposes,
the following will be used: Since the ideal of science is to achieve a systcm-
atic interrelation of facts } scientific method must be a pursuit of tkzs ideal
by experimentation, observation, logical arguments from accepted postu
lates r and a combination of these three in varying proportions. Therefore,
research and scientific method are closely related, if not one and the
same thing.
1.3 WHAT IS STATISTICS?
Statistics has often been classified as a method of research along
with, or in opposition to, .such methods as case studios, the historical
approach, and the experimental method. Since this classification fre
quently leads to confused and incorrect thinking, it is not wise. It is
better to regard statistics as supplying a kit of tools which can be
extremely valuable in research. This book will stress gaining an tinder-
standing of these tools and learning which tool should be xised in vari
ous situations arising in scientific research. Only when you know which
tool to xiso, how to use it, and how to interpret your results can you
hope to do productive research. To summarise: the science of statistics
has mxieh to offer the research worker in planning, analyzing, and inter
preting the results of his investigations, and tliis book is devoted to
an exposition of those methods and techniques that have proved useful
in many fields of inquiry,
As is the case with many words in the English language, the word
statistics is used in a variety of ways, each correct in its own sphere.
In the plural sense, it is usually taken to be synonymous with data.
However, to the statistician, there is another meaning of the* word.
This moaning is the plural of the word statistic, which refers to a quan
tity calculated from sample observations* (These terms will be defined
in considerable detail in later chapters.) In the singular sense, statistics
is a science, and it is in this sense the word will be employed most fro-
quently in this book. The science of statistics deals with:
(1) Collecting and summarizing data.
(2) Designing experiments and surveys.
(3) Measuring the magnitude of variation in both experimental
and survey data.
(4) Estimating population parameters and providing various
measures of the accuracy and precision of these estimates.
(5) Testing hypotheses about populations.
(6) Studying relationships among two or more variables.
1.4 STATISTICS AND RESEARCH
As indicated in the preceding section, statistics enters into research
and/or scientific method through experimentation and observation.
That is, experimental and survey investigations are integral parts of
scientific method, and these procedures invariably lead to the use of
statistical techniques. Since statistics, when properly used, makes for
more efficient research, it is recommended that all researchers become
familiar with the basic concepts and techniques of this useful science.
Because statistics is such a valuable tool for the researcher, it some
times gets overworked. That is, there are many cases where statistics
is used as a crutch for poorly conceived and/or executed research. In
addition, there are cases in which statistics is employed in good faith
but, unfortunately, insufficient attention is paid to the assumptions
required for a valid use of the methods employed. For these and other
reasons, it is essential that the user of statistics clearly understands the
techniques he employs. Consequently, in this book careful attention
will be given to both the methods and the underlying assumptions in
the hope that such an approach will lead to the proper application and
use of statistics in scientific research.
1.5 FURTHER REMARKS ON SCIENCE, SCIENTIFIC
METHOD, AND STATISTICS
In the preceding sections, your attention has been called to the close
connection that statistics has with experimentation, scientific method,
and research. However, in each case, the discussion was quite brief.
Because these various topics and their interrelationships are so impor
tant to the remainder of this book, a few additional remarks are justi
fied. To expedite the discussion, the following questions and answers
have deon devised:
What Is Logic?
Logic deals with the relation of implication among propositions, that
is, the relation betweeii premises and conclusions. In scientific method,
logic aids in formulating our propositions explicitly and accurately so
that their possible alternatives become clear. When faced with alterna
tive hypotheses, logic develops their consequences so that when these
4 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
consequences are compared with observable phenomena we have a
means of testing which hypotheses are to be eliminated and which
one is most in harmony with the observed facts.
What Is Science?
Science is knowledge which is general and systematic knowledge
from which specific propositions are deduced in accordance with a few
general principles. Although all the sciences differ, a universal feature
is "scientific method/' which consists of searching for general laws
which govern behavior and of asking such questions as: Is it so? To
what extent is it so? Why is it so? What general conditions or consider
ations determine it to be so?
What Is Scientific Method?
Scientific method is the pursuit of truth as determined by logical
considerations. The ideal of science is to achieve a systematic interrela
tion of facts; scientific method, using the approach of "systematic
doubt/' attempts to discover what the facts really are.
What Is Experimentation?
The function of experimentation is the elimination of untenable
theories* Experimentation is used to test hypotheses and to discover
now relationships among variables. It must be remembered, however,
that no hypothesis which states a general proposition can be demon
strated to be absolutely true; only probable inferences are possible.
What Part Does Experimentation Play in Scientific Method?
Experimentation is only a means toward an end. It is a tool of scien
tific method. Conclusions drawn from experimental data are frequently
criticized. Such criticisms are usually based on one or more of the
following arguments: (1) the interpretation ia faulty, (2) the original
aasumptiorm are faxilty, or (3) the experiment waa poorly designed or
badly executed. Obviously, careful attention should be given to the
design of the experiment HO that the procedures used are both valid and
efficient.
What Is Experimental Design?
Experimental design is the plan u^ed in experimentation. It involves
the assignment of treatments to the experimental units and a thorough
underst a nding of the analysis to be performed when the data become
available,,
What Is the Relationship Between Statistics and Experi
mental Design?
vStatintics enters into experimental design because, even in the best
planned experiments, one cannot control all the factors and because
1.6 APPLICATIONS OF STATISTICS IN RESEARCH 5
one wishes to make inferences based on the observed sample data. To
be of any practical use, these uncertain inferences must be accompanied
by probability statements expressing the degree of confidence which
the researcher has in such inferences. To make certain that such prob
ability statements will be possible, the experiments should be designed
in accordance with the principles of the science of statistics.
1.6 APPLICATIONS OF STATISTICS IN RESEARCH
Early applications of statistics were mainly concerned with reduc
tion of large amounts of observed data to the point where general
trends (if they existed) became apparent. At the same time, emphasis
in many sciences turned from the study of individuals to the study of
the behavior of aggregates of individuals. Statistical methods were ad
mirably suited to such studies, aggregate data fitting consistently with
the concept of a population.
The next major development in statistics arose to meet the need for
improved analytical tools in the agricultural and biological sciences.
Better analytical tools were needed to improve the process of interpre
tation of, and generalization from, sample data. For example, the
farmer is faced with the task of maintaining a high level of produc
tivity of field crops. To aid him, the agronomist conducts an endless
number of experiments to determine differences among yields of various
crop varieties, effects of various fertilizers, and the best methods of
cultivation. On the basis of the results of his experiments, he is ex
pected to make accurate and useful recommendations to the farm
operator. Clearly then, statistics, being a science of inductive inference
using probabilistic methods, should be of great value to the researcher
in agronomy.
In early agronomic experimentation, in order to compare a number
of fertilizers, it was thought necessary to devote only a single plot to
each treatment and determine yields in order to arrive at valid con
clusions concerning relative values of the treatments. However, the
agronomists soon found that the yields of a series of plots treated alike
differed greatly among themselves, even when soil conditions appeared
uniform and experimental conditions were carefully designed to reduce
errors in harvesting. For this reason, it became necessary to find some
mearivS for determining whether differences in yields were due to dif
ferences in treatments or to uncontrollable factors which also con-
tribxite to the variability of plot yields. Statistical methods were ap
plied, and their value in scientific investigation of agronomic practices
was soon proved.
Closely related to agronomy is the science of plant breeding. The
ultimate objective of any plant-breeding research program is the de
velopment of improved varieties or hybrids. A variety may be im
proved in many possible ways, e.g., in ability to use plant nutrients,
in disease or insect resistance, in cold tolerance, or in its suitability to
the needs or fancies of the grower and/or consumer. Plants are organ-
6 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
isms conditioned by genetic factors and by the environment in which
they grow. The plant breeder, therefore, utilizes the principles of genet
ics in attempting to improve inheritable characteristics of plant va
rieties, just as the producer attempts to obtain high production by
maintaining a favorable environment. However, results of past genetic
studies do not provide all the answers relative to the inheritance of
plant characteristics. Thus, plant breeders continually carry out basic
genetic research in each crop along with practical plant-breeding pro
cedures in order to ensure future progress.
Development of a superior new variety by hybridization is seldom
a haphazard occurrence. Usually the breeder has in mind the charac
teristics desired for his particular purpose or area. Growing many plant
selections to decide which excel in a quantitatively inherited character
requires growing thorn in a randomised, replicated field design. Choice
of design depends on the numbers involved; the uniformity of the soil;
the accxiracy and precision of the particular estimates deemed neces
sary to get the desired results; the time, effort, and money available;
and perhaps other factors. The data collected are then analyzed in
accordance with the plan of the experiment, which was designed to
make possible proper comparisons among the strains being tested. The
statistical methods employed must, of course, have a logical relation
ship to the biological processes under consideration, as well as to the
way in which the experiment was conducted, if they are to be useful.
After the data have been analyzed statistically, the results must be
interpreted in view of the assumptions made and of the existing
knowledge so that some conclusion may be reached with regard to
accepting or rejecting the hypotheses being tested. Selection of the
strain to be released as a variety, or of those to be tested further,
may then bo made with assxirance that the decision will, in all likeli
hood, bo a reasonable one.
Other research areas in which good use of statistical theories and
methods is made are poultry breeding, animal breeding, and animal
nutrition* Poultry brooding, for example, is concerned with the raising
of more efficient and more productive fowl. Increased egg production,
egg sixe, egg color, interior egg quality, more efficient meat production,
long life, disease resistance, and high fertility are some of the factors
with which the poultry breeder IH actively concerned. If a statistically
sound research program is adopted, the researcher will be able to reach
defensible conclusions and bring about more efficient use of resources.
One of the more important uses of statistics in breeding work is the
separation of environmental and hereditary effects* The literature of
the field i# full of reports dealing with this type of research, both with
poultry and domestic animals. For the reader interested in this par
ticular urea of research, we refer to such writers as Ilutt (25) and
Lush (29), 1
1 Numborn in pnr<*nfrhonoH designate rofarfmoow linted at mid of chapter.
1.6 APPLICATIONS OF STATISTICS IN RESEARCH 7
In the field of animal nutrition, many experiments have been devised
to discover the significance of various vitamins in the different phases
of animal production. In such investigations, several groups of animals,
as homogeneous as possible, are selected for experimentation. These
homogeneous groups are usually formed by considering such criteria as
age, weight, sex ? heredity, vigor, and previous nutrition. A check group
is chosen and fed a standard ration. The other groups are fed different
levels of the vitamin in question, one of them on a ration a great deal
higher than the standard ration for the vitamin and another on a ration
containing little, or none, of the vitamin. The remainder of the groups
are fed rations somewhere between the extremes. The animals are on
the randomly assigned rations for a given period of time, and the re
searcher records such data as daily gain in weight, economy of gain,
livability, etc. If the experiment has been properly designed in accord
ance with established statistical principles, conclusions of great value
to the farmer may then be drawn. Of course., much work of a more
complex nature than this simple example has also been done in animal
nutrition research. Consultation of technical journals in this field will
reveal many instances where statistics has been of great help.
In the past, many persons thought statistics had no place in the
so-called "exact sciences' 7 such as chemistry, physics, and the various
branches of engineering. These fields are concerned with exact measure
ment, with quantities that can. be measured with a ruler, thermometer,
flow meter, thickness gauge, telescope, or pressure gauge. Therefore,
the doubters asked, why use a "pseudo-science'' statistics that at
best merely estimates quantities? As the true meaning of statistics and
its application has come to wider attention, these persons have readily
admitted there is indeed a place for this important tool in the exact
sciences. In fact, it has become apparent that ail of these sciences
themselves are based on statistical concepts. For example, it is evident
that the pressure exerted by a gas is actually an average pressure an
average effect of forces exerted by individual molecules as they strike
the wall of a container. A similar situation is true in regard to tem
perature.
Since the popularly accepted theory is that all matter is made up of
small particles, it does not require much imagination to see that a
statistical approach is the logical one to adopt in investigations of the
ultimate nature of matter. Such particles are actually part of an almost
inconceivably large population one that is, for all practical pur
poses, our closest approach to the infinite population. All of these
particles exhibit individual behavior characteristics. With the com
paratively crude devices of the exact sciences we can generally only
note the results of group behavior an average effect and until re
cently science has been limited to this. But even in these crude applica
tions statistics plays its role. For instance, examine the chart of the
elements in any chemistry classroom. The atomic weights shown on
this chart are actually "weighted averages' 7 of the atomic weights of
8 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
individual isotopes of the given element, the "weights" being the fre
quency of occurrence of the element in a normal or naturally occurring
mixture.
Statistics has also invaded the fields of meteorology and astronomy.
The modern science of meteorology is to a great degree dependent upon
statistical methods for its existence. The methods which give weather
forecasting the accuracy it has today have been developed using mod
ern sample survey techniques. Thus, weather stations throughout the
United States are able to give us highly accurate predictions for their
individual areas. In addition, by suitable selection of gathering points
and proper treatment of the data, an over-all picture of the weather
for larger areas is pieced together. Again we may see statistical sampling
in action when we turn our attention to snow survey teams which de
termine the amount of snow present in a given area and thus the
quantity of water to be drained from that area following a thaw. In the
more theoretical aspects of meteorology, statistical inference and
analysis are being xised to develop new techniques for advancing the
field. In astronomy, statistics havS long played a major role. One hundred
years ago the uncertainty in the measurement of the semimajor axis
of the earth's elliptical orbit was 1 part in 20. Today statistical methods
have reduced this uncertainty to 1 part in 10,000.
Statistics is now playing an important role in engineering. For
example, such topic** as the study of heat transfer through insulating
materials per unit time, performance guarantee testing programs, pro
duction control, inventory control, standardisation of fits and toler
ances of machine parts, job analyses of technical personnel, studies in
volving the fatigue of metals (endurance properties), corrosion studies,
time and motion studies, operations research and analysis, quality
control, reliability analyses, and many other specialized problem?* in re
search and development make great use of probabilistic*, and Btatintical
methods*
Because the above problems are but a small portion of those to
which the science of statistics IB being applied in industry, the reader
can readily appreciate that the application of statistical methods to the
field of engineering is riot limited to a few areas but is general in nature.
As an indication of the wide scope of industrial statistics, P. L. Algor
of the General Electric Corporation has listed the following ten major
areas of application:
1* Defining the value of observations
2* Design of experiments
3. Detection of causes
4. Production quality control
5* (letting more out of the inspection dollar
6. Design specifications
7. Measurement of human attributes
<S, Operational research
9. Market research, including opinion polling
10. Determining trends* 3
1.6 APPLICATIONS OF STATISTICS IN RESEARCH 9
If applied statistics is to play a primary role in the future of engi
neering, or, to be more general, in that of industry, it is quite evident
that there is a great need for specific training of personnel entering the
field. This training is needed for the young engineer as well as for the
young businessman, since each must be capable of dealing with combi
nations of men and machines. Professor S. S. Wilks of Princeton Uni
versity has made this statement of the problem :
The statistical problems which the future scientist or engineer will en
counter will cut across traditional lines. Therefore, in order that he may be
properly equipped to deal with these problems, he should have a fairly
broad statistical training. The training should cover not only statistical
quality control methods as the term is now understood, but the design of
experiments, analysis of variance, and many other topics. It should be built
into the training of scientists and engineers, as calculus is now made part
of their basic education. 3
Agricultural engineering, which combines the practices of engineer
ing and agriculture, has also benefited greatly from the use of statisti
cal methods. In this field, statistics has helped the researcher with such
varied projects as the testing of weed-control machinery, certain eco
nomic aspects of farm electrification, comparison of various drying
methods for grain, determination of the effects of drying rate on pop
corn, irrigation research, roofing studies for farm buildings, and meth
ods of cultivation.
Statistics is also proving an important tool in food technology re
search. Foods exhibit to a marked degree what is widely called "bio
logical variation." Their constitution is heterogeneous, and their com
plexity is such that duplication is highly improbable. Food properties
are affected not only by the multiplicity of factors influencing their
growth but also by the infinite variety of processing and storage con
ditions to which they may be subjected. Thus, it is impossible to give a
general answer to a question such as "What is the moisture content of
corn?" Before attempting to answer, one would first have to ask "What
variety ... at what stage of its growth or processing cycle . . , where
was it grown?" and such questions. Having obtained the necessary
specifications, the food technologist might be able to quote an average
value* In short, he might specify a frequency distribution of moisture
content of sweet corn under the stated conditions.
This type of problem was encountered by Bard (6) in his investiga
tion of certain palatability factors tenderness, juiciness, and fiber
cohesivenevss -of canned beef as conditions of time and temperature of
processing were varied. In his work, a statistical approach dictated the
design of the experiment, and analysis of variance was freely employed
to delineate between variation due to raw material and that caused by
2 P. L. Alger, "The growing importance of statistical methods in industry/'
General Electric Review, Dec., 1048, p. 12.
3 S. S, Wilks, "Statistical training for industry," Analytical Chemistry, Vol.
19, Dec., 1947, p. 955.
10 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
processing treatments. Another example in food technology is provided
by Bernhard (7) in his comparison of several techniques of estimation
of the frequency of occurrence of insect fragments in cream-style
corn. The conventional methods utilize castor oil to separate the
insect fragments by flotation. Bernhard wished to compare the effi
ciency of castor oil and lard oil, each at three different temperatures
for each of four different times of mixing oil and food samples. Of course,
repeated samples of any one set of determination conditions could be
expected to yield variable results for the number of insect fragments
present. Thus, statistical methods were required to enable the variation
among mixing times, temperatures, and oils to be analyzed.
One of the most difficult areas of food research is that of evaluating
a food product in terms of consumer reaction. It is well known that
most objective tests of food acceptability (such as laboratory measure
ments of shear strength, etc.) must bo correlated with consumer pref
erence by means of taste-panel observations in order to achieve firm
standing. The problems of the "taste panel" are many. To what extent
is the taste panel representative of the entire population of tasters?
How is variation from sample to sample of a food product distinguished
from variation from tnstor to taster? How can subjective evaluation of
a particular property of food, for example, odor, be separated from
evaluation of another property, such as flavor? To what extent can or
should restrictions such as instruction*) to evaluate a narrow area be
placed upon the taster, in view of the fact that the heart of the taste-
panel system is the use of the integrated pattern of individual reaction
to a complex event?
These and many other sxich problems of food evaluation are not en
tirely solved. Kven the basic justification for the introduction of sta
tistical analysis is not always clear. For example, a group of tasters
may be asked to rank in order of merit five varieties of corn. In search
ing for a method of evaluating results of this type of problem, many
workers have followed the procedure of allotting a number to each rank,
e.g., 5 for first, 4 for second, etc. These figures are then treated an num
bers ami analyxed by analysis of variance to check for significant varia
tion among; the five varieties* Huch a procedure is not entirely valid
because analysis of variance can only be used with numbers, and the
ranking figures nre not originally wet down an qxmntitativo relative esti
mates of tanto reaction. However, in view of the lack of exact methods
of analysis, the technique mentioned can provide valuable assistance
to the research worker who deals with food products,
In the social sciences, statistical methods also find wide application*
Because of their vital interest in public opinion, the major political
parties have become acquainted with the statistician* In economic re
search, stat.lstieal methods are almost indispensable. Economic laws
refer to muss or group phenomena, and the determination of these laws
often depends upon the judicious use of statistical techniques.
In marketing research, an objective may be increasing eonsmmption
1.6 APPLICATIONS OF STATISTICS IN RESEARCH 11
of those foods shown by nutritional studies to be inadequately supplied
in the average diet. The initial role of statistics here is merely one of
finding consumption per capita and comparing it with some goal. Of
course, the nature of the distribution of consumption per capita is as
important as the average. Another objective is the analyzing of mar-
keting methods in order to find the least costly way of doing the job.
As a result, a smaller portion of society's efforts need be expended on
product handling.
Measuring demand is another of the many difficult tasks in eco
nomics. The research worker must have a knowledge of consumer pref
erences, supply of money, its distribution, etc. In measuring supply,
he must have an intimate acquaintance with marketing functions,
services, and costs and be familiar with trends in operational efficiency,
both physical and managerial. Data on these particulars can only be
digested and made available through statistical procedures.
In production economics probably the most important comparisons
are made when two or more characteristics are simultaneously studied
or measured. This involves statistical techniques known as regression
and correlation. These tools are invaluable to the economist. By using
them, one factor can be shown in its relationship with other factors. For
instance, if we made the hypothesis that net income per acre becomes
higher as farm size increases, we would want to find the influence of
farm size on net income per acre. We might then collect data for ten
units of each farm size, ranging from 40 acres to perhaps 480 acres with
40-acre increments. These data, if properly obtained in accordance
with the rules of statistical procedure, could then be analyzed to aid
the researcher in making a contribution to the theory of production
economics.
It is possible to go on almost indefinitely enumerating the fields
wherein statistics is being, or could be, applied. Statistics is utilized
for a systematic approach to problems in public health studies, epide
miology, demography, biological assay, psychology, education, sociol
ogy, and in various areas of home economics. Oddly enough, statistics
is not confined to the so-called scientific world, for it also is applied in
the arts. It has been used to aid in determining the authorship of cer
tain manuscripts by analyzing the length of sentences. Authenticity of
paintings has alo been established by analyzing the frequency of brush
strokes.
Although statistics is very much in the realm of an applied science,
it has its theoretical basis in mathematics. Development of the theo
retical branch of the science is as important as that of the applied
branch if progress in the field is to continue. Unfortunately, there is a
gap between statistical theory and application much the same as exists
in other sciences. This gap is steadily being closed, but the job is far
from completion. Thus it is not surprising to find statistics in use as
both a science and an art. It is a science because its methods are basi
cally systematic and of wide application. And it is an art because sue-
12 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
cess in its application depends on the skill, special experience, and
knowledge of the person using it. The research worker will become more
appreciative of this fact as he gains a greater understanding of statisti
cal methods and their uses.
1.7 SUMMARY
The scope of statistics might be summarized as concerned with: the
presentation and summarization of data, the estimation of population
quantities and the testing of hypotheses, the determination of the ac
curacy of estimates, the measurement and study of variation, and the
design of experiments and surveys. Inherently and inextricably in
volved in all of the above-mentioned areas is the process known as
methods of reduction of data, or the computational aspects of statistics,
The statistical method is one of the devices by which men try to
understand the generality of life. Out of a welter of single events, hu
man beings seek endlessly for general trends. Controlled, objective
methods by which group trends 'are abstracted from observations on
many separate individuals are called statistical methods. These meth
ods are especially adapted to the elucidation of quantitative data which
have been affected by many factors. Statistical methods are fxmda-
mentally the same whether employed in the analysis of physical phe
nomena, the study of educational measurements, the study of data
resulting from biological experiments, or the analysis of quantitative
material in economics. Agriculturists, biologists, chemists, physicists,
and other researchers all attempt to eliminate the many miusanee fac
tors which influence the variables under investigation and to concen
trate their attention upon one or two of the most powerful factors
affecting the phenomena being studied. Yet, many disturbances are
always present and thus statistical methods of analysis nre vitally
necessary* Wherever there is a mass of numerical data that admits of
explanation, the statistician shoxild consider itB analysis his field of
endeavor.
To utilise statistical methods to advantage, a person should:
(1) Be well versed in the subject matter of the field in which the
research is to be conducted*
(2) Know how to organize masses of data for efficient tabulation
and how to lay out economical routines for handling data and
computation.
(3) Know effective means of presenting data in tabular and
graphic form.
(4) Have some knowledge of the mathematical theory of statistics
in order to have assurance there is a fair correspondence be
tween his data and the assumptions underlying the formulas
ho XISOH.
(5) Bo acquainted with a variety of ntatistieal techniques, the
limitations and advantages of each, the assximptions upon
which they are based, the place each occxipicn hi a logical
1 .7 SUMMARY 1 3
analysis of the data, and the interpretations which can be
made from them.
Statistics, then, boils down to numerical results, the methods and
processes used in obtaining them, the methods and means for estimate
ing their reliability, and the drawing of inferences from these resultslv
During the past half-century, the thinking world appears to have
awakened to an unusually deep appreciation and respect for numerical
facts. There has been a growing tendency to reduce observations and
accumulated data to an orderly arrangement, making possible the
evaluation of results by means of a systematic method of analysis.
Formerly, many persons believed statistical analysis could be used
only in certain highly specialized fields. However, more and more
methods of statistical analysis are finding their way into scientific
workshops in all fields. This is due largely to the fact that some of the
enthusiastic supporters of statistical methods have worked faithfully
to develop and explain methods useful to and usable by those persons
not specifically trained in higher mathematics.
In the field of statistical analysis advancement has been rapid in
recent years. Many useful methods are now available for the analysis
of data arising from different sources. A clear grasp of simple and
standardized statistical procedures will go far to elucidate principles of
experimentation. However, one must remember that these procedures
are in themselves only a means to a more important end. As fundamen
tal and pervasive as statistical thinking is in the modern world, it must
not be considered an end in itself. The statistical method is a tool for
organizing facts so they are rendered more available for study. A sta
tistical study can only describe what is; it cannot determine what
ought to be, except insofar as it may throw light upon probable con
comitants and consequences of certain situations. It is fatuous to sup
pose the statistical method can provide mechanical substitutes for
thinking, although it is often an indispensable aid to thinking. Men
see increased prevalence of the statistical method in scientific studies;
and, sometimes, failing to grasp underlying reasons for this develop
ment, they assume the use of tables, formulas, and numerical sum
maries is a badge of respectability. As a result, some studies, truly
subjective in nature, are invested with a false show of objectivity.
Thus, a vast superstructure of computation is raised upon a foundation
inappropriate to such treatment. When such a picture is painted, it is
neither good statistics nor good philosophy.
Most statistical studies will not answer all the questions we would
like to have answered regarding a given problem. From the very nature
of statistical work, results are apt to be partial and fragmentary, rather
than complete and final. Therefore, the researcher must make up his
mind that questions must sometimes be left unanswered. He must also
on occasion freely admit his study has limitations. Any shortcomings in
his work and the danger of attributing more than claimed for his in
vestigation should be pointed out to his readers by the researcher.
14 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
It is also imperative that conclusions drawn from observational re
sults be based on a detailed knowledge of procedures employed in the
investigation. The interpretive function in statistical analysis is one of
the most important contributions of statistics, and the statistician
should plan experiments and investigations which will yield maximum
information and valid conclusions from scientific research data. In
ference from the particular to the general must be attended with some
degree of uncertainty, and research workers in all fields of science must
recognize the role statistics plays in this, the most important aspect of
research.
The role of statistics in research is, then, to function as a tool in de
signing research, in analyzing its data, and in drawing conclusions
therefrom. A greater and more important role can scarcely be en
visioned. In utility to research, statistics is second only to the mathe
matics and common sense from which it is derived. Clearly the science
of statistics cannot be ignored by any research worker even though he
may not have occasion to use applied statistics in all of its detail and
ramifications.
Problems
1.1 Discuss the following terms or phrases: (a) observation mid descrip
tion; (b) cause and effect; (c) analysis and synthesis; (d) assumption,
postulate, and hypothesis; (e) testing of hypotheses; (f) deduction
and induction.
1.2 What do you believe operations researchers mean by the phrase
"measure of effectiveness"?
1.3 Saaty (39), in a chapter entitled "8ome Remarks on Scientific Method
in Operations Research/' refers to: (a) the jxulgment phase, (b) the
research phase, and (c) the action phase. Give your interpretations
of these three phases. Then compare your views with those of Saaty.
1.4 Read Chapter 12, "Some Thoughts 'on Creativity," in 8aaty (H<>).
Then prepare a brief report on your reactions to his ideas*
1.5 Prepare a report on the pros and eons of: (a) individual research and
(b) interdisciplinary team research.
. ,6 DmcuHS the similarities and dissimilarities of pure and applied research.
K7 Prepare a report on the subject of "scientific method/ 1
1.8 Prepare a report on your interpretation of "the role of statistics in
research,"
1.9 By consulting the technical journal** in your area of specialization,
prepare and submit a list of references (properly documented) which
illustrate the use of statistical methods.
1.10 Submit a list of publications (bookn, monographo, papors, etc.) which
you believe would be worthwhile additions to the references* prenerited
with this chapter.
References and Further Reading
1, Algcr> P. L* The growing importance of Htatintical method** in industry.
General Ktertric Review, 51 (No. 12): II, 1948,
2. American Management Association. Getting the Moat from Product
and Development. Now York> 1055.
REFERENCES AND FURTHER READING 15
3 ^ Making Effective Use of Research and Development. New York, 1956.
4. . Engineering and Research in Small and Medium Size Companies.
New York, 1957.
5. Anderson, J. A. The role of statistics in technical papers. Trans. Amer. Assn.
Cereal Chemists , 3:69, 1945.
6. Bard, J. C. Changes in tenderness, fiber cohesiveness and moisture content
of canned beef due to thermal processing. Master of Science thesis, Iowa
State University, Ames, 1950.
7. Bernhard, F. L. Recovery and identification of insect fragments from cream
style corn. Master of Science thesis, Iowa State University, Ames, 1951.
8. Beveridge, W. I. B. The Art of Scientific Investigation. W. W. Norton and
Co., New York, 1951.
9. Buros, O, K. (editor) Research and Statistical Methodology Books and Reviews
1933-38. Rutgers University Press, New Brunswick, N.J., 1938.
10. . (editor) The Second Yearbook of Research and Statistical Methodology
Books and Reviews. The Gryphon Press, Highland Park, N.J., 1941.
11. . (editor) Statistical Methodology Reviews 194-1-50. John Wiley and
Sons, Inc., New York, 1951.
12. Bush, G. P., and Hattery, L. H. (editors) Teamwork in Research. American
University Press, Washington, D.C., 1953.
13. Chapanis, A, R, JE. Research Techniques in Human Engineering. The Johns
Hopkins Press, Baltimore, 1959.
14. Churchman, C. W. Theory of Experimental Inference. The Macmillan Com
pany, New York, 1948.
15 ^ Ackoff, R. L., and ArnofT, B. L. Introduction to Operations Research.
John Wiley and Sons, Inc., New York, 1957.
16. Cohen, M. R., and Nagel, E. An Introduction to Logic and Scientific Method.
Harcourt, Brace and Company, New York, 1934.
17. Cox, G. M. The value and usefulness of statistics in research. Lecture given
for the USDA Committee on Experimental Design. Washington, D.C.,
Jan. 11, 1951.
18. Flagle, C. D., Huggins, W. H., and Roy, R. H. Operations Research and
Systems Engineering. The Johns Hopkins Press, Baltimore, 1960.
19. Frecnlmati, P. The Principles of Scientific Research. Public Affairs Press,
Washington, D.C., 1950.
20. Good, C. V., and Scatea, D. EL Methods of Research: Educational, Psycho
logical, Sociological. Appleton-Century-Crofts, New York, 1954.
21. Goode, H. H., and Machol, R. E. System Engineering, An Introduction to the
Design of Large Scale Systems. McGraw-Hill Book Co., Inc., New York,
1957.
22. Gryna, F. M., Jr., McAfee, N. J., Ryerson, C. M., and Zwerling, S. (editors)
Reliability Training Text. Second Ed. Institute of Radio Engineers, Inc.,
New York, 1959.
23. Hawley, G. C)., and Ostle, B. Training for reliability and quality control.
Proc. of the Seventh National Symposium on Reliability and Quality Control in
Electronics, pp. 91-96, Jan. 9-11, 1961.
24. Hill way, T. Introduction to Research. Houghton MifTiin Co., Boston, 1956.
25. Hutt, F. B., Genetics of the Fowl. McGraw-Hill Book Company, Inc., New
York, 1949.
26. Jeffreys, It. Scientific Inference. Cambridge University Press, London, 1937.
27. JOVQIIS, W. S, The Principles of Science. Second Ed. Macmillan and Co.,
New York, 1877.
28. Johnnon, P. O. Modern statistical science and its function in educational
and psychological research, Sci. Monthly , 72:385, 1951.
29. Lush, J. L. Animal Breeding Plans. The Iowa State University Press, Ames,
1945.
30. Luszki, M- TO. B* Interdisciplinary Team Research: Methods and Problems.
Now York University Press, New York, 1958.
16 CHAPTER 1, THE ROLE OF STATISTICS IN RESEARCH
31. Miller, D. W., and Starr, M. K. Executive Decisions and Operations Research.
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1960.
32. Morse, P. M., and Kimball, G, K. Methods of Operations Research. John
Wiley and Sons, Inc., New York, 1051.
33. Ostle, B. Planning Experimental Programs. Humble Oil and Refining Co.,
Refining Department Technical and Research Divisions, Research and
Development Division, Baytown, Tex. RL. 44M.52, 6-20-1, Aug. 2O, 1952.
34. . Statistical problems in designing range management investigations.
Report on Meeting of Kconomics of Range Resource Development Com
mittee of the Western Agricultural Economics Research Council, Reno,
Nev., Oct., 1954.
35_ ._ Statistical problems in designing experiments to study the economics
of fertilizer application. Conf. Proc., Farm Management Research Com
mittee of the Western Agricultural lilconomics Research Council, Corvallis,
Ore., Jan., 1956.
3(} . Statistics in engineering. Jour. Engin, Educ., 47:(No. 5):410 14,
Jan., 1957.
37^ __ > and Tischer, R. G. Statistical methods in food research. Advances in
Food Research, 5:161-259, 1954.
38. Popper, K. R. The Logic of Scientific Discovery. Basic Books, New York, 1959.
39. Saaty, T. L. Mathematical Methods of Operations Research. McGraw-Hill
Book Co., Inc., New York, 1959.
40. Rasicmi, M., Yaapaii, A., and Friedman, I... Operations Research: Methods
and Problems. John Wiley and Sons, Inc., Now York, 1959.
41. Sehenek, H., Jr. Theories of Engineering Experimentation. McGraw-Hill
Book Company, Inc., New York, 1961,
42. Snedoeor, G. W. On a uniqxio feature of statistics. Jour. Amer. Stat* Assn.,
44:1, 1949.
43. ,____ % The statistical part of the scientific method, Ann. N.Y+ Acad. Sci.,
52:792, 1950.
44. Taton, R. (trans, by A. J. Pome.rans) Reason and Chance in Scientific Dis
covery. Philosophical Library, New York, 1957.
45. Walker, H. 3VL Statistical literacy in the social sciences. Amer. Stat. y
5 (No. 1):6, 1951.
46. Whitney, F. Li. The Elements of Research. Third Ed. Prentice-Hall, Inc.,
New York, 1950.
47. Wiesen, J. M., and Oatle, B. Some problems in the preparation of quality
specifications. Proc* of the Fifth National Symposium on Reliability and
Quality Control in Electronics, pp. 375-80, Jan/ 12-14, 1959.
48. Wilks, H. 8, Statistical training for induntry. AnaL Wusm., 19:953, 1947,
49. WilHon, K B. An Introduction to $rie,ntijtc /^search. McCrraw-Hill liook
(Company, Inc., New York, 1952,
50. Worthing, A. G., and (tofFner, J. Treatment of Experimental Data. John Wiley
and Sons, Ino., New York, 1943*
CH APTE R 2
MATHEMATICAL CONCEPTS
IT is DIFFICULT to achieve a clear understanding of statistical methods
without discussing, to some extent, the underlying theory. Since the
theory of statistics is intimately associated with the theory of prob
ability and, further, since probability is an important branch of mathe
matics, this implies that every student of statistical methods should
be willing to "use a little mathematics once in a while!" Consequently,
it seems desirable to present here a few basic mathematical concepts,
formulas, and techniques which may prove helpful to the reader. These
ideas will be presented as definitions and/or theorems 1 (without proofs).
SET THEORY
The subject of the theory of sets is fundamental in mathematics. In
this text, however, we shall be concerned only with a few basic concepts
which are useful in the theory of probability.
Definition 2.1 A set is a collection of elements. 2
Definition 2.2 The universal set is the set consisting of all elements
under discussion. (NOTE: The universal set is some
times referred to as a space.)
Definition 2.3 The null set is the set containing no elements at all.
Definition 2.4 Associated with each set, A, is another set, A', called
the complement of A and defined to be the set consist
ing of all the elements of the universal set which are not
elements of A.
Definition 2.5 For any two sets, A and JB, the union of A and B is the
set consisting of all elements which are either in A or
in B or in both A and B. The union of A and B is com
monly denoted by A^JB.
Definition 2.6 For any two sets, A and J5, the intersection of A and B
is the set consisting of all elements which are both in
A and R. The intersection of A and B is commonly
denoted by AC\B or by AB.
Theorem 2.1 If A and B arc two sots which have no common ele
ments, then the sot AJS is the null set.
A useful device for illustrating the properties of the algebra of sets
is the Venn diagram. In such a diagram, the points interior to a rec-
1 The expression "theorems" will bo xisod in a very broad sense to describe
various proportion, propositions, theorems, ote., which result from the definitions.
While not strictly correct, this procedure will materially reduce the number of
terms to bo absorbed by the reader.
a The term element will bo left undefined.
C17J
18
CHAPTER 2, MATHEMATICAL CONCEPTS
FIG. 2.1 A Simple Venn diagram.
tangle constitute the xmiversal set. Arbitrary sets within the universal
set (that is ? subsets of the universal set) will be represented, for con
venience, by the points interior to circles within the rectangle. In
Figure 2.1 the set A is shaded by vertical lines, the set B is shaded by
horizontal lines. Since A I) 9*0, that is, does not equal the null set, AB
appears as the erosshatehed area.
Probability theory, which we shall sximmari^e in the next chapter,
depends on the number of elements in a set. We will denote the number
of elements in any arbitrary set A by n(A).
Theorem 2.2 If A and B have no elements in common,
n(A^JB) n(-4)4-n(jR).
Theorem 2*3 If A and B have no elements in common, n(AB) 0,
Theorem 2,4 For arbitrary sets ,*1 and B, it is true that
NOTATION
As in all subjects, the system of notation employed in a matter of
concern to the reader. Since statistics is HO entwined with mathematics,
it in no surprise that problems of notation arise* In the remainder of
this book every attempt will be made to define and explain special
symbols and notation. However, at this point it seems appropriate to
mention some of the more frequently occurring signs and symbols.
Definition 2,7 The absolute value of a number, x, denoted by |xf ? is
its numerical value neglecting its algebraic nigtx* For
example, j ~~3| 3 and J3| 3,
Definition 2,8 # = y is read "x is equal to 37."
Definition 2.9 ;rp^// is read ".# is not equal to |/. ?>
NOTATION
19
Definition 2.10
Definition 2.11
Definition 2.12
Definition 2.13
Definition 2.14
x=z/ is read "x is approximately equal to y*
x<y is read "re is less than y."
x<y is read "x is less than or equal to y."
x>y is read "x is greater than y."
x>y is read "x is greater than or equal to y.'
Definition 2.15
Y, =
Y 2 +
- +
NOTE: The Greek capital letter sigma, )? is known
as the summation sign. Further, i is called the index
of summation, while 1 and n are known as the limits
of summation.
Theorem 2.5
Theorem 2.6
c Yi = c
t - where ^ is a constant.
Theorem 2.7
Theorem 2.8
+
, +
Theorem 2.9 ( ^ Y^\ - J Y* + 2 S S
NOTE: In this theorem the notation ]>D*-<y ^ s i n ~
terpreted to mean that we sum all possible products
YiY, letting i and j go from 1 to n, subject only to the
restriction that in any particular term,
Definition 2.16
t - - (FO(F 2 )
(F n ).
NOTE: In contrast to Definition 2.15, in which we
introduced 23 as ^e summation sign, we have here
20
CHAPTER 2, MATHEMATICAL CONCEPTS
introduced the Greek capital letter pi, TI? as the prod
uct sign.
Theorem 2.1O
i = (i)(2)
(n) = nl
NOTE: The symbol nl is called n factorial, or fac
torial n.
Definition 2. 17 ! 1 .
NOTE: This will prove useful later.
PERMUTATIONS AND COMBINATIONS
Permutations and combinations are concerned with the different
subgroups and arrangements that can be formed from a given set. A
permutation is a particular sequence (i.e., arrangement) of a given set
or subset of elements, while u combination is the set or subset without
reference to the order of the contahied elements.
Definition 2,18
If an event, A, can occur in n(A) ways and if a differ
ent event, B, can occxir in n(75) ways, then the event
"either A or R" can occur in n(A)+n(B) ways pro
vided A and B cannot occur simxiltaneously.
NOTE: You will notice the similarity between this
definition and Theorem 2.2,
If an event, A, can occur in n(A} ways and a sub
sequent event, fij can occur in n(B) ways, then the
event "both A and B )f can occur in n(A) -n(B) ways.
An r-pormutation of n things is an ordered selection or
arrangement of r of them.
An r-combination of n things is a selection of r of them
without regard to order.
The number of different permutations which can be
formed from n distinct objects taken r at a time is
JP(n, r)=tt(/i 1) * - (n r+l)n!/(n r)!
The number of different permutations which can be
formed from n objects taken n at a time, given that
n- are of type i t where i=* 1, 2, - * , A:, and ^n t - =n
is P(n\ niy n a , - M*) ^nl/niln^l * * - n&\
The number of different combinations which can be
formed from n dintinct objects taken r at a time ia
Definition. 2,19
Definition 2.20
Definition 2.21
Definition 2.22
Definition 2,23
Definition 2*24
Definition 2.25 <7(n, r) for r <0 and r>n.
SOME USEFUL IDENTITIES AND SERIES
In statistical work it in often necessary to sum a series of terms or
simplify a particular expression. A few of the more useful results are
given here for roady reference.
SOME IMPORTANT FUNCTIONS 21
Theorem 2.11 ( a + by =
r=0
Theorem 2.12 If, in Theorem 2.11, we let a=l and & = Z, we obtain
Theorem 2.13 If, in Theorem 2.11, we let a = q and & = p=l -q
where 0<p<l, we obtain
n
1 = ^JL, C(n, r)q n ~ r p r .
r
This is a very useful expression in probability and
statistics.
n 1
Definition 2.26 e x = exp (oc) ===
Theorem 2.14 (1 # n )/(l c
00
Theorem 2.15 i/(l x ) n = 1C C(n + i 1, z)^'-
Theorem 2.16 ]T) C(a, i) -C(J, c i) = C(a + &,<;).
i
n
Theorem 2.17 ^ i = n ( w + l)/2.
n
Theorem 2.18 y^ ^2 ... n ^ n _^ i)(2>z + l)/6.
Theorem 2.19 23 i^< = a?/(l x)* for 1 < x < 1.
i 1
SOME IMPORTANT FUNCTIONS
Some mathematical functions not always presented in courses in
elementary mathematics are of great interest to the statistician. Two of
these will be presented here for your convenience.
Definition 2,27 The gamma function, denoted by r(y>), is defined by
the integral
CHAPTER 2, MATHEMATICAL CONCEPTS
y- OO
T-T-? I 3CP~~~^'& " K (Jf2C
Jo
for p>0. An alternative form for this function is
r(p) = 2 f yip-ierSdy,
J
where the transformation used was x = y z +
Theorem 2*20 If, in Definition 2.27, we let p~n where n is a positive
integer, we obtain F(n) = (n 1) -r(ri l) = (n 1) !
Theorem 2.21 F(i) vV^ (^r) 172 .
Definition 2,28 The beta function, denoted by J3(p,q), is defined by the
integral
i
= I
^ o
for p>Q and #>CK An alternative form for this func
tion is
, <?>
/W /Ii
^
cos*- 1 Q de,
whore the transformation used was ^ = vsi
Theorem 2*22 /3(p, q) ==/5(r/, p).
Theorem 2.23 ft(p, ff) = r(p) 4
MATRICES
Many of the methods to be discussed in this book depend on the
theory of linear .statistical models. This theory is most expeditaoxisly
handled in terms of matrix algebra. Therefore, it is appropriate that
the reader be made aware of the basic concepts* As in the preceding
sections of this chapter, definitions and theorems will be stated without
discussion.
Definition 2.29 A matrix A of dimension rXc is a rectangular array
of elements a^ arranged in r rows and c columns:
Definition 2*30
Definition 2.31
If it is necessary to emphasise the dimension, we shall
write A rG instead of A.
If ,4 is of dimension nXl> it is called an nXl vector.
AB when and only when .4 and & are of the same
dimension and a,; 6^- for all i and j.
MATRICES
23
Definition 2.32
Definition 2.33
Definition 2.34
The product of a matrix A and a scalar (ordinary)
number k is a matrix B where &# = ka iy - for all i and j.
That is, kA =*= Ak = 5.
The sum of two matrices, -4 and Z?, can be defined only
when A and B are of the same dimension. Then
<A-hJ? = C where c 1 -y = a^+&^.
The product of two matrices, say AB, can be defined
only when the number of columns in A equals the num
ber of rows in B. Then AB = C where
Definition 2.35
Theorem 2.24
Theorem 2.25
Theorem 2.26
Definition 2.36
Theorem 2.27
Definition 2.37
Definition 2.38
Theorem 2.28
Definition 2.39
Definition 2.40
Definition 2.41
Definition 2.42
=1
NOTE: We must be very careful of the order of the
factors when multiplying one matrix by another.
Even if AB and BA are both defined, they are not
necessarily equal.
The transpose of a matrix A of dimension rXc is de
noted by A', where A f is a matrix of dimension. eXr
in which a'-/ = c&yi. That is, the rows of A' are the col
umns of A and the columns of A f are the rows of A.
= AA, A*
If r = c, A is called a square matrix,
For a square matrix A, we can write
~AAA, etc.
In a square matrix of dimension nXn, the elements
an, #22, - , a n , form the main diagonal and are
known as diagonal elements.
A square matrix which is symmetric with respect to
its main diagonal is called a symmetric matrix.
For a symmetric matrix, A' = A.
A symmetric matrix in which a^' = for all i^j is
called a diagonal matrix.
A diagonal matrix in which a=l for all i is called a
unit (or an identity) matrix, and will be denoted by /.
A matrix having all its elements equal to zero is called
the null matrix, and will be denoted by 0.
The determinant of a square matrix A of dimension
denoted by \A\, is defined by
where the second subscripts n, r%, - , r n run through
all the n\ possible permutations of the numbers
1, 2, - - - , n, and the sign of each term (either +
or ) is determined according to a well-defined rule.
24
CHAPTER 2, MATHEMATICAL CONCEPTS
NOTE: If A is of dimension 2X2, then
Theorem 2.29
Theorem 2.30
Theorem 2.31
Theorem 2.32
Theorem 2.33
Definition 2.43
Definition 2.44
Theorem 2.34
A =
For any square matrix A, \ A j = J A' J ,
If two rows (or columns) of a square matrix are inter
changed, the determinant changes its sign.
If two rows (or columns) of a square matrix are identi
cal, the determinant is 0.
If -4, B, and C are square matrices such that AB= C,
then \A\ - \B\ -|C|.
If a multiple of one row (column) is added to another
row (column) of a square matrix, the determinant is
unchanged.
For any arbitrary matrix -4 ? the determinant of any
square submatrix of A is called a minor of A.
For a square matrix A, the minor obtained by deleting
the ith row and yth column, multiplied by ( I)* 4 "*",
is known as the cofactor of a t -/. We shall denote the
cofactor of ay by cof a*/.
For a square matrix A of dimension nXn ? the de
terminant | .4 1 may be found by evaluating
** n
*> - 2: <
Definition 2,45
Definition 2.46
If, for a square matrix, \A\^0 9 then A is of rank n
and A is said to bo nonsingular.
For a nonsingular sciuare matrix -4, the inverse of A
is denoted by A 1 and is defined by
Theorem 2.35 For a nonmngulur square matrix A, it is true that
Theorem 2*36 For a nonsingular square matrix ,4, it is true that
i A i """"" * w\ \ A *~" * i
V.** J **"** V ** J *
LINEAR EQUATIONS
Many times in statistical work wo find it necessary to discuss sys
tems of linear equations such as:
+ a 22 * a + 4- a*x - y* ^^ ^
Hr
PROBLEMS 25
The matrix notation introduced in the preceding section gives us an
extremely concise method of representing such systems. For example
it is clear that *
AX= Y
is the same as Equation (2.1) if
(2.2)
A =
<Zin~
X =
and Y =
Theorem 2.37 If Jl in Equation (2.2) is nonsingular. then
X=A-iY. That is,
oc<i
j_ -
~rr 23 y*(cof a^
A iwi
forj = 1, 2, - - - , n .
NOTE: Another way of writing this is
matrix A in which a^
has been replaced by
i=l, 2, , n
X* =
That is,
W-i y
J = l,
Problems
2.1
2.2
2.3
Consider a box of resistors that are color coded (red, black, or yellow)
according to resistance rating. Suppose that all red (JB) resistors
and some^of the black (2J) resistors are manufactured by company E.
The remainder of the black resistors are manufactured by company F t
while the yellow (F) resistors are manufactured by company G. The
universal set consists of all the resistors in the box. Letting R stand
for the set of all red resistors, B stand for the set of all black resistors,
and so on, write as many equations and inequations as you can to
describe the relations existing among the various sets.
List all subsets of the set {X, Y, Z}.
Consider the space consisting of the 26 lower case letters of the alpha
bet. If the sets A, B, and C are defined as A { a, &, c, d e\
B~ {b, d,f, h,j} f and C- {c,/, i, I, m} , find:
26 CHAPTER 2, MATHEMATICAL CONCEPTS
(<*) AVJB, B\JC, A\JB\JC
(&) AB, J5C, ABC
(c)
(d) (A\JB) (A\JB f )
(e) (A^JB) (A'WB) (
2.4 How many different subsets arc there in a set containing n distinct
elements?
2.5 Draw Venn diagrams for the following and shade the indicated area:
(a) A^JA'BC
(6)
(c)
2.6 Referring to Problem 2.3, give the number of elements in each set
discussed.
2.7 If A"i=*4, A^aa 3, A T 'a = i, X" 4 = 7, 1^=8, Fo = 2, F 3 = 1, and r 4 = 3,
find:
(4
T, -Y*
*L
t 1
2.8 Given the following observations:
F m - 4 Fm - 3 Fun -
3 Fm 3 Fin* 9
1 Fm * F*M - 4
- 8 F 4ftl 14 K 4M -
* 22 F 4 t * 7 F 4 is -
find:
433
1 I
PROBLEMS 27
3-1 &-1
2.9 Evaluate: P(7, 3), P(5, 5), P(17, 2), P(7, 4), P(7,0).
2.10 Evaluate: P(ll; 2, 2, 5, 2), P(8; 5, 3).
2.11 Evaluate: (7(7, 3), C(5, 5), C(17, 2), (7(7, 4), (7(7,0), (7(8, 5), C(8, !
C(7, 8).
2.12 Using the binomial expansion (Theorem 2.11), and letting a = 6 =
verify that 2 6 = 32. Write out each term and show its value.
2.13 Expand (i+f) 4 .
2.14 Find
]C C"( 7 > y)'C(5, 4 a: y)-
vo
2.15 Evaluate:
r
(a) I ic s/3 e 2ic ^^
*/ o
r l
J
oo
I
^ o
2.16 Show that C(n, r)~C(n 1, r)+C( 1, r-1).
2.17 A lot contains 100 items. A single sample of two items is to be selected.
How many differently constituted samples are possible?
2.18 If
a
find: .4-f- J?, 4 B, and AB.
2.19 If
find -45 and
2.20 Find the transposes of the matrices given in Problems 2.18 and 2.19.
Also find the transposes of the solution matrices in each of those
problems.
2.21 Find the inverses of the matrices in Problem 2.18.
2.22 Transform the matrices in Problem 2.18 into diagonal form.
2.23 Evaluate \A\ by expanding by minors (see Theorem 2.34) for
rl -3 1-1
2 1 2
Ll 5 3J
Is A singular?
2.24 Solve the following sot of equations using determinants.
10
28 CHAPTER 2, MATHEMATICAL CONCEPTS
References and Further Reading
1. Cram6r, H. Mathematical Methods of Statistics. Princeton University Press,
Princeton, N.J., 1946.
2. Graybill, F. A. An Introduction to Linear Statistical Models. VoL I. McGraw-
Hill Book Company, Inc., New York, 1961.
3. Kemeny, J, G., Snell, J. L., and Thompson, G. L, Introduction to Finite
Mathematics. Prentice-Hall, Inc., Englewoocl Cliffs, N.J., 1957.
4. Riordan, J. An Introduction to Combinatorial Analysis. John Wiley and Sons,
Inc., New York, 1958.
5. Whitesitt, J. E. Boolean Algebra and its Applications. Addison-Wesley
Publishing Company, Inc., Reading, Mass., 1961,
CHAPTER 3
A SUMMARY OF BASIC THEORY IN
PROBABILITY AND STATISTICS
As INDICATED at the beginning of Chapter 2, a proper appreciation of
statistical methods is difficult without an understanding of the associated
theory. If we do not have sufficient grounding in the theory of prob
ability and statistics, the possibility of misapplication of methods
based on this theory is enhanced.
PROBABILITY
In general, statistics enters into scientific method through experi
mentation or observation. Any investigation, is only a means to an end.
It is a device for testing a stated hypothesis or for acquiring an amount
of knowledge however small from which a conclusion may be drawn.
Most statements resulting from scientific investigations are only in
ferences. They are uncertain in character. The measurement of this im-
certainty by use of the theory of probability is one of the most important
contributions of statistics.
Probability is just a measure of the likelihood of occurrence of a
chance event. A fairly simple definition of probability, generally re
ferred to as the classical definition of probability, is :
Definition 3.1 If an event can occur in N mutually exclusive and
equally likely ways, and if n of these possess a charac
teristic E, then the probability of E occurring is the
fraction n/N. This is customarily written P(E} ~n/N.
There is a natural relation between set theory and probability theory
which is easily recognized once we adjust to a change in language. In
probability, the universal set is called the sample space ? each subset is
called an event, and an element is referred to as a sample point. Then,
the definition of probability is:
Definition 3.2 The probability of occurrence of the event A is the
ratio of the number of sample points in the event A
to the number of sample points in the sample space.
Symbolically, P(A} ~n(A}/N where n(A} is the num
ber of sample points in the event A, and N is the num
ber of sample points in the sample space.
Some additional expressions encountered in probability and statistics
are the words experiment and outcome.
C291
30 CHAPTER 3, THEORY IN PROBABILITY AND STATISTICS
Definition 3.3 An experiment is any well-defined action.
Definition 3.4 Each possible result of an experiment is called an out
come (of the experiment) .
The tie-in between the two definitions just given and the ideas ex
pressed earlier is as follows: An outcome is a sample point, the totality
of outcomes is the sample space, and an event is a set of outcomes.
Definition 3.5 A random {chance) variable is a numerically valued
function defined over a sample space. It is a rule which
assigns a numerical value to each outcome of an experi
ment.
Definition 3.6 A discrete random variable is one which can take on
only a finite or a denumerable number of values.
Definition 3.7 A continuous random variable is one which can take on
a continuum of values.
NOTE : A one-dimensional continuous random variable
is most easily thought of as one w r hich can take on any
value within a specified interval along a straight line.
The definitions of probability advanced in the preceding paragraphs
are such that difficulties are sometimes encountered in their use. For
example, it is not always easy to tell if two events are equally likely.
Then, too, how do we handle the concept of an experiment that can be
performed infinitely many times?
Before formulating a new definition that will give us greater flexi
bility, let us examine some preliminary ideas. Consider a random ex
periment 8 that may be repeated many times under uniform conditions.
Each time the experiment is performed, observe whether an event E
does or does not take place. In the first n performances of 8, E will occur
a certain number of times, say/. We shall call the ratio f/n the relative
frequency of E in the first n performances of the experiment 8. It will
be observed that f/n will generally tend to become more or less con
stant for large n. This phenomenon is sometimes referred to as statisti
cal regularity. It is now conjectured that for given 8 and E we should
be able to find a number P such that as n, the number of performances
of 8, gets very large, the ratio f/n should be approximately equal to P.
Definition 3.8 "Whenever we say that the probability of an event E
with respect to an experiment 8 is equal to P, the
concrete meaning of this assertion will thus simply be
the following: In a long series of repetitions of 8, it is
practically certain that the (relative) frequency of E
will be approximately equal to P." 1
Theorem 3.1 For any event E, it is true that Q<P(E} <1.
Theorem 3.2 P(J5) +P(not E} = 1.
NOTE : Using the set notation introduced in Chapter
2, this would appear as P(E) +P(I?') = 1.
1 H. Cramer, Mathematical Methods of Statistics, Princeton University Press
Princeton, N.J., 1946, p. 148. '
Theorem 3.3
PROBABILITY
For arbitrary events A and B, P(A or B} =
31
Theorem 3.4
NOTE : For three events, this extends to
(5)+P(C) P(AB}
The extension to more than three events
can be made quite easily.
If A and B are mutually exclusive (i.e., have no ele
ments in common), then P(A or
NOTE: For three events that are pairwise disjoint
(i.e., mutually exclusive), this extends to P(A^JB^JC)
= P(A)+PCB)+P(C). The extension to more than
three events is obvious.
Let A be an event in an arbitrary sample space such
that P(A} 7^0. Let B be any event in the same sample
space. Then, the conditional probability that B occurs,
knowing that A has occurred, is defined by P(B\ A)
Definition 3.9
Theorem 3.5 For arbitrary events A and B, P(A and J?) =P(AJ3)
Definition 3.10
Theorem 3.6
Theorem 3.7
NOTE: For three events, this extends to P(ABC}
= P(A) -P(B\A) -P(C\AB). It should be realized that
other permutations of the factors and the letters are
possible. The extension to more than three events is
obvious.
Two events A and B are said to be statistically inde
pendent if P(A\B)=P(A) and P(B \ A) =PCB). This
is equivalent to saying that A and B are statistically
independent if P(AB) = P(A} -P(B}.
NOTE: Three events (A, .B, and C) are mutually in
dependent if P(A|J3)=PCA), P(A|.BC)=P(A),
P(ABlC') ~P(AB), and so on for all possible events.
This is equivalent to saying that A, jB, and C are
mutually independent if A, B, and C are pairwise in
dependent [that is, PCAJ3)=P(A)-P(B), P(AC)
= P(A)-P(C), and P(J5C) =P(J5) -P(C) ], and if
P(ABC} -P(A) -P(B) -P(C).
If A and B are statistically independent, P(AJ5)
NOTE: For three events that are mutually inde
pendent, this extends to P(ABC} =P(A) -P(B) -P(C).
The extension to more than three events is obvious.
For any events A and B,
P(A ^J J5) = 1 P(not
-!-[{!-
-P(not B \ not A)
P{l P(B\ not A}}].
32 CHAPTER 3, THEORY IN PROBABILITY AND STATISTICS
Theorem 3.8 If A and B are statistically independent, Theorem 3.7
becomes
P(A VJ B) = 1 - P(not /I) P(not B)
Theorem 3.9
Theorem 3.10
Definition 3.11
Theorem 3,11
NOTE : This can easily be extended to k events that
are mutually independent by writing
P(E l
= 1 - [{1
{1 -
Let 7/i, 7/2, - - , 7/ 7i be mutually exclusive events
whose union is the sample space. Let E be an arbitrary
event in the same sample space such that
Then
Referring to Theorem 3.9 and invoking the fact that
7^), it is seen that
P(7/ l
n //o
+"p(//7J
^| 77.)
77 n . This
and similar results hold for 77$, - *
theorem in known as JB ayes' theorem.
ConRidor an experiment with only two possible out
comes, that i t /I and A'. If at each performance of
the experiment (i.e., each trial), P(/t) remains the
Hume, then the repeated trials are known as Bernoulli
trials.
NOTE: When clincussing HeruoxiIH trials, it in CUH-
tomary to refer to one of the two possible ut<iome
an a success and to the other as a failure.
Let fo(^; n, p) denote the probability that n Bernoxilli
trialn will result in exactly x HUCCOSSOS and n j? faii-
tireH when the probability of a success at each trial is p
and the probability of a failure at each trial in # I
p. Then 6(x; n, p) (^(n ? as)p*ff ft ""*.
NOTE: Probabilities giveti by &(#:; n, p) are often re
ferred to as binomial probabilities. Kvaluation of bi
nomial probabilities can be a tedious task. However,
tables (10, II) are available and can be xined to good
advantage,
PROBABILITY DISTRIBUTIONS
33
MATHEMATICAL EXPECTATION
Definition 3.12 Consider the function 6(05; n, p) introduced in Theo
rem 3.11. The expected value of x, which will be de
noted by E[x], is defined by E[x}-=np. That is, the
expected number of successes in n trials is defined to
be np, even if it may be impossible to observe such a
number.
PROBABILITY DISTRIBUTIONS
Definition 3.13
Definition 3.14
Definition 3.15
Theorem 3.12
Theorem 3.13
For any random variable X, we will denote the
P(X<x} by F(x}. Further, F(x) will be referred to as
the cumulative distribution function (c.d.f.) or, simply,
as the distribution function (d.f.) of the random vari
able X.
If X is a discrete random variable, we will define
the probability function (p.f.) of the random variable
X to be /(#)== P(.X === re) .
If X is a continuous random variable, we will define
the continuous probability density function of the ran
dom variable X to be f(x) dF(x)/dx.
For a discrete random variable X, F(x} = ]Cy** /(?/)
NOTE: If F(x) is defined, f(x) may be obtained by
differencing. However, close attention must be given
to equality and inequality signs.
For a continuous random variable X,
F(*) - f * f(y)dy.
*/ 00
Theorem 3.14 F(x) has the following properties:
(1)
(2)
(3)
if
Theorem 3.15
/(a?) has the following properties:
(1) /te)^0.
(2) 2^ /(#) = 1 if X is a discrete random variable,
alia: ?
or
f(x)dx = 1
Theorem 3.16
if X is a continuous random variable.
For a discrete random variable X,
P(a < X < 6)
-F(a) =
34
CHAPTER 3, THEORY IN PROBABILITY AND STATISTICS
Theorem 3.17 For a continuous random variable X,
P(a < X < 6) = F(b) - F(a) - f f(x)dx.
J a
Definition 3.16 For two random variables X and Y, we will denote
P(X<x, Y<y) by F(x, y). Further, F(x, y) will be
referred to as the joint cumulative distribution function
of X and F,
Definition. 3.17 If X and Y are discrete random variables, the joint
probability function of X and Y will be denoted by
Definition 3,18 If -X" and Y are continuous random, variables, the joint
probability density function of X and Y will be de
noted by
dxdy
Theorem 3.18 For discrete random variables X and F,
Theorem 3.19 For continuous random variables X and F,
f** /-i/
F(x, y) == I ds I f(sj t)dt.
Theorem 3.20
Theorem 3.21
F(x, T/) has the following properties:
111 ft \ """ OO ?/ ) EST ft ( *JT ' CO ) ssss /'[ CO "- OO ) ?sa O
v^.*/*^ ;^// * V**^? / *\ ? / v '*
/O\ t/T/ -^s *>*. N 1
(^; /* (, 00 ; OO; as 1.
(ti) F(c&, ?/) ^^O/)? which in the marginal cumu
lative distribution function of F,
(4) F(*r, <^o) /'\(a;) ? which in the marginal cutnula~
tivr, distribution function of A*"*
^C^j ?/) h* 1 ^ the following properties:
(1)
(2)
- 1 or
Kit X tUt |/
n whether A" and F are discrete or
Theorem 3*22 If X and F arc discrete, then
EXPECTED VALUES
35
P(a < X < b, c < F <
= F(b, d) - F(b, c)
- F(a, d) + F(a, c)
Theorem 3.23
If X and Y are continuous, then
P(a <X<b,c<Y<d) = F(b, d)
- F(a, d}
, c)
F(a, c}
/b s* d
dx I f(oc, y)dy.
.j. +s n
Definition 3.19 Associated with the marginal c.d.f.'s in Theorem
3.20, we have marginal p.d.f/s (or p.f.'s) denoted by
/i(V) and/ 2 (y), respectively.
Definition 3.20 Conditional p.d.f.'s (orp.f.'s) and c.d.f.'sare defined as
follows :
(i)
(2)
(3)
(4)
Theorem 3.24 If X and Y are statistically independent, f(&,y}
=/iO) -/ 2 (2/) and F(x, y] =F x (x) -F 2 (?/).
NOTE: All the definitions and theorems given for
two random variables may easily be extended to three
or more random variables.
EXPECTED VALUES
To aid in the description of probability distributions, it is helpful to
know something about their properties. Of special importance are those
properties associated with the concept of mathematical expectation.
The expected value of any function of a random variable is defined
as the weighted average (weighted by the probability of its occurrence)
of the function over all possible values of the variable. Since expected
values are used so much in statistics, a special notation has been de
veloped. The symbol E[ - - ] will be used to denote the expected value
of whatever appears within the brackets. For example, the expected
value of a function B(X} will be denoted by E[0(X)].
Definition 3.21 &[0(X)] = ^ ()'/(), * discrete
all a;
/oo
6(x)
BO
x y x continuous.
Definition 3.22 E[6(X, F)] = 23 X) #<>> 3>)/O, y)> * and y discrete
all x all]/
36 CHAPTER 3, THEORY IN PROBABILITY AND STATISTICS
/OO X OO
dx I Q(x 7 y)f(x, y}dy^ x and y continuous.
^-00 *J 00
Theorem 3.25 For expected values, the following properties hold:
(1) E[c] =c where c is a constant.
(2) E[c6(X, Y)]==cE[0(X, F)].
[k ~~| X*
T^ c<Q-(X Y} = y^ c>F\9 (Y V}1
/ * 6tt7^^-V , * / X / C'l^iyA*^ > */.!
i-1 J t_l
Definition 3.23 The /cth moment with respect to the origin of X Ls
denoted by A^ J^pf*]-
Definition 3.24 MI is known as the mean, and it is commonly denoted
by/*.
Definition 3.25 The /cth moment about the moan, or the A;th central
moment, of X is denoted by MA- = A T [(^Y /*)*].
Definition 3.26 /xa is known as the variance, and it is commonly de
noted by or 2 .
Definition 3.27 The positive square root of the variance, &, is known
as the standard deviation.
Theorem 3.26 <r*=*iJi!i~-tA***K[X*] (ft[X])*.
Definition 3.28 When dealing with two random variables, the product
moments of X and F are defined by /*J t , = J$[X r Y*].
Definition 3.29 The central product moments are defined by
- Mv>)*i where a v = / LY 1 and
Definition 3.30 MIA is known as the covariance of J\T and F, and it is
commonly denoted by <r XK .
NOTE: The variance of X, <r^ t is sometimes written
as &XX* Similarly, ^, <r rr * These alternative nota
tions show the close relation between variances and
eovarianeoB.
Definition 3-31 The product moment correlation between -Y and F is
defined by PXY^VXY/VX&Y* ** should be rioted that
Theorem 3,27 J
Theorem 3*28 If X and F are statistically independent,
OTHER DESCRIPTIVE MEASURES
Definition 3,32 The value of a? such that F(JT) p m called the 100^
fracttte of the distribution of the random variable A".
Definition 3*33 When p=tK5 in Definition 3.32, the corresponding
vahie of x is known as the median of the distribution.
Definition 3*34 The mode of a distribution is thut value of ^ for which
is a maximum.
PROBLEMS 37
SPECIAL PROBABILITY DISTRIBUTIONS
Certain distributions occur so often in statistical problems that they
merit special attention. Some of these are tabulated in Tables 3.1 and
3.2. Since most applications involving these distributions require the
use of probabilities associated with the distributions, it is convenient
to have available adequate tables of such probabilities. Accordingly,
tables for the Poisson, standard normal, chi-square, "Student's" t, and
F distributions are presented in Appendices 2 through 6. Each of these
tables is given in cumulative form, that is, in terms of the cumulative
distribution function, so that the reader will have to learn only one
method of reading the tables.
Problems
3.1 A sample of 3 TV sets is selected from a lot of 30 sets. If there are 5
defective sets in the lot, what is the probability the sample will con
tain no defectives? 3 defectives? 1 defective and 2 nondefectives?
3.2 A buyer will accept a lot of 10 TV sets if a sample of 3, selected at
random, contains no defective sets. What is the probability of accept
ing a lot of 10 that contains 5 defectives?
3.3 An electrical circuit consists of 4 switches in series. Assume that the
operations of the 4 switches are statistically independent. If for each
switch the probability of failure (i.e., remaining open) is 0.02, what is
the probability of circuit failure?
3.4 Rework the preceding problem for the case where the circuit consists
of 4 switches in parallel.
3.5 Defects are classified as type A, B, or C, and the following probabilities
have been determined from available production data: P(A)=0.20,
P(#)=0.16, jP(C)=0.14, P(.AB)=0.08, PC-AC) =0.05, P(BC)-0.04,
and P(ABC} =0.02. What is the probability that a randomly selected
item of product will exhibit at least one type of defect? If an item
exhibits at least one type of defect, what is the probability that it
exhibits both A and B defects?
3.6 An electrical assembly consists of two parts connected in series in the
order: A followed by B. The probability that part A is defective is
0.025 and the probability that part B is defective is 0.011. What is the
probability of having a defective assembly? A nondefective assembly?
An assembly that fails only because part B is defective?
3.7 Suppose the probability that a certain piece of air-borne electronic
equipment will not be in working order after its first flight is 0.40,
and the probability of failure drops to one-half its previous value
after each succeeding flight. (Assume no repair and replacement,)
What is the probability the equipment will be in working order after
three flights? After four flights given it has survived two flights?
3.8 Consider a four-engine aircraft (two on each wing) where the prob
ability of an engine failure is 0.05. Assume that the probability of one
engine failing is independent of the behavior of the others. What is
the probability of a crash if the plane can fly on any two engines?
If the plane requires at least one engine operating on each side in order
to remain in the air?
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PROBLEMS 41
3.9 Suppose 3 defective dry cells are mixed in with 7 nondefectives, and
you start testing them one at a time. What is the probability that you
will find the last defective on the sixth test?
3.10 Three operators (A, B, and <7) alternate in operating a certain ma
chine. The number of parts produced by A, B, and C are in the ratio
3:4:3 and, of the parts produced, 1 per cent of A's, 2 per cent of B's,
and 5 per cent of C*B are defective. If a part is drawn at random from
the output of their machine, what is the probability it will be de
fective?
3.11 Referring to Problem 3.10, what is the probability that, if a defective
part is selected, it was produced by A? by B? by C?
3.12 Iif(x 9 ?/)=exp { (rc+2/)} for x>0, y >0, find:
(a) /,(x), (b) My}, (c) Fi(x), (d) F*(y} 9 (e) f(y\x), (f) f(x\y),
(g) F(x, 2/), (h) F(y\x), (i) F(x\y). .
3.13 If /(a, $/) =3z for <y <rc, <x <1, find the same functions as asked
for in the preceding problem.
3.14 If f(x 9 y) =24?/(l x y} over the triangle bounded by the axes and
the line x + y = 1, find the same functions as asked for in Problem 3.12.
3.15 During the course of a day, a machine turns out either 0, 1, or 2
defective items with probabilities |, f, and , respectively. Calculate
the mean and variance.
3.16 Given that the number of accidents occurring at a particular inter
section between 10:00 P.M. and midnight on Saturday is 0, 1, 2, 3, or 4
with probabilities 0.90, 0.04, 0.03, 0.02, 0.01, respectively, determine
the expected number of accidents.
3.17 Suppose that the life in hours of a certain type of tube has the p.d.f.
/(#) = a /x* } #>500, and /0*0 =0, x <5QQ. Find the c.d.f. Determine
the mean and variance. What is the probability a tube will last at
least 1,000 hours?
3.18 A submarine carries three missiles. Assuming the only error is in one
direction (e.g., a range error but no sideways error) and that a hit
within 40 miles of the target is considered a success, compute the
probability of a successful operation (i.e., an operation in which at
least one hit is a success) if all three missiles are launched and the
error p.d.f. is:
/(#) = (lOO+aO/10,000 - 100 <x <0
(100 aO/10,000 0<x<100
= elsewhere.
3.19 Referring to the previous problem, the submarine can carry eight
missiles of a smaller sisse. However, in this case a hit must be within
15 miles to be successful. Assuming the same p.d.f., should the light
or heavy missiles be used?
3.20 A service station will be supplied with gasoline once a week. Its weekly
volume of sales in thousands of gallons is predicted by the p.d.f.
f(x') = 5(1 x) 4 for <x <1. Determine what the capacity of its under
ground tank should be if the probability that its supply will be
exhausted in a given week is to be 0.01.
3.21 Show that the correlation between two random variables is if they
are statistically independent.
3.22 Let X have the marginal density /i(#) =1 for % <x <, and let the
conditional density of Y given X be
42 CHAPTER 3, THEORY IN PROBABILITY AND STATISTICS
= 1; x <y <1 x, <x
= 0; elsewhere.
Find the correlation, between X and F. Discuss the relationship be
tween eon-elation, and statistical independence.
3.23 A process is producing parts that are, on the average, 1 per cent defec
tive. Ten parts are selected at random from the process and the
process is stopped if one or more of the ten are defective. What is the
probability that the process will be stopped?
3.24 In inspecting 1,000 welded joints performed by a certain welder using
n specific process, 150 defective joints were discovered. If the welder
is about to weld 5 joints, what is the probability of getting no defective
joints? of one? of two? of two or more? Discuss any assumptions you
make in solving this problem.
3.25 A large number of rivets is used in assembling an airplane. It has
been determined that the probability distribution for the number of
defective rivets is Poisson with X = 2. Find the probability that the
number of defective rivets in a plane will be no more than two.
3.26 Suppose there is an average of 1 typographical error per 10 pages in
a certain book. What is the probability that a 30-page chapter will
contain no errors?
3.27 A telephone vswitchboard handles, on the average, 600 calls during the
rush hour. The board eaix make a maximum of 20 connections per
minute. What is the probability the board will be overtaxed in any
given minute during the rush hour?
3.28 Assuming a normal distribution, find:
(a) P( 3 < Y < 1) ; given ;x = ; tr == 1 .
(6) 7>( 3<F<0.r>); given M 0, <rl.
( C ) p( 8<F <0); Rivon/A 2, <r*^4
02) P(4<F<50); given M* 0.1 , <r*4
O) J^FSrS); Rivon/A-0, <r a l
CO /* ( Y < - 3) ; gi ven M 2, cr* - 4.
3.29 AnBuming a clu-nquare (liHtributum, find:
c)
) for vlf>
(r) P(23.8 <x s <3(K4) for v 24
(d) P(x
3.30 ARBuminp a ^-cliHtri{)utu>n, find:
(a) />C|| > 2.01 5) for p-5
(6) P(>2X)15) for ^-5
(c) ;>( 1.341 <2. 121) for p1
(rf) /*(^<l.r>) for 20.
3*31 AHHuming an /^-diHtribution, find:
for Vi-11, v t *tt
(r) W > 7,79) for ^ , v a - 11
(rf) 7* (0.221 </^S2.62) for n6,
REFERENCES AND FURTHER READING 43
3.32 The finished diameter on armored electric cable is normally distributed
with mean 0.77 inch and standard deviation 0.01 inch. What is the
probability the diameter will exceed 0.795 inch? If the engineering
specifications are 0.78 0.02 inch, what is the probability of a defec
tive piece of cable?
3.33 If the p.d.f. for the life of a certain type of component is /(#) = (1/100)
exp { rc/lOO} for x >0, what is the probability that a randomly
selected component will last 400 hours? That it will last 400 hours
given that it has already survived 200 hours? If an assembly uses
three of these components in series, what is the probability that an
assembly incorporating three randomly selected components will not
fail because of component failure?
3.34 The hazardrate is defined as /(#)/{ 1 F (or) Mf/O) = (1/0) exp { x/6}
for x >0, what is the hazard rate?
References and Further Reading
1. Anderson, R. L., and Bancroft, T. A. Statistical Theory in Research. McGraw-
Hill Book Company, Inc., New York, 1952.
2. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
3. Cramer, H. Mathematical Methods of Statistics. Princeton University Press,
Princeton, N.J., 1946.
4. Derman, C., and Klein M. Probability and Statistical Inference for Engineers.
Oxford University Press, New York, 1959.
5. Hald 7 A. Statistical Theory with Engineering Applications. John Wiley and
Sons, Inc., New York, 1952.
6. Irluntsberger, 13. V. Elements of Statistical Inference. Allyn and Bacon, Inc.,
Boston, 1961.
7. Kemeny, J. G., Snell, J. L,., and Thompson, G. L. Introduction to Finite
Mathematics. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1957.
8. Mood, A. M. Introduction to the Theory of Statistics. McGraw-Hill Book
Book Company, Inc., New York, 1950.
9. Mosteller, F., liourke, H. E. K., and Thomas, G. B. Probability and Statistics.
Addison- Wesley Publishing Company, Inc., Reading, Mass., 1961.
10. National Bureau of Standards. Tables of the Binomial Probability Distribution.
Applied Mathematics Series 6. U.S. Govt. Print. Off., Washington, D.C.,
1949.
11. rtomig, H. G., 5Q1OO Binomial Tables. John Wiley and Sons, Inc., New
York, 1953.
12. Wadsworth, G. P., and Bryan, J. G. Introd^lction to Probability and Random
Variables. McGraw-Hill Book Company, Inc., New York, 1960.
13. Whitesitt, J- El. Boolean Algebra and its Applications. John Wiley and Sons,
Inc., New York, 1961.
CH APTE R 4
ELEMENTS OF SAMPLING AND
DESCRIPTIVE STATISTICS
IN THIS CHAPTER we shall discuss the basic ideas of sampling and the
presentation of sample data. Certain useful statistics of a summarizing
nature will be defined and efficient methods of calculation outlined.
To begin the discussion of sampling, the reason for taking samples
should be mentioned. The reason is usxially one of the following:
(1) Due to limitations of time, money, or personnel, it is impossible to
study every item in the population; (2) the population, as defined, may
not physically exist; (3) to examine an item may require that the item
be destroyed.
Before proceeding to the actual mechanism of obtaining samples and
the analysing of data therefrom, it will be wise to define some terms
frequently encountered*
4.1 THE POPULATION AND THE SAMPLE
In statistical work it is important to know whether we are dealing
with a complete population of observations or with a sample of ob
servations selected from a specified population.
A population is defined as the totality of all possible values (measure
ments or counts) of a particular characteristic for a specified group of
objects. Such a specified group of objects is called a universe. Obvkmsly
a universe can have several populations associated with it. Some exam
ples of universes and populations are :
(1) The employees of Arizona State University as of 5:00 P.M. on
December 4, 1902.
(2) Associated with the preceding universe are many populations,
for example, the population of blood types, the population of
weights, the population of heights, etc,
(3) The univerae of all single-dwelling units in Tempe, Arizona, on
December 31, 1902,
(4) Associated with thin universe of single-dwelling tmits are such
populations as the number of rooms per unit, the number of
people residing in each unit, and so on.
(5) A universe may contain only one object, such an a piece of
steel pipe, and the population consists of all possible measure
ments of its inside diameter.
(6) A universe might consist of all vacuum tubes of a specific type
manufactured by a given manufacturer under similar condi
tions,
C44I
4.2 TYPES OF SAMPLES 45
(7) Populations associated with the preceding universe are:
lengths of life, function on test, etc.
These examples should suffice to impress upon the reader the impor
tance of clearly defining the population under investigation.
The concept of a sample, as opposed to a population, is very im
portant. A sample is just a part of a population selected according to
some rule or plan. The important things to know are: (1) that we are
dealing with a sample and (2) which population has been sampled.
If we are dealing with the entire population, our statistical work will
be primarily descriptive. On the other hand, if we are dealing with a
sample, the statistical work will not only describe the sample but also
provide information about the sampled population.
4.2 TYPES OF SAMPLES
There are several types or classes of samples encountered in practice.
The characteristics which distinguish one type from another are : (1) the
manner in which the sample was obtained, (2) the number of variables
recorded, and (3) the purpose for which the sample was drawn. The
last two characteristics listed are easily understood in any practical
situation although No. 3 is frequently not clearly stated and perhaps
even forgotten. The manner of obtaining the sample is very important
and will be discussed further.
Samples may be grouped into two broad classes when their method
of selection is considered, namely, those which are selected by judg
ment and those which are selected according to some chance mecha
nism. Samples selected according to some chance mechanism are
known as probability samples if every item in the population has a
known probability of being in the sample. In particular, if each item in
the population has an equal chance of occurring in the sample, then the
sample is known as a random sample.
Why are random samples preferred to subjectively selected samples?
An answer to this question may be formulated as follows: A good
sample is one from, which generalizations to the population can be
made ; a bad sample is one from which they cannot be made. To general
ize from a sample to a population, we need to be able to deduce from
any assumptions about the population whether the observed sample is
within the range of sampling variation that might occur for that popu
lation under the given method of sampling. Such deductions can be
made if, and only if, the laws of mathematical probability apply. The
purpose of randomness is to insure that these laws do apply. If we had
equally well-established and stable laws of personal bias, subjective
sampling could be used.
We can sample from different populations in various ways:
(1) A random sample may be drawn from a population specified
by a continuous probability density function. In this case, the
46 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
question of sampling with or without replacement does not
arise*
(2) A random sample may be drawn from an infinite population
specified by a discrete probability density function. Again, the
question of with or without replacement does not arise.
(3) A random sample may be drawn from 11 finite population
(specified by a discrete probability density function) where the
sampling is performed with replacement. Sampling with re
placement effectively makes the population infinite.
(4) If sampling from a finite population is performed without re
placement, we no longer have a random sample as defined
earlier. Sometimes, a "random" sample for this situation is de
fined as one in which each set of n objects has an equal chance
of being the sample of size n.
Other types of samples of a specialized type are sometimes en
countered. Two of these are:
Stratified Random Sample
The population is first subdivided into subpopulations or strata.
Then a simple i^andom sample is drawn from each stratum.
Systematic* Random Sample
Consider the N units in the population to be arranged in some
order. If a sample of sixe n is required, take a unit at random from
the firt k^N/n units and then take every fcth unit thereafter.
Having defined the various types of sampling frequently encountered,
the following caution is noted: The methods of analysis will not be the
same for each type of sampling, (treat care must bo exereised to use the
proper method of analysis; fuihtrc to do HO oan load to serious errors in
judgment when the decision-making stage is reached.
4,3 SAMPLING; FROM A SPECIFIED POPULATION
How do we go about .selecting a sample from a specified population?
Some examples will serve as explanation: (I) Suppose the population
consists of only two values*. One of them can be selected at random by
tossing ati uubiaHcd coin. (2) Consider a population consisting of 100
items. One hundred numbered tickets (corresponding to our population
of items) can be placed in a howl and tickets selected in a chance
manner. (3) In the previous example, the sample values could have been
selected using a table of random numbers.
To Illustrate the use of a table of random numbers, consider the
problem of obtaining a sample of n**r* batteries from a lot of A^ 25,
First, number the batteries: 01, 02, , 25. Second, refer to a table
of random numbers such an given in Appendix 7 and proceed through
the following steps.
(t) Select by any method one of the four pages of tabled values.
(2) Without direction, bring a pencil point down on the printed
page BO an to hit a random digit.
4.4 PRESENTATION OF DATA 47
(3) Read this digit and the next three to the right, for example,
2167.
(4) Let the first two of these specify the row and the last two the
column.
(5) Go to this point in the table of random numbers and read the
specified digit and the next one to the right. This reads 73.
However, the only possible numbers of use in the specified
problem are 01, 02, - - ,25. Thus, it is necessary to run down
the column until five suitable numbers are observed. In order
the numbers observed are 73, 48, 54, 01, 18, 38, 60, 70, 44,
30, 41, 86, 23, 64, 31, 71, 68, 64, 13, 12. The numbers specifying
the five batteries to be included in the sample have been
underscored.
(6) Appropriate changes should be made in step No. 5 to handle
different problems.
4.4 PRESENTATION OF DATA
Having obtained a random sample from a specified population, some
way of reducing it to an understandable form is called for. To illustrate
the usual techniques for presenting such data, consider the data in
Table 4.1.
In this form the data are, to say the least, confusing. It is not easy to
visualize any pattern in the observed values, nor is it easy to estimate
the average function time. We find it convenient, therefore, to arrange
the values in a frequency distribution as in Table 4.3. To accomplish
this, we first make use of a tally sheet as shown in Table 4.2. Inciden
tally, Table 4.3 provides us with an array, that is, the values arranged
in order of magnitude.
Upon examination of Table 4.3, we note that all the observations are
greater than or equal to 59 milliseconds and less than or equal to 70.5
milliseconds. That is, we have established the range of our data. Fur
ther, we can roughly estimate the average function time to be 65 milli
seconds.
However, since it takes too long to scan all the values in Table 4.3,
the data are still in rather cumbersome form. To remedy this, it is
customary to condense the data even more by tabulating only the fre
quencies associated with certain intervals, usually referred to as class
intervals. To set up class intervals, a good working rule is to have no
fewer than 5 and no more than 15 intervals. Also, the limits of the class
intervals should be chosen so that ther^ is no ambiguity in assigning
observed values to the classes. This latter requirement is most easily
satisfied by: (1) selecting class limits which carry one more decimal
place than the original data, or (2) proper use of inequality and equality
signs. We shall adopt the second of these two procedures in this text.
Using class intervals of width 1 millisecond, we get the data in the form
of Table 4.4. In this table we have used the letter X to represent the
various function times in milliseconds. To interpret the values and
48 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
TABLE 4,1-Function Times of 201 Explosive Actuators
Measured in Milliseconds
(Hypothetical Data)
64.0
61,5
69.0
65.25
69.0
66.0
63.5
65.25
66.25
67.25
67.25
62.5
61.75
63,5
63.75
66.5
66.0
65.5
65.25
66.5
64,5
67.75
64.5
68,0
63 . 75
68.0
70.5
68.0
65.0
62.0
62 . 75
61.5
60.0
65.75
66.0
62.0
65 . 75
60.75
63.75
62,0
70.25
64.75
68.5
65.0
66,5
64.0
67.0
67.0
63.0
64.0
67.0
63.25
65.25
67.5
65,0
67.5
64.5
68.0
63 . 5
68,75
63.0
66.25
67.0
65.25
64,0
65.25
63.0
67.0
65,5
62.0
64.5
66.25
65.0
63.75
67.5
65.5
64.75
67.0
68.0
59.0
64.5
67.0
67,75
63.25
63.25
65.5
64.0
67.0
64.5
67,5
65,0
61,0
64.5
63.0
66.5
66.0
65.0
61.25
69.5
64,0
68,0
64.5
66.5
64,25
65.0
62.25
63.5
63.0
67,0
65 . 25
65,0
65 .
65.25
65.25
63,0
65 . 5
65,0
62,0
64,0
62.5
64,75
61.5
62 . 75
68.5
63.5
63,0
64,5
67,0
61.75
66.25
64.75
65.5
62.75
68,5
61,5
63,0
65,5
65.5
63.0
65.5
66.75
69.5
65.25
63 . 5
66.0
62.25
62,5
61.5
68,0
63 . 75
66.0
64.0
67,0
67.75
65.25
67.75
68.0
63.5
63,25
63,0
61.75
69,0
65.0
62.5
62,0
64,75
64.0
66.75
66,0
64.5
64.25
62.5
66.5
66,75
64.5
60.0
65,0
66.0
64.5
66.25
65*75
65.5
64,5
62.0
65,25
64.25
63.0
64.0
66.75
65 . 25
63,75
67.0
61.0
70.0
70.0
65 , 5
65 25
64.5
67,5
65.75
70.0
frequencies, we proceed as follows: One actuator had a function time
of more than 58 milliseconds but less than or equal to 59 miHinoconda;
two actuators had a function time of more than 59 miIliecondB tmt leB
than or equal to 00 milliseconds; ami so on. Pleane note that we have
less information available in Table 4,4 than in Table 4.3, This IB be
cause we no longer know the individual values but only in which clans
interval they fall. But the loa in accuracy is balanced to &ome extent
by the gain in conciseness. The column headed "relative frequency"
tella us what proportion of the total observations fall in each class. The
valuen are foxmd by dividing each elans frequency by the total fre
quency.
4.4 PRESENTATION OF DATA
49
Conforming to the adage that a "picture is worth ten thousand
words/ ' we often represent our distribution by a chart or frequency
histogram. This is illustrated by Figure 4.1. The dotted line pictures a
frequency polygon. Note that the frequency histogram is formed by
erecting rectangles over the class intervals, the height of each rectangle
agreeing with the class frequency if the left-hand scale is read, and with
the class relative frequency if the right-hand scale is read. The fre
quency polygon is formed by joining the midpoints at the tops of the
rectangles.
It is also to be noted that the frequency histogram and polygon, as
well as the frequency distribution, give us not only an estimate of the
average value but also an idea of the amount of variability present in
the data.
Another convenient way of tabulating data is to prepare a cumu
lative frequency distribution showing the number of observations less
than or equal to a specified value. The figures are obtained by adding, in
cumulative fashion, the frequencies recorded in Table 4.4. This is
illustrated in Table 4.5. The graph which arises from this table is shown
TABLB 4.2-Tally Sheet for Data of Table 4.1
Function
Time
(MS)
Tally
. Fre
quency
Function
Time
(MS)
Tally
Fre
quency
59.0
1
1
65.0
-t*44- 1~H4- 11
12
^Q 9 c;
fC OC
-4*4_li -^-4_l_l 1111
1 4-
D y . Z.O
^O d
\JO . Z.O
1^^ ^
jrTTtHfc- JL JTJL J.- JL JL JL JL
-1-4-4J -4-4-U 1
JLrr
ov . o
59.75
oo . o
65.75
JL JL JL JL"I~-A 1
1111
4
60.0
11
2
66.0
-H44. Ill
8
60.25
66.25
-H44-
5
60.5'
66.5
H44^1
6
60.75
1
1
66.75
1111
4
At n
1 1
x:7 fk
+4-tJ_ -4-4>iJ t 1
1 ?
O JL . U
61.25
1 1
1
1
\j i \j
67.25
JL JL Jl JL- XT. JL JU J 1
11
JL ^
2
61.5
-ir4-
5
67.5
-H44-
5
61.75
111
3
67.75
1111
4
62.0
-H4-3L 11
7
68.0
-Hb44- 111
8
62.25
11
2
68.25
62.5
-H4-4.
5
68.5
111
3
62.75
Ill
3
68.75
1
1
63.0
-fr-H4~-H44~ 1
11
69.0
111
3
63.25
1111
4
69.25
63 . 5
-W44- 11
7
69.5
11
2
63.75
-1HH4 1
6
69.75
AA. n
J M-4-l_-1H~4~4-
10
7O
1 1 1
3
V>*x VJ
64.25
i. A 1 JL' 1 IjfTTC
111
JL \J
3
/ \j . \j
70.25
j * i
1
1
f\A. X
-4J.J1 M-4-4-J. 1111
1 4.
7O S
i
1
Q~r * O
64.75
J/TTTC I . J. 1 I"T 1 JL 1 JL
-H4JL.
JL Tt
5
/ \J . vJ
i
TABLE 4.3-Frequency Distribution for Data of Table 4.1
Number of Actuators
Function Time
Exhibiting Given Function
Relative Frequency
(MS)
Time = Frequency (f)
(r. f.)
59.0
1
0.005
60.0
2
0.010
60.75
1
0.005
61.0
2
0.010
61.25
1
0,005
61.5
5
0,025
61.75
3
0.015
62.0
7
0.035
62 . 25
2
0.010
62.5
5
0.02S
62 . 75
3
0.015
63.0
11
0.055
63.25
4
0.020
63.5
7
0.035
63 . 75
6
, 030
64.0
10
0.050
64.25
3
0.015
64.5
14
0.070
64.75
5
0,025
65.0
12
0.060
65.25
14
0.070
65 . 5
11
0.055
65 . 75
4
0.020
66.0
8
0,040
66.25
5
0.025
66,5
6
0.030
66.75
4
0.020
67,0
12
0,060
67.25
2
0.010
67.5
5
0,025
67 . 75
4
0,020
68.0
8
0.040
68 . 5
3
0,015
68.75
1
0.005
69.0
3
0.015
69.5
2
0,010
70.0
3
0.015
70.25
1
0.005
70.5
1
0.005
Totals
201
1.005*
Total exceeds 1 .000 because of errors of rounding*
C503
TABLE 4.4-Frequency Distribution (Using Class Intervals) for Data
of Table 4.1
Function Time
(MS)
Number of Actuators
With Function Time
In Specified Class
Interval = Frequency (f)
Relative Frequency
(r. f.)
58 < X < 59
1
0.005
59 < X < 60
2
0.010
60 < X < 61
3
0.015
61 < X < 62
16
0.080
62 < X < 63
21
0.104
63 < X < 64
27
0.134
64 < X < 65
34
0.169
65 < X < 66
37
0.184
66 < X < 67
27
0.134 ,
67 < X < 68
19
0.095
68 < X < 69
7
0.035
69 < X < 70
5
0.025
70 < X < 71
2
0.010
Totals
201
1.000
-4O
30
g20
Ul
ct
1O
-'V
0.20
0.15 >:
Ul
O.1O 2l
ui
0.05
O.OO
58 60 62 64 66 68 7O
TIME (IN MILLISECONDS)
FIG. 4.1 Frequency histogram and polygon plotted from Table 4.4.
C511
52 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
TABLE 4.5-Cumulative Frequency Distribution Formed From Table 4.4
Function Time
(-30
Number of Actuators With
Function Time Less Than
or Kqual to the Specified
Value == Cumulative
Frequency (c.f.)
Relative Cumulative
Frequency (r.c.f.)
58
0.000
59
1
0.005
60
3
0.015
61
6
0.030
62
22
0.109
63
43
0.214
64
70
0.348
65
104
0.517
66
141
0.701
67
168
0.836
* 68
187
0.930
69
194
0.965
70
199
. 990
71
201
1 ,000
in Figure 4.2 and is quite helpful in interpreting the observed data.
Note the cumulative (ogive) curve is plotted by joining the right-hand
endpoints at the tops of the rectangles, This curve (see clotted line) in
formed as just mentioned because it represents the cumulative fre
quency up to and including the upper clans limit.
4.5 CALCULATION OF SAMPLE STATISTICS
If a satnple is to be described in any reasonable manner, it m desirable
to calculate certain representative values which {summarize a great deal
2OOr~
t2O
u.
>
80
2E 4O
O
O
eo 62 64 66
TIME (IN MHJJSECONOS)
7O
o.eoS
&
40.403:
o
FIG. 4.2 Cumulative frequency histogram and polygon
plotted from Table 4*5*
4.6 THE ARITHMETIC MEAN 53
of information. Not all of the representative values to be described in
the following pages are of equal importance. However, we have gone
into considerable detail in defining them all so that the reader will be
aware of their existence, uses, advantages, and disadvantages.
4.6 THE ARITHMETIC MEAN
It is not surprising that the ordinary arithmetic mean is the most
common of these representative values. The sample mean, denoted by
X , is defined as the arithmetic average of all the values in the sample.
The formula for calculating the sample mean is
X (X, + - - - + Xn)/n = X Xt/n = Z) X/n (4.1)
i=l
where there are n observations in the sample,
Example 4.1
Given the sample values 3, 4, 2, 1, and 4, calculate the mean.
The above example illustrates the method of computing the arith
metic mean. It is to be noted that the arithmetic mean is affected by
every item in the sample and is greatly affected by extreme values.
Two interesting properties of the arithmetic mean are: (1) the sum of
the deviations from the arithmetic mean is zero, and (2) the sum of the
squares of the deviations from the arithmetic mean is less than the sum
of the squares of the deviations from any other value.
As might be expected, the arithmetic mean has both advantages and
disadvantages. Its advantages are: (1) it is the most commonly used
average, (2) it is easy to compute, (3) it is easily understood, and (4) it
lends itself to algebraic manipulation. The one major disadvantage of
the arithmetic mean is that it is unduly affected by extreme values and
may therefore be far from representative of the sample.
Before proceeding to a second representative measure for describing
samples, it will pay us to look at methods of calculating the arithmetic
mean when our data are in the form of a frequency distribution. If for
each different value of X we have a frequency/, then the sample mean
is given by
2- = '
/I+/2+ * +fd
n
ai
where there are d different values of X.
54 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
Example 4.2
_ The data in Table 4.3 are of the typo just described. Then
X = {(1)(59.0) + (2) (60.0) + - - - + (1)(70.5) }/20l = 13,071.75/201
= 65.034 milliseconds,
Many times oxir data appear in frequency tables where we no longer
know the actual values of the observations but only to which class in
terval they belong. In these instances, the best we can do is to approxi
mate the sample mean. To obtain this approximation we assume that
the values in a particular class interval are xmiformly distributed over
the interval. 1 This permits us to use the midpoint for each observation
in the interval when calculating the mean. Thus, if we denote the mid
point of the tth interval by *, and there are k intervals, the sample
mean is approximately
"V s r^ ^^ l +* ' + /&& ^ .-, ^ A.^^^ >. v
A = - - - ---.- _ _-,-.._-_-_ _ ^- ^4.o)
y i -j- - * + jk n n
Example 4.3
Considering tho data of Table 4,4, it is seen that 3fs { (I)(58.5)
+ (2) (59.rO + - + (2)(70.f>) }/201 - 13,032.5/201 - 4.38 milli-
Short-cut methods of ealexilation for use when machines are not
available are summarized in liquations 4.2a and 4.tta:
(w) (4 , 3a)
n
where JTo and fo are arbitrary origins, # A*" X^ z"= ({ f u )/, and
t(j is thci width of a class intorvah
Example 4.4
-Y / ^ fZ
to
3
_20
20
5
10
30
8
40
2
10
la
_ ?,
-90
In the above table, A'o 8 "^. Therefore
writorn anHXtmt^ that all the valuon in an inti h rval are concentrated at
the midpoint.
4.8 THE MEDIAN 55
Example 4.5
Class Interval f i fi
$<X<\.5
3
10
2
6
15<X<25
5
20
1
5
25<X<35
8
30
35<X<45
2
40
1
2
18
9
Thus ^^30+ ( 9/18) (10) =25.
4.7 THE MI ORANGE
Another representative value of importance, especially when a quick
average is needed, is the midrange. The midrange is defined as
^ -^-min ~T" -^-max , ^ ,.
MR = (4.4)
where X^^ is the smallest (minimum) sample value and X max is the
largest (maximum) sample value. It must be realized that even though
the midrange is quick and easy to compute, it is often inefficient be
cause all information contained in the intermediate values has been
ignored. Also it can be quite unrepresentative if either the smallest or
largest value is decidedly atypical of all the data.
4.8 THE MEDIAN
A representative value frequently employed as an aid in describing a
set of data is the median. The sample median, denoted by M, is the
[(n+l)/2]th observation when the values are arrayed in order of mag
nitude. Theoretically, one-half the observations should have a value
less than the median and one-half the observations should have a value
greater than the median. However, in practice it does not always work
out quite this way due to clustering of the observations (see Example
4.6). Regardless, the median is important as a measure of positioner
location.
Example 4.6
If we consider the data of Table 4.3 where n = 201, the median is the
(201 + l)/2 = 101st item in the array. Counting down the frequencies in
Table 4.3 we find the 101st item to be 65. Thus M = 65 milliseconds.
Example 4.7
Given the sample (2, 3, 4, 6, 6, 7), the median is the (6 + l)/2 =3. 5th
observation in the array. To avoid ambiguity, it is agreed that the
median will be halfway between the third and fourth observations in
the array. Thus Af = 5.
When data are grouped in class intervals as in Table 4.4, the median
56 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
cannot be located exactly. However, if we assume that the observations
in each class interval are uniformly distributed over the interval, a close
approximation to the median may be obtained. The first step is to lo
cate the class in which the median belongs : This is done by adding up
the class frequencies until we find the class which contains the
[(w-f-l)/2]th observation. Of course, if a cunrulative frequency distribu
tion has been, formed as in Table 4.5, the median class is easily located.
Then, the sample median may be approximated using the equation
M &
+ 1
c
2
(4-5)
where
L M = lower limit of the median class,
n = number of observations in the sample,
,JJ=ssum of the frequencies in all classes preceding the median class,
f M == frequency in the median class, and
t# = width of the median class.
Example 4,8
Considering the data of Table 4.4, we sec that
7QV
64.01 milliseconds.
/ini . 7Q\
M & 64 + ( ) (1)
\ 34 /
It is possible to approximate the median graphically from a cumula
tive frequency (ogive) curve using the relative curmiiative frequency
(r.c.f.) scale. This will be illustrated in Section 4.9.
The median, a measure of position, is affected by the number of items
bxit not by the magnitxide of extreme values. Two characteristics of the
median which are of interest are: (1) the sum of the absolute values of
the deviations from the median is less than the sum of the absolute
values of the deviations from any other point of reference, and (2) theo
retically the probability IB that an observation selected at random
from a set of data will be IOHH than (greater than) the median.
Some advantages and disadvantages of the median with which one
should be familiar if he wants to make proper une of this statistic will
now be mentioned. The advantages are: (1) it in easy to calculate, and
(2) it is often more typical of all the observations than is the arithmetic
moan sinco it is not affected by extreme values. The disadvantages* are:
(1) the items mtint be arrayed before the median can be obtained and
(2) it does not lend itself to algebraic manipulation*
4.9 PERCENTILE, DECILE, AND QUARTILE LIMITS
In this section we shall consider locating various values which divide
i he population or sample into groupn according to the magnitude of the
4.9 PERCENTILE, DECILE, AND QUARTILE LIMITS 57
observations. The median (see Section 4.8) was obviously one such
value since it divided the array into two groups, each containing 50 per
cent of the observations. We now wish to determine other such values.
Let us consider the most general case first. If we want to locate a
value, say P#, such that p per cent (0 <p < 100) of the observations are
less than P p and 100 p per cent of the observations are greater than P p ,
we call Pp the upper limit of the pth percentile, and approximate P& by
the [p(n+ l)/100]th observation in the sample array if we start counting
from the smallest value. For example, P$7 is the upper limit of the 67th
percentile and is approximated by the [67(n+l)/100]th observation in
the sample array. Similarly, P 6 e is the upper limit of the 66th percentile
and is approximated by the [66(n+l)/100]th observation in the sample
array. If we refer to the 67th percentile, we mean the interval from P 6 e
to P 6 7 in general, the pth percentile is the interval from P p ~i to P p .
(NOTE: Percentile limits are special cases of the fractiles introduced
in Definition 3.32.)
Example 4.9
If we consider the data of Table 4.3, what is the upper limit of the 80th
percentile? P 8 o is approximately the 80(201 + 1)/100 = 161.6th observa
tion in the array which is 67 milliseconds.
Example 4.10
What is the upper limit of the 35th percentile in the sample given in
Example 4.7? P 3S is approximated by the 35(6 + l)/100 = 2.45th obser
vation in the array. To avoid ambiguity, we agree to set P$$ forty-five
one hundredths of the way from the second to the third observation
in the array when we count from the smallest value. Thus P 3 s is 0.45 of
the way between 3 and 4, that is, jP 35 = 3.45.
The reason for the word percentile should now be clear : if we locate
Pi> Pz, - - , PQQ, we have (theoretically) split our array into 100 parts
(percentiles) , each containing 1 per cent of the observations.
The meaning of such terms as decile limits and quartile limits (see
the heading of this section) is now almost obvious. The decile limits
DI, D 2; - - ; Z>9 theoretically split our array into ten parts (deciles),
each containing 10 per cent of the observations. The quartile limits
Qij Qa, and <2a theoretically divide our array into four parts (quartiies),
each containing 25 per cent of the observations. No particular methods
of calculation will be presented for decile or quartile limits since they
are only special cases of percentile limits. This is clear once we observe
that
Pio * Di P4o = Dt P 75 = Q 3
P 2 o = Dz Pfeo == >6 = Qa = M Pso =* >s
P 2 s = Qi Peo = >e Poo = D g
P 30 = >3 ^70 #7
58
CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
In Section 4.8 we mentioned the possibility of estimating the median
from, the graph of the relative cumulative frequency distribution. To
illustrate this technique we shall undertake the location of percentile
limits in general. Consider the cumulative frequency curve of Figure
4,2 which we reproduce here as Figure 4.3. The procedure is as follows:
Pp being the upper limit of the pth percentile which says that p per cent
of the observations are less than or eqxial to P p , all we have to do is
locate p/100 on the relative cumulative frequency scale, draw a hori
zontal luie from this point to the ogive curve, and from here drop a
vertical lino down to the horizontal axis,
illustrated in Figure 4.3 for P$i and P&$.
200
thus locating P p . This is
UJ
9
^ 12O
u.
UJ
> 80
4O
/v-bc
QC
6O 62 64 66 68 7O
TIME (IN MILLISECONDS)
FIG- 4*3 Cumulative frequency (ogive) "curve*' plotted from Table 4.5,
4.1O THE MODE
Another valxio of aid in describing a sample is the mode* The mode, is
defined as the value which occurs most frequently in the sample- The
mode of the sample will he denoted by MO, It should ho obvious that,
the mode will not always be a central value; in fact, it may often he un
extreme value. Then too, a sample may have more than one mode.
We should, at this time, distinguish i>et\veen an ahtwlute made und a
relative mode. An absolute mode in what wo defined above (there may, of
coxmto, be more than one absolute mode) ; a relative mode is a value
which occurs more frequently than neighboring values even if it is not
an absolute mode.
Example 4.11
<Hvon a sample eotmiHting of the values ft, 7,
nay thore IK no nuKlc or tlu v rc an* fiw modern
only once,
1, 4, and *i we may
e otich vahu* oecurn
4.1 THE MODE
59
Example 4.12
Considering the data of Table 4.3, we see there are two absolute
modes, 64.5 and 65.25 milliseconds, since each of these values occurs 14
times and no other value occurs that frequently.
Example 4.13
Given a frequency histogram like that shown in Figure 4.4, we would
say there are two relative modes: one in Class A and one in Class J5.
However, the mode in Class A is the only absolute mode.
A B
FIG, 4.4 Example of a bimodal frequency histogram.
If our data are grouped in class intervals, it will be impossible to
locate the mode exactly. Under such circumstances, the best we can do
is to approximate the value of the mode. As was the case when approxi
mating the median, the first step is to locate the modal class. This is
accomplished quickly by picking out the class interval which shows the
highest frequency. The sample mode is then approximated by
MO ^ Z MO +
(4-6)
where
&MO = lower limit of the modal class,
di = the difference (sign neglected) between the frequency of the
modal class and the frequency of the preceding class,
^ 2 = the difference (sign neglected) between the frequency of the
modal class and the frequency of the following class, and
w=* width of the modal class.
Example 4.14
Consider the sample given in Table 4.4. The modal class is from 65 to
66 milliseconds with a frequency of 37. Therefore,
MO = 65 + C ") (1) = 65.23 milliseconds.
\o "~r~ It) /
60 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
Since the mode is, by definition, the most typical value, it is often
considered the most descriptive of the representative values discussed
so far. However, its importance diminishes as the number of observa
tions becomes limited.
4-11 THE RANGE
All the representative values discussed in the preceding sections have
been some sort of average or measure of position. It must be clear,
though, that they are not sufficient by themselves to describe most
populations or samples adequately. This statement may be verified
easily if we consider two sets of data which have the same mean, the
same median, and the same mode but which differ greatly in the amount
of variation present in each set of data. It would seem then that some
measure of the variation, or dispersion, among the individual values is
also needed. Several such measures have been devised, and we shall
mention four of these in this and succeeding sections.
A measure of dispersion, to be suitable, should be large when the
values vary over a wide range (and there are quite a few extreme
values) and should be small when the range of variation is not too great.
The simplest measxire of variation is one that has been mentioned
before, that is, the range. If we denote the smallest (minimum) sample
value by Xmin, and the largest (maximum) sample value by -Xn> ft *, the
sample range is given by
The sample range, though easy to obtain, is often termed inefficient
because it ignores all the information available from the intermediate
sample values* However, for small samples (n< 10), the efficiency (rela
tive to other measures of variation yet to be defined) is quite high. For
a more explicit discussion of the efficiency of the range relative to the
standard deviation the reader is referred to Section 4.12. Thus we find
the sample range enjoying a favorable reception and wide use, because
of ease in computation, in such applications as statistical quality con
trol where small samples arc the rule rather than the exception,
Example 4.15
For the sample given in Example 4.7, we obtain .R=*7- 2 = 5.
4-12 THE STANDARD DEVIATION AND VARIANCE
Perhaps the best known and most widely used measure of variability
is the standard deviation. Of almost equal importance is the square of
the standard deviation, this quantity being known as the variance. We
shall explain, both of these measures by defining the variance*
The sample variance, denoted by s 2 , is defined by
4.12 THE STANDARD DEVIATION AND VARIANCE 61
- Xy + - + (X n - Xy}/(n - 1) (4 g)
Sometimes it is convenient to let x^ = Xi X, that is, it is simpler to
denote deviations about the mean by lower-case letters. Then,
However, the best form for machine calculation purposes is
n^ 2 / n \ 2
L, -Xt C 2.4 x *
-1 \ t1 S
n
n 1
The sample standard deviation is then defined as the positive square
root of the variance, namely,
s = v^ 5 "". (4.11)
The use of n 1 (instead of n) when defining the sample variance
may seem peculiar to the reader, since we implicitly used a divisor of N
when defining the population variance. Our reason for using n 1 is
this: In general, one prefers unbiased estimators 2 to biased estimators,
and the use of n 1 gives us an unbiased estimator of <r 2 . If n were used,
the resulting function of the sample observations would produce biased
estimates of the unknown population variance biased because, on the
average, the estimates would be too small. Thus the student of statistics
must resign himself to remembering that, while the population variance
is defined using a divisor of N, the sample variance requires a divisor
of n 1. Incidentally, we refer to n 1 as the degrees of freedom associ
ated with the sample variance (and standard deviation) ,
Example 4.16
For the sample (13, 5, 8, 5) we see that
$* - {(13 - 7.75) a + (5 - 7.75)2 + (8 - 7.75)* + (5 - 7.75)*} /3
- {(S.25) 2 + 2(~ 2,75) 2 + (0,25) 2 }/3 - 14.25 and
thus s = VI 4.25 = 3.775.
If we had used the formula recommended for machine calculation,
the same value of s 2 would have been obtained:
a An estimator is a statistic, that is, a function of the sample values, which will
provide us with numerical estimates of a parameter.
62 CHAPTER 4, SAMPLING ANO DESCRIPTIVE STATISTICS
s * =* { 132 + 52 + 32 + 52 _ ( 13 + 5 + 3 + 5)2/4} /3
^ 283 - (31) 8 /4 ^ 283 240.25 = 42.75 =
If the sample data appear in a frequency distribution, the following
forms are appropriate for calculation. When no class intervals are in
volved (as in Table 4.3),
n 1
or
n 1
where -ST is an arbitrary origin, n y^/, and %***X-~XQ*
Example 4.17
ConBidor "Pablo, 4.6. Utfirig Kqxiation 4.12, we obtain
$* = {12,700 - (45C)) 2 /1&}/17 - 1450/17 - 85.3.
Utnng Kqxiation 4.12a,
3 * ** {1900 - (- 90)/18}/17 - 85,3.
TABLE 4.6-tll ust ration of the Use of Equations 4.12 and 4J2a
, .
\ ^" " *"* <&tj
-Y
/
.AY
/-Y*
Z
&
f&
10
3
30
300
20
60
1 , 200
20
5
100
2,000
10
50
500
30
8
240
7,200
40
2
80
3,200
10
20
200
Totals I H 450 1 2 , 700 . . <>0 1 , <)()0
When claMH intervals are involvc^l, tho appropriut<i formuta^ arc:
,.?^-<2:/v.
n 1
and
i
In which w ivS tho width of a claws interval and * ( f )/t^ where fn in
an arbitrary origin*
4.12 THE STANDARD DEVIATION AND VARIANCE
TABLE 4.7-Illustration of the Use of Equations 4.13 and 4.13a
Totals
18
450 12,700
9
63
Class
Interval
/
*
/*
f?
i
fi
/*
5
< X <
: 15
3
10
30
300
2
6
12
15
< X <
: 25
5
20
100
?,
,000
1
5
5
?.S
< X <
: 35
8
30
240
7
,200
3.S
< X <
: 45 . . .
2
40
80
^
,200
1
2
2
19
Example 4.18
Consider Table 4.7. Setting
be verified by evaluating
s* = {12,700 (450) 2 /18}/17
, it is seen that $ 2 = 85.3. This may
or
s z =
[{19- (-
It was mentioned in Section 4.11 that, for small n, the range is
reasonably efficient relative to the standard deviation. By this state
ment was meant that if one wishes to estimate <r, it can be done using
either R or s. When sampling from a normal population, the efficiency
of the sample range relative to the sample standard deviation as an
estimator for the population standard deviation is given in Table 4.8.
As an example of the use of this table, if a person desires to use R
TABLE 4.8-Efficiency of Range (R) Relative to Standard Deviation O) as
an Estimator of cr for a Normal Population
Sample Size (n)
Relative Efficiency
<r/JS(#)
2
1.000
0.886
3
0.992
0.591
4
0.975
0.486
5
0.955
0.430
6
0.933
0.395
7
0.912
0.370
8
0.890
0.351
9
0.869
0.337
10.
0.850
0.325
12
0.815
0.307
14
0.783
0.294
16
0.753
0.283
18
0.726
0.275
20
0.700
0.268
30
0.604
0.245
40
0.536
0.231
50
0.490
0.222
64
CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
rather than s, he would estimate a- by calculating
# = (R) { Value of a/E(K) for given n] .
(4.14)
4.13 THE COEFFICIENT OF VARIATION
The coefficient of variation has been explained by statisticians in
different ways. However, attention usually is called to the rather obvi
ous fact that things with large values tend to vary widely, while things
with small values exhibit small numerically small, that is variation.
Thus, to afford a valid comparison of the variation among large values
and t'he variation among small values, such as the variation among
salaries of industrial executives and the variation among the wages of
day laborers, the variation is expressed as a fraction of the mean, and
frequently as a percentage. This measure of relative variation is called
the coefficient of variation and is defined as
CV - s/ T .
In percentage form, this becomes
100 CV 100(V X) per cent.
(4.15)
(4.16)
TABLE 4.9-Special Form for Calculating and Presenting Sample Statistics
n
*
x
*
'
(T.xy/n
-
,^ ,u^
....
I> 9
* 2
$
-
A max
Am In
-
-
R
SPECIAL NOTES
FORMULAE
s* - 2>V( -
R BK A"**
largest observation smallest observation.
4.14 SUMMARY 65
The coefficient of variation is, of course, an ideal device for comparing
the variation in two series of data which are measured in two different
units; e.g., a comparison of variation in height with variation in weight.
Example 4.19
For the sample given in Example 4.16 we see that CV = 3. 775/7. 775
= 0.4871, and in percentage form 100 CV = 48.71 per cent.
4.14 SUMMARY
The greater part of this chapter has been devoted to outlining meth
ods of calculation for various statistics; i.e., functions of sample values,
which are useful in statistical inference. Not all of the statistics dis
cussed are used in everyday applications. However, a select few are used
so often that it is convenient to have a standard form for calculation
and presentation of results. One such form is presented in Table 4.9.
66 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
Problems
4.1 Plot a frequency histogram and polygon for the following data. Make
approximate eye-estimates of the arithmetic mean, median, and mode.
WEEKLY WAGES OF 188 FEMALE EMPLOYEES OF
A SHOE MANUFACTURING COMPANY
$20 . 15
$25.00
$40.39
$25.49
$25 . 70
24.15
22.54
23.80
29.60
18.74
25.62
23.89
28.37
26.00
16.70
26.00
27.82
24.80
26.52
28.09
27.84
25.80
25.88
25.04
24.98
22.97
23.20
23.24
29.00
24.55
25.48
20.88
21.70
25.76
26.20
28.00
28.92
27.92
25.80
22.45
28.24
25.70
22.75
21.40
27.10
31.37
26.77
26.00
18.64
27.39
24.53
24.25
28.28
30.32
23.00
28.13
26.23
21.55
28.04
25 . 58
22.78
26.88
26.64
22.83
23.45
25 . 20
29 . 29
25.62
23.40
26.12
27.08
24.40
25.49
30.48
27.03
26.11
21.80
20.85
26.79
26.25
22.04
22.54
21.85
25.65
27.50
29.48
25.20
26.00
22.69
25 . 78
21.77
24.32
26.00
22.52
17.50
26.52
20.48
22.92
23,96
26.00
22.00
22.44
26.00
26.35
25.64
22.48
27.25
24.19
23.75
28.94
21.85
22.99
22.33
24.18
25.65
23.12
22.71
26.48
23.23
23.44
31.00
25.38
25.83
18.60
33 . 80
30.61
22.00
29.72
23.28
25 . 65
23 . 80
26.90
24.55
23.12
29.24
26.00
22.68
24.04
32.60
22.15
25.15
22.53
25.12
23.72
22.99
25.70
27.98
26.34
23.08
24.24
28.00
27,14
23.13
26.38
24.00
26.03
31.60
24.79
24.73
27.48
30.23
22.47
34.99
22.09
19.30
24.55
26.67
24.08
25 . 78
23.42
30.60
28.32
22.28
24.73
25.65
29.15
27.74
23.69
28.83
25.64
22.48
25.20
23.84
25.68
28.24
30.72
28.92
25.73
PROBLEMS 67
4.2 Plot a frequency histogram and polygon for the data given below.
PER CENT SILICON IN 236 SUCCESSIVE CASTS OF PIG IRON
1.13
1.00
0.96
0.67
0.77
0.65
0.83
0.92
0.80
0.94
0.96
0.76
0.34
0.60
0.79
0.73
0.85
0.62
0.60
0.66
0.84
1.00
0.99
0.96
0.60
0.32
0.87
0.89
0.70
0.91
1.20
1.00
0.97
1.00
1.08
0.85
0.71
0.72
0.74
0.96
0.92
1.00
0.67
0.77
0.74
1.32
0.85
0.94
0.94
0.89
0.98
0.87
0.97
0.94
0.60
0.72
0.72
0.65
0.88
1.00
1.09
0.60
0.72
0.88
1.17
1.00
0.75
0.73
0.91
1.11
1.45
1.45
0.87
0.64
0.60
1.00
0.81
1.14
0.68
0.74
0.36
0.85
1.17
1.00
0.82
0.77
0.67
0.70
0.68
0.89
0.93
1.13
1.00
0.80
1.00
0.86
0.73
0.66
0.79
0.51
0.60
0.89
1.00
1.18
0.82
0.60
0.76
1.07
0.84
0.93
0.73
0.60
0.79
0.61
1.14
1.33
1.00
0.80
0.71
0.95
0.87
0.83
0.65
0.64
0.85
0.78
0.86
0.60
0.92
0.87
1.00
0.91
0.72
0.79
0.70
1.00
0.81
0.80
0.81
0.87
0.60
0,86
0.94
1.00
0.97
0.70
0.37
1.00
1.00
0.99
0.84
0.72
0.48
1.50
1.50
1.00
0.99
0.80
0.85
0.84
1.00
0.91
0.60
0.68
0.75
0.47
0.73
0.97
0.92
0.60
0.82
1.14
0.87
0.70
O.80
0.95
0.61
1.02
1.45
0.93
0.57
0.60
0.61
0.69
0.81
1.00
1.25
0.90
0.60
0.82
0.84
0.92
0.71
0.94
0.87
0.84
0.94
0.97
0.90
0.99
0.97
1.06
1.10
0.89
0.69
0.86
0.61
0.38
0.89
0.97
0.87
0.71
0.33
0.80
0.64
0.26
1.16
1.25
0.66
0.56
1.12
0.73
0.62
0.78
0.68
0.61
1.00
1.11
1.00
0.81
0.70
0.85
1.00
1.50
1.18
0.94
68 CHAPTER 4, SAMPLING AND DESCRIPTIVE STATISTICS
4.3 A random sample of 201 women students was obtained and their
heights and weights were recorded as follows:
HEIGHTS AND WEIGHTS or 201 WOMEN STUDENTS AGED 18
UNIVERSITY OF BRITISH COLUMBIA, 1944-45*
5-4
139
5-5
1414
5-44
99
5-2
1164
5-1J
1184
5-3A
5-4
158
5-4
123
5-84
151
5-4
1104
VJ '-' jj
108
S-Tk
146&
5-44
152|
5-64
118
5-64
123
5-54
5-31
5-2|
1274
1414
127
v *
5-64
5-7
5-32
1174
1394
122
5-94
5-5
5-5
142|
119
1351
5-3
5-94
5-14
1194
1344
114
5-44
5-54
5-3 1
115
117
1124
M* ** 4^
5-51
175
4-11
1101
5-5
1301
5-71
1484
5-54
149
5-84
148
124
5-2 1
128
5-3
125|
5-6
128
5-3
130
5-1
1074
5-1J
1044
5-41
1124
5-24
1024
5-5
129
5-14
104
5-1*
107J
5-24
1234
5-54
1184
5-3
1104
5-44
1341
5-61
1184
5-6
142
5-8
152
5-3
5-5
5-8
1324
1254
5-54
5-54
5-14
1371
1084
1261
5-2^
5-7
5-34
120
143
126|
5-2
5-54
5-54
964
122
1494
5-2
5-2
5-41
114
1441
1334
5-34
126
5-7
1301
5-2
1101
5-9
134
5-7
152|
v/ v *
5-5
140
5-84
1304
5-4J
115
5-74
1214
5-74
1304
5-5
131
5-54
125J
5-5
111
5-6
128
5-81
151
5-64
5-7
1224
162
5-21
5-4
110
117
5-44
5-61
128|
1434
5-44
5-5
1274
129
5-54
5-64
125
1404
5-3
1324
5-3J
1164
5-10
1514
5-6
117
5-7
1394
5-41
5-4*
112*
125
5-2 1
5-44
974
126
5-10
5-54
1104
125J
5-104
5-7
1404
1381
5-34
5-74
108
1574
* * *f
5-21
1754
5-0
126
5-74
1414
5-54
1274
5-6
1184
5-3
105|
5-54
121
5-64
1644
5-3 4
112
5-44
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5-0
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5-2
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** *
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,*
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s-f
11 4 i
* hi* *,**..*
^ ^xm-******-,
Source: U.B.C, Students* Health Service,
Plot a frequency histogram and polygon for (a) the heights and
the weights.
PROBLEMS 69
4.4 Plot a cumulative frequency curve for the data of Problem 4.1.
Estimate the median from this curve.
4.5 Plot a cumulative frequency curve for the data of Problem 4.2. Esti
mate the median from this curve.
4.6 Plot a cumulative frequency curve for (a) the heights and (b) the
weights given in Problem 4.3. From these curves, estimate the median
height and median weight.
4.7 Given the samples listed below, calculate for each the mean, median,
mode, midrange, range, variance, standard deviation, and coefficient
of variation:
(a) 5, 19, -3, 7, 1, 1
(i) 5, -3, 2, 0, 8, 6
(c) 6, 9, 5, 3, 6, 7
(<Z) 1,3,2, -1,5
() 10, 15, 14, 15, 16
(/) 0, 5, 10, -3
() 8, 7, 15, -2,
4.8 Suppose that F = 100 and s 2 = 15. What would the values of 7 and s 2
become if each original observation were (a) increased by 10 units,
(b) multiplied by 10 units?
4.9 Given the observations:
2
10
3
10
4
20
5
50
6
30
Calculate the same statistics as asked for in Problem 4.7.
4.10 Calculate the same statistics as asked for in Problem 4.7 for each
of the following sets of data:
(a) Problem 4.1
(6) Problem 4.2
(c) Problem 4.3.
4.11 A bag of potatoes was sampled for quality, five potatoes being selected
at random from the bag. Among the observations recorded were the
weights of the potatoes: 17, 15, 10, 12, and 11 ounces. Calculate
7, s 2 , s, and JR. What property (using the word rather loosely) is
common to the sample range and the sample variance (or standard
deviation)?
4.12 Given that n~25, 222/ 2=== 600, and Y = 204, calculate the variance,
standard deviation, and coefficient of variation.
C H APTE R 5
SAMPLING DISTRIBUTIONS
CERTAIN SAMPLING msTRtBUTioNS pertinent to methods to be pre
sented in later chapters will be discussed in this chapter. The law of
large numbers, TchebychefPs inequality, and the central limit theorem
will be given. Various approximations to exact sampling distributions
will also be considered.
5.1 SAMPLE MOMENTS
In the preceding chapter, the calculation of several different sample
statistics was outlined. Of particular importance are those statistics
known as sample moments. They are defined by
(5.1)
and
n
~ In (5.2)
where 7c = 0, 1, 2, - . In particular, it is noted that, wj -ST. The
reader will see the similarity of the above definitions to the definitions
of /4 <md M* given in Definitions 3,23 ami 3.25, It may be shown that
J(m) = /4 for all k but K(m k ) docs not equal p M except for A*=0, 1. For
this reason, m f k is known as an unbiased estimator of MiC^ O, 1, - )
while m^ is a biased estimator of /* A (fc=ss2, 3, * - ) In<ucleutally, this
last remark is just a restate on t of the reason given in the preceding
chapter for using $* rather than w 2 as an estimator of cr 2 .
5.2 VARIANCE OF THE SAMPLE MEAN
It has jxmt boon stated that K(^}y.. That in, in nampling from a
specified population, the expected value (or average) of all poasiblo
Bample meaiiB is the popxtlation mean. However, we realise that it ia
equally important to know nomothing about the variation among all
possible values of the nample mean. To investigate thin variation, con
sider the variance of the sample mean,
n
After Borne algebraic manipulation, it in wen that
2
<r x J
1703
<* -*>}>
5.3 TCHEBYCHEFF'S INEQUALITY 71
~ M) 2 + n(n -
{a-* 2 + (n - !)[(*< - M )(* y - /*)]}/. (5.4)
Two cases must now be distinguished: (1) random sampling and (2)
sampling without replacement from a finite population. For these two
cases, we obtain, respectively,
and
^2 AT *j
(5.6)
n N - 1
A 7 " being the size of the population.
The preceding result is very important. It says that, no matter what
the population (as long as it has a finite variance), the distribution of
the sample mean becomes more and more concentrated in the neighbor
hood of the population mean as the sample size is increased. That is,
the larger the sample size, the more certain we become that the sample
mean will be close to the (unknown) population mean. This result will
be expressed more precisely in the following section.
5.3 TCHEBYCHEFF'S INEQUALITY
A useful inequality is that due to Tchebycheff, namely
P( | X - fJL | > ka) < 1/k*. (5 . 7)
This inequality is often expressed in the following alternative form:
P(\ X \ < ka) > 1 1/k 2 (5 . 8)
or
P(fji - k<r < X < VL + k<r) > 1 - I/* 2 . (5.9)
TchebychefPs inequality shows how <r may be used as a measure of
variation. It can be applied in a wide variety of cases for it assumes only
the existence of M and <r 2 . That is, no assumption is made concerning
the form of the population but only that the mean and variance exist.
If we restrict our attention to unimodal distributions, the inequality
may be sharpened. Under such a restriction, we obtain
P(\ X MO | > kB) < 4/9k* (5.10)
where MO is the mode and J5 2 = cr 2 + (MO ^) 2 . An alternative form is
72 CHAPTER 5, SAMPLING DISTRIBUTIONS
where A = (/* MO)/cr. It should be noted that if the distribution is
not only unimodal but also symmetric, that is, M MO, then Equations
(5.10) and (5.11) reduce to
P( | X AC | > *<r) < 4/9k*. (5. 12)
5.4 LAW OF LARGE NUMBERS
It is now possible to give precise formulation to the law of large
numbers. Invoking TchebychefFs inequality with respect to the sample
mean, we have
(5,13)
or
P(ju - cr/V^" < X < fjL + <r/vX> > 1 l/ 2 - (5.14)
Setting K~k<r/*\/n, it is seen that
(5. 15)
Thus, when sampling from any population with a finite variance, the
sample size may be chosen large enough to make it almost certain that
the sample mean will be arbitrarily close to the population moan. This
is what is known as the law of large numbers.
In reliability and quality control work, much attention is given to
the number of defective items in a sample of size n. Thus it will be of
interest to see how the law of large numbers provides information in
such a case. If we assume random sampling from a binomial population
m which M~P and <r 2 ~pCL p), then, as n gets large,
^ 1 (5 * 16)
where x is the number of defective items observed in a sample of size n
and e is an arbitrarily small positive quantity. That is, as n increases,
we become more and more certain that the observed fraction defective
will be a good estimate of the true fraction defective in the population.
5,5 CENTRAL LIMIT THEOREM
Without doubt, the most important theorem in statistics is the can-
tral limit theorem. It is important not only from the theoretical point
of view but also becaxise of its impact on statistical methods. Since a
proof of this theorem Is beyond the scope of this text, it will be stated
without proof. Hero, then, is the theorem :
// a population has a finite variance of <r% and mean M? then the dis
tribution of the sample mean approaches the normal distribution with
the variance <r*/n and mean jj, as the sample size n increases*
t
Note that nothing Is said about the form of the sampled population.
5.7 THE HYPERGEOMETRIC DISTRIBUTION 73
That is, no matter what the form of the sampled population, provided
only that it has a finite variance, the sample mean will be approxi
mately normally distributed. This is indeed a remarkable theorem.
5.6 RANDOM SAMPLINC FROM A SPECIFIED
POPULATION
Suppose a random sample of n observations is obtained from a given
population. The joint probability density function for (JSTi, X 2 , - - - , X n }
will be represented by g(x^ , x). Now, it will be remembered that
a random sample implies statistically independent observations. Fur
ther, for statistically independent variables, the joint probability
density function may be expressed as the product of the marginal
densities. Thus,
' gnOn) (5-17)
and, since each observation came from the same population,
where f(x) is the probability density function describing the sampled
population. It should be noted that Equation (5.18) gives the joint
probability density function of the sample in the order drawn. Inci
dentally, the function
n /(**)
i=i
is often referred to as the likelihood function.
5.7 THE HYPERGEOMETRIC DISTRIBUTION
In many instances, the type of sampling performed in industrial ap
plications is the selection of a sample of n items out of a lot of N items.
This selection is usually done in such a manner that the sampling is
without replacement. Thus we have a "random" sample only in the
specialized sense that every possible group of n items in the lot has the
same chance of comprising the sample. In such a case, if x represents
the number of defective items in the sample,
/(*) - C(D 7 x)-C(N - D,n- x)/C(N, n) ;
a-a,a+l, ...,-!,&
a = max (0, n N + 2?)
b = min (D, n)
where D represents the number of defective items in the lot.
The distribution specified by Equation (5.19) is known as the hyper-
74 CHAPTER 5, SAMPLING DISTRIBUTIONS
geometric distribution. It is the distribution underlying practically all
acceptance sampling by attributes where an item of product is classified
as either defective or nondefective. The reader should become very
familiar with the hypergeometric distribution and be competent in
evaluating probabilities associated with it.
Using the theory of earlier chapters, it is seen that
M = E[X] = nD/N (5.20)
and
<r* = E[(X M) 2 ] = nD(iV D)(N n)/N~(N 1). (5.21)
Thus, as expected, the average number of defective items in a sample
is equal to the size of the sample multiplied by the fraction of defective
items in the lot.
5,8 THE BINOMIAL DISTRIBUTION
Suppose that a random sample of size n is selected from an infinite
binomial population described by
*(y) - P"(i - p) 1 -*; y = o, i (5.22)
< p < 1
or that a random sample of wize n is selected (using sampling with re
placement) from a finite population of N items, D of which arc defec
tive. In the latter case we can, therefore, let p~f)/N.
Both of the sampling situations described above lead to the same
sairxpling distribution of j? where x represents the number of defective
items in a sample of n items. This distribution is described by the p.f -
/(*) C(, *)p*(l - p "-*; * 0, 1, , n (5.23)
< p < 1.
Using theory already developed, it can easily be shown that
M ^ /<;[A*1 np (5,24)
and
<r* /4(-Y - M)^J np(l ^) npg (5,25)
where # 1 ~- p. Those results will prove usefxil in later work,
Probabilities associated with the binomial density of Equation (5*23)
or with its cumulative* form have boon published by the National
Bureau of Standards (4), Robertson (7), and Hornig (8), Reference will
be made to nueh tables as the need arises.
5,9 BINOMIAL APPROXIMATION TO THE
HYPERGEOMETRIC
Under certain conditions it in permismhle to UHC the binomial distri
bution aa an approximation for the hypergeometrie contribution. This
5.10 POISSON APPROXIMATION TO THE BINOMIAL 75
approximation is usually invoked to simplify numerical calculations.
To see how the approximation is justified, consider the hypergeometric
distribution
/(#) = C(D, x)-C(N D,n x)/C(N, n). (5.26)
Writing this out in detail, we obtain
IN N 1 N ( x - 1) N x
N - D - 1 N - D - (n - x - I)'
N x 1 j\f ~ ( n i)
(5.27)
Setting D = pN and dividing the numerator and denominator of each
factor inside the braces by N, it is seen that
1 - l/N 1 O - 1)/7V 1 x/N
q l/N q (n ~ x 1)/N\
1 - (n -
where g= 1 p. Letting A 7 " get very large, it is clear that
/O) > C(n, oc)p*q"-*. (5 . 29)
That is , if JV is large, the hypergeometric distribution may be ap
proximated by the binomial distribution. The question of how large N
should be relative to n before using the binomial approximation is one
which must be answered. First, since tables of logarithms of factorials
are not available for k greater than 2000, calculation of the hyper
geometric will be extremely tedious for such cases. Second, and perhaps
more to the point, Burr (1) has said that if the lot size N is at least
eight times the sample size n, it will be satisfactory to use the binomial
as an approximation to the hypergeometric. However, since Burr's
statement is only a general comment with no reference to the magni
tude of the error involved, it seems only fair to say that each individual
case must be considered on its own merits.
5.10 POISSON APPROXIMATION TO THE BINOMIAL
In instances when we do not have access to published tables of the
binomial distribution, it becomes necessary to find some way of ob
taining the required probabilities without excessive calculation. In
such cases, we usually seek some form of approximation to the binomial
which involves less computation or is associated with tables that are
more readily available. Two such approximations involve, respectively,
the Poisson and normal distributions. The first of these will be dis
cussed in the present section, while the normal approximation will be
examined in Section 5.11.
76 CHAPTER 5, SAMPLING DISTRIBUTIONS
If p is very small (less than 0.1) and n is quite large (greater than 50),
it is sometimes convenient to approximate the binomial p.f. by the
Poisson p.f. in which jj. np. To see how this approximation is justi
fied, consider the following argument. In
(5.30)
f(x) = C(n,
n(n !)-( x -f- 1)
set p = jji/n. Then,
n(n - 1) - - (n - x
xl
/ M \y /* y
f _ J f ! _ _ 1
\ n / \ n /
/ w \ /n 1\ /n x+ 1\ ^
"" w/ v w / " * * v^ / ~^
*
n
If we let n *>co and p ^0 such that np^p. remains constant,
/(*) -+ (1)(1) - - (t) -- e-*(O - ^ Y- (5.32)
\ xl
which is the Poisson probability function.
Therefore, if np is large relative to p and n i& largo relative to np ? the
Poisson may bo used as a reasonable approximation to the binomial.
All that is necessary is to net p in the Poisson distribution equal to np
of the binomial distribution we are attempting to approximate. In
other words, the means of the two distributions have been equated.
5.11 NORMAL APPROXIMATION TO THE BINOMIAL
The binomial distribution may also be approximated by the normal
distribution. As in the preceding section, the sample i2o should be
reasonably large before the approximation IB employed,
To illustrate the nature of the approximation, ccmsidor Figure 5.1.
Hero, the binomial distribution for n^ 10 and p^^ is pictured by the
ordinates at the various values of x. If rectangles of width one are
erected as shown, the area of the histogram equala 1, This is just an
alternative way of expressing the fact that the sum of the ordinates
equals 1. Umng areas under the normal curve, probabilities associated
with various x values may be closely approximated.
5.11 NORMAL APPROXIMATION TO THE BINOMIAL
01 23456 789 1O
X= NUMBER OF FAILURES IN SAMPLE OF SIZE 1O
FIG. 5.1 Binomial distribution for n = 10 and p = V 2 (solid line
ordinates), area representation (dotted line rectangles)
and the normal approximation.
In order to evaluate probabilities associated with a normal distribu
tion, the mean and variance must be known. To specify the mean and
variance of the approximating normal, let v = np and cr 2 = np(l p)
= npq where np and npq are the mean and variance, respectively, of
the binomial distribution to be approximated. Then, for any integers a
and b (a<6) in the closed interval (0, n), the approximation takes the
form :
P{a <: X < b}
P{a < X <b]
P{a < X < b}
.,{
(a ) np
P<- ^= ^Z
-v/npg
(a
np
\/Vfcp<?
O ) ^P
-\/npq
< ^
^ Z
\o -f- tJ ^P
(*
^/npq
+ i) ~
np
*
-Vnpq
np
-^/npq
or
- np
-\/npq
(5 . 33)
(5.34)
(5.35)
(5 . 36)
Other illustrations could be given, but the foregoing, together with the
examples which follow, should be sufficient. The important thing to
note is that ^ is added to or subtracted from the limit so as to include
or exclude a or 6, the proper choice being indicated by the nature of the
inequality. This adding or subtracting of is often referred to as a
"correction for continuity. "
Example 5.1
A random sample of 100 observations is drawn from a binomial
population in which p=0.2. Evaluate P {10^X^25}. We say that
78 CHAPTER 5, SAMPLING DISTRIBUTIONS
4 "~~ 4
- P{- 2.62 < Z < 1.38}
= G(1.38) G( 2.62) - 0.91621 - 0.00440
= 0,91181.
Example 5.2
Referring to Example 5.1, evaluate P { 10 < A"<25 } . We have
4 ~ 4 >
= P{- 2.37 < Z < 1.38} = G(1.38) - G(- 2.37)
= 0.91621 0.00889 =* 0.90732.
Example 5.3
Referring to Example 5,1, evaluate P {X >2G } . Proceeding as before,
P{X>26} c*
1 G(1.62) * 1 0,94738 = 0.05262.
It is reasonable to ask what error is involved in using the approxima
tion just described. Mood (3) has said that, if npq>25, the error is less
than Q.l5/^/npq. However, we should realize that, for a given n, the
normal curve gives a better approximation when p is close to than
when p is close to or 1. On the other hand, if n is large enough (say
100 or more), the approximation will be satisfactory for most values of
p. If p is very close to or 1, the approximation will be lews reliable in
the tails than near the center of the distribution. Thus, in reliability
work, where very small values of p are frequently encountered, the
normal approximation may not be too good and one should use either
the Poisson approximation or calculate exact probabilities.
5.12 THE MULTINOMIAL DISTRIBUTION
If a random sample of size n is taken from the multinomial popula
tion described by
< pi < 1 (5.37)
0, 1
t 1 l
a multinomial distribution is obtained. This distribution is defined by
== Pi*2> p* r s < i < I 5
Pi - 1
i**l
5.13 THE NEGATIVE BINOMIAL DISTRIBUTION AND THE GEOMETRIC DISTRIBUTION 79
Xi = 0, 1, - - - , n
k
where x+ is the number of items occurring in the class associated with p t -.
The number p t is the probability of any item being assigned to the ith
class and it is, of course, the fraction of the total population belonging
to the fth class. For example, an item of product may be assigned to
one of four classes: good, minor defect, major defect, or critical defect.
Then, the n sample items would be classified into the four groups upon
inspection. The number falling in the first group would be denoted by
o?i, the number in the second group by x 2 , and so on.
5.13 THE NEGATIVE BINOMIAL DISTRIBUTION AND
THE GEOMETRIC DISTRIBUTION
A sampling distribution encountered fairly often in industrial appli
cations is that known as the negative binomial distribution. Suppose p is
the probability of a defective item and g = 1 p is the probability of a
nondefective item. If random sampling is being carried out, it is fre
quently of importance to know the probability that the rth defective
unit will occur on the (x+r)th unit sampled.
To obtain the probability just described, it is noted that: (1) the last
unit must be defective and (2) in the preceding x+r 1 units sampled
there must be exactly r I defective units. Then,
= {C(x + r - l,r - l)p-HT} -p
*; * - 0, 1, . (5.39)
Another way of saying this is that the probability of the rth defective
unit occurring on the mth unit sampled is
s (m) = C(m 1, r l)p r ^- r ; m = r, r + 1, - - . (5.40)
It is sometimes of interest to know the probability of the rth defective
unit occurring on the rth or (r + l)st or ... or nth unit sampled. This is
given by
n n
X) C(m l,r l}p T q m ~ r = ^ C(n, w)p m g w - m (5 .41)
wr mT
and the last expression may be found by consulting tables of the cumu
lative binomial distribution.
If in Equation 5.39 we let r=l, the negative binomial distribution
simplifies to the geometric distribution.
SO CHAPTER 5, SAMPLING DISTRIBUTIONS
5.14 DISTRIBUTION OF A LINEAR COMBINATION OF
NORMALLY DISTRIBUTED VARIABLES
Suppose we consider
U = i^aiXi (5.42)
i1
where, for the moment, all that is known is that the a* are constants
and the Xi are variables. It is clear that
(5.43)
< 1
and
where yu* is the mean of X+, of is the variance of -X",-, and o\-/ is the co-
variance of Xi and X 3 . If all the -X\-are mutually (pairwise) independent,
4 -!>?** (5-45)
since cr^ equals if X^ and X$ are statistically independent.
Consider now the case where X* is a random sample from a normal
population with mean n* and variance erf (i= 1, - - - , n). In this case it
may be shown that U is also normally distributed. If each X^ is
randomly selected from the same normal population, that is, from a popu
lation 1 A^(ju, <r)> then U is normally distributed with mean
and variance
r '.
i1
5.15 DISTRIBUTION OF THE SAMPLE MEAN FOR
NORMAL POPULATIONS
In Sections 5.1, 5.2, and 5.5, it was stated that;
(2) <r| <r^/n, and
1 Tho notation N(t* f <r) stands for "normally distributed with mean
standard deviation <r/'
5.17 CHI-SQUARE DISTRIBUTION 81
(3) regardless of the form of the sampled population (provided it
has a finite variance), the distribution of the sample mean is
asymptotically normal with mean // and variance a 2 /n.
In the present section it is stated (without proof) that if a random
sample is taken from a population N(p,, <T), then the sample mean will
be distributed N(n, o-f^/n} for all values of n. It should be clear that
the probability density function for ~X is
*\/ j yif
~ - e -n(x-M) 2 /2^ (5.46)
and that Z= Vn(X jLt)/cr is N(Q, 1).
5.16 DISTRIBUTION OF THE DIFFERENCE OF
TWO SAMPLE MEANS
If a random sample of n^ observations is obtained from a population
with mean ^ and variance <r\ and if a random sample of n% observa
tions is obtained from a population with mean _M_ 2 and variance v\, what
can be said about the distribution of U = Xi X^ where Xi is the mean
of the first sample and -XT 2 is the mean of the second sample?
Regardless of the form of the populations sampled, it is true that
^V-* 2 = A**! "~ Vx 2 = MI M2 (5.47)
and
2 2
= <4 + 4 = + ( 5 - 48 >
x l -T- x t
If, howeyer,_the sampled populations are both normal, it is also true
that U = Xi X% is normally distributed with mean and variance given
by Equations (5.47) and (5.48). In this situation,
(5.49)
/ o~i
V ~^
is normally distributed with mean and variance 1.
If the populations are not normal but both sample sizes are suffi
ciently large, the central limit theorem may be invoked to achieve an
approximate normal distribution for the difference of two sample means.
5.17 CHI-SQUARE DISTRIBUTION
One particular distribution arises quite frequently in applied work
and is known as the chi-square distribution. When referring to the chi-
square distribution, the parameter v is called the degrees of freedom.
The probability density function for chi-square with v degrees of free-
82 CHAPTER 5, SAMPLING DISTRIBUTIONS
dom is given by
f f u \ = !LLl f_L_ ; u > (5 . 50)
-^ J 2*/*rO/2)
where w is used rather than x 2 (chi-square) for ease in writing. The
cumulative chi-square distribution is tabled in Appendix 4 for all in
tegral values of v from 1 through 100.
5.18 DISTRIBUTION OF THE SUM OF SQUARES OF
INDEPENDENT STANDARD NORMAL VARIATES
If a random observation is obtained from a normal population with
mean M and variance <r 2 , then the variable
Z 2 = (X /-OVV 2 (5.51)
is distributed as chi-square with 1 degree of freedom. Now, consider
the variable
u = :fc csr< - M*)Vo- (5-52)
where the J5T* are independently and normally distributed with means /x
and variances erf. Then, U is distributed as ehi-quare with k degrees
of freedom.
It is clear that, if a random sample of size n is obtained from a nor
mal popxilatkm with mean M and variance a- 2 ,
U - (X* - ju) 2 A* (5.53)
t^i
is distribtited as ohi-Hquaro with n degrees of freedom.
5.19 DISTRIBUTIONS OF THE SAMPLE VARIANCE
AND STANDARD DEVIATION FOR NORMAL
POPULATIONS
It can be proved that the mean and variance, 3T and s 2 , of a random
sample from a normal population are statistically independent* Further,
it is readily shown that the variable
IT - (n - 1W* - S (-Y, - ??)*/<?* (5-54)
*'n
is <listribiited ns (^hi-scixiare with v^n~~*\ degrees of freedom.
From the preceding result, the distributions of m^ \/m^ 2 , and s
can be obtained. These are:
1 / w, \ ( i)/ 2
( ) ,,(->/-/-' (5.55)
5.21 DISTRIBUTION OF F 83
1 / n V
(wiz) =
^ y /VJ 1\
r (V)
\ , ,, v ^, /o^2 / r- c*/C\
-1 (Vw 2 ) n ~ 2 e~' im2/2<r , (5.56)
V
n <
\2cr
1 /^ 7 _ 1 \ (n 1)/2
s t) = _ - - ( - - -) ( 5 2)(n-3)/V-<"-l>* 2 / 2 * 2 , (5.57)
} n - 1\ \ 2o- 2 /
r
and
/^- 1 \
_ 1 \ Cn
(5.58)
5.20 DISTRIBUTION OF "STUDENT'S" t
Consider two independent random variables, Z and U, where Z fol
lows a standard normal distribution and U follows a chi-square dis
tribution with v degrees of freedom. Form the ratio
t = Z/VW^- (5.59)
Then, the probability density function of t is
_ OT< , <0 o (5.60)
and it is referred to as the ^-distribution with v degrees of freedom. This
distribution is extremely useful in many problems of statistical in
ference. A table of cumulative percentage points of t is given in Ap
pendix 5.
5.21 DISTRIBUTION OF F
Given two independently distributed chi-square variates, U with vi
degrees of freedom and V with ?, degrees of freedom, it may be shown
that
(5.61)
is distributed as F with vi and v* degrees of freedom. The probability
density function is
84 CHAPTER 5, SAMPLING DISTRIBUTIONS
+ ;
/
\
f(F) = - - - - - ( ) -- . (5.62)
J "" 2
Of particular interest in applied statistics is the fact that when two
random, samples are obtained, one from each of two normal populations,
the ratio
is distributed as F with ^i = ni 1 and v^ us 1 degrees of freedom.
This will find application when analyses of variance are discussed later.
Appendix 6 gives certain percentage points of the ^-distribution.
5.22 ORDER STATISTICS
Observations on a chance variable usually occur in random order.
However, in certain cases, observations ordered according to magnitude
are encountered. This can happen in two ways: (1) the observations
were obtained in random order but were subsequently reordered ac
cording to magnitude, and (2) the observations naturally became
available in order of magnitxide. As an example of the latter, consider
the life testing of a group of vacuum tubes. The first observation to
arise is that associated witlx the weakest tube (i.e., the txibe with the
shortest life), the second observation is associated with the next weak
est tube, and so on. Since such data occur fairly often in xudxistrial
applications, some sampling distributions associated with order sta
tistics will now be discussed.
Consider a population specified by /(a;), a<x < 6. Denote the smallest
and largest values in a random sample of n observations from this
population by u and v, respectively. Then it may be shown that
v) - /?(*)>-*; a^u^i^b. (5 , 63)
The marginal p*d,f /s of u and v are
gl (u) n/O)[l *X0] n ~ l ; a ^ u b ' (5.64)
and
] n ~ l ; a ^ v ^ b. (5.65)
These distributions are very useful when dealing with problems involv
ing extreme value.
Order statistics are also valuable when dealing with the sample range,
ft ea D u = -JTmax -Xmin- If Ht Equation (5*68) we let v u+ R> we obtain
, K) n(n - !)/()/( + R)[F(u + -R) - F(u)}^. (5-66)
PROBLEMS 85
Then
& R
= f
+J n
g(u, K)du; < R < b a. (5.67)
It should be noted that if, instead of dealing with the joint distribution
of the range and the smallest sample value, we deal with the joint dis
tribution of the range and the largest sample value, namely,
g(v, R) - n(n - !)/(* - K)f(v)[P(i^ - F(v - r}]\ (5.68)
then
/*
g(v, R)dv; < R < b - a. (5.69)
u-f-72
Equations (5,67) and (5.69) will, naturally, produce the same result.
Example 5.4
If /(re) =!, < <1, then g(R) =n(n l)jR n ~ 2 (l R) where <JB<1.
Example 5.5
There is no simple expression for the distribution of the range when
sampling from a normal population. Pearson (5) gave the values of the
mean arid standard deviation for ranges from a standardized normal
distribution. Pearson and Hartley (6) evaluated the probability integral
of the range for sample sizes of 2 to 20. Incidentally, the mean and
standard deviation of the range when a standard normal population has
been sampled are denoted by d% and d s , respectively. That is, when
sampling from any normal population, (JL R = d%cr x and cr R = d^or x . Selected
values of d* and da are given in Appendix 8.
Problems
5.1 How large a sample should be taken if we want to be 95 per cent sure
that 3T will not fall farther than cr/2 from ju?
5.2 A book of 400 pages contains 400 misprints. Estimate the probability
that a page contains at least three misprints.
5.3 A lot contains 1400 items. A sample of 400 items is selected. If no
more than two defective items appear in the sample, the lot will be
accepted. Evaluate the probability that the lot will be accepted,
assuming that the lot is 1 per cent defective.
5.4 The width of a slot on a forging is normally distributed with mean
0.900 inch and standard deviation 0.003 inch. The specifications are
0,900 0.005 inch. What percentage of forgings will be defective?
5.5 Referring to Problem 5.4, samples of size 5 are obtained daily and
their means computed. What percentage of these sample averages will
be outside specifications?
5.6 The diameters of some shafts and some bearings are each normally
distributed with, standard deviation equal to 0.001 inch. If the shaft
has a mean diameter of 0.500 inch and the bearing has a mean diame
ter of 0.503 inch, what is the probability of inter ference?
86 CHAPTER 5, SAMPLING DISTRIBUTIONS
5.7 Three resistors are connected in series. Their nominal ratings are 10,
15, and 20 ohms, respectively. If it is known that the resistors are
normally distributed about the nominal ratings, each having a stand
ard deviation of 0.5 ohm, what is the probability that an assembly
will have a resistance in excess of 46.5 ohms?
5.8 Rework Problem 5.7 assuming that the standard deviation is 5 per
cent of nominal in each case.
5.9 A "1-poimd" box of candy is machine packed to contain 32 pieces of
candy. If the weights of the pieces of candy are normally distributed
with a mean of 0.5 ounce and a standard deviation of 0.05 ounce, what
are the probabilities that a customer receives:
(a) loss than 1 pound, (b) less than 15 ounces, (c) more than 1 pound,
(d) more than 16.2 ounces, (c) exactly 1 pound?
5.10 Referring to Problem 5.9 and assuming the standard deviation re
mains unchanged, how should you change the mean of the process
so that only 1 customer in 100 will receive lews than the advertised
weight?
5 J 1 A factory assembles stoves at the rate of 500 per week. On the average,
5 per cent of the stoves are found to be defective, when inspected
following final assembly. What is the probability that next week's
production will contain less than 20 defective wtovew?
5.12 Review all parts of the book pertaining to the Pomstm, normal, chi-
square, t y and F distributions, and be certain that you know how to
une the tables in Appendices 2 through (K
References and Further Reading
1. Burr, T. W. Engineering Statistics and Quality (Control. McGraw-Hill Book
Company, Inc., New York, 1953,
2. Ijiebermau, CK J, ami Owen, I"). B, Tables of the Hyper geometric Distribution.
Stanford University ProwH, Stanford, Calif., 19W),
3. Mood, A. M. Introduction to the Theory of Statistics. McGraw-Hill Book Com
pany, Inc., Now York, 1950.
4. National Bureau of Standards Tables of the Binomial Probability Distribution*
Applied MathmoticH HerioH 6. U.S. Govt, Print. Off., Washington, D.C., 1949,
5. Pearson, K. S., The percentage limits for the dintributkm of range in Hamplen
from a normal population, ttiomctrika, 24: 404 -17, Nov., 1932.
0. - , and Hartley, II. ()., The probability integral of the range in wampleH
of TV observation** from a normal population. Biometrika^ 32:301-10, April,
1942.
7. Robertson, W II, Tablvs of the Binomial Distribution Function for tftmall
Value of p. Bandia Corporation Monograph fcSCR-443, Albuquerque, N* Mex,
Jan., I960.
8. Ilomig, H, G. S&-10Q Binomial Table*. John Wiley and Bono, Inc., New
York, 1953.
C H APTE R 6
STATISTICAL INFERENCE: ESTIMATION
IN THIS CHAPTER, general concepts associated with, that part of statisti
cal inference referred to as "estimation and prediction" will be ex
amined. Examples dealing with particular populations frequently
encountered in applied work will also be given.
6.1 SOME PRELIMINARY IDEAS
In general, we do not know the values of the parameters of the dis
tribution function or the values of the population mean and variance.
In practice we obtain a random sample from the specified population,
assuming that we know the form of the distribution (normal, binomial,'
etc.). From the sample we attempt to estimate the true but unknown
values of the population parameters. At this point criteria should be
stated by which we may judge, or evaluate, different estimators of a
parameter. First, let us define an estimator as some function of the
sample values which will provide us with an estimate of the parameter
in question. Now, let us set down certain desirable properties of a good
estimator which may be used as criteria to distinguish between good
and bad estimators. Other criteria may be found in the literature, but
the three given here are perhaps the most important from a practical
point of view.
(1) An estimator is said to be unbiased if the expected value of the
estimator is equal to the population quantity being estimated.
That is, if is an estimator of 0, is said to be unbiased if the aver
age of all possible values of is Q.
(2) Let @ be an estimator of 6 calculated from a random sample of
size n. If, as n gets very large (i.e., approaches N where N is the
number of items in the population), the probability that will
be very close to 6 approaches 1, or certainty, then is called a
consistent estimator of 6. In other words, if we take a larger and
larger sample, we expect to get an estimate which is very close
to the true value, and the probability that we will do so is very
great.
(3) If 0i and 2 are two different (but both unbiased) estimators of 6
with variances o^ and <r$ 2 , respectively, and if <r\<<r\, then
we prefer <?i to $ 2 . That is, in general, we prefer the estimator
(out of the class of all unbiased estimators) which has the mini
mum variance.
Estimates of the type discussed above are of a special kind known as
point estimates. There is a second class of estimates, however, known as
C873
88 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
interval estimates. These are very important in statistical methodology
and, if at all possible, we obtain an estimate of this type. Let us illus
trate the difference between point and interval estimates by a short
example.
Example 6.1
If I wish to estimate the average weight of the people in a class
room, I could take a random sample of five people, record their weights,
and average them. The resulting average (suppose it turned out to be
160 pounds) would be my point estimate. However, this is not suffi
cient for our purposes. If *I say that the true average weight of all the
people in the room (they are my population) is between and^ 300
pounds, I am very confident of myself in fact, I am almost certain of
my statement. Rut if I make my interval much smaller and in prac
tice the interval should be as small as possible my degree of confidence
m my interval estimate will become less. For example, if I say 1 think
that the average weight of all people in the room is between 158 and
162 pounds, my degree of confidence may bo quite small,
If I wish to be ahlo to evaluate my degree of confidence for any
interval estimate, it is customary to make certain assumptions con
cerning the distribution of the observations being obtained. Several
examples of such confidence intervals -will be studied later in this
chapter.
6.2 METHODS OF OBTAINING POINT ESTIMATORS
Several principles of estimation, leading to routine mathematical
procedures, have been proposed for obtaining "good'* estimators.
These include:
(1) The principle of momenta
(2) Minimum chi-sqxiaro
(3) The method of leant squares
(4) The principle of maxinuim likelihood.
The application of these principles in particular canes will lead to
estimators which may differ and hence poHsesn different attributes of
"goodness." A principle much in line, yielding estimators with many
desirable attributes of "goodness" and obtained by easily applied
routine mathematical procedures, is that of maximum likelihood de
vised by H. A, Fisher (7, 8}* This important principle of estimation
will be used in the remainder of the chapter.
The procedure for determining the maximum likelihood estimate of
a population parameter 6 is as follows;
(1) Determine the density function of the sample, 0(*Yt, -X"* - * -
A r n ; 0). Note that in Section 5.0,
4** I
6.4 CONFIDENCE INTERVALS: GENERAL DISCUSSION
89
was referred to as the likelihood function*
(2) Determine
= log
I 1
This step is not essential. However, since likelihood functions
are products, and since sums are usually more convenient to deal
with than products, it is customary to maximize the logarithm
of the likelihood rather than the likelihood itself,
(3) Determine the value of 6 which will maximize L by solving the
equation
6.3 MAXIMUM LIKELIHOOD ESTIMATORS
Rather than burden the reader with the details of obtaining maxi
mum likelihood estimators, the results for four of the more common
distributions are presented in Table 6.1.
TABLE 6.1-Maximum Likelihood Estimators Associated With Certain
Distributions
Distribution
Parameter
Maximum Likelihood Estimator
Binomial
7->
p =f/n = observed relative frequency
Poisson
X(=M)
X=jLt=jr= 53 x/n
Normal
a(=l*)
a = = X = 21 X/n
Exponential
b\ = a*)
0(=/i)
^=^^ m ^ ^Z(X~T^/n
0==X= ] X/n
6.4 CONFIDENCE INTERVALS: GENERAL DISCUSSION
A point estimate of a parameter is not very meaningful without some
measure of the possible error in the estimate. An estimate of a param
eter 6 should be accompanied by some interval about , possibly of the
form 6 d to +d, together with some measure of assurance that the
true parameter 6 does lie within the interval. Estimates are often given
in such form. Thus, the activated life of a thermal battery may be
estimated to be 300 20 seconds with the idea that the life is unlikely
to be less than 280 seconds or greater than 320 seconds. The develop
ment engineer engaged in research on capacitors may estimate the
mean life of a certain type of capacitor under stated conditions to be
300 50 hours with the implication that the correct average life very
probably lies between 250 and 350 hours. The failure rate for a specific
component might be estimated as being less than 0.02 with the feeling
90 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
that the true failure rate is most likely no greater than the stated limit.
In this last case, the point estimate might have been anywhere between
and 0,02.
Confidence intervals enable us to obtain a useful type of information
about population parameters without the necessity of treating such
parameters as statistical variables. It should be clearly understood that
we are merely betting on the correctness of the rule of procedure when
applying confidence interval techniques to a given experiment. It will
be observed in the following sections that this technique may be ap
plied to various familiar population parameters such as the mean and
variance.
An examination of the following sections will reveal that the method
for finding confidence intervals consists in first finding a random
variable, call it Z, that involves the desired parameter but the dis
tribution of which does not depend upon any other unknown param
eters. Next, two numbers, Z\ and Z* 9 are chosen such that
P{Z 1 < Z < Z*} = T (6.1)
where y * s the desired confidence coefficient, such as 0.95. Then the
two inequalities are manipulated so that the probability statement
assumes the form
P{L < < U] - y (6.2)
whore L and U are random variables depending on Z but not involving
0. Finally, we substitute the sample values in fj and U to obtain a
numerical interval which is the desired confidence interval, It is clear
that any number of confidence intervals can be constructed for a pa
rameter by choosing %i and #2 differently each time or by choosing dif
ferent random variables of the Z type,
The above has been concerned with what is called a two-sided con
fidence interval. However, we sometimes do not care how much our
estimate may err in one direction provided that it is not too far off in
the other. For example, xve may he estimating a standard deviation
which we hope will be small. We would be concerned only about an
upper limit and hence would want an interval of the form
p[e < U} y. (6.3)
The theory of one-sided intervals is basically the same as for two-sided
intervals.
6.5 CONFIDENCE INTERVAL FOR THE MEAN OF A NOR-
IVIAL POPULATION
All the necessary statistics are now available to make possible an
excellent scheme for estimating the mean of a normal population. It has
already been Haul that 3T in an xmhianed estimate of M- However, it is
possible to learn a little more about the estimate, namely, whether 3T
6.5 CONFIDENCE INTERVAL: MEAN OF NORMAL POPULATION 91
is close to M or likely to be far removed from ju- Making use of the f-dis-
tribution, the following statement can be made: 1
P\X o.975(n-l)S^- < M < X + ^
=* P<X *0.975(n-l)-4 : = < <X + /0.975Cn~l) ~= = 0.95 (6.4)
v -\/n
where io.arscn o is a numerical quantity extracted from the table in Ap
pendix 5 under the column labeled 0.975 and for n 1 degrees of free
dom. The above statement (Equation 6.4) is read: the probability,
before the sample is drawn, that the random interval
will cover, or include, the true population mean /z, is equal to 0.95.
Thus, if a random sample is obtained from a normal population with
mean M and variance cr 2 , and the two quantities
L = X 2o.975( D-Sjg- (6.5)
and
U = X + o.975(n-l)Sjc (6.6)
are computed, it can be said that one is 95 per cent confident that the
true mean ju will be in the interval (Z/, C7). One does not say that the
probability is 0.95 that M lies between L and U but only that one is
95 per cent confident that AC does lie between L and U. This distinction
is made because M either does or does not fall between L and U; the
probability is either or 1 for /UL is a constant and does not possess a
probability distribution. The distinction made above is a subtle one
and the concept may not be fully appreciated at this time. However,
it is a distinction that must be made.
Example 6.2
Consider the estimation of the mean breaking strength of some par
ticular material. We take at random a number of samples, for this
example, six, and subject them to test, recording the pressure at which
they break. These values might be as follows:
2206 Ibs. 2203 Ibs.
2209 Ibs. 2206 Ibs.
2205 Ibs. 2207 Ibs.
Averaging these values, we obtain a point estimate, that is, one value,
of 2206 Ibs. This means that, from our sample, a reasonable estimate of
the true (population) average breaking strength of the material is
2206 Ibs. However, we do not have any measure of our degree of con-
1 The symbol sg is known as the standard error of the mean, and it is clearly an
estimator of <rg as defined by Equation (5.5).
92 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
fidence in this estimate. If we are willing to assume a normal distri
bution, we can find:
L =5 X o.975(-l)<Sj.
2206 2.571(0.8165) = 2203.9 Ibs.
U = X + 0.975(71-1)^
= 2206 + 2.571(0.8165) = 2208.1 Ibs.,
where 0.975(5) =2.571 was obtained from the table in Appendix 5 and all
other values were calculated from our sample. We can now say that we
are 95 per cent confident that the true population mean breaking
strength lies between 2203.0 Ibs, and 2208.1 Ibs. A 99 per cent confi
dence interval can be found in a similar manner using 0.995(5) =4,032.
It should be noted that, in general, a WOy per cent confidence interval,
<y <1, may be obtained by using tfui-foo/ajc D * n Equation (6,4).
Rather than proceed as in Example 6.2, we might have wanted only
a lower confidence limit. That is, we might have no interest in an upper
limit on breaking strength, since ordinarily no harm can result from the
material being too strong. The statement needed (assuming a 0,95 con
fidence coefficient) is then
< M = 0.95. (6.7)
Note that here io.osu-i) is used instead of o.975(u~i) since we want the
entire 0.05 error risk to he on one side of the limit rather than to be
split equally beyond two limits. Thus, we would obtain
L X 2o.95< l)$j
2206 2.015(0.8165) = 2204.4, (6.8)
and could then state that we are 95 per cent confident that the true
population mean breaking strength is above 2204.4 pounds* In general,
the lower limit woxild be // 3T 7 ( n ~i>S.
It should be clear that if only a lOOy per cent upper confidence limit
is desired, the procedure woxald be to calculate
U J + *-y (-!)$. (6.9)
6.6 CONFIDENCE INTERVAL FOR THE MEAN OF A NON-
NORMAL POPULATION
A question which might logically arise is "What can we do if we want
a confidence interval estimate of the mean of a nonnormal population?"
The central limit theorem discussed in Chapter 5 provides us with an
answer which is often satisfactory. That is, unless the distribution is
rmxch different from normal and the sample si#e in extremely small, the
distribxition of sample means will be nearly normal so that the normal
theory may be applied with only a small error,
However, if the error introduced by the approximate procedure sxig-
gestod in the preceding paragraph cannot be tolerated, we always have
recourse to exact methods associated with the partictilar population
distribution involved. No attempt will be made to list all the different
6.7 CONFIDENCE INTERVAL: VARIANCE OF NORMAL POPULATION 93
situations. Rather, we shall state only that the basic approach is always
the same as outlined in Section 6.4. If the need arises for an exact answer
for a nonnormal distribution, the reader is referred to many such ex
amples in the literature. If the particular case in question cannot be
located in this manner, a mathematical statistician should be con
sulted.
6.7 CONFIDENCE INTERVAL FOR THE VARIANCE OF A
NORMAL POPULATION
Using a technique similar to that outlined in Section 6.5, a confidence
interval for estimating the variance of a normal population can be
found. This time, however, the chi-square distribution will be used to
obtain the confidence interval specified by
\
J
'
{
Y
(n 1) X [(l~T)/2] (n 1)
= T (6.10)
(n 1) [U Y)/2] (n 1
and this is read: the probability, before the sample is drawn, that the
random interval (Z/ 3 Z7), where
__ (6 _ u)
-1) X[ (l4 _ y ) /2 ]
and
U- - . , (6.12)
* [ Cl-Y) /2] (n-1) * C (1 Y) /21 Cn 1)
will include the true population variance a- 2 is equal to y. Or, as it is
more often phrased, we are 100-y per cent confident that the true popu
lation variance cr 2 will be in the interval (I/, C7) . For the example used
in Section 6.5, we find the 90 per cent confidence interval for a- 2 to be
(1.8, 17.5).
As with means, we can determine a one-sided confidence interval.
This would be defined by
-\
L
'
T (6.13)
1 T) (n 1) '
if an upper limit is desired. Although a lower limit is conceivable, it
would seldom be of interest.
If we are interested in a confidencejuaterval for estimating cr rather
than cr 2 , the confidence limits Z/ = VZ and U f =\/U, where L, and U
are the confidence limits for cr 2 , may be computed. It should be noted
that this is not the exact solution. However, it is sufficiently accurate
for most purposes,
94 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
6.8 CONFIDENCE INTERVAL FOR p, THE PARAMETER
OF A BINOMIAL POPULATION
It has already been suggested in Section 6.3 that the best point esti
mate of p is
p = f/ n = observed relative frequency. (6.14)
If a two-sided, lOOy per cent confidence interval estimate of p is de
sired, the following two equations must be solved for p:
CO, x)p*(l - p)--* - (1 - T)/2 (6.15)
(6.16)
The solution of Equation (6.15) is L, while the solution of Equation
(6.10) is ?/. If /==0, L is taken to be 0; if /=n, J7 is taken to be I. You
may then state that you are IQOy per cent confident that the true
value of p is between Ij and 7.
Example 6.3
Consider an industrial process producing parts which arc classified
either as defective or mmdefeetive. In a random sample of 200 items,
6 arc found to be defective. Thus p =0/200 0.03. To obtain L and U
such that we would have 95 per cent confidence in the limits, we would
substitute 200 for n, for /, and 0.95 for y in liquations (6.15) and
(6.16) and solve lining tables of the binomial distribution* However,
due to the nature of the tables, only approximate answers would be
possible. That IB, the tables are not published for small enough incre
ments of p to permit an exact solution. Interpolation would ho necessary.
Because the computation involved in solving Equations (0.15) and
(0,10) in tedious, several attempts have boon made to provide conven
ient tablet* uncl graphs for the research worker to use. For example,
Uald (12) has published comprehensive tables for certain sample sixes.
More condensed tables are given in Rnodooor (20). (Copper atul Pear
son (5) published charts which arc very helpfuL ( 1 alvert (4) gives
charts and nomographs from which one-sided tipper eonficlenee limits
can be read with reasonable accuracy. Mxieiich (15) has constructed a
compact and easily used slide rule which extends the charts provided
by Oalvert.
As pointed out in Sections 5.10 and 5.1 1, it is often possible to ap
proximate the binomial distribution by the Poisson or normal distri-
btition. These approximations can sometimes be used to advantage in
establishing confidence intervals for binomial probabilities* However,
the details will not be discussed here.
6.9 CONFIDENCE INTERVAL: DIFFERENCE OF TWO MEANS 95
6.9 CONFIDENCE INTERVAL FOR THE DIFFERENCE BE
TWEEN TH E M EANS OF TWO NORMAL POPU LATIONS
Many practical problems in statistics involve the comparison of two
sample means. When the two random samples from which the means
are computed can be assumed to have come from normal populations,
confidence limits for the true difference, that is, for the difference
between the means of the two populations may be computed.
Case I : erf = cri
If it can be assumed that the two normal populations have equal
variances, that is, if we can assume a common variance a 2 , then the
ratio
~y "V "V" r V : /~^V" V \. / N. 1 /9
-A. 1 -A-2 -"^ 1 -^L 2 / .A l -^-2\ / W-l^Z-2 X 4 -'*
+ nj (6.17)
is distributed as Student's t with ni+n 2 -2 degrees of freedom if s 2 is
calculated by means of the formula
(6.18)
In Equation (6.18) the expressions T^aff and 2^x1 represent the sum
of the squares of the deviations about the means in the first and sec
ond samples, respectively. Also, s 2 is often referred to as the pooled
estimate of variance.
Under the assumptions stated above, 100^ per cent confidence limits
for /xi ^2 may be found by calculating
+ n 2 2 HI + n% 2
* \, * +js ' i, \ A ~r f / 1 **t \' f i~T~'"Z ** / ~ Ji, i Ji. 2 *" \ " s
Example 6.4
There are two methods of measuring the jnoisture content of heat-
processed beef. For Method 1 we obtain ^"1 = 8^6, s? 109.63, and
ni 41. For Method 2 the comparable results are: Jf2 = 85.1, s! = 65,99,
and ri2 == 31. Thus,
J 2 = (40(109.63) + 30(65.99) }/70 = 90.93,
and ^-x a M {(90.93/41) + (90.93/31) } ^ = 2.27.
Finally, assuming an 80 per cent confidence interval is desired, we
obtain L=* 3. 5- (1.294) (2. 27) ^0.6 and J7 = 3. 5 + (1.294) (2. 27) ^6.4.
Case II :
If there is reason to believe that the two populations have different
variances, the procedure just discussed is not appropriate. What, then,
can be done? If we are willing to assume that Si crf and sl = crl, an
96 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
approximate 100-y per cent confidence interval may be found by calcu
lating
i/i + Jl/2) 1/a (6.20)
where 3 [(14 _ T)/; >] is the 100(1 + ^)72 fractile of the standard normal
distribution. However, because of the doubtful validity of the assump
tion that the sample variances equal the population, variances, this
procedure provides only a very crude estimate of the true mean differ
ence. Consequently, the procedure should be used with extreme cau
tion.
Case III: Paired Observations
If two samples of equal size are obtained (that is, if ?&i = n2==n) and
if the observations in one sample can logically be paired with the ob
servations in the other sample, a modified procedure applies. By pair
ing, it is meant that the observations (Xi, X%, , X n ) and the
observations (Fi, Y 2 , - - , F n ) are associated as follows:
Xi is related to Y\
Xi is related to F$
X n is related to Y n *
In the language of a later chapter, the variables X and Y are said to be
correlated. When such a correlation in assumed to exist, an appropriate
procedure is to calculate the differences, /-> X Y, and then estimate
/ji D ^fj>x~~VY Confidence limits are then given by
where 4=*4>/n and $%** { 22/> 2 - ( J^D^/n} /(n-l).
Example 6.5
It IB desired to compare the prices of Delicious and Melntoeh apples.
On a certain day> prices (per box) were obtained from a random selec
tion of eleven markets. ABBUming (I) prices to be normally distributed
and (2) the price of one variety in a market would he influenced by the
price of the other variety in the name market, the method of paired
observations will he tincd* The data ure given in Table 6.2* Calculation
yields 75^CU4, * -0.0018, and ^-(U)018/1K Therefore, a 95 per
cent confidence interval for ^/> is specified by
_ 0.14=F (2.228) (.OOlS/i !}'** ($Q.tI, $0.17).
Although, not Illustrated in Kxample 6.5, it should be obvious that some
6. TO CONFIDENCE INTERVAL: VARIANCES OF TWO POPULATIONS
97
of the differences could be negative. This is so because the differences
are defined as D = X Y, not as D = \XY\. Actually, it does not mat
ter whether XY or Y X is used as long as the same choice is used
throughout a given problem.
Rather than burden the reader with excessive repetition, we shall
only remind him that one-sided confidence limits are also possible. All
that is necessary is a change in the value of t in Equations (6.19) and
(6.21), or a change in the value of z in Equation (6.20).
TABLE 6.2-Price per Box of Delicious and Mclntosh Apples
Market
Delicious
Mclntosh
Difference
1
$2. 15
$2 32
$0 17
2
2.16
2 34
0.18
3
2.13
2.30
0.17
4
2.25
2.40
0.15
5. . . .
2.20
2 34
0.14
6
2.18
2.20
0.02
7
2.27
2.42
0.15
8 . .
2.21
2,36
0.15
9
2.23
2.36
0.13
10
2.16
2.30
0.14
11 . .
2.20
2.34
0. 14
6.10 CONFIDENCE INTERVAL FOR THE RATIO OF THE
VARIANCES OF TWO NORMAL POPULATIONS
The problem, of estimating the ratio of two population variances (or
standard deviations) is also frequently encountered. If the two popu
lations are normal, the /^-distribution may be used to provide the de
sired confidence intervals. The procedure is to calculate
and
U
=(f)
-(3)
r-l.na 1),
(6.22)
(6.23)
and these limits define a 100-y per cent confidence interval for
Should only aix upper (or lower) limit be desired, it can easily be found
by using F y in Equation (6.23) or FI_ T in Equation (6.22). One other
useful result is the following: If only an abbreviated F-table is avail
able (e.g., one that contains only the upper percentage points), the
identity
98 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
-^("i^s) ~ - (6.24)
& U-~?OO2,*'l)
permits the calculation of F-values at the left-hand tail of the distri
bution.
Example 6.6
Using the data given in Example 6.4, 99 per cent confidence limits
for oi/V? are found to be:
L = (65.99/109.63) (0.416) = 0.25
U = (65.99/109.63) (2.52) = 1.52.
If a confidence interval for the ratio of the standard deviations of
two normal populations is desired (that is, if we wish to estimate
cra/o-x), it is appropriate to calculate Z/=vX and tf'===vT7 where L
and U arc defined by Equations (G.22) and (6.23).
6-11 TOLERANCE LIMITS: GENERAL DISCUSSION
One common method used by engineers to specify the q uality of
manufactured product is the method of tolerance limits. When such
limits are quoted, it is expected that a certain percentage of the product
will have a quality between the stated limits. For example,, suppose
electrical gaps are judged by the characteristic;, " transfer time." It is
then desirable to be able to quote two limits, A and /?, such that we
are fairly certain that, say, 98 per cent of all gaps produced will exhibit
transfer times between A and 1$> Such limits (dearly provide us with a
measure of the quality of the product under consideration. For certain
weapon applications, it is convenient to be able to set a one-sided toler
ance limit. An example of such a case is the following: 90 per cent of all
Type XYZ batteries will yield an activated life of at least 200 seconds.
In general, then, tolerance limits are limits within which we are highly
confident will lie a certain percentage of the individuals of a statistical
population.
To apply tolerance limits in u satisfactory manner, certain conditions
must be met. In summary, the conditions upon which tolerance limits
are based are the following:
(1) All assignable causes of variability must be detected and
eliminated BO that the remaining variability may be con
sidered random.
(2) Certain assumptions must be made concerning the nature of
the statistical population tinder study*
6.12 TOLERANCE LIMITS (TWO-SIDED; ONE-SIDED)
FOR NORMAL POPULATIONS
Tolerance limits considered in this section are based on the assump
tion that the parent population may bo described by a normal dintri-
6.12 TOLERANCE LIMITS FOR NORMAL POPULATIONS 99
but ion. If the true mean and standard deviation are known, tolerance
limits are formed by adding to and subtracting from the mean some
multiple of the standard deviation. That is, if & and o- are known, toler
ance limits take the form jj, zo- where z is selected from Appendix 3
and depends only on the proportion of the population to be included
within the calculated limits. For example, the limits M 1.645<r include
90 per cent of a normal population with mean JJL and standard devia
tion cr. One-sided tolerance limits may, of course, be obtained by con
sidering n+za or JULZO- as the problem requires.
In a practical situation, ^ and cr are unknown. Only estimates, ~X
and s, are available. While it was true that the limits /z1.645<r will
include 90 per cent of the population, the same statement cannot be
made concerning J^it 1.645s. Just what proportion of the population
will lie between X Ks depends on how closely "X and s estimate M and
cr. Note that K is used here to represent the constant used with 'X and
5 in contrast with the z used with ^ and cr.
Since X and s, and hence ~XKs } are random variables, it is im
possible to state with certainty that j + Ks will always contain a
specified proportion, P, of the population. That is, it is impossible to
choose K so that the calculated limits will always contain a specified
proportion, P, of the population. However, it is possible to determine
K so that in many random samples from a normal population a certain
fraction y of the intervals Qc.Ks) will contain 100P per cent or more
of the population. When this notation is used, P is referred to as the
coverage and y as the confidence coefficient. This terminology is used
since we are lOOy per cent confident that the tolerance range specified
by jKs will include at least 100P per cent of the normal population
sampled.
Intuitively, it is reasonable to expect that values of K used with 3T
and s will be larger than values of z used with M and <r. It is_also clear
that if K is taken large enough, then the probability that X Ks will
contain at least 100P per cent of the population may be made very
close to 1. However, the smaller K is taken, the more meaningful and
useful the tolerance range becomes. The engineer is thus faced with a
decision: make broad statements with little risk of error or make pre
cise statements (i.e., a narrow tolerance range) with greater risk of
error. The problem, statistically speaking, becomes that of finding the
smallest value of K consistent with a specified confidence coefficient y,
proportion P, and sample size n.
We must not forget that one-sided tolerance limits are frequently
more appropriate than two-sided tolerance limits. That is, it is often
desirable to specify a single limit such that a given percentage of the
population will be less than (or greater than) this limit. Such a limit
is known as a one-sided tolerance limit and is usually of the form
~X+Ks (or IX Ks"), Both one-sided and two-sided tolerance limits
for normal populations will be discussed in the following paragraphs.
Table 6,3 is an abbreviated table of K factors for two-sided tol-
too
CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
erance intervals. Values of K taken from this table give a 95 per
cent confidence that at least a fraction P will be included in the
interval X + Ks. Table 6.4 is an abbreviated table of K factors for
one-sided tolerance intervals. Values of K taken from this table give
a 95 percent confidence that at least a fraction P will be above (below)
2T K$(~X + Ks). Much more extensive tables can be found in Bowker
and Lieberman (2), Eisenhart, Hastay, and Wallis (21), Owen (19),
and Weissberg and Beatty (23).
Example 6.7
Using the data of Example 6.2, find tolerance limits such that you
arc 95 per cent confident of including at least 99 per cent of the sampled
population. These limits are given by
3T Ks 2206 5.775(2) ~ (2194.45, 2217.55).
TABLE 6.3-Two-Sided Tolerance Factors
(Factors K such that the probability is 0.95 that at least a proportion P
of the distribution will be included between l?Ks where 3T and $ are
computed from a sample of size n.}
P
Yl
0,7500
0.9000
. 9500
0.9900
5
3.002
4.275
5.079
6.634
6
2,604
3.712
4.414
5.775
7
2.361
3 . 369
4.007
5 . 248
8
2.197
3.136
3.732
4.891
9
2 , 078
2,967
3 , 532
4.631
10
1.987
2 , 836
3,379
4.433
17
1.679
2.400
2.858
3,754
37
1 . 450
2,073
2.470
3 , 246
145
1 .280
1 .829
2.179
2.864
oo ,
1 .150
1 . 645
1,960
2.576
The foregoing discussion of tolerance limits and the K factors given
in Tables 6.3 and 6.4 depend squarely on the assumption of a random
sample from a normal population. If tolerance limits are calculated
using these tables when the sampled population is definitely non-
normal, considerable error is possible.
6-13 D1STRIBUTION-FREE TOLERANCE LIMITS
Sometimes it is desirable to Bet tolerance limitn that do not depend
on the assumption of normality. That is ? we recognize that it in riot
always possible to justify the assumption of a normal distribution. If
we are dealing with a statistical variable that can be described by a
continuous distribution, one very simple set of dwtribution^fr^a toler
ance limits is specified by X m i n and -Sfmax, the smallest and largest
PROBLEMS 1O1
TABLE 6.4-One-Sided Tolerance Factors
(Factors K such that the probability is 0.95 that at least a proportion P
of the distribution will lie above (below) ~X Ks(X+Ks) where X and
s are computed from a sample of size n.}
P
0.7500
. 9000
0.9500
0.9900
5
2.150
3.412
4.212
5.751
6
1.895
3.008
3.711
5.065
7
1.733
2.756
3.400
4.644
8
1.618
2.582
3.188
4.356
9 ....
1 .532
2.454
3.032
4.144
10
1.465
2.355
2.911
3.981
17
1,220
2.002
2.486
3.414
37
1 .014
1.717
2.149
2.972
145
0.834
1.481
1.874
2.617
CO ....
0.674
1.282
1.645
2.326
values in a random sample of size n. Clearly, the confidence in such
limits will depend on n. Persons interested in reading further on
this topic are referred to Murphy (16), Ostle (17), Owen (18), and
Wilks (24).
Problems
6.1 As a physicist or chemist, you would soon become acquainted with
such "constants*' as Planck's constant and Euler's constant. To con
sider a specific case, Planck's constant is defined as "the quantum of
energy radiated from black bodies -5- frequency of radiation. 33 Suppose
you were attempting to find the value of this constant by experi
mental methods. You ran 6 experiments and obtained the following
estimates of h (Planck's constant) :
6.53 X 10~ 27
6.54 X 10~ 27
6.58 X 10~ 27
6.56 X 10~ 27
6.55 X 10~ 27
6.55 X 10- 27
What inferences can you make about the true value of h? Be careful
to state explicitly any assumptions you make.
6.2 Given that n = 9, 7 = 20, and
JC y* = 288,
calculate a 95 per cent confidence interval for ju- on the assumption
that you have a random sample from a normal population. Interpret
this confidence interval.
102 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
6.3 From a random sample of 100 aptitude test scores drawn from a
normal population, the 95 per cent confidence interval for ju is calcu
lated to be (45, 55). Fifty other random samples, each of size 100, are
drawn from the same population, but only 10 of their means fall
within the above limits. Is it not correct to expect 95 per cent of such
sample means to be between 45 and 55? Explain your answer.
6.4 A random sample of 25 observations from a normal population had
a mean of 20 and a sum of squares of the deviations from the mean of
2400, Compute and interpret the 90 per cent confidence interval for
the population mean.
6.5 It has been reasonably well established that a particular machine
produces nails whose length is a random variable with a normal dis
tribution- A random sample of 5 nails yields the following results:
1 . 14 inches
1 . 15 inches
1 . 14 inches
1.12 inches
1.10 inches
Calculate 99 per cent confidence limits for /*.
6.6 The density of each of 27 explosive primers was determined, with the
sample average being 1 .53 and the wampic standard deviation being
0.04, [Determine a 90 per cent upper confidence limit for /*.
6.7 The firing of 101 rockets yielded an average range (i.e., distance
flown) of 3000 yards and a standard deviation of 40 yards. Determine
an 85 per cent lower confidence limit for M.
6*8 Using the data of Problem 0,1, compute a 05 per cent confidence
interval for or 2 .
6.9 Using the data of Problem 4.11, compute a 90 per cent confidence
interval for <r.
6.10 If in a sample of 14 holts the estimate of the population standard
deviation of their lengths was # ,,021, what are the OS per cent con
fidence HmitR for the standard deviation of the population (<r)? What
assximptionB nmnt he made to determine these limits?
6.11 Using; the data of Prohlem 6.10, determine a 00 per cent upper con
fidence limit for <r*
0.12 Uninpr the data of Problem 0.5, determine a 07,5 per cent upper confi
dence limit for cr.
6A3 Using the data of Problem 0.0, determine a 00 per cent upper confi
dence limit for <r,
6.14 In 1054 the mean earnings of 68 physicians in communities from
10,000 to 25,000 was $13,944, with #$40ia. Find the 00 per cent
confidence limits for the population standard deviation. State your
assumptions*
6.15 In a random natnple of 400 farm operators* 05 per cent were owners
arid 35 per cent were mmownern. Determine 95 per cent confidence
limits for the true percentage of farm owners in the population of
operators? sampled.
0,10 In a random sample of 600 light bulbs, 12 were defective. Determine
a 95 per cent upper confidence limit for the true fraction defective.
6.17 Uning the data of Problem 4.1, and the results found in Prohlem
PROBLEMS 1 03
4.10, determine: (a) 95 per cent confidence limits for /x and (b) 80 per
cent confidence limits for a 2 . State all assumptions.
6.18 Using the data of Problem 4.2 and the results found in Problem 4.10,
determine: (a) a 99 per cent upper confidence limit for M and (b) a 95
per cent upper confidence limit for <r. State all assumptions.
6.19 Using the data of Problem. 4.3 and the results found in Problem
4.10, determine 50 per cent confidence limits for each of the means
and 50 per cent upper confidence limits for each of the standard devia
tions. State all assumptions.
6.20 You are engaged as a testing engineer in an electrical manufacturing
plant. One of the products being produced is an electric fuse, and the
most important characteristic of this fuse is the length of time before
it "blows" when subjected to a specified load. A testing program was
undertaken and the following sample data (in seconds) were obtained.
Day 1 Day 2
42
69
45
109
68
113
72
118
90
153
Place a 90 per cent confidence interval on the true difference between
the means of the two different days' productions. Assume that each
day's production may be represented by a normal population. State
all other assumptions which you make and interpret your numerical
answer.
6.21 Given that
7i - 75, ! 9, ylt - 1482, F 2 60, n* - 16, ytj = 1830,
i-i j-i
and assuming that the 2 samples were randomly selected from 2
normal populations in which erf = erf, calculate an 80 per cent confi
dence interval for JLAI- jus-
6.22 Two barley varieties have been grown at a number of locations over
several years in an area and their general adaptability is under dis
cussion. Which variety would you select for the area on the basis of the
following yields in bushels per acre?
Trebi 41.2, 19.3, 45.5, 63.9, 63.8, 44.2, 42.5, 53.0.
Svanota 39.4, 30.8, 44.5, 51.5, 41.1, 26.5, 35.7.
Place confidence limits on the difference between the means.
6.23 Two varieties of tomato were experimented with concerning their
fruit-producing abilities. The study was done in a greenhouse and,
because of extreme variations (among locations within greenhouses)
of temperature, light quality, and light intensity, the experimental
plants were placed in pairs (one of each variety) at several locations.
The following data were obtained:
1O4 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
WEIGHTS or RIFE FRXTITS FOR Two VARIETIES OF TOMATO
(in pounds)
Location
Variety
Difference
A
B
A B = D
1
3 . 03
3.10
2.35
3.86
3.91
2.65
1.72
2.30
2.70
3 , 60
2.28
2.68
2.17
3.56
3 . 73
1.48
1.85
1.86
2.76
2.68
.75
.42
.18
.30
.18
1.17
.13
.44
.06
.92
2
3
4
5
6.
7
8
9
10. . ....
Total
29.22
25.05
4.17
0.24
6.25
6.26
6.27
6.28
6*29
6.30
6,31
Determine 90 per cent confidence limits for the true difference between
the expected weights of the two varieties. State all assumptions.
ITwing the data of Problem 6.20, obtain 95 per cent confidence limits
for orj/crl.
Using the data of Problem 6.21 and ignoring the assumption used
there, namely, that cr?cri ? obtain 99 per cent confidence limits for
Using the data of Problem 6,22, obtain 95 per cent confidence limits
for <r s /<r T .
Using the data of Problem 6.2, determine with 95 per cent confi
dence: (a) 95 per cent tolerance limits and (b) an upper 99 per cent
tolerance limit.
Using the data of Problem 6,4, determine with 95 per cent confidence:
(a) 76 per cent tolerance limits and (b) a lower 90 per cent tolerance
limit*
Using the data of Problem 6.6, determine, with 95 per cent confi
dence, a 99 per cent upper tolerance limit on the densities.
Using the data of Problem 6*7, determine, with 95 per cent confi
dence, a 90 per cent lower tolerance limit on the ranges,
Consider tho following definitions:
(a) If the expected value of an estimator does not equal the true value
being estimated, tho difference between tho expected value and
the true value ia known aa the bias of the estimator,
(6) If an estimator has bias, it IB Baid to bo accunde*
(c) If an estimator has a small bias, it is Baid to bo relatively accurate.
(d) If an estimator has a large bias, it i& said to bo inaccurate*
($) The precision of an estimator is a measure of the repeatability of
the estimator. Therefore, precision may be expressed in terms of
the variance of an estimator, with a large variance signifying lack
of precision and a small variance signifying high prtuttaton* Obvi
ously, absolute precision implies a variance, an ideal seldom (if
REFERENCES AND FURTHER READING 105
ever) achieved. (JNTOTE: Sometimes a measure of precision is
referred to as a measure of reliability. Because the word "relia
bility" has another meaning in engineering, this is unfortunate.
However, as with many expressions, the phrase is now a part of
the language of statistics and will therefore continue to be used.)
It should be observed that an estimator may be: (1) both precise and
accurate, (2) neither precise nor accurate, (3) precise but not accu
rate, or (4) accurate but not precise.
(a) Discuss the foregoing concepts and definitions relative to the con
tents of Section 6.1.
(6) Discuss these ideas taking cognizance of costs and other economic
and physical limitations which continually plague the researcher,
(c) Discuss the accuracy and precision of the various estimators that
have been introduced so far in this text.
References and Further Reading
1. Anderson, R. L., and Bancroft, T. A. Statistical Theory in Research. McGraw-
Hill Book Company, Inc., New York, 1952.
2. Bowker, A. H., and Lieberman, G. J. Engineering Statistics. Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959.
3. Brownlee, K. A, Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
4. Calvert, R. L. The Determination of Confidence Intervals for Probabilities of
Proper, Dud, and Premature Operation. Sandia Corporation Technical
Memorandum SCTM 213-55-51, Sandia Corp., Albuquerque, N, Mex., Oct.
17, 1955.
5. Clopper, C. J., and Pearson, E, S, The use of confidence or fiducial limits
illustrated in the case of the binomial. Biometrika, 26: 40413, 1934.
6. Dixon, W. J., and Massey, F. J. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
7. Fisher, R. A. On the mathematical foundations of theoretical statistics.
Philosophical Transactions of the Royal Society. Series A, Vol. 222, 1922.
g p _ Theory of statistical estimation. Proceedings of the Cambridge
Philosophical Society, Vol. 22, 1925.
9. Freund, J. E. Modern Elementary Statistics. Second Ed. Prentice-Hall, Inc.,
Englewood Cliffs, N.J., 1960.
IQ 1 Livermore, P. E., and Miller, I. Manual of Experimental Statistics.
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1960.
11. Hald, A. Statistical Theory With Engineering Applications. John Wiley and
Sons, Inc., New York, 1952.
12. . Statistical Tables and Formulas. John Wiley and Sons, Inc., New
York, 1952.
13. Huntsberger, D. V- Elements of Statistical Inference. Allyn and Bacon, Inc.,
Boston, 1961.
14. Mood, A. M. Introduction to the Theory of Statistics. McGraw-Hill Book
Company, Inc., New York, 1950.
15. Muench, J. O. A Confidence Limit Computer. Sandia Corporation Mono
graph SCR-159, Sandia Corp., Albuquerque, N. Mex., April, 1960.
16. Murphy, R. B. Non-parametric tolerance limits. Ann. Math. Stat., 19:581-89,
Dec., 1948.
17. Ostle, B. Some remarks on the problem of tolerance limits. Industrial
Quality Control, 13(No. 10):11~13, April, 1957.
18. Owen, D. B. Distribution-Free Tolerance Limits. Sandia Corporation Techni
cal Memorandum SCTM 66A-57-51, Sandia Corp., Albuquerque, N. Mex.,
June, 1957.
106 CHAPTER 6, STATISTICAL INFERENCE: ESTIMATION
19. . Tables of Factors for One-Sided Tolerance Limits for a Normal Dis
tribution. Sandia Corporation Monograph SCR-13, Sandia Corp., Albu
querque, N. Mex., April, 1958.
20. Snedecor, G. W. Statistical Methods. Fifth Ed, The Iowa State University
Press, Ames, 1956.
21. Statistical Research Group, Columbia University. Selected Techniques of
Statistical Analysis. (Edited by C. Eisenhart, M. W. Hastay, and W. A,
Wallis.) McGraw-Hill Book Company, Inc., New York, 1947.
22. Wadsworth, G. P., and Bryan, J. G. Introduction to Probability and Random
Variables. McGraw-Hill Book Company, Inc., New York, 1960.
23. Weiswborg, A., and Boatty, G. H. Tables of Tolerance- Limit Factors for
Normal Distributions. Battelle Memorial Institute, Columbus, Ohio, Dec.,
1959.
24. Wiika, S. S. Statistical prediction with special reference to the problem of
tolerance limits. Ann. Math. Stat. y 13:400-409, 1942,
C H APTE R 7
STATISTICAL INFERENCE:
TESTING HYPOTHESES
7.1 GENERAL CONSIDERATIONS
A HYPOTHESIS is defined by Webster as "a tentative theory or supposi
tion provisionally adopted to explain certain facts and to guide in the
investigation of others." A statistical hypothesis is a statement about
a statistical population and usually is a statement about the values of
one or more parameters of the population. For example, the following
could be taken as hypotheses: (1) the probability of a 1 on a toss of a
certain die is f , (2) the mean height of American adult males is 5 feet
8.4 inches, (3) the mean length of a certain brand of 6-inch rulers is
5.99 inches and the standard deviation is 0.02 inch.
It is frequently desirable to test the validity of such hypotheses. In
order to do this, an experiment is conducted and the hypothesis is
rejected if the results obtained from the experiment are improbable
under this hypothesis. If the results are not improbable, the hypothesis
is accepted. For example, we might test hypothesis (1) above by toss-
ing the die 600 times. Intuitively, it is evident that if 600 1's are ob
tained, the result is improbable under the hypothesized probability of
^, and the hypothesis should be rejected. On the other hand, if 100 1's
were observed, this result would not be improbable and the hypothesis
would undoubtedly be accepted. When results such as these are ob
tained, intuition (combined with common sense) is sufficient to decide
whether to accept the hypothesis. However, in actual practice, experi-
meiital results do not usually lead"lu au^li obvi0'usi?5ilt5tn^iong; hence
the leraedTlfo^^ It shoulcTBeT pointed out
that although we accept or rejec^aT^ypothesIs^we^have not proved or
disproved the hypothesis.
In testing hypotheses, there are two types of errors which can be
made. These are called:
Type I error the rejection of a hypothesis which is true.
Type II error the acceptance of a hypothesis which is false.
To aid the reader in comprehending the nature of statistical hypotheses,
decisions, and the various types of error, Table 7.1 has been found
helpful.
When setting up an experiment to test a hypothesis, it is desirable
to minimize the probabilities of making these errors. In order to make
it easier to talk about these errors and their probabilities, the proba
bility of making a Type I error is designated as a. and the probability
of making a Type II error is designated as /S. It should also be noted
[1071
108 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
that 100 a (in per cent) is commonly referred to as the significance level.
What constitutes suitably small values of a and ? This is not a ques
tion which can be answered unequivocally for all situations. Obviously
the values of a and /3 should depend on the consequences of making
Type I and II errors, respectively. For example, if we are considering
the purchase of a lot of batteries (or some other very critical item) for
use in weapons, we might hypothesize that the lot is of satisfactory
quality. (Actually we should state this hypothesis in more precise
terms.) If this hypothesis is true and we reject it, no great harm has
been done since we can always wait for the next lot (assuming that \ve
are not in a hurry). Consequently a. can be relatively large (perhaps
0.25 or larger). Oil the other hand, if the hypothesis is false and we
accept it, the result may be a large number of dud weapons. Since this
is very undesirable, should be quite small, (maybe 0.01 or leas). It
should be pointed out that the supplier might feel differently about
these probabilities*
TABLE 7.1-Definition of the Types of Errors Associated With Tests
of Hypotheses
Decision
True Situation
Hypothesis is true
Hypothesis
is false
error
Accept the hypothesis
No error
Type I error
Type II
No error
Reject the hypothesis* ,
An important consideration in discussing the probabilities of Type
II errors is the "degree of falseness" of a false hypothesis. In a given
experiment, if the hypothesis is false but is nearly true (such as hypoth
esising that a probability is J when actually it is 1.0001/2), ft could be
quite large. However, if the hypothesis is grossly false, (such as hy
pothesizing that a probability is f when it is actually 1), /9 should be
much smaller. For a given experiment testing a specific hypothesis,
the value of 1 /3 is known as the power of the test. Since the power
depends on the difference between the value of the parameter specified
by the hypothesis and the actual value of the parameter where the
latter is unknown> 1/9 should be expressed as a function of the true
parameter. Such a function is known as a power function and is ex
pressed as 1 /8(0) where 6 represents the true parameter value. The
complementary function, /3(0), is known as the operating characteristic
(OC) function.
Before proceeding further with the details of testing hypotheses, a
few more remarks of a general nature are in order- It is good practice
not only to state the hypothesis to be tested (denoted by //) but also
to state the alternative(s) to // (denoted by A). This ia not only good
procedure; it also aids in the determination of the regions of acceptance
7.1 GENERAL CONSIDERATIONS
1O9
and rejection when considering the sample space of all possible values of
the test statistic. Incidentally, the rejection region is frequently referred
to as the critical region. Using the notation of this paragraph, it is seen
that <x = P (reject H\H is true) and @ = P (accept H\ A is true).
Example 7.1
Consider a simple hypothesis, H:IJ,=/JL QJ against a single alternative*
A:fjL=fjLij where we are dealing with a normal population with known
variance, cr 2 . Let the decision to reject H (accept A) or to accept H
(reject A) be based on a single observation obtained at random from the
population under examination. If the random observation is less than
C (see Fig. 7.1), H will be accepted; if the random, observation is greater
than or equal to (7, H will be rejected. That is, X^C constitutes the
rejection or critical region. The probabilities a. and /3 are represented by
the shaded and cross-hatched areas, respectively. Clearly, besides de
pending on the choice of C, a depends on the hypothesis under test
(frequently called the null hypothesis) while /3 depends both on the null
hypothesis and on the alternative hypothesis.
DISTRIBUTION
ASSUMING I-T
IS TRUE
DISTRIBUTION
ASSUMING A.
IS TRUE
ACCEPT J
REJECT L-
FIG. 7. 1 Graphical illusfraHon of the acceptance and
rejection regions in Example 7.1.
Example 7.2
Modify Example 7.1 to the following extent: Consider H:JJL=JJL O
versus the composite alternative A :/z >MO- In this situation a is the
same as before but j3 is now better denoted by /3(/i) =P (accept H\JJL).
Clearly /3(/x) changes as we think of the "alternative distribution" in
Figure 7.1 taking all possible positions for which ni >/xo- Thus, an OC
curve similar to the one shown in Figure 7.2 is generated.
Example 7.3
Consider a further modification of Example 7.1, namely, T:/* = Mo
versus the alternative Arpt^Ma. The acceptance and rejection regions
might be as shown in Figure 7.3, namely, reject if 3C<Ci=Mo -ka or
if -Xr>C2==Mo + &<r and accept if Ci <X <C 2 . Only the distribution of
the test statistic under H is shown. The distribution under A may be
visualized if the reader thinks of sliding the distribution shown to the
left and to the right. For this situation, an OC curve similar to the one
in Figure 7.4 would result.
FIG. 7.2 Type of OC curve to be expected in situations
similar to Example 7.2,
DISTRIBUTION
ASSUMING Ji
IS TRUE
Ci
P* tirr*T i-i-*--*
/So
Arv^fTPT w ....
c a
. ., ._-*.. ocr.ipr'T w ... *
X
FIG. 7,3 Graphical illustration of the acceptance
and rejection regions in Example 7,3,
FIG. 7-4 Type of OC curve to be expected in situations
similar to Example 7.3*
T.2 ESTABLISHMENT OF TEST PROCEDURES 111
7.2 ESTABLISHMENT OF TEST PROCEDURES
When establishing a test procedure to investigate, statistically, the
credibility of a stated hypothesis, there are several factors that must
be considered. Assuming a clear statement of the problem has been
formulated and that an associated hypothesis has been stated in mathe
matical terms, these are :
(1) The nature of the experiment that will produce the data must
be defined.
(2) The test statistic must be selected. That is, the method of ana
lyzing the data should be specified.
(3) The nature of the critical region must be established.
(4) The size of the critical region (that is, ot) must be chosen.
(5) A value should be assigned to 0(8) for at least one value of
other than the value of 8 specified by H. This is equivalent to
stating what difference between the hypothesized value of the
parameter and the true value of the parameter must be de
tectable, and with what probability we must be confident of
detecting it.
(6) The size of the sample (i.e., the number of times the experi
ment will be performed) must be determined.
It should be clear that these steps will not always be taken in the order
listed. Not all of the steps are independent, and frequently it is neces
sary to reconsider (several times) the various steps until a reasonable
test procedure is formulated. More will be said on this subject later.
For now, some explanatory examples will probably be of more value
than additional generalizations.
Example 7.4
With respect to a specific coin we have H:P (heads) =p = 0.5 and
A :p5*=0,5. The experiment will consist of tossing the coin some number
of times, counting the number of times heads occurs, and rejection of
// will take place if either a very small or very large proportion of heads
are observed. Let a. = 0.05. Ignore /?(>) for the moment. Consider n = 5
and the rejection region to consist of either no heads or all heads. Then,
P (rejection | p=0. 5) =tSr = 0.0625. Since this is greater than ct = 0.05,
a larger number of tosses is required. Let us try n = 6, keeping the same
rejection region. Now we have P (rejection |p = 0.5) = 0.03125 which is
less than a. = 0.05. Thus an acceptable test procedure has been devel
oped. (NOTE: The probabilities of rejection given were, of course, cal
culated using f(x) = C(n,
Example 7.5
In Example 7.4 we derived the test: "Toss the coin six times and
reject // if either zero or six heads occurs; otherwise, accept H." Clearly,
other rejection regions might have been chosen together with different
112 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
values of n, as long as P (rejection \p = 0.5) < ex. What we found in
Example 7.4 was the smallest value of n for the specified rejection
region. The reader should investigate some of these other possibilities.
Example 7.6
Now consider @(p) for the test derived in Example 7.4. For selected
values of p the approximate values of /3(p) = l p 6 (1 p) 6 are given
in Table 7.2. (NOTE: Only approximate values are given because the
exact answers involve an unnecessary number of decimal places. For
example, for p = 0.5,
1 -P (rejection | p = 0.5) = 1 -0.03125 = 0.96875^0.97.)
If one did not consider the derived test to be discriminating enough (as
evidenced by the OC curve), the discriminatory power could be in
creased by: (1) changing the sample size and the definition of the crit
ical region or (2) concocting an entirely different test procedure and test
statistic. It is clear that we are faced with just this situation in the pres
ent case. The test derived in Example 7.4 is good for detecting two-
headed or two-tailed coins (nearly as good as looking at both sides of
the coin) but is poor for detecting slightly, or even moderately, biased
coins. Thus a modified or new test is required.
TABLE 7.2-Selected Values of the OC Function for Example 7.6
P
Approximate Values of
(p)
0.1
0.47
0.2
0.74
0.3
0.88
0.4
0.95
0.5
0.97
0,6
0.95
0.7
0.88
0.8
0.74
0.9
0.47
1,0
Example 7.7
Consider the following modification of Example 7.4, namely,
H :p>0.5 and A :p <0.5. The experiment will remain the same but the
regions of acceptance and rejection will change* Obviously, the occur
rence of many heads does not tend to deny //, so the rejection region
will be only that region in which few heads occur. That is, a one-tailed
test (like a one-sided confidence limit) is required. Proceeding as before,
it is found that a possible test is: "Toss the coxa five times. If no heads
occur, reject //; otherwise, accept." This gives <** 0.031 25.
7.4 NORMAL POPULATION; H^^po VERSUS Ai}*, >
7.3 NORMAL POPULATION; H: M ~ Mo VERSUS
Suppose that we wish to know if a random sample could be from a
normal population with mean ^o- More specifically, assuming normal
ity, the hypothesis ^J: M = Mo w iu be tested relative to the alternative
A IM^/XQ. For a chosen a 9 the procedure is to compute
t = (X - MO) A s = Vn(X - MO) A (7.1)
and reject H if t< a-./2>c w -i) or if <> a --/2Xn--i>; otherwise, accept H.
Example 7.8
A metallurgist made four determinations of the melting point of man
ganese: 1269, 1271, 1263, and 1265C. Are these in accord with a
hypothesized value of 1260C? Here the hypothesis is H: M 1260, the
alternative is A :ju^l260, and
/ = ^ """ Mo = 1267 1260 __
Jjf "~ 1.862 ~~ 3 " 83
is computed. Since 0.375(3) ==3.182 (a 5 per cent significance level is
assumed), H is rejected and it is concluded that the hypothesized value
is incorrect. By using a 5 per cent significance level, it is recognized that
the probability of Type I error will be no greater than 0.05. That is,
there is a maximum risk of 5 per cent in rejecting the hypothesis that
M 1260 if the hypothesis is really true.
It would also be of interest to examine the OC curve for the test pro
posed above. However, it would be necessary to prepare OC curves for
many values of <x and n. Also, the formula for $ associated with
"Student's" fr-test involves the noncentral it-distribution which must
be considered beyond the scope of this text. Thus, the reader is referred
to examples of such curves given in Bowker and Lieberman (3).
7.4 NORMAL POPULATION; JET:,z</*o VERSUS A: M >/x , OR
H:M>A*O VERSUS A:jm<fjL*
A more common situation than that considered in Section 7.3 is the
case of a one-sided alternative. For example, a manufacturer produces
wire cable which must have a breaking strength not less than 1500
pounds. A new and cheaper process for making the cable is discovered
and he wishes to change to the new process, provided that cable so pro
duced will have a mean breaking strength greater than 1500 pounds.
Thus he could formulate the hypothesis Hi !*,<}** = 1500 pounds as
opposed to the alternative ,A:/x>1500 pounds. The hypothesis H
would be rejected if a sample of the new cable presented sufficient evi
dence that M actually exceeds 1500 pounds.
In general terms, when testing the hypothesis H:^<^ G versus
the procedure is to calculate
114 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
* = (3 s - MO) A* = Vn(X - MO) A (7.2)
and reject H if 2><(i xni); otherwise, accept H.
If the hypothesis JT:M>MQ versus A:M<MO is under investigation,
the test statistic is calculated as in Equation (7.2), but now H is
rejected only if t< (!_) CTI -_I).
Example 7.9
A manufacturer of television sots purchases tubes from one of the
few large suppliers of such specialized material. He will not purchase
tubes, however, unless it can be demonstrated that the average length
of life will exceed 500 hours. A random sample of 9 tubes is subjected to
a "life test" and the following values are obtained : T =* 600 and s 2 2500.
It is assumed that the "lengths of life" (measured in hours) are normally
distributed. Shall the hypothesis //:^<500 be accepted? For this ex
ample, *= (GOO 500)/16.67 = 6,00 far exceeds 0.95(8) =1.860, and the
null hypothesis is rejected. As can be seen, a 5 per cent significance level
was used. This means that the maximum risk of rejecting //:/z<500
when // is really true is 5 per cent. Therefore, the manufacturer of tele
vision sets will undoubtedly purchase tubes from this supplier.
The reader is again referred to Bowker and Lieberman (3) for sample
OC curves related to these tet procedures,
7,5 NORMAL POPULATION; H:<r 2 =*=<rg VERSUS
Suppose that we have a sample of sixe n drawn randomly from a
normal population and some predetermined value of the variance is to
be substantiated or refuted; i.e., we wish to test the hypothesis,
//:<r 2 = <r as opposed to the alternative A :<r 2 ^<TQ. If a probability, <x,
of making a Type I error has been chosen, i.e., a significance level of
100 <* per cent has been selected, the hypothesis // will be accepted if
xV/.Xn-l) < Z) (X - DVorJ < XV^C^' < 7 - 3 )
Otherwise, // will be rejected,
Example 7*10
Consider the data given in Example 7.8* Do these values mipport the
hypothesis that, if repeated measurements arc assumed to be normally
distributed, the true variance of all such measurements IB equal to 2?
Here the hypothesis in //:<7 2 2 and the alternative IB A :cr 2 p^2. It
IB determined that S(-V 3T)*M-20. Since xS.onw> -0.216 and
Xo.> ""9-#5 we Bee. that // is rejected. You will note that once again a
probability of Type I error equal to 0.05 was ehcmen. That is, we have
run a maximum risk of 5 per cent of rejecting a true hypothesis,
Jf wo winh to determine the CO csurve for this teat, the required
values can bo ealcsulated using
i < X 2 < (^o/o-^X
7.7 BINOMIAL POPULATION; H:p = p VERSUS A:p ^ p
115
The reader is referred to Bowker and Lieberman (3) for examples of
OC curves associated with this test.
POPULATION;
VERSUS A:o
H:a*<<r VERSUS A:<r*>o%,
7.6 NORMAL
or H*r*>o
It is usually more realistic to consider the hypothesis that the popu
lation variance is less than or equal to some particular value than to
consider the hypothesis that it equals some value. This is so because,
in general, a small variance is considered to be desirable. In such a case,
the hypothesis H:<T*<CTQ is formulated as opposed to A:cr 2 >cr|. The
hypothesis H will be rejected only if % 2 = XX-3T 3f) 2 /cro>x 2 (i-<*)(n_i)-
Should jfif:cr 2 >cr (as opposed to A:<r 2 <j7p) be under investigation,
the rejection region would be x 2== T^ (X 3T) VcrS < y ,._, } .
Sample OC curves may be observed in Bowker and Lieberman (3)
for a: = 0.05 and <x = 0.01.
Example 7.11
Consider the data of Table 7.3 which were obtained from a random
sample of 80 bearings. To test (using a. = 0.05) H : cr 2 < 0.00005 versus
A : or 2 > 0.00005, we calculate
X 2 = ]T *2/CK00005 ^ 0.000474/0.00005 = 9.48.
Since this does not exceed xJ 96C79) = 100.7, we are unable to reject H.
TABLE 7.3-Number of Bearings Observed With the Indicated Diameters
Diameter (Inches)
Number
3.573
4
3.574
2
3.575
9
3.576
12
3.577
12
3.578
10
3.579
9
3.580
9
3.581
8
3.582
4
3.583
1
Total
80
7.7 BINOMIAL POPULATION; H:p=p Q VERSUS
The coin-tossing experiment discussed in Example 7.4 illustrates the
type of problem to be considered in this section. However, in practice,
one is usually given the sample size and asked to determine the rejec
tion region, rather than (as in Example 7.4) being asked to find the
116 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
smallest sample size that is consistent with a specified rejection region.
For example, for a fixed sample size, n, the acceptance and rejection
regions are determined by solving
C(, *)po(l - Po)""" = /2 (7.4)
x0
and
C(w, *)pS(l - po)-"" = a/2 (7.5)
for Z/ and Z7. The acceptance region defined by these two equations is
the set of positive integers between, but not including, L and C/.
Unfortunately, it is usually impossible to find integral values of L
and U to satisfy Equations (7.4) and (7,5). Therefore, it is customary
to choose those values of L and U which make the value of each of the
summations as large as possible without exceeding a/2. Occasionally,
the restriction of being less than or equal to a/2 will be relaxed if, by
so doing, the probability of rejecting a true hypothesis will be only
slightly larger than the chosen a.
Example 7.12
In a certain cross of two varieties of peas, genetic theory led the in
vestigator to expect one-half of the seecLs produced to be wrinkled and
the remaining one-half to be smooth. Taking of0.01 and n4() l deter
mine L and C7, and thus define the acceptance and rejection regions,
Using Equations (7,4) and (7,5) with po0.5y we obtain L**l\ and
f/aa29. Therefore, the acceptance region consists of those values of x
for which 11 <x <29.
Without adequate tables, the procedure discussed BO far in this sec
tion i not very palatable to the researcher. Consequently, some ap
proximate procedures which lend themselves* to easy calculation will
bo investigated.
In Section 5.11 the normal distribution wan HUggented as a posnible
approximation to the binomial distribution. If such an approximation
is used, Equations (7,4) and (7.5) are replaced by
ssss.-^ > a/2 (7.6)
PO) *
and
( (U 0.5) npo)
P<Z > == s ^--==r=.> = /2 C7.7)
^ ~ ' " po) f / ^ ;
where Z is a standard normal variate- These equations may then be
solved for L and t/*
7.7 BINOMIAL POPULATION; H:p = p VERSUS A:p =7^ p 117
Example 7.13
Using the normal approximation, find L and 17 for the situation
described in Example 7.12. Since a. = 0.01, we see that
{ (L + 0.5) - 40(0.5) }A/40(0.5) (0.5) - - 2.575
and
{(/ - 0.5) 40 (0.5)} A/40 (0.5)(0.5) = 2.575.
Thus, L = 11. 4 ^11 and E7 = 28.6^29.
Rather than proceed as indicated in the preceding paragraph, it is
common to take the number of events (x) occurring in the class associ
ated with p and calculate
(oc + 0.5) np .. .
Z = J " , torx<np Q (7.8)
V^po(l p)o
or
(x - 0.5) - np Q
Z = - - for oc > np Q . (7.9)
Then, the hypothesis H:p = p Q will be rejected if Z^Zo./z or if Z>2i_ a / 2
where z a and ^i_/2 are found in Appendix 3.
Example 7.14
Consider the situation described in Example 7.12. A random sample
of 40 seeds segregated into 30 wrinkled and 10 smooth. Using Equation
(7.9), _
Z = {(30 - 0.05) - 40(0.5) }/V40(0.5) (0.5) ^3 > 2.995 = 2.58.
Therefore, H:p=*Q.5 is rejected.
Another useful approximation is available because the square of a
standard normal variate is distributed as chi-square with one degree of
freedom (see Section 5.18). When the chi-square approximation is used,
the test statistic is
t 1
where
0i = x
0% = n x
The hypothesis H:p = p<> will be rejected if x 2 >x?_* a:) ; otherwise,
118 CHAPTER 7, STATISTICAL INFERENCE; TESTING HYPOTHESES
will be accepted. It should be clear that O stands for observed and that
E stands for expected in Equation (7.10).
Example 7.15
It will be instructive to rework Example 7.14 using this method.
Thus,
x s (| 30 _ 20 [ - 0.5) a /20 +( | 10 - 20 | - 0.5) 2 /20 = 9.025 > X 2 99(1) 6.63.
Therefore, as in the preceding example, //:p==0.5 is rejected. (NOTE:
X 2 = 9.025 ==J 2 ^(3)*.)
7.8 BINOMIAL. POPULATION; H~p<p Q VERSUS
OR H:p>p Q VERSUS Aip<p*
In many practical situations, the hypothesis H:p<p^ is more ap
propriate than //: p~po* An example of this would be any hypothesis
concerning the per cent of defective items in a production lot. When
dealing with this type of problem,, the researcher may use only the
exact procedure or the normal approximation, The chi-squarc approxi
mation may not be used because it effectively adds together the areas
under both tails of the standard normal curve when only a one-tailed
test is appropriate.
Only the case If:p^po versus yl:p>po will be discussed in detail.
The discussion for the case H:p*>pQ versus A :p<.p^ would proceed in
a similar fashion, the only change being which tail of the distribution
is used for the rejection region. The value of U which defines the ac
ceptance and rejection regions is determined by solving
C(n, *)po(l - po) w == <* (7,11)
for U, As before, it will be necessary to settle for that value of U such
that the value of the summation closely approximates a. The rejection
region, then, consists of the positive integers greater than or equal to
U* If the normal approximation, is used, calculate
Z { O - 0,5) - npoJ/VnpiC^Vo)- (7-12)
Example 7*16
Prom past experience it has been determined that a qualified operator
on a certain machine turning out 400 items per day produces 20 or
fewer defective items per day, A new operator in hired to run the same
machine and the hypothesis is made that he IB a qualified operator,
Taking ^M).03, determine f/, and thus define the acceptance and
rejection regions. Here, the hypothesis is // :p <0.05 and n**400, Using
Kquatkm (7*11), wo find that t/29. Thus, if the new operator pro
duced more than 28 defective items in a run of 400, we would reject the
hypothesis that ho is a qualified operator*
Example 7.17
Using the normal approximation, test the hypothesis //:p;<0*05
T.9 TWO NORMAL POPULATIONS; H:M* = J-t* VERSUS At^ ^ M* 119
versus A:p>Q.Q5, given that = 32/400 = 0.08. Let <x = 0.03. Using
Equation (7.12),
Z = {(32 - 0.5) - 400(0.05) } A/400(0.05) (0.95) s* 2.6 > Z t97 = 1.89.
Thus, #:? <0.05 is rejected.
7.9 TWO NORMAL POPULATIONS; H:^^^ VERSUS
The methods to be described here are closely allied with those dis
cussed when obtaining confidence limits for MI ^2. Consequently, it is
recommended that the reader review the earlier material. Without
further preamble, we shall present and illustrate the appropriate pro
cedures.
Case I: <rf = of
In this case the procedure is to calculate
/ - (3*1 - X 2 )A^_^ 2 (7.13)
where
and
x + w 2 ~ 2)
- 2), (7.15)
and to reject H:^^^ if *< aa/2)cn 1 H-n 2 ~2) or if _ a
Clearly, some simplification of the formulas will occur if ni = n 2 .
Example 7.18
Wire cable is being manufactured by two processes. We wish to
determine if the processes are having different effects on the mean
breaking strength of the cable. Laboratory tests were performed by
putting samples of cable under tension and recording the load required
to break the cable. Using the data given in Table 7.4, and letting
CK = 0,05, test the hypothesis H:/jLi=fj,% versus A :^ix^ 2 . Calculations
yield
!Fi 8.17, ^ 2 11.29, s 2 = 5.29, and t = 2.44.
Since <. 975(11) = 2.201, the hypothesis H is rejected.
Example 7.19
Two rations (feeds) are to be compared with respect to their effect
on the weight gains of hogs. Ten animals are available, and five are fed
feed No. 1 while the other five are fed feed No. 2. Using <2 = 0.10 ; test
the hypothesis that the two feeds are equally effective in causing hogs
to gain weight. The data obtained are given in Table 7.5. Calculations
yield t= 1.58. Since .95(8) = 1.860, we are unable to reject -ff:/zj=rj.
120
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
TABLE 7.4-Critical Values of the Load (Coded Data)
Process No. 1
Process No. 2
9
14
4
9
10
13
7
12
9
13
10
8
10
TABLE 7.5-Gains in Weight (in Lbs.)
Feed No. 1
Feed No. 2
1
4
2
3
4
9
5
10
8
9
Case II: <rf
When this situation prevails, that is, when we are unwilling to as
sume that erf equals a\, a reasonably good approximate procedure is
as follows. Compute
*' - (x, -
+
and reject if
or if
where
t f
(7.17)
(7.18)
Example 7.20
As an illustration, Example 7 AS will be reworked on the assumption
that of does not equal cr|. Thus, Xi^S.17, ^11,29, $f=5.4,
sl*6.2, w> lS 0.9, to a 0.74, *'- 2.4, < J 2.571, ^ a 2.447, and the
weighted average of ti and ts is 2.52. Conclusion: accept H*
7.9 TWO NORMAL POPULATIONS;
Case III: Paired Observations
The procedure in this case is to calculate
t =
= p* VERSUS A:JJ*
121
j (7-19)
and to reject H:fj.i = v>2 (or H f ifj, D = AH M2 = 0) if t< <!-/ 2) c~i> or if
>(!-/ 2)<n-i). Here, of course, n is the number of pairs of observa
tions. Or, in other words, n is the number of differences, D~XY.
Also, it should be clear that > = X F.
Example 7.21
In a Brinell hardness test, a hardened steel ball is pressed into the
material being tested under a standard load. The diameter of the spher
ical indentation is then measured. Two steel balls are available (one
from each of two manufacturers) and their performance will be com
pared on 15 pieces of material. Each piece of material will be tested
twice, once with_each ball. The data obtained are given in Table 7.6. Cal
culations yield Z> = 8, s 2 ^121.6, and ^2.81. Using c* = 0.05, it is seen
that 2.81 >t. 975CU) =* 2. 145, and thus we reject the hypothesis that the
two steel balls give the same average hardness indication.
TABLE 7.6-Data Obtained in a Brinell Hardness Test
Sample No.
Diameters
D = X Y
X
F
1
73
43
47
53
58
47
52
38
61
56
56
34
55
65
75
51
41
43
41
47
32
24
43
53
52
57
44
57
40
68
22
2
4
12
11
15
28
5
8
4
1
10
2
25
7
2
3
4
5
6
7
8
9
10
11
12
13
14
15
As has been stated before, the OC curves associated with tests of
significance must be examined if one is to be certain that the suggested
test procedure is discriminating enough. Once again we shall beg the
question, and refer the reader to Bowker and Lieberman (3) for samples
of such curves.
172
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
7.10 TWO
NORMAL
OR jHT:i
POPULATIONS;
VERSUS A:/J
<MZ VERSUS
By now, the technique for one-tailed test procedures should be clear.
Consequently, only a brief discussion will be given. With reference to
the three cases discussed in the preceding section, the same test sta
tistic will be calculated here as was calculated there. The only differ
ence will be in the selection of the critical values of t from the table in
Appendix 5. As in other examples of one-tailed tests, the values will
be chosen so that all of a. (rather than a/2) will be at one end of the
distribution. Some OC curves are again available in Bowkcr and
Liebcrman (3). The tests are summarized in Table 7.7.
TABLE 7.7-One-Sided Test Procedures for Comparing the Means of Two
Normal Populations
Hy-
po thesis
Assumption
Statis
tic
Equa
tion
Rejection Region
Mi ^Ma
Cl-al
/
7 A3
t >*<>-<*) (m-Hn^-a)
Mi <M2
CT? 7*01
t'
7.16
t' > weighted average using
100(1 a) per cent points
M/><0
paired observations
t
7,19
>2<l.~- <*)<n~ 1)
Mi >M'2
a?~c-8
I
7.13
/< J<lorXH nj-.*)
Mi >Ma
a* *d
t'
7.16
'<the negative of the
weighted average using
100(1 a) per cent points
Ml>>0
paired observations
t
7.19
< ~(l~~)(n^ l>
Example 7.22
Two pieces of moat, one a control and the other treated to tenderize
the. fibers, are to be tested. Tenderness will be measured by the force
needed to shear samples of meat. (Lower shear force values indicate
more tender moat.) trivon the data in Table 7,8, and letting o:^ 0.025,
test the hypothesis 7/:ju r >^ versus A : ^ T j<fjL a , ^O^l^nltitlonB yield
TABLK 7.8-Shear Force Values for Tenderness Test
Control
50
44
24
50
41
43
Treated
46
40
32
23
54
51
7.12 TWO NORMAL POPULATIONS; H:0f ^ a* VERSUS A-.o\>ol 123
Since t. 975(12) =2.179, we reject H and conclude that the treatment does
improve the tenderness of the meat.
Further examples could be given. However, rather than take up
space for such a purpose, we will rely on problems to illustrate the
other cases.
7.11 TWO NORMAL POPULATIONS; flr:of=of VERSUS
As in Section 6.10, the F-ratio will be the appropriate statistic. That
is, the procedure will be to calculate
F = s\/ si (7.20)
)n ^^ or if F>F a
larger sample variance
and reject H if F<F^ m(ni ^ )n ^^ or if F>F a _ a/ 2>< nr -i,nj^L>. Alterna
tively, we can calculate
(7,21)
smaller sample variance
and reject only if F > F &*/%) ^ iy v $ where z>i and z> 2 represent, respec
tively, the degrees of freedom associated with the numerator and de
nominator. OC curves may be obtained by calculating
/3 = P{ (a-2/CTj),F( a /2)(ni I,n 2 1) < -P < (<^2/Vi)F (1 a /2) (Wi I,n 2 1) } -
Sample OC curves are given in Bowker and Lieberman (3).
Example 7.23
Using the data of Example 6.4 and letting a: = 0.05, test the hypothesis
J/:oi =<r| versus A 10^7*0$. It is seen that F = 109.63/65.99 = 1.66
with jfi 40 and v 2 = 30 degrees of freedom. Since F. 975(40,30) =2.01, we
are unable to reject H.
7.12 TWO NORMAL POPULATIONS; JHTiofrSof VERSUS
A:oi><ri, OR H:a%>02 VERSUS A:oi<oi
As was done in Section 7.10, only a summary of the test procedures
will be given. This appears in Table 7.9. No examples will be given,
but some of the problems at the end of the chapter will provide an
opportunity to apply the indicated method. As in other sections, solv
ing of problems is strongly recommended as an aid in increasing an
understanding of the various methods.
TABLE 7.9-One-Sided Test Procedures for Comparing
The Variances of Two Normal Populations
Hypothesis
Statistic
Equation
Rejection Region
3i3
F
F
7.20
7.20
l?r^ir"
124 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Reference is again made to Bowker and Lieberman (3) for those who
wish to examine OC curves associated with the tests of this section.
7.13 MULTINOMIAL DATA
Many times, our sample elements may be assigned to any one of sev
eral different classes, or categories, rather than simply to one or the
other of two classes as in Section 7.7. In such a situation we must work
with the multinomial distribution rather than the binomial distri
bution.
A common problem is to test the hypothesis
H:pi = p i0 (i = 1> 2, - - - , K)
where there are k classes. Of course,
k k
^ Pi = 52 P = i-
t~l i~l
A simple test procedure is available by means of the ehi-square approxi
mation. In this case, the degrees of freedom equal h1, that is, one
less than the number of classes (or parameters). The procedure is to
calculate
x 2 - Z (0< - RWRi (7.22)
* i
where O* represents the number observed in the ith cla^s and JKt^npiQ
reresents the number expected in the ith class if // Ls true. Clearly,
i n* Then, if X*>XU~~)<A-~I ^ e hypothesis // is rejected.
Example 7.24
In a particular genetic experiment, the observations were classified
as follows:
Class A99
Class B< 33
Class C 24
Class D 4
but genetic theory called for a 9:3:3:1 ratio. Using a 5 per cent mg~
nificanee level, do the data support the theory? Calculation yields
x i . (99 - 90) a /^> + (33 - 30)V3D -f- (24 - 3Q}*/30 + (4 - I0)/10 - 6,0.
This is less than xtascs) "7.81, and thus we are unable to reject the hy
pothesized theory.
7.14 PO1SSON DATA
There are several processes which give rise to observations distri
buted according to the Poisson probability function
/(#) <r*X*/#l; a; - 0, 1, 2, - - . (7,23)
7.14 P01SSON DATA 125
Some examples are: (1) radioactive disintegrations, (2) bomb hits on a
given area, (3) chromosome interchanges in cells, and (4) flaws in ma
terials.
Obviously, many hypotheses and alternatives could be considered
and discussed. However, for purposes of illustrating the methods of
analysis, only two will be examined.
To test the hypothesis H:\<\ Q versus A : X > X , it would be appropri
ate to obtain, for a sample of one,
P = i ~ p(p _ i) (7.24)
where F(x) is read from Appendix 2 under the assumption X = X . If
P<&, the hypothesis H would be rejected.
Example 7.25
A random sample of two phonograph records shows 1 and 4 de
fects per record, respectively. Assuming a: = 0.01, test the hypothesis
H:\ <0.5 versus A :X >0.5. (NOTE: This is testing the hypothesis that
the average number of defects per record is less than or equal to J.)
Since we have a total of 5 defects from 2 records, we make use of the
fact that w~ xi+x$ also follows a Poisson distribution with parameter
X' = nX = 2A. Consulting Appendix 2 for X' = 2A = 2() = 1, we see that
F(w 1)=^(5 1)= J P(4 : ) =0.996 and thus P = 1 F(w l) i =0.004.
Since this is less than oc = 0.01, the hypothesis H:\<0.5 is rejected in
favor of the alternative A :X >0.5.
The second situation to be examined is of interest from a methodo
logical point of view since it combines the assumption of a Poisson dis
tribution with the chi-square method of analysis. Essentially, it is a
comparison of several Poisson distributions to see if the parameters
(that is, the X's) differ significantly. The procedure is best illustrated
by an example.
Example 7.26
Suppose a phonograph record manufacturing company is investigat
ing 5 different production processes. Four records are selected at ran
dom from those produced by each process and the number of defects
per record is counted. The data are given, in Table 7.10. Chi-square is
then computed for each process, using the observed process average as
the expected number of defects per record for that process. Each of these
chi-squares has 3 degrees of freedom. Using the additive property of
chi-square, it is noted that the total, 9.70, has 15 degrees of freedom.
It can be verified that none of these 6 values of chi-square is significant
at the 1 per cent level. Thus, there is little question about the uni
formity of records produced by the same process. However, if the chi-
square representing the variation among processes is calculated, that is
X 2=, [(64-35,2) 2 -4-(28-35.2) 2 -l-(32 35.2) 2 + (32-35.2)2+ (20 35.2) 2 ]/35.2 = 32.18,
we see that x 2==:: 32.18 >x%o(4) = 13.3. Therefore, the hypothesis of no
differences among processes is rejected. It might be concluded that some
processes (probably numbers II, III, IV, and V) will allow production
of product containing fewer defects.
126
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
TABLE 7.10-Number of Defects per Record From XYZ
Manufacturing Company
Number of Defects
Process
Process
^ (0 _ R .y
X fn 7 - -"' - - -I,-
Process
per Record
Totals
Means
i ^-^ 77
j-1 J&
I
11, 16, 17 20
64
16
42/16=2.62
II
5 7 5 11
28
7
24/7 3.43
III
11, 9, 7, 5
32
8
20/8 =2.50
IV
8 10, 7 7
32
8
6/8 = .75
V. . .
5654
20
5
2/5 = .40
Total
176
9.70
7.15 CHI-SQUARE TEST OF GOODNESS OF FIT
One thing that is often done, with no justification other than saying
it appears reasonable, Is to assume that the variate under discussion
follows a particular distribution. For example, data are frequently
assumed to be samples from a normal population, and you may well
question this assumption. At this time, one procedure useful in check
ing on the validity of such assumptions will be presented.
The procedure is to make a comparison between the actual number
of observations and the expected number of observations (expected
under the "uwwiunption") for various values of the variate. The ex
pected numbers are usually calculated by iivsing the assumed distribxi-
tion with the parameters set equal to their sample estimates. The chi-
sqxiare statistic will be calculated according to Kquatiou (7.22) and the
degrees of freedom will be /c -p 1, where p represents the number of
parameters estimated by sample statistics. For example, if a normality
assumption were xnuler test, & and a- 2 would be estimated by 5T and s* 2 ,
and the degrees of freedom would be Ai 3, where k represents the num
ber of class intervals used in fitting the distribution. If the assumption
of a Poisson distribution were being tested, X /x would be estimated by
X) and the degrees of freedom would be A? 2.
Rather than continue the discussion in general terms, an example
involving the Poisson distribution will be studied.
Example 7.27
The* data given in Table 7.11 show the number of "senders** (a type
of automatic equipment used In telephone, exchanges) that were in xise
at a given instant. Observations were made on 3754 different occasions.
The expected numbers were calculated from/(.r) s*e, x A*/x! where A wan
set eqxial to 3TI()-44. Hince x 2 a43.43 >X*gg(<w) 37.6, the hypothesis
of a PoiBHon diHtrihution with ju* 10.44 is rejected.
One point to bo noted in Kxamplc 7.27 wan the combination of the
entries of the top two linen of the table to form a ningle elana. Thin was
7.15 CHI-SQUARE TEST OF GOODNESS OF FIT 127
TABLE 7.11-Number of Busy Senders in a Telephone Exchange*
Number
Observed
Frequency
Expected
Frequency
Deviation
(O-JS) 2
Busy
(0)
OB)
(O-E)
JE
o\
11\
-t- 3 74
11 01
1
>
Si
1 15}
2
** )
14
5 98
+ 8 02
10 76
3. . .
24
20 82
-1- 3 18
4-Q
4
57
54 33
+ 2 67
i ^
5
111
113 44
2 44
05
6
197
197 38
38
00
7. ..
278
294 38
16 38
01
8
378
384.16
6 16
10
9
418
445 63
27 63
1 71
10
461
465 24
4 24
03
11
433
441 . 56
8 56
17
12
413
384. 15
+ 28.85
2 17
13
358
308.50
+49.50
7 94
14
219
230.05
11.05
.53
15
145
160.11
15.11
1 43
16
109
104.47
+ 4.53
.20
17
57
64. 16
7.16
.80
18
43
37.21
+ 5.79
.90
19... . .
16
20.45
4.45
97
20
7
10.67
3.67
1.26
21
8
5.31
+ 5.69
1.36
22
3
4.51
1.51
.51
Total
3754
3753.77
+ 0.23
X 2 = 43.43
* Source: Thornton C. Fry, Probability and Its Engineering Uses (New York: D. Van
Nostrand Company, Inc., 1928), p. 295,
done because the expected number on the first line was too small. The
reason for avoiding such expected numbers is that they lead to large
chi-square values (perhaps even significant values of chi-square) which
do not reflect a departure of "observed from expected" but only the
smallness of the "expect ed." In other words, if some expected numbers
are too small, the chi-square statistic will be a poor indicator of the
validity of the hypothesis under test. Some authors say that "too
small" means less than 3; others say less than 5. Since not everyone is
agreed on the interpretation of what is too small, you should feel free
to use any reasonable definition. Personally, I favor the value "3."
128 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
7.16 BINOMIAL POPULATION; MORE THAN ONE SAMPLE
A situation which occurs frequently in experimental work is the fol
lowing: A hypothesis is to be tested and several experiments are con
ducted to produce data which bear on the problem. When this situa
tion prevails, it is natural to think of combining the experimental
results. For example, if the hypothesis H : p = po is being tested relative
to the alternative Aip^ps, it is quite common to have available k
samples (perhaps of different sizes) as a result of k replications, or
repetitions, of the basic experiment.
How should the data from the several samples be combined? There
are two ways this can be done, and each will be discussed and then il
lustrated in Example 7.28. It will be noted that the analysis performed
involves the chi-square distribution and depends on the previously
mentioned additive property of chi-square. Actually, several chi-square
values are calculated, and each of these contributes a different item of
information relative to the hypothesis under test.
You will note that a chi-square value (with 1 degree of freedom) is
found for each sample. Each of these values can be interpreted as
in Section 7.7. As the next step in the analysis, wo may calculate
x 2===: Xi+X2 + " + X& with k degrees of freedom* This value will be
referred to as the pooled chi-square, and it is clearly a pooling or accum
ulation of the bits of evidence provided by the k independent samples.
This value may now bo used to assess the validity of the hypothesis
under test. An alternative way of pooling the information from several
samples is to lump the original data into one large sample arid compute
the total chi-square (with 1 degree of freedom) associated with this
super sample. One other statistic should also be obtained, namely, the
heterogeneity chi-square. This quantity, which has fc 1 degrees of
freedom, is found by subtracting the total chi-square from the pooled
chi-square. It is used to measure the lack of consistency among the
several samples.
Example 7.28
Consider again, the hypothesis tested in Example 7.12. Now, instead
of I sample, 8 separate experiments give rise to 8 samples as shown in
Table 7.12. Assuming c*0.01, it is seen that: (1) no 1 of the 8 samples
leads to rejection, (2) the super sample of 1600 observations yields
X**2.66 which is not significant, and (3) the pooled chi-square is sig
nificant, Why do we get these seemingly contradictory results? The
pooled chi-square is significant because we have accumulated enough
evidence from each sample to indicate that the hypothesis //:p0,5
should be rejected. The reason the total chi-square did not give the
same answer is that in 3 samples smooth seeds predominated while in
5 samples wrinkled seeds predominated* This effect was hidden (i.e.,
the majorities in opposite directions tended to cancel out) when the
data were lumped into one large sample. Attention is called to the
previously mentioned lack of consistency among the 8 samples by the
significant heterogeneity chi-square.
CONTINGENCY TABLES
129
TABLE 7.12-Chi-Square Analysis Combining Data From Several Samples
of Smooth and Wrinkled Peas
Sample
Number
Sample
Size
Number
Wrinkled
Number
Smooth.
On-Square (x*)
d.f.
1
100
60
40
4 00
i
2
200
108
92
1 28
i
3
180
80
100
2 22
i
4
208
118
90
3 77
i
5
300
165
135
3 00
1
6
182
106
76
4 Q4
1
7
230
105
125
1 7S
i
8
200
90
110
2 00
1
Pooled x 2 - - -
8
22.94 = y^ x 2
8
Total
1600
832
768
"*" 7 * ^-/ A. .
<-l *
2.56
1
Difference
20 38
7
7.17 CONTINGENCY TABLES
Suppose n randomly selected items are classified according to two
different criteria. The tabulation of the results could be presented as
in Table 7.13, where O^ represents the number of items belonging to
TABLE 7.13-An rXc Table
Rows
1.
2.
Columns
2c
O r
the ?")th cell of the rXc table. Such data can be used to test the hy
pothesis that the two classifications, represented by rows and columns,
are statistically independent. If this hypothesis is rejected, the two
classifications are not independent and we say there is some interaction
between the two criteria of classification.
The exact test for independence is difficult to apply. However, if n,
the sample size, is sufficiently large, a reasonably good approximate
procedure is to calculate
X 2 =
(7.25)
130 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
where
Oij = observed number in the (i/)th cell,
= expected number in the (f/)th cell,
^j = observed number in the ith row, and
y i
= observed number in the/th column.
The value of chi-square given by Equation (7.25) has *>=(? l)(c 1)
degrees of freedom. If x a ^x*i-->i:<r--i)Ce-i>]> the hypothesis of inde
pendence should be rejected.
Example 7.29
A company has to choose among throe proposed pension plans. One
hypothesis that the company wislics to investigate is: Preference for
plans is independent of job classification. It asks the opinion of a
sample of the employees and obtains the information presented in
Table 7.14. The expected numbers for each ceil arc calculated and
appear in Table 7.15. Calculation then yields x 2== ll <X*99(6) ~ 10*8 so
the hypothesis cannot be rejected. Thus, it is concluded that the em
ployees' choices of pension plans are quite probably independent of their
job classifications.
TABLE 7,14-Classification of Employees by Job and
Pension Plan Preference
Number of
Employees
Favoring
Classification
Plan A
Plan B
Plan C
Total
Factory employees ...*....
160
30
10
200
Clerical employees . . * .,,.*.
140
40
20
200
Foremen and supervisors ....
Executives
80
70
10
20
10
10
100
100
Total
450
100
SO
600
TABLE 7JS~~Expected Number of Observations
Classification
Wan A
Plan B
Plan C
Factory employees * ,
150
100/3
50/3
Clerical employees ,
ISO
100/3
50/3
Foremen sine! supervisors
75
100/6
50/6
Executives ..*<,,
75
100/6
50/6
7.18 SPECIAL APPROXIMATE METHODS FOR 2X2 TABLES
131
If we are presented with an A 7 '- way contingency table, that is, one in
which the individual elements are assigned to the cells of the table by
N different criteria, the hypothesis of mutual independence of the N
criteria may be tested by a simple extension of the rules formulated for
the rXc table. As usual, we shall compute the sum (over all cells) of
"(observed expected) ^/expected," where the expected value in any
cell is given by the product of the marginal (border) totals associated
with the row, column, etc., in which the cell is located divided by n^" 1 .
The resulting statistic is approximately distributed as chi-square with
"(r l)(cl) . " degrees of freedom, where there are r rows, c col
umns, etc., in the YV-way table. Other hypotheses may also be tested
in such tables, for example, see Mood (12), but we shall not discuss
these at this time.
7.18
SPECIAL
TABLES
APPROXIMATE METHODS FOR 2X2
If the contingency table consists of two rows and two columns, as in
Table 7.16, a short-cut method of computing chi-square is available.
The appropriate formula is
/& (7.26)
where n = a+b + c+d and k= (a+&) (c+d) (a+c) (6 + d). This will give
the same numerical value of chi-square that would be obtained if the
procedure of Section 7.17 were followed. It should be clear that the
chi-square statistic thus obtained will have only 1 degree of freedom.
TABLE 7.16-A 2X2 Table
Ai
A*
Total
JBi
d
I
a+b
Bi
c
d
c+d
Total
a+c
b + d
n
As in Section 7.7, a correction for continuity may be used to sharpen
the approximation. This is accomplished by calculating
= ( | ad be | n/2y/k
X
~ 0.5)
(7.27)
It must be remembered that this correction should not be applied to
rXc tables in which r>2 and
Example 7.30
A random sample of 250 men and 250 women were polled as fo their
desires concerning the ownership of television sets. The data in Table
7.17 resulted. Calculation by either method yielded
<? = 13* >
L .99(l>
6.63.
132
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Thus, the hypothesis that desire to own a television set is independent
of sex is rejected.
TABLE 7.17-Results of Sample Poll on Television Ownership
Classification
Men
Women
Total
Want television
80
170
250
Don't want television
120
130
250
Total
200
300
500
7.19 THE EXACT METHOD FOR 2X2 TABLES
It should be noted that an alternative way of looking at a 2X2 table
is to consider the two fractions, pi = a/(a+6) and P2=c/(c-|-rf) 7 as esti
mates of pi and p<2> the parameters of two binomial populations. In this
frame of reference, a comparison of pi and p% should yield evidence rela
tive to the hypothesis H:pi~p^ If we wish to test fl:pi~pz versus
Aipi^pz, we may use the approximate method of the preceding sec
tion or, if we choose, an equivalent test based on the normal approxi
mation,
In this case, however, the exact test procedure is not too difficult to
apply, especially if a digital computer is available. Thus, it seems ap
propriate to indicate the nature of the exact method.
It can be shown that the exact probability of observing pi = a/(a+b)
and p^^c/(c+d) when pi~pz is
+
I
alblcldlnl
(7.28)
To obtain the final probability to be used in assessing the validity of
//:y>i = p2 it is necessary to add to Pi the probabilities of more diver
gent fractions than those observed. Assuming; Pi<.p% (and the table
can always be arranged to make this so), the next more divergent situa
tion would be the one in which a and d are each decreased by xmity, and
6 and c are each increased by unity. For this array, we calculate
(7.29)
The cell entries are again changed, following the same rule as before,
and PS is calculated. Continue in this manner until P a +\ Is calculated.
Then, if
(7.30)
*' I
is less than or equal to ix, the hypothesis H:pi = pz should be rejected.
7.20 SEVERAL NORMAL POPULATIONS; H:}!* = M . . . = I** 133
Example 7.31
Robertson (13) reported on the analysis of an experiment involving
the evaluation of a silicon dip as a protection for vacuum tubes. The
data shown in Table 7.18 were obtained. Using the procedure outlined
above, he found P = P X +P 2 +P 3 = .0957. He then concluded that "the
failure rate for protected tubes is fust barely significantly less than that
for unprotected tubes. 77 Apparently a 10 per cent significance level had
been decided upon prior to the analysis.
TABLE 7.18-Success-Failure Results From an Experiment
on 690 Vacuum Tubes*
Failures
Nonfailures
Protected
*2
338
Unprotected . . .
7.
343
/
* Source: W. H. Robertson, "Programming Fisher's exact method of comparing two per
centages," Technometrics, Vol. 2, No. 1, pp. 103-7, Feb., 1960.
7.20 SEVERAL NORMAL POPULATIONS; H:^=^ 2 =^
In Section 7.9 a &-test was proposed for testing H:JJL^ = ^ versus
Aipir^fjiz under the assumption that cri = cr|. Now we wish to propose
a procedure for handling the situation in which we have k normal pop
ulations, fc>2.
Intuitively, it seems reasonable that the validity of the hypothesis
H : MI = M2 = - - = &k should be assessed by comparing the sample esti
mates of pii, Hz, , Mfc. That is, it is to be expected that any suggested
test procedure will involve a comparison of F a , F 2 , , 7*;. (NOTE:
The choice of Y rather than X as the symbol denoting the character
istic was prompted solely by the desire to agree with symbolism to be
used in certain techniques that will be presented later in the book.)
If the assumption is made that o-f == of = - - - = a>, that is, if
homogeneous variances are assumed, the appropriate test procedure is
to calculate
_ * 1
* = - (7.31)
k n< y fc
y^ v f F-- T^ 2 / ^T (* n
--C-^ ^r v ^ u * *J / Z-j \i *}
where
F*7 jth observation in the ith group (sample) ; i = 1, - - , k (7.32)
J = 1, - - - , n*
n* = number of observations in the ith group (7 . 33)
134
CHAPTER r, STATISTICAL INFERENCE: TESTING HYPOTHESES
ii/ni = mean of the observations in the ith group (7 . 34)
Y =
Then, if F>Fci-.
of all observations.
ya ) where *>i = k 1 and
k
X- - i),
z 1
(7.35)
the hypothesis H : /*i == pc 2 = - - =M^ would be rejected. (NOTE: The
reader may easily verify that, if k = 2 y the procedure outlined in this
section is algebraically equivalent to that of Section 7.9.)
Before presenting an example, a convenient tabular form for carry
ing out the specified test procedure will be indicated. Invoking the
identity
km
^
or, in abbreviated form,
G
vv
K/
(7.36)
(7.37)
the necessary calculations are conveniently presented as in Table 7.19.
(NOTE : Such a table is usually referred to as an analysis of variance
table.) Actually, the labor involved in calculating the various sums of
squares may be materially reduced if we use the following algebraically
equivalent forms :
TABLE 7,19-Tabular Presentation of the /Mest for the Equality of
Means of k Normal Populations Under the Assumption
of Homogeneous Variances
Source of
Variation
Among groxips . , .
Within groups. . . .
Total
Degrees of
Freedom
Sum of
Squares
Mean S^
1
ib ( - 1)
G yv
G. /*** / / 1
v^i/j// t"
V-fTw,/
k
y* y
*-d
F- Ratio
G/W
7.20 SEVERAL NORMAL POPULATIONS; H:JA,i = \*>* . . . = M. A
135
Myy =
= sum of the squares of all the observations, (7.38)
/ &
it (7.39)
and
In the above equations,
(7.40)
(7.41)
"^- = total of the observations in the ^th group, (7.42)
i = total of all observations,
(7.43)
and
k
= total number of observations in all the groups combined. (7.44)
Example 7.32
Consider the data of Table 7.20. Using Equations (7.38) through
(7.41), the results shown in Table 7.21 were obtained. Since
F = 72 > ^.99(3,16) = 5.29,
the hypothesis H :/xi=/jt 2 =
cance level.
is rejected at the 1 per cent signifi
TABLE 7.20-Sample Data From Four Normal Populations To Be Used
in Example 7.32
Groups
1
2 3
4
45
35 34
41
46
33 34
41
49
35
44
44
34
43
Observations
33
41
42
44
41
41
Totals
184
68 170
378
Means
46
34 34
42
136
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
TABLE 7.21-Analysis of Variance Using the Data of Table 7.20 To Test
the Hypothesis Hi^i =M2 = M3 = ^4
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
Meatx . . .
1
32 000
32 000
Among groups
3
432
144
72
Within groups
16
32
2
Total
20
32,464
^
7.21 SEVERAL
H :oi = erf =
NORMAL POPULATIONS;
= erf
In Section 7.11 an .P-test was proposed for testing the hypothesis
Hi <r? = <T2 versus A:<rf=^cr|. At this time, we wish to consider the
situation in which we have k normal populations, &>2. Several test
procedures have been proposed for handling this type of problem, but
only the method due to Bartlett (2) will be presented in this book.
As in Section 7.20 7 the sample observations will be denoted by
Y a 0&= 1, - - , 7c;j = l, - - 3 n t ). Other symbols will also be defined as
in the preceding section and, in addition, we will denote K^ "F- by
2/ij. Thus, in agreement with an earlier definition,
nt
Using this notation, the mechanics of Bartlett's procedure are as
shown in Table 7.22. If, in this table, x 2 ^x*i-.o<*--i:>> the hypothesis
r:a-f = cr|== - . - = cr| would be rejected. (NOTE: The researcher
will find it necessary to compute the corrected value of chi-square only
if the uncorrected chi-square falls close to and above the tabulated
value, and then only if he wishes to obtain a very accurate evaluation
of the exact probability of Type I error.)
Example 7.33
Consider the data of Table 7.23. Following the procedure indicated
in Table 7.22, the results presented in Table 7,24 are obtained. It is
seen that x 2sa= 2.81 <x^95(s) SBB 7,81 J and thus the hypothesis of homoge
neous variances may not be rejected at the 5 per cent significance level.
(WOTE: There was no need to calculate the corrected value of chi-
square in this example; the computations were carried out only to illus
trate the method.)
7.22 SAMPLE SIZE
A question frequently asked of statisticians is,
is needed for this experiment?
How large a sample
The question is deceptively simple,
7.22 SAMPLE SIZE 137
TABLE 7.22-Computations for Bartlett's Test for Homogeneity of Variance
Sample
Z2
%'
Degrees of
Freedom
!/<*/.
2
^i
logiQSf
(d./.) logios?
1
A 2
7 'Vi,'
'Tl\ 1
l/( n * 1)
2
c.
loeTin^i
(**. . 1^ lnefin?i
2
J~l
71 2
EAf,-
^2 1
i/c n _ _ i\
2
Crt
Ififfi r9<>
f'yt.n 1 ) lOffi a.V
k .
J-l
n &
Zyi
TZjfc 1
l/(nk 11
2
c
IrtOt, n C
( <vt i ^. 1 ) 1 /-) nr- n c t
j-i
Sum
PP",
fc
Efo- 11
^ i
^ 1*7* I" 1 ) lrt<TinC*
KK i/i/
t-1
t-l Wt ~ 1
i-1
/ fc
Pooled estimate of variance = s* = TF V2/ / E ( w * "
/ i-i
log
. 10 [B - i <*, - i) io gl0 5< "]
I i I -J
Correction factor - C
Corrected
1 + [l/3(* - l)]f Z) - - -- 1 / S (< - 1)1
L i-i Wi 1 / i.i J
Note: log 10 - 2.3026
but the answer is hard to find. Before the statistician can provide any
thing better than an "educated guess/' he must retaliate with several
questions, the answers to which should enable him to attack the prob
lem with some hope of reaching a valid answer. Frustrating as this may
be to the researcher, it frequently serves a very good purpose, for it
forces the researcher to give serious thought to several aspects of his
TABLE 7.23-Four Samples From Normal Populations
JS <?! === C r 2 :== 0"3 ^^ 04
1
2
3
4
48
42
33
78
49
39
42
69
67
51
46
60
75
57
47
52
53
75
50
63
33
45
50
35
138 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
TABLE 7.24-Computatlons for Bartlett's Test: Data from Table 7.23
Sample
Zrf
Degrees
of Free
dom
1/d.f.
4
logics?
(dj.) Iog 10 5?
1
1113.0
5
.2000
222.6
2.34753
11.73765
2
820.8
4
.2500
205.2
2.31218
9.24872
3
173.2
4
.2500
43.3
1 . 63649
6.54596
4. , . .
1330.0
7
.1428
190.0
2.27875
15,95125
Sum
3437.0
20
.8428
43.48358
Pooled estimate of variance = s z = 3437/20 = 171.85
B
<X- - 1) = (2.23515) (20) = 44,7030
(2.3026) (44.7030 - 43.48358) = 2.80784
Correction factor = C 1 4- [1/3(3) ] (.8428 1/20) 1.0881
Corrected
2.80784/1.0881 2.5805
problem. To illustrate, some of the questions that might be asked by
the statistician are:
(1) What is your hypothesis? What are the alternatives?
(2) What are you trying to estimate?
(3) What significance level are you planning to use? What confi
dence level?
(4) How large a difference do you wish to be reasonably certain of
detecting? With what probability?
(5) What width confidence interval can you tolerate?
(6) What do you expect the variability of your data to be?
When answers to these and other questions are provided by the re
searcher, the statistician can be of help in determining the needed
sample size.
Before you get the impression that all is lost, let me hasten to assure
you that the picture is not all black. In some cases, fairly simple
formulas arc available for estimating the required sample size. Also, if
OC curves are available for the test procedure to be used, the reqxiired
sample size may be determined upon examination of these curves.
Tables have also been provided for certain procedures and four of these
are reproduced in Appendices 9 through 12 for your use. If all of these
three approaches (that is, formulas, OC curves, or tables) fail to meet
your demands, a professional statistician should be consulted.
Example 7.34
Consider testing the hypothesis //r^^Mo versus A :^F^Q at the 5 per
cent significance level. If <r is estimated to be 0.8 and a difference
5 (M MO| =1.2 is to be detected with probability 0.9 (this is equiva-
7.22 SAMPLE SIZE 139
lent to setting /3 = 0.1 at ^==^ ~ 1.2 and at M=Aio+1.2), how large a
sample is needed? Setting D = 1.2/0.8 = 1.5 and consulting Appendix 9,
it is found that n = 7.
Example 7.35
Consider testing H :JJL <MO versus A :ju >/zo at the 1 per cent significance
level. If or is estimated to be 1.2 and 5=ju ju 0.9 is to be detected
with probability 0.95, how large a sample is needed? Setting
D = 0,9/1.2 = 0.75
and consulting Appendix 9, it is found that n = 3l.
Example 7.36
Consider testing H ://i <pc 2 versus A :^i >Ma at the 2 per cent signifi
cance level. If <r is estimated to be 1.0 and 5=/z 3 MS 1.6 is to be de
tected with probability 0.99, how large should the two samples be? Set
ting D = 1.6/1.0 = 1.6 and consulting Appendix 10, it is found that
n 1 ==n2 = 16.
Example 7.37
Consider testing Hip. 3=/x 2 versus A I/JLI 7*1*2 at the 1 per cent significance
level. If cr is estimated to be 1.5 and <5 = [MI Ma| =1.8 is to be detected
with probability 0.95,, how large should the two samples be? Set
ting Z> = 1.8/1.5 1.2 and consulting Appendix 10, it is found that
ni = 2 = 27.
Example 7.38
Consider testing H:CT Z <O-Q versus A'cr*>o$ at the 5 per cent sig
nificance level. If a value of cr 2 = 4oo is to be detected with probability
0.99, how large a sample is needed? Using R=4 and consulting Ap
pendix 11, it is seen that 15 <v <20. Crude interpolation suggests
*> = 19 or n =
Example 7.39
Consider testing H:O-*>CTQ versus A:o*<o% at the 5 per cent signifi
cance level. If a value of cr 2 0.33 o% is to be detected with probability
0.99, how large a sample is needed? Since Appendix 11 is constructed
for values of R>1, a slight change in procedure (from Example 7.38) is
required. The table in Appendix 11 is entered with of ==/?== 0.01,
^'=* o: = 0. 05, and R' = 1/R = 3. Thus, it is noted that 24 o <30. Crude
interpolation suggests ^ = 26 or n==^+l=27. (NOTE: Although the
roles of a. and /? were interchanged when Appendix 11 was consulted,
the actual test would be carried out at the original value of ex. which, in
this example, was 0.05.)
Example 7.40
Consider testing H:<ri>a% versus ^L:<jf<cr| at the .5 per cent sig
nificance level. If a value of (r! = 4crf is to be detected 'with probability
0.99, how large should the two samples be? Using J? = 4 and consulting
i4O CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Appendix 12, it is noted that 30 Oj,=*> 2 <40. Crude interpolation sug
gests that i>i = if 2 == 34 or n\ = n 2 = 35.
7-23 SEQUENTIAL TESTS
In all the test procedures described thus far, the sample size has been
decided upon in advance. As has been inferred, the determination of
the proper sample size is often difficult . However, given the necessary
information (e.g., an estimate of the variability to be encountered and
statements concerning the allowable risks associated with incorrect
decisions) , the required sample size may be specified (see Section 7.22).
The reader should realize, though, that there is a certain "cost 77 at
tached to such an approach. That is, there is an implicit assumption
in the fixed (predetermined) sample size approach that a sample of the
specified size will be taken, and observations recorded for each sample
unit, regardless of whether all the observations are needed to reach a
decision. In view of this and in the hope of achieving economies due to
reduced sample sizes, it seems desirable to seek a test procedure in
which the sampling may be terminated as soon as it is possible to reach
a decision to either accept or reject the hypothesis under test. For cer
tain specific cases, namely, those which involve a simple hypothesis
T:0 = 0o and a single alternative A :0 = 1? such a test has been devised.
It is known as the sequential probability ratio test. In the remainder of
this section, the general nature of this procedure will be described and
certain specific applications illustrated.
The sequential method of testing proceeds as follows: Sample units
are randomly selected one at a time (i.e., sequentially) and, after each
observation is obtained, one of the following decisions is made:
(1) Accept J/:0 = #o (i.e., reject A:d^&i).
(2) Reject H:Q = 9v (i.e., accept A:0^di).
(3) Obtain an additional observation.
To determine which of these three decisions is appropriate, the ana
lyst should calculate
-R.-II4 (7 - 46)
*-i /o(#*)
where /o(#) is the probability function (or probability density func
tion) under the assumption that H:Q^Q Q iB true and/i(x) is the proba
bility function (or probability density function) under the assumption
that A :&***&i is true. Then, depending on the value of /?*, one of the
three decisions previously listed is reached by proceeding according to
the following rule:
(1) If Rn<&(l~-c*) y accept H (i.e., reject A).
(2) If jB*>(l0)/, reject H (i.e., accept A).
(3) If /(! ) <B n < (1 ~-*j8)/a, obtain an additional observation.
7.23 SEQUENTIAL TESTS 141
In the above, a. and /5 are, respectively, the preassigned risks of:
(1) rejecting H when H is true and (2) accepting H when A is true. If
an and r n are used to denote the acceptance and rejection values, re
spectively, for a test statistic, the decision rule may be restated in the
following form:
(1) If the value of the test statistic is less than or equal to a n ,
accept H (i.e,, reject ^4).
(2) If the value of the test statistic is greater than or equal to r nt
reject H (i.e., accept A}.
(3) If the value of the test statistic is greater than a n and less
than r n , continue sampling.
It should be clear, of course, that the sample size is a variable in a
sequential procedure as contrasted to its role as a (predetermined)
constant in the classical test procedures. Thus, in addition to examin
ing the power of a sequential test procedure by studying its OC func
tion, it is appropriate that its "cost" be assessed by considering the
average size of sample required to reach the decision to accept or to
reject. This analysis is usually made in terms of the ASN function,
where the letters ASN stand for average sample number. Rather than
go into details concerning the ASN function and the savings due to
reduced sample sizes, let us be content with the general statement that
the potential savings are considerable, in some cases as much as 50
per cent.
Considerable space could be devoted to a detailed discussion of the
sequential probability ratio test for each of the commonly encoun
tered situations. However, it is doubtful if such discussions would serve
any useful purpose. Accordingly, the tests have been specified in Table
7.25.
Example 7.41
Consider a binomial popxilation and the hypothesis J E J:p = 0.10
versus the alternative A:p~0.20. Let <x ==0.01 and /?== 0,05. Then log
[/3/(l )] = 2.986 and log [(1 ~/3)/a] =4.554. If we represent a
sample unit possessing the characteristic associated with p by the
symbol d and a unit not possessing this characteristic by g (e.g., defec
tive and nondefective units, respectively), then the sequence
gggdgdggdgggddgdgddgd
would terminate at this point with the decision to reject H and accept A.
Example 7.42
Consider a normal population with known standard deviation, cr = 10.
Test the hypothesis f:/x = 50 versus the alternative A:jjL = 7Q. Let
a ===0,01 and /?== 0.01. Then log [/?/(! a) ] = 4.595 and log [(1 /8)/a]
4.595. If sequential sampling yielded, in the order shown, the fol
lowing values of X (60, 75, 65, 70), the sampling would terminate at this
stage with the decision to reject H and accept A.
I
c3
|
w
!
o
s
I
2
PH
o
?x
I
Is
/
CQ.
!V
^
I
.
ecu
S
o
CO
.A
PP
25
CO
s
8
Cj <!.*
n H'
3 "S
8 S
1
><
1X1
5
.1
pq
A
-
A
to* =3.
I 1 A
r^fl 6ft ' X
a A
it ii
ex* ex
b t>
II II
b b
II If
b b
II II II II II II
fcq *nj tej ^ tej ^
-S
SJ
c
PROBLEMS 143
Problems
7.1 A company engaged in the casting of pig iron must be concerned with
the per cent of silicon in the pig iron. The data given below constitute
a random sample of the production records. Using a. ==0.02 and assum
ing normality, test the hypothesis that the process average is 0.85
grams of silicon per 100 grams of pig iron.
NUMBER OF GRAMS OF SILICON INT
100-GRAM SAMPLES OF PIG IRON
1.13
0.87
0.80
0.92
0.85
0.81
0.60
0.97
0.97
0.48
0.92
1.00
0.94
0.92
0.72
0.61
1.17
0.81
0.87
0.71
0.36
0.97
0.68
0.89
0.73
1.16
0.82
0.68
0.79
1.00
7.2 Consider these observations to represent the average hourly earnings
during May, 1940, of a random selection of 50 male workers in a speci
fied industry.
EARNINGS
(in Cents per Hour)
35
65
68
77
81
52
82
74
73
71
68
79
73
70
67
82
61
77
84
56
29
53
61
83
92
99
80
62
50
64
76
47
59
64
72
55
63
107
48
70
55
70
43
66
85
79
90
39
88
86
(a) What is your best estimate of the average hourly earnings for all
male workers in the industry?
(6) How good is your estimate in (a) above? What is its standard
error?
144 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
(c) Establish confidence limits for your estimate in (a) above. Write
out your statement about these confidence limits in words. State
your assumptions clearly.
(d) Is your estimate in (a) above in agreement with the hypothesized
true value of 68 cents per hour for average earnings in May,
1940? Explain your answer.
(e) What additional data would you need to estimate the total earn
ings in the industry for the month of May?
(/) Test the hypothesis that ^ <80.
7.3 Test the hypothesis that the mean life (in years) of wooden telephone
poles is less than 8 years. State any assumptions you make about the
following data:
LENGTH OF LIFE OF 1000 WOODEN
TELEPHONE POLES
Life
(in years)
Number of Poles
Replaced
.5 but under 1 ,5
4
1.5 but under 2.5
7
2.5 but under 3.5
15
3 .5 but under 4.5
32
4.5 but under 5.5
30
5.5 but under 6.5
57
6.5 but under 7.5
61
7.5 but under 8.5
73
8.5 but under 9.5
96
9.5 but under 10.5
104
10 . 5 but under 11,5
103
11.5 but under 12,5
95
12.5 but under 13.5
91
13.5 but under 14.5
73
14.5 but under 15.5
64
15.5 but under 16.5
38
16.5 but under 17.5
30
17.5 but under 18.5
18
18.5 but under 19.5
5
19.5 but under 20.5
1
20.5 but under 21.5
1
21 .5 but under 22.5
2
Total 1000
7.4 A consumer panel report on the economic and geographic distribution
of the purchases of a particular product reveals among other things
that the nation's families bought, on the average, 17.5 Vbs. of that
product in 1949, This estimate was based on returns from a supposed
random sample of 122*5 families, and the standard deviation of indi
vidual family purchases in this sample was found to be 7.5 lb& From
sales and inventory records, it is determined that average purchases
PROBLEMS 1 45
per family in 1948 must have been at least 18.5 lbs. ; or 1 pound more
than the sample estimate for 1949. Could this difference of 1 pound
be due to sampling variation, or does it indicate that average con
sumption of the product by families had decreased in 1949 from the
1948 level of consumption? What assumptions did you make?
7.5 Using the data in Problem 6.1, test the hypothesis #:/* = 6.55X10- 27
versus A: M 5^ 6. 55X1 0- 27 . Let a> = 0.01.
7.6 Using the data in Problem 6.5 and letting <x = 0.025, test H:fj,<l.I2
inches versus A ip, >1.12 inches.
7.7 Using the data of Problem 6.6 and letting a = 0.05, test H:jji>1.55
versus A :& <1.55.
7.8 Using the data of Problem 6.7 and letting = 0.25, test H:fj,<29QQ
yards versus A :/JL >2900 yards.
7.9 Using the data of Problem 6.1 and letting a: = 0.01, test H:cr<0.0l
X10- 27 versus A :cr>0.01 X10~ 27 .
7.10 Using the data of Problem 6.5 and letting a: = 0.10, test H :cr* <0.0001
versus A :or 2 >0.0001.
7.11 Using the data of Problem 6.6 and letting ot. = 0.005, test #:cr>0.05
versus A:cr<0.05.
7.12 Using the data of Problem 6,7 and letting <x = 0.01, test T:o- = 50
yards versus A:cr^50 yards.
7.13 In making a certain cross., a geneticist expected a segregation of 15
A's to 1 B. In a random sample of 800 he observed 730 A's and 70
B's. Do the data support the expected ratio? Why?
7.14 In a random, sample of 400 farm operators, 65 per cent were owners
and 35 per cent were nonowners. Test the hypothesis that in the pop
ulation of farm operators 60 per cent are owners. Use a probability of
Type I error equal to ,05.
7.15 A manufacturer of light bulbs claims that on the average 1 per cent
or less of all the light bulbs manufactured by his firm are defective.
A random sample of 400 light bulbs contained 12 defectives. On the
evidence of this sample, do you believe the manufacturer's claim?
Why? Assume that the maximum risk you wish to run of falsely reject
ing the manufacturer's claim the true fraction defective is .01 has
been set at 2 per cent.
7.16 A sampler of public opinion asked 400 randomly chosen persons from
some specified population whether they favored candidate A or B]
220 voted for A and 180 for B. Using a probability of Type I error
equal to .05, do you think that opinion in the population may have
been equally divided? Why?
7.17 A supermarket is to be built in a new location. The question arose as
to whether provision should be made for individual customer service
at the meat counter, or whether a self-service counter with all meats
ready-cut and packaged would adequately serve customers in the new
area. The management decision was that individual customer service
would not be supplied unless 40 per cent of the prospective customers
desired such service. A random sample of 160 prospective customers
showed only 50 respondents desiring individual service. Does it appear
that the proportion of preference in the population of prospective
customers equals or exceeds the critical level set by management?
7.18 In a triangular test for selecting judges to compose a taste panel, a
prospective judge was successful in selecting the odd sample 11 times
146 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
in 15 trials. Would you select him for the panel? How many would he
have to pick correctly to be chosen? What is the probability of Type I
error if we accept the above judge for our panel? Construct the com
plete table of probabilities, showing them also in cumulative form, for
n-^15.
7.19 Retail sales data indicate that | of the families in the WOI-TV area
have television sets. A random sample of 900 families from the area is
to be taken, (a) What is the expected number of television families for
the sample? (b) The sample yields 360 families with television sets.
Indicate at least two methods by which we may obtain approximate
confidence limits for the population proportion of families owning tele
vision sets, (c) Is the observed number, 360, in "reasonable" agree
ment with the expected number?
7.20 Eighty out of 1000 randomly chosen cases of diphtheria resulted in
death. What methods or techniques are available for using these
results to tost the hypothesis that the true percentage of fatality is
10 per cent? State whether the tests are exact or approximate.
7.21 A botanist observed 350 seedlings for the purpose of studying chloro
phyll inheritance in corn. The seed came from self-fertilized hetero
zygous green plants. Hence, green and yellow seedlings were expected
in proportions of 3 green to 1 yellow. The sample showed 120 green
and 30 yellow seedlings. Is this sample in agreement with expectation?
7.22 A metropolitan newspaper was considering a change to tabloid form.
A random sample of 900 of its daily readers was polled to secure
readership reaction to such a change. Of this sample, 541 persons
opposed the change in format for the paper, (a) Is it likely that more
than 50 per cent of the readers are in favor of the change? (b) Describe
two or more procedures for obtaining confidence limits for the popu
lation proportion opposed to the change.
7.23 From a keg containing 1000 bolts, a random sample of 20 bolts has
been presented to you for testing. One hundred per cent of the bolts
in the sample successfully pass the test. Of all the bolts in the keg,
what is your estimate of the percentage that will pass? What limits
would you place on the reliability of your estimate; that is, what con
fidence*, statement would you make about the true percentage of all
the bolts that will pass the test?
7.24 After a survey of opinion is made, point and interval estimates are cal
culated. The investigator states that the 95 per cent confidence inter
val is from (>0 per cent to 75 per cent of the population in favor of a
law. Describe precisely the moaning of this statement.
7.25 IMng the data of Problem .20 ami letting a = 0.005, test ff:^^<f^i
versus A :/ia >Mx-
7*2Ci lining the data of Problem 6.2)1 and letting a 0.05, tost //r^i Ms
versus ^1 in\ *&%.
7.27 UniTig the data of Problem 0,22 and letting a0.10, test //r/xi^Ma
vorfcms A :/xi y^Ma-
7.2cS Wo are told that the moan yields of two corn hybrids wore 75 and cS5
buaholB per acre, respectively, and that each had boon tritul in 10
fields Holoctod at random from ftomo population of fiolcln. Further,
uHHmning that cr'f cri, wo are told that the ntamlard error of each of
the* above means wan 3. Tost the hypothesis that /*i -"/AS-
PROBLEMS
147
7.29 The diameter of a cylinder was measured by 16 persons. Each person
made three determinations using a micrometer caliper and three
determinations using a vernier caliper. Following are the averages of
the three determinations (in inches) , for each caliper, made by the
16 persons.
Micrometer
Vernier
Micrometer
Vernier
Micrometer
Vernier
1.265
1.265
1.270
1.269
1.264
1.267
1.265
1,267
1.267
1.273
1.266
1.272
1.267
1.267
1.268
1.270
1.266
1.273
1.266
1.266
1.267
1.270
1.268
1.267
1.268
1.267
1.267
1.267
1.265
1.268
1.265
1.267
7.30
Is there any difference between the means of the populations of meas
urements represented by the two samples? The method to be used is
determined by the fact that each person used both calipers. Do you
think the difference is attributable to imperfections of the calipers or
to the difficulty of setting the vernier caliper?
The following are the lengths in millimeters of 6-year-old white crap-
pies from East Lake, Lucas County, Iowa, in 1948. Measurements
were made by William Lewis and T. S. English.
Males
Females
228
217
219
231
225
219
230
' 217
222
214
224
220
225
220
221
225
221
228
222
233
239
225
234
222
227
223
223
222
223
234
241
223
225
253
220
233
213
224
235
281
224
212
218
235
231
231
220
224
264
251
321
223
246
247
214
241
272
Is there any difference between the lengths of male and female crap-
pies of this age group in East Lake in 1948?
7.31 In order to test two methods of teaching spelling, 40 pupils were ran
domly assigned to two classes and one method was tried on each class.
At the end of the trial a test was given. Following are the scores on the
tests :
148
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Method
A
Method
B
10
48
20
57
20
50
27
60
25
51
35
63
30
52
40
64
33
54
41
65
37
56
50
67
41
57
50
67
43
65
54
73
46
73
56
83
46
86
57
95
Test the hypothesis that the two methods of teaching spelling are
equally effective. State all your assumptions.
7.32 Using the data of Problem 6.23 and letting <x = 0.01, test H:fj, D ^Q
versus A :&& ?^0.
7.33 A certain stimulus administered to each of 9 patients resulted in the
following increases in blood pressure: 5, 1, 8, 0, 3, 3, 5, 2, 4 mm. Hg.
Can it be concluded that the stimulus will be in general accompanied
by an increase in blood pressure?
7.34 Suppose an investigator of group differences in I.Q. finds, for inde
pendent random groups A and B of 11 subjects each assumed to be
from normal populations of same variance, a difference in sample
means of
TA - 7* 3.9 I.Q. points
and an estimated standard error of the mean difference of 2.0, He
selects lOOcx as 5 per cent,
(a) For the data as given, what hypothesis might he test? Perform
the required test and state your conclusions.
(6) Suppose group A had been given special coaching designed to
"increase" I.Q,, while group B had been maintained as a con-
troL What hypothesis might he test? Perform the required test
and give the resulting inferences.
7.35 In examining the resistance to crushing offered by kernels of a single
ear of corn, we choose at random two lots of 10 kernels each with the
following results:
CRUSHING RESISTANCE
(in points)
Lot I
Lot II
8
18
8
20
14
20
15
20
16
22
16
24
17
27
18
28
18
30
20
31
PROBLEMS
149
Using the method of paired observations, we find the difference be
tween the two means to be significant. We draw four more sets of two
samples, each time with a significant difference. This seems surpris
ing, since all the samples were taken from the kernels of the same ear.
Can you explain the results?
7.36 (a) For the data given below, test the hypothesis that the true mean
tensile strength of the product of the C and C Manufacturing
Company is greater than the corresponding value for its competi
tor. State all your assumptions.
(6) Ignoring any assumption about variances you may have found it
necessary to make in (a) above, test the hypothesis that the two
population variances are equal.
TENSILE STRENGTH or SCREW
DRIVER OP 34 VALVE CAPS
PRODUCED BY THE C AND C
MANUFACTURING COMPANY
Test
Tensile
Strength
in
Pounds
Y
Test
Tensile
Strength
in
Pounds
Y
1
130.1
19
153.5
2
132.3
20
154.1
3
133.4
21
154.7
4. ...
135.5
22
155.4
5
137.7
23. . . .
156.7
6
139.3
24
157.5
7
140.4
25
158.4
8
144.2
26
159.4
9
145.0
27. . . .
160.7
10
146.7
28
161.9
11
147.4
29
163.1
12
148.3
30
164.8
13. . . .
149.7
31
169.3
14. ...
150.6
32
171.2
15. . . .
151.1
33
174.0
16
151.8
34
180.7
1 *7
1 CO 1
1 / . . . .
18
I 3 . X
152.7
Total
5183.7
TENSILE STRENGTH OF SCREW
DRIVER OF 36 VALVE CAPS
PRODUCED BY A COMPETITOR
OF THE C AND C MANUFAC
TURING COMPANY
Test
Tensile
Strength
in
Pounds
Y
Test
Tensile
Strength
in
Pounds
Y
1. ...
65.7
20
149.4
2. ...
101.3
21
151.0
3
103.0
22
153.3
4
103.6
23. ...
155.2
5
107.2
24
157.6
6
115.9
25
160.7
7
117.4
26
164.3
8. ...
122.6
27. ...
166.1
9
126.5
28
168.8
10
129.1
29. ...
170.4
11
132.3
30. ...
180.6
12. ...
134.6
31. ...
184.6
13
135.2
32
188.8
14. . . .
136.7
33
192.9
15
138.3
34
196.0
16. . . .
142.1
35
200.4
17
143.4
36
204.8
18
147.2
19. . . .
148.2
Total
5295.2
7.37 Using the data which follow, test the hypothesis that the true mean
crushing strengths of air-dried and green Douglas fir wood are the same.
State all your assumptions and interpret your results.
ISO
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
CRUSHING STRENGTHS OF 248 SAMPLES OF AIR-DRIED DOUGLAS FIR, SIZE
2" BY 2" BY 8". TESTED BY FOREST PRODUCTS LABORATORY DOMINION
GOVERNMENT, AT UNIVERSITY OF BRITISH COLUMBIA, 1945
N 1
4713
N 1
5641
N 9
7145
E 7
6508
S 5
8413
W 3
7446
2
5516
2
5550
E 3
6200
8
6828
6
7690
5
7941
3
5956
3
7433
4
7501
9
6098
7
8484
6
8159
4
5652
4
7097
5
8086
10
6359
8
8139
7
9316
5
5951
5
7865
6
8055
S 1
5208
9
7595
8
9515
6
7178
6
8045
7
8042
2
4648
10
7021
9
8171
7
6630
7
7408
8
8678
3
7153
11
6416
10
9001
8
6284
8
7344
9
6710
4
6504
W 3
6657
N 3
8161
9
6246
9
7518
10
7512
5
6562
5
8264
4
7820
10
4689
10
7280
S 1
6438
6
7105
7
7268
6
8560
11
4825
E 4
7174
3
6074
7
7114
8
8101
7
8222
12
4697
5
7234
4
7170
8
6263
9
7066
8
8387
E 3
5757
6
8452
5
7306
W 3
5530
10
7301
9
7500
4
6661
7
8709
6
7760
4
6632
N 1
5961
10
2181
5
6098
8
7710
7
7049
5
6429
2
6254
11
7655
6
5867
9
7609
8
6863
6
6912
3
7247
E 3
7373
7
5573
10
6731
9
6987
7
7053
4
7480
4
7949
8
6282
S 3
6342
10
6511
8
6370
5
8512
5
8199
9
5536
4
6924
W 3
7025
9
7413
6
8911
6
8547
10
4941
5
7712
4
6775
10
6335
7
8988
7
8464
S 2
4003
6
6805
5
7754
N 1
6584
8
9330
8
8594
3
4789
7
7539
6
7495
3
7518
9
9899
9
7092
4
4889
8
7630
7
7990
4
7106
10
9025
10
7433
5
5304
9
7501
8
6149
5
7135
11
8920
S 1
6444
6
5350
10
7531
9
6774
6
7596
E 3
6419
2
6545
7
5601
11
6096
10
7137
7
7573
4
8403
3
7320
8
5932
12
6983
N 1
4858
8
7521
5
8220
4
7886
9
5245
W 3
6212
3
6148
9
7261
6
9501
5
8173
10
5585
4
6530
4
5388
10
6364
7
9250
6
7844
11
4313
5
7800
5
5883
11
6905
8
9479
7
7613
12
4924
6
7713
6
5930
E 3
7608
9
9985
8
8469
W 3
5196
7
7759
7
6252
4
6793
10
9686
9
7675
4
4810
8
7253
8
5920
5
7734
11
8849
10
7371
5
6641
9
6898
9
6260
6
6465
S 3
6693
W 3
7113
6
4625
10
7403
10
6403
7
7499
4
6338
4
7283
7
6704
N 2
6144
11
6644
8
7703
5
5976
5
8337
8
5555
3
6717
12
5841
9
7470
7
8495
6
8509
9
6813
4
7021
E 3
6650
10
7178
8
9184
7
7510
10
6061
5
8096
4
5802
S 1
6201
9
9485
8
8361
11
4959
6
7608
S
7287
3
7878
11
8507
9
7485
12
5618
7
8025
6
6379
4
7155
12
8270
10
8522
14
3958
8
8115
PROBLEMS
151
CRUSHING STRENGTHS OF 248 SAMPLES OF GREEN DOUGLAS FIR, SIZE 2" BY
2" BY 8". TESTED BY FOREST PRODUCTS LABORATORY DOMINION
GOVERNMENT, AT UNIVERSITY OF BRITISH COLUMBIA, 1945
N 1
2428
W13
2343
N 7
3639
E 3
3446
S 3
3412
S 7
4088
2
2173
N 2
2603
8
3645
4
2892
4
3904
8
4377
3
2896
3
2911
9
3487
5
3629
5
4030
9
4267
4
2980
4
3158
10
3351
6
3442
6
4212
10
4256
5
3378
5
3553
E 3
3591
7
3412
7
4423
11
4109
6
3167
6
3659
4
2849
8
3477
8
4575
12
3325
7
3208
7
3800
5
3911
9
3474
9
4318
W 3
3297
8
3342
8
3645
6
2591
10
3007
10
3829
4
3606
9
2982
9
3505
7
2769
S 1
2493
11
3933
5
3534
10
3301
10
3834
8
4097
2
2505
12
4608
6
4159
11
2330
E 3
2506
9
3203
3
3449
W 4
3340
7
4393
12
2651
4
2818
10
3179
4
3224
5
3887
8
3992
E 3
2478
5
3775
S 1
2668
5
3485
6
4097
9
4049
4
2665
6
3318
2
2766
6
3667
7
3440
N 1
2813
5
3033
7
3686
3
3280
7
3343
8
4503
2
2574
6
3205
8
3705
4
3295
8
3431
9
3806
3
3286
7
3282
9
3543
5
3844
W 3
2643
10
3939
4
3310
8
3229
10
3848
6
4022
4
3039
N 1
2902
5
3610
9
3137
S 3
2778
7
3575
5
3510
2
2869
6
3637
10
2693
4
2743
8
3784
6
3469
3
3610
7
3871
S 1
2128
5
3541
9
3621
7
3635
4
3547
8
3757
2
2200
6
3580
11
3698
8
4016
5
4012
9
3716
3
1977
7
3803
W 3
3032
9
3777
6
3919
E 3
3105
4
2498
8
3787
4
3132
10
3642
7
4585
4
3172
5
2732
9
3623
5
3781
N 3
3257
8
4553
6
3679
6
2920
10
3848
6
4141
4
3426
9
4235
7
3854
7
3102
11
3530
7
3730
5
4001
10
4495
8
3670
8
3050
12
3296
8
4162
6
3993
11
3694
9
3386
9
3230
W 3
2845
9
3559
7
4201
12
3492
10
3368
10
3053
4
3015
10
3532
8
4555
E 3
3173
S 1
2688
11
2993
5
3384
N 1
2296
9
3914
4
3879
3
3089
12
2518
6
3671
2
2458
10
3931
5
3751
4
3212
W 3
2938
7
3794
3
2794
E 3
3769
6
4197
5
3618
4
2272
8
3863
4
3075
4
3622
7
4110
7
3551
5
3144
9
3712
5
3166
5
4168
8
4061
8
3752
6
2904
10
3553
6
3255
6
4246
9
4589
9
3474
7
3314
N 1
2607
7
3233
7
4282
10
3762
10
3556
8
3448
2
2591
8
3600
8
4118
11
2733
W 3
3181
9
3468
3
3042
9
3471
9
3928
S 4
3071
4
3163
10
3289
4
2450
10
3735
S 1
3095
5
3886
9
3733
11
2456
5
3444
11
3329
2
3218
6
3873
10
3823
12
3078
6
3593
7,38 Two lots of steers, 10 head in each lot, were used in a 90-day feeding
trial. Lot 1 received standard ration A* Lot 2 received special ration
K. Steers on ration A gained 1.84 Ibs. per head per day, while the ani
mals fed K gained at the rate of 2.36 Ibs. per head per day. Two
questions were of interest,
(a) Will daily gains on ration K exceed 2 Ibs. per day? The variance
of the mean gain, 2.36, was found to be .0144.
(6) Is ration K better than standard ration A in producing gains?
The variance of the mean gain, 1.84, for lot 1 was .0256; thus,
152
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
7.39
7.40
7.41
we see that the pooled sum of squares for daily gain of the two lots
is 3.60,
Answer the two questions with the information given above. Why do
we use twice the pooled variance in examining the difference in gains
between the two lots, whereas in answering question (a) we use the
variance without such modification?
A sample of rural families and of urban families was taken to study
differences in coffee purchases by the two groups. The data obtained
are listed below in terms of pounds per family purchased annually.
Family No.
Rural
Urban
1
12.1
8.3
2. ...
6.8
9.3
3
9.1
9.2
4
11.1
11.1
5
11.4
10 7
6. .
13.3
4.6
7
9.8
9.8
8
11.3
7,9
9
9.4
8.5
10
10.2
9.1
11
9.7
12
6.2
Would you attribute the difference in coffee consumption observed in
those samples to normal sampling fluctuation, or is there a real dif
ference between rural and urban coffee consumption? Select your own
level for control of the Type I error and draw your conclusion accord
ingly* What is the specified population from which these data provide
you a sample? State your assumptions.
It has been suggested that the resistance of wire C is greater than the
resistance of wire D. The following data (in ohms) were obtained from
tests made on samples of each wire :
C
D
0.140
0.135
0.138
0. 140
0.143
0.142
0.142
0.136
0.144
0.137
0.139
Assuming that trc* 3 *^? test (using a=0.0l) the hypothesis
H:^ c <fjL against the alternative Aifj. c >fM I> . State your conclusion
and interpret the results.
If the estimate of the population standard deviation from one sample
of 45 is 12, and a corresponding estimate from another sample of 45
is 18, arc these samples consistent with the hypothesis that they are
from normal populations with the same variance?
PROBLEMS 153
7.42 Two methods of determining moisture content of samples of canned
corn have been proposed and both have been used to make determi
nations on portions taken from each of 21 cans. Method I is easier to
apply but appears to be more variable than Method II. If the varia
bility of Method I were not more than 25 per cent greater than that of
Method II, we would prefer Method I, Based on the following sample
results, which method would you recommend?
ni = n 2 = 21, Pi = 50, F 2 = 53, &?=720, S 2/1 = 340. (Hint: Test
H : af = 1 .25<ri against A :<ri>l. 25o-f . Under this hypothesis
(sf/1.25)/sl is distributed as F( VjV ), where ^1=^2 = ^1 I=n 2
1=20.)
7.43 The amount of surface wax on each side of waxed paper bags is be
lieved to be normally distributed. However, there is reason to believe
that there is greater variation in the amount on the inner side of the
paper than on the outside. A sample of 25 observations of the amount
of wax on each side of these bags was obtained and the following data
recorded:
Wax in Pounds per Unit Area of Sample
Outside surface
Inside surface
^=0,948
]CX 2 -=91
7 = 0.652
2:1^=82
Conduct a test (using a: = 0.05) of the hypothesis HKTQ^O^ against
the alternative A:o% <cr|.
7.44 Using the data of Problem 6,21 and letting a. = 0.05, test H:o- 1 =cr 2
versus A :<ri 7^0- 2 .
7.45 Using the data of Problem 6,22 and letting a: = 0.01, test .Z7:crj. = cr 2
versus A :<ri r^cr^
7.46 Using the data of Problem. 6.20 and letting 01 = 0.01, test H:cri<cr 2
versus A :o-\ >cr z .
7.47 Using the data of Problem 6.23 and letting oi = 0.05, test H:& A <cr B
versus A :cr^ ><r B .
7.48 A child psychologist, analyzing personality differences in children by
a protective technique, classified the responses of a group of 99 pre
school children into three major types: static form of response, 23;
outer activity, 51; inner activity, 25. Do these data differ significantly
from a chance distribution of responses? Use a; ==.01.
7.49 A random sample of 147 women college students were interviewed
with regard to their habits concerning the purchase of clothing. The
source of each individual's income was also determined. Given the
data below and letting a. = 0.10, test the hypothesis that women pur
chase clothing without planning in the following proportion:
Frequently 10 per cent
Seldom 80 per cent
Never 10 per cent
154
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Source of Income
Numbers Who Purchased Clothing Items
Without Planning
Frequently
Seldom
Never
Earned all of spending money. . .
Earned part of spending money .
Had regular allowance ,
2
8
4
15
14
17
12
25
27
5
7
11
Money given, as needed
7.50
7.51
7.52
7.53
Referring to the data of Problem 7,49 and letting oc = 0.01, test the
hypothesis that frequency of purchasing clothing items without plan
ning is independent of source of income.
An experimenter testing three chemical treatments applied each to
200 randomly selected seeds, and then conducted germination tests.
The following results were obtained:
Number
Chemical
Germinating
Not
Germinating
A.
190
10
B
170
30
C
180
20
Test the hypothesis that the percentage of seeds germinating is inde
pendent of the chemical used.
An experimenter feel different rations to three groups of chicks. As
sume that the chicks were assigned to the rations (groups) at random
and that all other management practices for the three groups were the
same. A record of mortality is given below. Would you attribute the
differences among the mortality rates of the three groups to rations?
Why?
Ration
Number
Lived
Died
A
87
94
89
1$
6
11
B
C
I selected a random sample of students at Arizona State University
and asked their opinions; on a proponed radio program. The results arc
given below. The same number of each sex waa included within each
class group, that is, freshmen and sophomores each consisted of 100
men and 100 women, while juniors and seniors each consisted of 50
PROBLEMS
155
men and 50 women. Test the hypothesis tnat opinions are independent
of the class groupings.
7.54
7.55
7.56
Num
ber
Class
Favoring
Program
Opposed to
Program
Freshmen
120
80
Sophomores
130
70
Tuniors
70
30
Seniors ....
80
20
An agency engaged in market research conducted some of its sampling
by mail. For one survey these results in terms of response to succes
sive mailings were obtained:
Response No.: 1st
2nd
3rd
4th Original Mailing
Returns:
150
60
40
20
1000
Another agency obtained the following results in a mail sampling of a
similar population:
Response No,: 1st
2nd
3rd
4th Original Mailing
Returns:
200
30
50
25
800
Does it appear that the two mail samplings were homogeneous in
eliciting replies from the two populations?
In a large city the division of the voting strength between two candi
dates for mayor appeared to be about equal. The campaign manager
for candidate A polled a random sample of 2500 voters two weeks
before the election. In this sample 1313 of the voters indicated, they
would vote for A. If the sample is representative of the population of
voters in this city, is it likely that A will be elected? Establish 99 per
cent confidence limits for the proportion of voters favoring A. ^
An opinion-polling agency reported the distribution of a sample in tins
manner :
Republicans
Democrats
Independents
Total
400
450
150
1000
A newspaper poll in the same area yielded this distribution in terms
of declared political opinion of respondents :
Republicans
300
Democrats Independents Total
325 75 700
156
CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
Are these two samples homogeneous with regard to division of politi
cal opinion?
7.57 The following data were obtained from a random sampling of the rec
ords of a specific company.
NUMBER OF BREAKDOWNS
7.58
7.59
7.60
7.61
7.62
Machine
A
B
C
D
Total per Shift
Shift 1
10
6
12
13
41
Shift 2
10
12
19
21
62
Shift 3
13
10
13
18
54
Total
per machine
33
28
44
52
157
Test the hypothesis that the number of breakdowns on each machine
is independent of the shift. Use ct = 0.05.
Road tests gave the data shown below regarding tire failures. Letting
<x = 0.05, test the hypothesis that left-right tire wear is independent of
front-rear tire wear.
NUMBER or FAILURES
Front
Rear
Totals
Left
Right
115
125
65
95
180
220
Totals
240
160
400
A car rental firm has a particular car that has experienced 13 break
downs in the past year. Using a Poisson distribution and letting
<x = 0.01, test H:jj,<lQ versus ^L:M>10.
Ignoring the correction for continuity in Equation (7.10), that is,
dropping the adjustment of 0.5, show that x 2 = (a rb) 2 /r(a + b)
where a and b are the observed numbers in the two classes and r
equals the hypothesized ratio of type A to type B.
Work the preceding problem using the correction for continuity and
show thatx 2 =(|a~-r&j (r + l)/2) 2 /V(>+&).
Rework the problems noted below, using the method described in
Section 7.20:
(a) 7.26
(6) 7.27
(c) 7.28
W) 7.30
(e) 7.31
(/) 7.37
(<7) 7.39
7.63 Given the following data (three random samples from three normal
PROBLEMS
populations) and assuming homogeneous variances, test the hy
pothesis jH r :jui=/z 2 =M3. Let o: = 0.10.
Sample 1 Sample 2 Sample 3
48
72
48
24
24
12
36
48
24
48
7.64 Assuming homogeneous variances, test the hypothesis that the four
normal populations, from which the following random samples were
obtained, have the same mean. Let c* = 0.025.
Sample 1 Sample 2 Sample 3 Sample 4
95
45
95
20
50
40
130
55
105
95
15
50
10
65
135
80
60
45
125
7.65 Using the data of Table 7.23 and assuming homogeneous variances,
test the hypothesis H:^i=jU2= = M3=M4- Let a. = 0.01.
7.66 Using the data of Table 7.20, test the hypothesis H:<r? =<r| =o- ==oi.
Let <*=0.05.
7.67 Letting OL = 0.10, test the hypothesis of homogeneous variances for
each of the following problems: (a) 7.63 and (b) 7.64.
References and Further Reading
1. Anderson, R. L., and Bancroft, T. A. Statistical Theory in Research. McGraw-
Hill Book Company, Inc., New York, 1952.
2. Bartlett, M. S. Some examples of statistical methods of research in agricul
ture and applied biology. Jour. Roy. Stat. Soc. (Suppl.), 4:137, 1937.
3. Bowker, A. H., and Lieberman, G. J. Engineering Statistics. Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959.
4. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
5. Davies, O. L. (editor). The Design and Analysis of Industrial Experiments.
Second Ed. Oliver and Boyd, London, 1956.
6. Dixon, W. J., and Massey, F. J. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
7. Freund, J. E. Modern Elementary Statistics. Second Ed. Prentice-Hall, Inc.,
* Englewood Cliffs, N.J., 1960.
g 9 Liver more, P. E., and Miller, I. Manual of Experimental Statistics.
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1960.
9. Hald, A. Statistical Theory with Engineering Applications. John Wiley and
Sons, Inc., New York, 1952.
10. . Statistical Tables and Formulas. John Wiley and Sons, Inc.,
York, 1952,
158 CHAPTER 7, STATISTICAL INFERENCE: TESTING HYPOTHESES
11. Huntsbcrger, D. V. Elements of Statistical Inference. Allyn and Bacon, Inc.,
Boston, 1961.
12. Mood, A. M. Introduction to the Theory of Statistics, McGraw-Hill Book
Company, Inc., New York, 1950.
13. Robertson, W. H, Programming Fisher's exact method of comparing two
percentages. Technometrics, 2 (No.l): 103-7, Feb., 1960.
14. Snedecor, G. W. Statistical Methods. Fifth Ed. The Iowa State University
Press, Ames, 1956.
15. Statistical Research Group, Columbia University. Selected Techniques of
Statistical Analysis. (Edited by C, Kisenhart, M. W. Hastay, and W, A.
Wallis.) McGraw-Hill Book Company, Inc., New York, 1947.
16. Wadsworl.li, Cr. P., arid Bryan, J. CT. Introduction to Probability and Random
Variables. McGraw-Hill Book Company, Inc., New York, 1900.
C H APTE R 8
REGRESSION ANALYSIS
THE METHODS OF ANALYSIS studied thus far in this text have been con
cerned with data on only one characteristic associated with the experi
mental units. That is, in any given problem we have been working with
only one variable. However, as you will realize, many problems involve
more than one variable. Consequently, it is necessary that techniques
developed for analyzing multivariate problems be studied. Some of
these techniques will be investigated in this chapter.
8.1 FUNCTIONAL RELATIONS AMONG VARIABLES
When we possess information on two or more related (or concomitant}
variables, it is natural to seek a way of expressing the form of the
functional relationship. In addition, it is desirable to know the strength
of the relationship. That is, not only do we seek a mathematical func
tion which tells us how the variables are interrelated, but also we wish
to know how precisely the value of one variable can be predicted if we
know the values of the associated variables. The techniques used to
accomplish these two objectives are known as regression methods and
correlation methods. Regression methods are those used to determine the
"best" functional relation among the variables, while correlation
methods are used to measure the degree to which the different variables
are associated.
More specific statements will be forthcoming in succeeding sections.
For the moment, it will suffice to say that the functional relationships
will, in general, be represented mathematically by
, X p |0x, ,O (8.1)
where
77 = the response (or dependent} variable
Jft = the ith independent variable (i=l, - - - , p),
0j=the jfch parameter in the function (j = 1, ,?),
and <t> stands for the assumed form of the function. Equation (8.1) is
sometimes written as
77 = <(^i, , X*). (8.2)
When this abbreviated form, is used, one should always remember that
the parameters belong in the expression; they have been omitted solely
to achieve brevity. In the language of statistics, a function such as
specified by Equation (8.1) is known as a regression function. However,
in some areas of application, a more natural (and common) expression
[159]
16O CHAPTER 8 r REGRESSION ANALYSIS
is response function. In this book both expressions will be used, the
choice being dictated by the context.
8.2 A WORD OF CAUTION ABOUT FUNCTIONAL RELA
TIONS
In any analysis it is hoped that the postulated (assumed) function
represents some basic, or causal, mechanism associated with the ex
perimental units and the factors under investigation. However, science
is not always so far advanced that the basic variables and the basic
mechanisms of a process are known with certainty. In such cases, the
methods of regression, and correlation may still prove useful as analytic
and predictive tools.
Because of the frequent uncertainty about basic variables and basic
mechanisms, a word of warning must be sounded relative to the inter
pretation of analyses involving concomitant variables. This warning
is: J^lst because a particular functional relationship has been assumed
and a specific computational procedure followed, do not assume that a
causal relationship exists among the variables. That is, because a func
tion has been found that is a good fit to a set of observed data, we
are not necessarily in a position to infer that a change in one variable
causes a change in another variable.
In summary, the only person who can safely say that the basic
variables are those used and that the basic mechanism operates in
accordance with the selected mathematical function is a pcrsoxi well
trained in the subject matter field in which the experiment was per
formed. The statistical analysis (in this instance, a regression and/or
correlation analysis) is only a tool to aid him in the analysis and inter
pretation of data.
8.3 THE CHOICE OF A FUNCTIONAL RELATION
How does an analyst go about choovsmg a particular functional rela
tionship as representative of the population under investigation? Two
methods arc employed. Those are: (1) mi analytical consideration of
the phenomenon concerned, and (2) an examination of scatter diagrams
plotted from the observed data. While the first method is preferred,
the second should not be underrated. If little is known about the basic
mechanisms involved, the use of scatter diagrams can be quite helpful.
8-4 CURVE FITTING
Once we have decided on the type of mathematical function that
best seems to fit, or represent, our concept of the exact relationship
existing among the variables, the problem of choosing a particular
member of this family of functions arises. That is, a certain function
has been postulated as being an expression of the true state of affairs
in the population, and it is now necessary to estimate the parameters
of this function. The determination of these estimates and thus the
specification of a particular function is commonly referred to as curve
fitting.
8.5 THE METHOD OF LEAST SQUARES 161
How do we go about fitting a curve to a set of data? That is, how are
the estimates of the parameters obtained? Again we are faced with the
problem of choosing among several methods of estimation. The ap
proach taken should, of course, provide us with the "best 77 estimates,
Since this is simply another part of the general problem of estimation,
the criteria by which estimators are selected will be similar to those
outlined in Chapter 6.
8.5 THE METHOD OF LEAST SQUARES
To proceed to the method of estimation, it should be noted that
there are several methods outlined in the literature, all of which give
acceptable answers. For our purposes it will be sufficient to discuss the
method of least squares. By the use of this method excellent results may
be obtained. In fact, if the usual assumption of normality is made, the
method of least squares becomes equivalent to that of maximum
likelihood.
To study the method of least squares, assume that we are consider
ing a certain characteristic (77) which is related to or depends on certain
other characteristics (X\, - , X^) according to the relationship
97 = <^(-X"i, - - , 2 p \ 0i, , 6g). (8.3)
Both the form of the function and the values of the parameters must
be determined. (NOTE: In practice, the form is usually assumed to be
known and thus the problem reduces to estimating the parameter
values.)
The reason that the parameter values cannot be determined without
error is that the observed values of the dependent variable will seldom
agree with the expected values. That is, even if we can control the X
values (or measure them without error) the observed value of the
dependent variable, denoted by Y, will not equal the expected value, 77.
This is expressed by
Y n + e = *(Xi, -, X, | lf ,*,)+ e (8.4)
where 6 stands for the error made in attempting to observe 77. Many
factors contribute to the value of e, but it seems reasonable to as
sume that it (c) is a random variable with mean and variance o-^.
Under these conditions we must be content with estimating the un
known parameters, namely, the 0's and cr^.
To see how the method of least squares operates, consider the data
of Table 8.1. Denoting the estimator of 0/ by j(j= 1, - * , g), form, the
n differences
F 2 - *(X, - , X,* I *!, - , 0) - F 2 - F 2
, - - - , X pn | $x, - - - , ff ) = Y n - t
162 CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.1 Symbolic Representation of n Observations on p + 1 Variables
Dependent
Variable
Independent Variables
F
Xi A 2 * X p
Fi
% s a
Y n
"\^ "V V
-<A, ITT. -A. 2n. -^yn
The values of the j(j ' = 1, * 7 #) are then determined by minimising
the sum of the squares of the deviations specified by Equation (8.5).
That its, the j arc found by minimizing
-
(8.6)
This is a familiar problem in calculus: *S is differentiated with respect
to each of the estimators, and each partial derivative is set equal to 0.
Symbolically,
dS
(8.7)
and this system of equations must be solved for the estimates, that is ;
for the $/.
8-6 GRAPHICAL INTERPRETATION OF THE METHOD
OF LEAST SQUARES
In order to portray graphically the concepts of the method of least
squares, it will be convenient to restrict ourelve to the case of two
related variables, This is because cases involving three or more vari
ables present great difficulties in graphic presentation. A two-variable
scatter diagram might appear as in Figure 8.1
FfG. 8,1 Example of a scatter diagram, often
referred to as a scattergrarru
Suppose that the functional relationship assumed to exist between
8.6 GRAPHICAL INTERPRETATION OF METHOD OF LEAST SQUARES 163
77 and X is that specified by the mathematical model
v = 0! + 6 2 X. (8.8)
The associated statistical model is
Y = X + <9 2 3T + e. (8.9)
The method of least squares would then be used to obtain a regression
(or estimating or predicting} equation. 1 Since Equation (8.8) is the
equation of a straight line, the regression equation would also turn
out to be a straight line such as shown in Figure 8.2. The resulting
Y
FIG. 8.2 Example of a scatter diagram with a straight line inserted
showing the vertical deviations whose sum of squares is to be
minimized by the proper choice of straight line.
regression equation would be denoted by
Y = 0! + 0*X (8.10)
where 8j estimates Oj(j= 1, 2) and Y estimates both Y and 77.
If a more complicated functional relationship had been assumed,
for example,
(8.11)
the problem would be somewhat more complex to handle mathemati
cally bxit the principle would be unchanged. The parameters would still
be estimated by minimizing the sum of the squares of the vertical
deviations about the appropriate curve. The reader's attention is
directed to Figure 8.3 for an illustration of a situation associated with
the mathematical model specified in Equation (8.11).
Y
X
FIG, 8,3 Example of a scatter diagram with a second degree polynomial
inserted showing the vertical deviations whose sum of squares is
to be minimized by the proper choice of parabola.
1 The terms regression equation, estimating equation, and prediction equation
are used interchangeably.
164 CHAPTER 8, REGRESSION ANALYSIS
8.7 SIMPLE LINEAR REGRESSION
Let us now consider in detail the case in which the postulated func
tional relationship is of the form 2
77 - /5 + ftiX (8.12)
or
F = fto + foX + e. (8.13)
The problem is, of course, to estimate (3o and /?i from the observed
sample data. That is, estimates of /3 and ft, denoted by 60 and 61,
must be found. Using the method of least squares, the estimates 6
and 61 are determined by solving the normal equations*
The solutions are
rs i ^
**
and
i = 7 j^ (8.16)
where a; = -X" 5T and y== F 7\ These estimates are then used to give
\is the regression equation
f J + i^. (8.17)
The presentation of a worked example will be deferred until Section 8.9
8.8 PARTITIONING THE SUM OF SQUARES OF THE DE
PENDENT VARIABLE
Regression computations may also be looked upon as a process for
partitioning the total sum of squares, ^,Y*, into three parts, each of
* The symbols $o> ft, 60, and 61 arc \isod here rather than 0o #i o, and 0\. This
ia in conformanco with general usage. Some authors prefer ct f ft, a, and 6 rather
than #g, A, baj and 61. Howovor, the latter aro hotter suited to extensions to three
or raoro variables.
* The phraeo "normal oquationfl" IB \xsed to describe the equations roaxilting
from the least gquaros differentiation. It has no connection with the normal
distribxition,
8.8 PARTITIONING THE SU/W OF SQUARES OF DEPENDENT VARIABLE 165
which is meaningful and useful. Prior to this chapter, the total sum of
squares was shown to be the sum of two quantities, the corrected 4
sum of squares and the correction for the mean :
Total S.S. == Correction for the mean + Corrected S.S.
= S.S. due to the mean + S.S. of deviations about the mean. (8.18)
That is,
2 . (3-19)
Using regression methods, the corrected sum of squares may also be
subdivided into two parts, the sum of squares due to (simple linear)
regression and the sum of squares of the deviations about regression:
Corrected S.S.
= S.S. due to regression + S.S. of deviations about regression. (8 . 20)
That is,
Substituting Equation (8.21) in Equation (8.19) gives
]T) F 2 = (Z YY/n + 6x 2: xy + Z (F - *") 2 - (8.22)
Expressing this in words,
Total S.S. = (S.S. due to the mean) + (S.S. due to regression)
+ (S.S. of deviations about regression). (8.23)
More properly, this result should be stated as
Total S.S. = (S. S. due to 6 ) + (S.S. due to Z>i 1 6 )
+ (Residual S.S.). (8.24)
A more extensive discussion of this type of manipulation and of the
associated notation is given in Sections 8.15 and 8.16. For the present
it is recommended that the reader make an effort to learn the notation
and the manipulative skills involved. The acquisition of such knowl
edge will prove most helpful in the remainder of this book.
Graphically, each of the indicated partitions of the total sum of
squares can be associated with the sums of squares of segments of the
F-ordinates. This is illustrated in Figure 8.4, where the ordinate Y Q ,
associated with X e , is partitioned according to the identity
Y - Y + (f - Y) + (Y - ?) (8,25)
In words, this says that
4 The expression "corrected sum of squares' 7 is used to represent the total sum
of squares minus the adjustment (or correction) for the mean. That is, it is
simply a synonym for the sum of the squares of the deviations about the mean.
166
CHAPTER 8, REGRESSION ANALYSIS
F(G. 8.4 Diagram to illustrate the partitioning of
the total sum of squares.
Observed Y = (Contribution due to the mean)
+ (An additional contribution due to regression)
+ (Deviation from regression).
(8.26)
If we carry through the algebraic manipulations without error, Kqua-
tion (8.22) may be derived from Equation (8.25)- The proof of this is
left as an exercise for the reader.
When each partition of 23^ 2 is associated with a corresponding por
tion of the total degrees of freedom, the technique is known as analysis
of variance. Such results are usually presented in tabular form,
referred to as an analysis of variance table. This is illustrated in
Table 8.2. The first line of this table is frequently omitted and the
total line expressed as ]C// 2sssa S^ 2 (5D^) 2 / n ' with n 1 degrees of
freedom. However, in this book the results will always be presented in
the form unod in Table 8.2.
TABLE 8.2 General Analysis of Variance for Simple Linear Regression
Source of
Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Due to &o
Due to &i| 60- . -
Residual . , , ,
1
1
n 2
( r YY/U
b\ 23 xy
V (Y J>)2
( z; nv
61 H *y
y; fy _ #)/( 2)
Total
71
T\ K 2
8.9 A PRACTICAL EXAMPLE
167
8.9 A PRACTICAL EXAMPLE
To illustrate the methods discussed in the preceding sections, con
sider the data in Table 8.3. Following the methods outlined in the
TABLE 8.3-Schopper-Riegler Freeness Test of Paper Pulp During Beating
Hours of
Beating
X
Schopper-Riegler
(in degrees}
Y
Hours of
Beating
X
Schopper-Riegler
(in degrees}
Y
1
17
8
64
2
21
9
80
3
22
10
86
4
27
11
88
5
36
12
92
6
49
13
94
7
56
Source: O. L. Davies, Statistical Methods in Research and Production, Oliver and Boyd,
Edinburgh, 1949, p. 161. By permission of the author and publishers.
preceding sections, we obtain
136 + 916i = 732
916o + 8196i = 6485
which yields 1? = 3*962+7A78X as the regression equation. It is also
observed that
F 2 = 51,712
Y = 41,217.23
> 2 = 10,494.77
y = 10,177.59
^ 317.18
and these results are presented in analysis of variance form in Table
8.4. (NOTE: A convenient form to use in performing the calculations
is given, in Table 8.5.)
The estimated function is pictured in Figure 8.5. Examination of
Figure 8.5 will suggest that a cubic equation (i.e., a third-degree
polynomial) would be a better fit to the observed data. However, a
discussion of the desirability or appropriateness of fitting a different
function and of methods of fitting other than a simple linear function
will be deferred until later.
168 CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.4-Analysis of Variance of the Schopper-Riegler Data of Table 8.3
Source of Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Due to 60
1
41,217.23
41,217.23
Due to b\ | 60
1
10,177.59
10,177.59
Residual
11
317.18
28.83
Total
13
51,712.00
TABLE 8.5-Suggested Form for Calculation and Presentation of Results
in Simple Linear Regression
n =
x -
Z F -
7 =
^c; 2
Z XY
Z ry
Z xy
Z -^ 2
Source of
Variation
Due to 60-
Due to bi\
Residual .
Degrees of
Freedom
Sum of
Squares
Mean
Square
Total
8.10 ASSUMPTIONS NECESSARY FOR ESTIMATION AND
TESTING! HYPOTHESES IN SIMPLE LINEAR RE
GRESSION
Before we can construct confidence Intervals or specify teat pro
cedures, certain assumptions are generally made. Thus, in addition to
the assumption of "no error' * in the independent variable made in
Section 8.5, the usual assumptions are:
8. ID ASSUMPTIONS IN SIMPLE LINEAR REGRESSION
169
(1) For a given X, the Y's are normally and independently dis
tributed about a mean n Y \ x = r) = /3Q+/3iX with variance
a Y\x~ a ir-x- The assumption concerning the mean is equiva
lent to assuming that e is normally and independently dis
tributed with mean 0, where e is defined by Equation (8.13).
(2) The variance o^ lx is the same for each X and can therefore
be denoted by o-^. The subscript E is used because o^, is the
variance of the errors denoted by e. It (cr|.) is commonly re
ferred to as the variance of the "errors of estimate."
The preceding assumptions are summarized by
V = /3 4-
i= 1,
3= 1,
, *
(8.27)
where
ii = the number of values of Y associated with the ith value of X,
it = n = the total number of values of F (or of X), and the e t -y
* 1
are normally and independently distributed with mean zero and stand
ard deviation 0-%. This last phrase is frequently abbreviated to "the
are NID (0,
10O
80
^. _
ujo:6O
ceD
Q-
O
20
3.962+ 7 478 X
J X
O 2 4 6 8 1O 12
HOURS OF BEATING
FIG. 8.5 Plot of data in Table 8.3 with the least squares line inserted
170 CHAPTER 8, REGRESSION ANALYSIS
8 11 ESTIMATES OF ERROR ASSOCIATED WITH SIMPLE
LINEAR REGRESSION ANALYSES
Granting the assumptions of Sections 8.5 and 8.10, and further as
suming that the failure of the assumed model to fit the observations
exactly is solely a function of the errors, the mean square for devi
ations about regression (i.e., the residual mean square) can be used as
an estimate of a-%. Symbolically,
residual mean square = ] (F - Y}*/(n - 2) = 4. (8.28)
We must always remember, though, that such an estimate can be
badly inflated if the assumed mathematical model is inadequate. More
will be said on this subject a little later.
Once we have determined the variance estimate $%, it is a straight
forward matter to obtain estimates of the variances of various statistics
calculated in the regression analysis. Those of general interest will be
given without derivation.
Estimated variance of the regression coefficient &i
2 2 / x ^ o /'Q O Q"\
o - ^ c / s 3C* t \O . J-t i'y
Estimated variance of estimated 'mean Y for given X
(8.30)
Estimated variance of predicted individual Y for given X
Estimated variance of the regression coefficient
These may then be used to test hypotheses about, or to provide interval
estimates* of , various unknown parameters.
8.12 CONFIDENCE AND PREDICTION INTERVALS IN
SIMPLE LINEAR REGRESSION
In most linear regression problems the estimator of greatest impor
tance is the slope 61. This is, of course, an estimator of X . To provide
a lOOy per cent confidence interval of /3i, we compute
- ii =F
where $ 6l is defined by Equation (8.29).
8.12 CONFIDENCE AND PREDICTION INTERVALS 171
Using the data in Table 8.3 and the results of Section 8.9, it may
be verified that
Example 8.1
he data in
that
s^ = 317.18/11 = 28.83
and
4 x = 28.83/182 = 0.1584.
Thus, a 95 per cent confidence interval for /Si is determined to be
(6.602, 8.354),
If a 100-y per cent confidence interval estimate of 0o is needed, we
have only to compute
L _
= ft + *-2-2*5 (8.34)
where s &o is defined by Equation (8.32). No example will be given;
however, some of the problems will require use of Equation (8.34).
It is also possible that we might wish to determine a confidence
region for the simultaneous estimation of /Jq and jSi. Making use of the
fact that
is distributed as tf* and: that (n 2)sJ/<r^ is distributed as x^ n _ 2) it
is seen that
is distributed as F with ?i = 2 and ^2 = '^ 2 degrees of freedom. The
boundary of the lOOy per cent confidence region is then determined
by solving
[w(6o /3o) 2 + 2nX(b Q /3 )(&i 1) + (&i - i) 2 23 x *\/ 2s l
=== Fy(2, n 2) (8.37}
for /So and /Si.
Another estimation problem of importance in simple linear regres
sion is associated with 5^ = 6o + &iX\ As you will remember, 3?" = &o + &i-X"
is an estimator of /x r j X ==^ = /3 + jS 1 -X". Further, by the assumptions of
Section 8-10, 77 is the mean of a normal population. Thus, it should not
be surprising that a lOOy per cent confidence interval estimate of 77 is
provided by
\ -= & :r / *~ (8 38)
rrl ^ [(1-H-Y) /2] (n 2)F V -^ /
C//
where s$> is defined by Equation (8.30).
It should be noted that ^ = &o+&i^ is also a predictor of F" /3 +
172
CHAPTER 8, REGRESSION ANALYSIS
That is, Y can also be used to predict an individual Y-value
associated with, a given X- value. (NOTE: This is in contrast to the
preceding paragraph where was used to estimate the mean of a nor
mal population.) When !F is used to predict an individual value rather
than a mean value, a 100-y per cent prediction interval is provided by
L> \-
U f ]
(8.39)
where s$ is defined by Equation (8.31).
The nature of the confidence and prediction intervals specified by
Equations (8.38) and (8.39) is illustrated in Figure 8.6. The most
FIG. 8.6 Graphical representation of the confidence and prediction
intervals specified by Equations (8.38) and (8,39).
noticeable feature of Figuro 8.0 is the curvature of the confidence and
prediction limits. That is, our estimates arc most precise at the average
value of X and may bo almost useless at values of X far removed f rom
j?. By "almost useless/' we mean that the confidence and prediction
intervals may turn out to be so wide an to render them of little value.
To state the preceding conclusion in a positive rather than a negative
fashion, any estimate of the mean value of Y for a given X or any
prediction about an individual Y associated with a given X will be
mont meaningful for those values of X near 3?.
As a corollary to the preceding paragraph, it is clear that if the esti
mation of o is of prime importance, the values of X should have been
selected (ptior to collecting data) BO that 3?=0. The reason for this
statement should be clear. It is, of course, that by so choosing the X-
8.12 CONFIDENCE AND PREDICTION INTERVALS
173
values, the narrowest confidence and/or prediction interval will occur
at X = 0, and it is at this value of X that Y = bo.
Following up the line of thought started in the preceding paragraph,
one might wonder if choosing the X- values in accordance with the ex
pressed recommendation is best for all purposes. For example, if the
estimation of pi rather than /5 is of prime importance, jshould the
values of the controlled variable still be selected so that X = 0? The
answer is, "Definitely not." If, then, our only interest lies in 1 (and it
frequently does), how should the values of X be chosen? In this case
the appropriate recommendation is: select two values of X (as far
apart as is reasonable) and obtain random observations on the Y-
variable at only those two X- values. By following this rule, the standard
error of 61 will be made as small as possible subject to the (uncon
trollable) magnitude of S E . In other words, if we proceed as indicated,
the confidence interval for 0i should be kept "small." (NOTE: The
reader can verify, heuristically, the wisdom of this approach by
noting that widely divergent -XT- values will increase ^C# 2 , the denomi
nator of Equation (8.29), and thus decrease the size of s^.)
FIG. 8.7 Illustration of the danger of extrapolation.
Another fact which should not be lost sight of is that predicting
values of Y for a given X value is even more hazardous than already
indicated if we attempt such a procedure for an X value outside the
range of the chosen values of X used in obtaining the sample regression
line. That is, extrapolation beyond the observed range of the independ
ent variable is very risky unless we are reasonably certain that the
same regression function does exist over a wider range of X- values
than we have in our sample. A simple illustration will suffice to point
out the possible trouble. Suppose we have values of X and Y which
plot (see dots) as in Figure 8.7. In the given range of -XT, a straight line
appears to be a good fit to the data and we might be tempted to project
our regression line farther in both directions. However, it is entirely
possible that if we had chosen a wider range of -XT-values and observed
the associated F-values (see circled dots), a second degree polynomial
might have been indicated as the true form of the regression function
rather than the straight line we have drawn. You can readily see that
predicting values of Y using an extrapolation of the straight line could
lead to serious errors. Therefore, the research worker is advised to
174 CHAPTER 8, REGRESSION ANALYSIS
act with caution whenever he makes predictions which involve going
outside the observed range of the independent variable.
One further remark and we shall proceed to the subject of testing
hypotheses. Although only two-sided confidence and prediction limits
have been discussed in this section, the reader will realize that one-
sided limits should be used if the problem calls for such a procedure.
If only an upper or lower limit is required, the researcher should make
the same changes in procedure as outlined in Chapter 6 but continue,
of course, to use the statistics specified in Section 8.11.
8.13 TESTS OF HYPOTHESES IN SIMPLE LINEAR RE
GRESSION
Sometimes the researcher is interested in determining whether the
estimated slope 61 is significantly different from some hypothesized
value of /Si^say /5(, That is, he wishes to test the hypothesis H:fBi = f3{
against the alternative A :/3i^/3. The appropriate test statistic is
where s b is defined by Kquation (8.29). The hypothesis // would then
be rejected if
t =>1 <l-a/2)(n-2> (8.41)
or if
t < (i /2)<tt 2). (8.42)
A common value of /5{ is since this reflects the hypothesis that Y is
independent of X (in a linear sense) ; that is, that X is of no value in
predicting Y if a linear approximation is used.
Example 8.2
Referring to Example 8.1 and letting a = 0.01, test the hypothesis
//:#!(). Calculation yields = (7.478 0)/0.08 18.788 >* .o9fc(n>
= 3.106. Thus, the hypothesis is rejected.
It is worth noting that the tent of the hypothesis //:$i can also be
performed directly from the analysis of variance table. To illustrate
the nature of this alternative procedure, consider Table 8.G. Because
of the assumptions made in Section 8.10, It is possible to demonstrate
that the expected values of two of the mean squares are as shown in
Table 8.6. Thus, if //:/? a = is true, both the "mean square due to
&IJ.&D" and the "residual moan square 77 are estimates of the same quan
tity, namely, <r^/It seems logical, therefore, to examine the ratio
i mean square due to bi &o
77 _ _ ._ _r w ____ ' (&
f . i A f " ""'"" " 9 ^
,/ residual mean square
and if this ratio is significantly larger than 1, some doubt would be
cast upon the validity of the hypothesis //:#L*=0. Since it may be
8.13 TESTS OF HYPOTHESES
175
TABLE 8.6 General Analysis of Variance for Simple Linear Regression
Showing the Expected Mean Squares
Source of
Variation
Degrees
of Free
dom
Sum of
Squares
Mean Square
Expected Mean
Square
Due to 60
1
( 2Z Y) 2 /n
( 51 V) 2 /n
Due to bi\b . .
Residual
1
n 2
t>i 2Z xy
52 (F - F) 2
bi 2Z xy
*iv_ ^ sr
^ = V (F Y) 2 /(n 2)
jr + J&? S *
ffl
Total
n
5Z Y*
demonstrated that the F-ratio specified by Equation (8.43) follows an
^-distribution with v\ = 1 and v 2 ^=n 2 degrees of f reedora, the hy
pothesis H:/3i = Q will be rejected if F^
n _ 2 ).
Example 8.3 ^u /-
Referring to Table 8.4, it is seen that F = 10,177.59/28.83 = 353.02.
Since this exceeds Fo. 99(1,11) = 9.65, the hypothesis .fiT:/3i = is re
jected. This is the same conclusion reached in Example 8.2. (NOTE: the
fact that t%F(i, v ) is the connecting link between the equivalent test
procedures.)
Other test procedures in simple linear regression are concerned with
such hypotheses as:
(1) H:/3 Q = /3o;
(2) H:v Yl x~x
(3) H:/3 Q = j3o
and fa =
Rather than discuss these in detail, we shall only indicate the appro
priate test procedures. For the three cases just mentioned, the respec
tive test statistics are:
where s bo is defined by Equation (8.32),
(8.44)
(8.45)
Q , and
(8.46)
The test procedures are summarized in Table 8.7.
Again we shall do no more than remind the reader of the possibility
of one-sided test procedures. By this time the method of procedure in
where
p =
is defined by Equation (8.30) and evaluated at X =
- pfo* + 2nX(b - /
176 CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.7 Summary of Test Procedures in Simple Linear Regression
Hypothesis
Statistic
Equation
Rejection Region
00 = 00
t
8.44
/, -> f,f\ Q. /*>\ f n ft\
or
1 ^ "" ^(L^/2)Cn- 2 )
01 = 0{
t
8.40
t > t i
or
^ 5= ^U a/2)(n 2)
= u
t
8.45
t > *
J | X"" < X'o
k L "/ */ ^T* */
or
^ 5: ^(1 /2) (n 2)
0o = 0o and 0! = 0(
F
8.46
jC ^" * (lu . ft) (^ 71 *)
such cases should be obvious once the two-tailed tests have been
specified,
8.14 INVERSE PREDICTION IN SIMPLE LINEAR REGRES
SION
The equation f^ = &o+&i-XT may sometimes be used to estimate the
unknown value of X associated with an. observed Y value. For example,
suppose that in addition to the data of Table 8,3, we have a Schopper-
Riegler reading of K = 60 but the hours of beating, X, are unknown.
How shall this unknown value be estimated? The procedure is as fol
lows. Compute
= (Fo - 60) /6 1
(8.47)
where F is the observed value of Y for which we desire to estimate the
associated X value. A lOOy per cent confidence interval for the true
but unknown X value is defined by
L]
U)
where
and
+
_
. - T). +
"n?lY
^V n )
(8 . 48)
(8.49)
(8.50)
_> (8.51)
If, as in frequently tho case, one has several (say m) values of Y
8.15 THE ABBREVIATED DOOLITTLE METHOD 177
associated with the unknown X, Equations (8.47) and (8.48) are modi
fied to read
X = (To - 5o)/6i (8.52)
and
U) D D 'V \ nrn
where
{8 . M)
(8.54)
(- 2)4+ ZICFo,- F )
(*',) 2 = - - , (8.55)
n +- m 3
t = ^(H-T)/2](n+7n-3), (8.56)
and B and D are the same as before. Since, in practice, m is usually
quite small relative to n, the computational labor may be reduced
materially by using s% rather than (s^) 2 . This leads, of course, to an
approximate solution but one which is sufficiently accurate for most
situations.
Example 8.4
Consider the problem posed at the beginning of this section. Using
Equation (8.47) and the results of Section 8.9, we obtain j = 7.494,
Using Equations (8.48) through (8.51), a 95 per cent confidence interval
estimate is determined to be (5.85, 9.15).
8.15 THE ABBREVIATED DOOLITTLE METHOD
Before proceeding to the consideration of regression problems of
greater complexity than simple linear regression, let us digress long
enough to study the mechanics of a method for solving a set of simul
taneous linear equations whose coefficients form a symmetric matrix.
This digression will be well worth while for several reasons, namely :
(1) In most regression problems, the postulated mathematical
model is linear in the unknown coefficients.
(2) The method is well suited to programming for high-speed
computers as well as being useful when only desk calculators
are available.
(3) It incorporates self-checking features which permit verifica
tion of the accuracy of the arithmetic calculations at each
stage.
Several methods of solving sets of simultaneous linear equations (or
178
CHAPTER 8, REGRESSION ANALYSIS
of inverting matrices) appear in the literature. Each of these is a vari
ation of a basic procedure, the variations being introduced to accom
plish a particular aim of the person proposing the special technique.
In this book, only one method will be discussed. It is known as the
Abbreviated Doolittle Method.
To illustrate the Abbreviated Doolittle Method, let us consider the
problem of estimating
77 = 0oX Q + faXt + fcXz + /3*Xz + /?4^4 (8.57)
where XQ is a dummy variable which always takes the value 1 (i.e,,
JXTo^l). The method of least squares would lead to
F = 6
and the coefficients
equations
(Z A"o-Y )fto + (Z
(8.58)
= 0, 1, 2, 3, 4) would be found by solving the
(23 -^^0)60 + (23
(Z -X-aA' )6o + C23
(23 -Y 8 AT )6o + (23
(23 A r 4 Jr )6o + (23
i + (23
i + (23
0^4) *4 == 23 ^
+ (23 ^i.v 8 )&n
1-^064 = 23 -v
+ (23 ^aA^)J 3
r^064 =- 23
+ (23 ^a-Yg)^
varoj4 = 23 A r
(8,59)
t 4- (Z -^4X2)62 + (Z Xt
= Z
If the data, consisting of n observations on each of the variables, are
written in matrix form, namely,
Y =
LFJ
and X
-Y
l2
then Equations (8,59) may be written as
= X'Y
(8.60)
where S' = [6 , &i ? 62, b%, b*]. To simplify the writing, we shall denote
X* X by A and X f Y by G. In this notation Equation (8.60) appears as
AB =:
(8.61)
8.15 THE ABBREVIATED DOOLITTLE METHOD 179
If we denote A~~ l by C, then
C = A~i = (X'X)-^ (8.62)
and
B = A- 1 *? - (JTJr)- 1 *? = CG. (8.63)
The Abbreviated Doolittle Method will now be used to obtain:
(1) the b's,
(2) the sums of squares associated with the sequential fitting of
the & ? s,
(3) the estimated variances of the &'s,
(4) the estimated co variances between pairs of &'s, and
(5) the elements of the inverse of the X' X matrix, that is, the
elements of C.
The mechanics of the forward solution using the Abbreviated Doo
little Method are summarized in Table 8.8. A discussion of the steps
involved is best given in two parts, one associated with the first section
of the table and one associated with all the succeeding sections:
First Section [Rows (0) Through (4)1
(1) In the front half of the table are entered the elements of the
matrix of coefficients defined by X r X } omitting those obvious
from symmetry. That is, we have entered a# ^JEVXTy recog
nizing that an = a,y .
(2) In the column headed "constant terms" are entered the ele
ments of the vector X f Y. That is, we have entered gt=^L,XiY.
(3) In the back half of the table are entered the elements of
the identity matrix, again omitting those obvious from
symmetry.
(4) In the check column are entered the sums of all entries in the
corresponding rows, including those elements omitted because
of symmetry.
Succeeding Sections [Rows (5) Through (14)]
(1) Each entry in a given row is generated according to the in
struction specified for that row. (See the first column of the
table.) This applies to the front half, the constant terms, the
back half, and the check column,
(2) The sum of all the entries in a row (with the exception of the
entry in the check column) should equal (within rounding
error) the entry in the check column. The advantage of the
checking procedure should be obvious: If an arithmetic error
has been made, it will be corrected before calculations are
started on the next row.
(3) Steps such as those described are continued until a row is
reached in which only a single B pa appears. With the calcula-
^s
,jcj
4>
OJ
CJ
^-^-
**
t CO
tf.5
^=
CO -*t
/latrix Associated y
M-H
w
PQ
CD CD CD CD --H
CD CD CD '^H
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to
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~d
CD -r-^
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^ pq 4
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4.
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C
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1 i
n it
*****
4 s
^^
^=j
^p<f
^ pq"
ittie Method Illustrated on tl
tr Equations Specified by Eqi
^
o
o
S S3 S3 S3
S 53 8
HI g
3 S
S 3
"nj< pY"\
53 9
^< -^i
^ pq*
S5 SS
^ pq
^5 ?5
"^ PQ
5
*o S
o
Q* ^
t-3
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l
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rxj
> <^
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2
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CD
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XT'
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oc?
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*T*
^S**
*>*
O
.
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r5
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1
I
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**""*"*
X *s
^""N
ot,
^
!*
^
a
O
s 31
CO
^3 ^
"^ J
"X X
x^ ^q
S
^^
1 ^
t ^--v
t ^s
1 ^
HH
c"Y"t
CD H-O
* 4 t*
CS CT^
f*O v*
-^ T t
^
, . . . .
tl li
11 It
11 11
11 11
11 11
H
&^'^iz?<3r
SS^
di-
8.15 THE ABBREVIATED DOOL1TTLE METHOD 181
tion of all entries in this row and the satisfaction of the
"check," the forward solution is complete.
The next step in the analysis is the completion of the backward
soUition. This will be performed in two parts, one to determine the b's
and the other to determine the c^ values.
Determination of the b's
The forward solution of the Abbreviated Doolittle Method has pro
vided us with the following set of equations :
(1) 6 + (-BoOftx + 0802)62 + (03)63 + (04)64 = B Qy
(1) &! + ( 12 )6 2 + (^13)63 + (14)64 = Siy
(1) 6 2 + (23)63 + (24)64 = 2,, (8.64)
(1) 6 3 + (34)64 = 3*,
(1) 6 4 =
Solving these in reverse order (hence the name "backward solution"),
we obtain:
64 ==
63 =
6 2 = Biy 6 4 2 4 6 3 2 3 (8.65)
b = B l J 4 J5i 4 6 3 i3 6 2 is
Determination of the c,/
(1) Since C= A^ = (X f X)~~ l is the inverse of a symmetric matrix,
it will be symmetric. This reduces the number of calculations
to be performed since Cji = c i:f .
(2) All Cij values may be calculated using the equation
4L-L (8.66)
Ar
in which some of the A ' values may be or 1 and some of the
B f values may be 0. It should be noted that some of the c^
values may be read directly from the forward solution, namely,
40 ^ 40
41
41 =
C42 ^ 42
43 = 43
44 = 44
182
CHAPTER 8, REGRESSION ANALYSIS
(3) If we choose to ignore the symmetry mentioned in (1) and
calculate all the c^ independently, a check can be made on the
arithmetic by comparing c^ and c/-.
(4) A final check could be made by seeing if CA equals /. It
should. However, rounding errors may cause minor discrepan
cies.
Having completed both the forward and backward solutions using
the Abbreviated Doolittle Method, the next step is to indicate how to
obtain the analysis of variance table and the standard errors associ
ated with the various statistics. Using the formula
&!,---, &1-1 = AiyBiy, (8.67)
S.S. due to bi\
the various sums of squares may be evaluated easily once the forward
solution of the Abbreviated Doolittle Method has been completed.
The analysis of variance may then be recorded as in Table 8.9. If we
do not wish to record the reduction in the residual sum of squares as
sociated with the sequential fitting of each additional b, it is proper to
note that the
S.S, due to regression =
(8.68)
il
and this pooled sum of squares possesses 4 degrees of freedom, (NOTE:
The sum of squares due to 60 is still recorded separately since it
is actually CEltY^^/n, that is, it is the correction for the moan.) The
estimated variances and covariances associated with the regression
TABLE 8.9-Analysis of Variance Associated With the Multiple Regression
Problem Discussed in Table 8.8
Source
of Variation
Degrees of
Freedom
Sum of
Squares
Mean Square
Due to &o
t
./I {>/ lJ (Vy
A. Qi/^Ot/
Due to hi
6
1
,/l \y t^ly
A ly^ii/
Due to b%
60, b\ ...,,..
1
"** SV"^2l,/
A ty&ty
Due to b%
1 &0> &ly 62
1
*** 3i/jCf 3^
A^ 3t/*^3j/
Due to 6 4
Resklual .
&0, &1, 6'2, &3- - -
1
w 5
AtyBtu
Y\ (F V
A 4j/^4|/
^ ^ y- (K l^Wte 5)
Total
n
V F 2
8.15 THE ABBREVIATED DOOLITTLE METHOD 183
coefficients are given by
s l* = c ^s (8.69)
and
From these, we may obtain
4 = (JTCJr)4 (8.71)
which must be evaluated at the particular set of .XT- values for which an
estimate is desired,
Example 8.5
Consider the data given in Table 8.10. The Abbreviated Doolittle
Method is applied to these data in Table 8.11, where the X's and Y
have been coded so that the elements of X f X and X'Y are approxi
mately of the same order of magnitude. (This is done to facilitate the
computations.) In this case, the coding is as follows: divide Xo by 10,
X^ by 100, X 2 by 10, X z by 1000, X 4 by 1000, and Y by 100. Then
Equation (8.65) is used to obtain JVlOO = 0.681468(X /10)
+ 0.227227(Jf i/100) + 0.055349(X 2 /10) 1 ,495563(X 3 /1000) + 1 .546520
(AV1000) and Equation (8.66) yields
2052.64 135.410 53.6729 538.760 2.25605 ~
135.410 20.0032 0.273033 22.3301 0.549940
53.6729 0.273033 2.73843 18.3161 -0.928040
538.760 22.3301 18.3161 171,128 -11.6657
2.25605 0.549940 -0.928040 -11.6657 8.32196 .
where the elements of C reflect the coding explained above. The analysis
of variance of the coded data is presented in Table 8.12.
Example 8.6
The example of Section 8.9 is reworked in Table 8.13 using the Ab
breviated Doolittle Method. Again we get !F = 3.962 + 7.47S-XV Also,
coo ^ 0.3460, coi = 0.0384, and c n = 0.0055. It can be verified that the
sums of squares agree (within rounding error) with the values reported
in Table 8.4.
Although the Abbreviated Doolittle Method was explained with
reference to Equation (8.57), it should be noted that it applies equally
well to any situation where the equation is linear in the unknown coeffi
cients. For example, the following are typical of cases frequently en
countered for which the technique will prove useful :
(1) n - /So + PiXi + + faX k ,
(2) 77 = /3 + /3i^i + PnXl + /3mXr, and
(3) 77 = /?o + /SxXi + faX 2 + faiXl + (3^x1 + ffuXiX*.
Some of these will be considered in the sections and chapters to follow.
TABLE 8.10~Crude Oil Properties and Actual Gasoline Yields
Crude Oil
Gravity,
API
Crude Oil
Vapor
Pressure,
PSIA
Crude Oil
ASTM
10% Point,
F.
Gasoline
End
Point,
F.
Gasoline
Yield
Per cent of
Crude Oil
Xi
X*
X 3
A"*
Y
38.4
6.1
220
235
6.9
40,3
4.8
231
307
14.4
40.0
6.1
217
212
7.4
31,8
0.2
316
365
8.5
40.8
3.5
210
218
8.0
41.3
1.8
267
235
2.8
38.1
1.2
274
285
5.0
50.8
8.6
190
205
12.2
32.2
5.2
236
267
10.0
38.4
6.1
220
300
15.2
40.3
4.8
231
367
26.8
32.2
2,4
284
351
14.0
31.8
0.2
316
379
14.7
41.3
1.8
267
275
6.4
38.1
1.2
274
365
17.6
50.8
8.6
190
275
22.3
32.2
5,2
236
360
24.8
38.4
6.1
220
365
26.0
40.3
4,8
231
395
34 . 9
40.0
6.1
217
272
18.2
32.2
2,4
284
424
23.2
31.8
0.2
316
428
18,0
40.8
3.5
210
273
13.1
41.3
1.8
267
358
16.1
38.1
1.2
274
444
32.1
50.8
8.6
190
345
34.7
32.2
5.2
236
402
31.7
38.4
6.1
220
410
33 . 6
40.0
6.1
217
340
30.4
40.8
3.5
210
347
26.6
41.3
1,8
267
416
27. B
50.8
8.6
190
407
45.7
Source: Nilon H. Prater, "Estimate Gasoline Yields from Crudes/' Petroleum Refiner
(now Hydrocarbon Processing and Petroleum Refiner) , Vol. 35, No. 5, pp. 23638, May,
1956. Copyright 1956, Gulf Publishing Co., Houston, Texas. By permission of the author
and publishers.
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1 86
CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.12~Analysis of Variance Associated With
the Regression Analysis of the Data in Table 8.10
Source
of Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Due to 60
1
1.236772
1.236772
Due to 61
6
1
0.021625
0.021625
Due to Z>u
60, &i
1
0.030985
0.030985
Due to 63
&GJ b\j bz
1
0.002921
0.002921
Due to 64
&o, &i 7 &>, 63. . .
1
0.287399
0.287399
Residual .
27
0.013477
. 000499
Total
32
1 .593179
TABLE 8.13-Solution of the Example of Section 8,9
by the Abbreviated Doolittle Method
Front
Half
Row
60
61
Terms
Back
Half
Check
(0)
13
91
732
1
837
(1)
819
6485
1
7396
(2)
13
91
732
1
837
(3)
1
7
56.3077
0.0769
64.3846
(4)
182
1360.9993
6.99790
1
1537,0014
(5)
1
7.4780
0.03845
. 0054945
8.44506
8.16 SOME ADDITIONAL REMARKS WITH REGARD TO
GENERALIZED REGRESSION ANALYSES
There are a few additional items which merit discxission at this
time, for they have a great deal to do with the analysis of any sot of
data when the regression technique is employed.
The first item to be discussed is notation. By now the reader may
have been woiidering what the significance is of such symbolism as
61(60 ttnd 62(60, 61. This notation is closely allied with the "conditional"
concept in probability. In fact, the notation is \ised in exactly the
same manner, that is, to indicate a condition or restriction. In the
present context, the notation calls attention to the fact that sxims of
squares associated with various coefficients are obtained in a definite
(seqxieutial) order. Jn particular, the sum of squares due to 6 is found
first and is the same as finding the correction for the mean. (NOTE: 60
itself is not equal to the mean; it also depends on the nature of the
mathematical model used to represent the data.) After finding the sum
8.16 GENERALIZED REGRESSION ANALYSES 187
of squares due to & , we find the sum of squares due to 61 ; hence the
symbolism &i| & which is read, "&i given that 6 has been determined. "
Referring to Table 8.9, the next sum of squares recorded was that
"due to 6 2 1 &o, &i" which implies that 62 was found after & and 61 had
been determined. The remaining symbols in Table 8.9 may be explained
in a similar manner.
The next item to be discussed is that of testing various hypotheses
associated with the regression function. Each hi may be used to test
the hypothesis H:(3i = Q. This is accomplished by computing
t = bi/s s Vc^ (8.72)
with v = n q degrees of freedom. In Table 8.9, g = 5, and thus
n q = n 5. An equivalent test procedure is provided by
2
F (&s/^-)/( res idual mean square). (8.73)
It should be noted that the various J^-tests defined by Equation (8.73)
are not all independent since the X variables are correlated. However,
the tests provide useful information if interpreted wisely. Each of the
mean squares reported in Table 8.9 may also be used to form J^-ratios
which provide additional important information. These /^-ratios,
defined by
mean square due to b* \ bo, &i, - - - , bi-i
p , ^g m 74)
residual mean square
will assess the significance of the additional reductions in the residual
sum of squares achieved by fitting the b's in the particular order adopted
by the analyst. The italicized words emphasize an important point : The
order of fitting the coefficients has a decided effect on the analysis. As
we shall see later, if the variables -X"i, X 2 , etc., represent successive
powers of a single variable X (i.e., the mathematical model is a poly
nomial in -XT), then a natural order is available. In other cases, the
order is a result of a decision on the part of the analyst to write the
terms of the model in a specific order when setting up the Doolittle
solution. One may, of course, make use of
S.S. due to regression = s, A iv B^ (8.75)
and calculate
mean square due to regression
p , . ~
residual mean square
^ , _ , ( 8 - 76 )
188 CHAPTER 8, REGRESSION ANALYSIS
in order to assess the over-all significance of fitting the regression
equation. Other tests which aid in determining the order of fitting and
the choice of variables to be retained in the regression equation are
available. However, discussion of these is not warranted in this book.
Instead, the reader is referred to other sources, such as Hader and
Grandage (10), for the pertinent details.
While not discussed at this time, it should be clear to the reader that
confidence interval estimates may be obtained through xise of tech
niques discussed in Chapter 6, The appropriate standard errors are
defined in Section 8.15. For further details the reader is again referred
to Hader and Grandage (10).
8-17 TESTS FOR LACK OF FIT
In Section 8.11 the assumption was made that the failure of the
model to fit the observations exactly was solely a function of the
errors. This assumption is seldom true, although it may be nearly so
in many cases. To check on its validity, one must have some measure
of error other than that provided by the residual sum of squares* The
only way to obtain such a measure is to insist that the experiment be
repeated some number of times at at least one value of the independent
(or controlled) variable. In addition, it is also wise to insist on running
the experiment at as many different values of the controlled variable
as is feasible. In the example considered in Section 8.9, the latter
recommendation was followed but no repetition of the experiment at
any value of the controlled variable was undertaken. This enabled us
to make a visual judgment about lack of fit but no statistical analysis
was possible.
To examine the statistical test for lack of fit, consider the data
reported by Hunter (11). These data are reproduced in Table 8.14,
Introducing the dummy variable, -ST , which is identically equal to 1,
and using the Abbreviated Doolittle Method, the simple linear re
gression equation is determined to be ^=1.76+2.86^3. The associ
ated analysis of variance is given in Table 8.15.
In this example, however, it is possible to subdivide the residual
sum of squares into two parts: one part being an estimate of experi
mental error and the other a measure of the lack of fit of the linear
model. The reason such a subdivision is possible should be clear: The
researcher was careful to provide some replication in the performance
of the experiment. To actually perform this subdivision of the residual
sum of squares, it is easier to calculate the experimental error sum of
squares and then obtain the lack of fit sum of squares by s\ibtraction.
The experimental error sum of squares is foxand by pooling the sums of
squares of deviations about the mean for each value of the independent
variable, that is,
Z (Z Y*~ (Z YY/n] (8.77)
all X
8.17 TESTS FOR LACK OF FIT 189
TABLE 8. 14-Percentage of Impurities at Different Temperatures
Coded Temperature
Per Cent of Impurities
Temperature (C.)
-STi
Y
200
1
6.4
200
1
5.6
200
1
6.0
210
2
7.5
210
2
6.5
220
3
8,3
220
3
7.7
230
4
11.7
230
4
10.3
240
5
17.6
240
5
18.0
240
5
18.4
Reprinted from: J. S. Hunter, "Determination of Optimum Operating Conditions by
Experimental Methods, Part II- 1, Models and Methods/' Industrial Quality Control, Vol.
15, No. 6, pp. 16-24, Dec., 1958. By permission of the author and editor.
TABLE 8.15-First Analysis of Variance for the Data of Table 8.14
Source of Variation
Degrees
of Freedom
Sum of Squares
Mean Square
Due to 60
1
1281.3333
1281.3333
Due to 61 1 60
1
228.5715
228.5715
Residual
10
40 . 3952
4.0395
Total
12
1550.3000
where the expression within the braces is calculated separately for each
value of -X". In the example,
Experimental Error Sum of Squares =
{(6.4) 2 + (5.6) 2 + (6.0)* - (6.4 + 5.6 + 6.0) 2 /3}
+ |(7,5) 2 + (6.5) 2 ~ (7.5 + 6.5) 2 /2}
+ { (8.3)* + (7.7)2 _ (8 .3 + 7.7)2/2}
+ { (11. 7) 2 + (10.3) 2 (11.7 + 10.3) V2}
H- { (17.6) 2 H- (18.0) 2 + (18.4) 2 - (17.6 + 18.0 + 18.4) 2 /3} = 2.3000.
Thus, it is now possible to record the results as in Table 8.16. In this
table, the experimental error mean square is a pooled estimate of o* B
that is uncontaminated by any inadequacy of the assumed linear
190 CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.16-Second Analysis of Variance for the Data of Table 8.14
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
Due to &o
1
1281.3333
1281.3333
Due to 61 1 &o
1
228.5715
228.5715
Lack of fit .
3
38.0952
12.6984
38.64**
Experimental error. . . .
7
2.3000
0.3286
Total
12
1550.3000
** Significant at the 1 per cent level.
model. To test the hypothesis that the linear model is appropriate (i.e.,
no real lack of fit exists), it is legitimate to obtain the ratio F=* 12.6984
/0.3286 = 38.64 with ^i==3 and *> 2 = 7 degrees of freedom. Since this
exceeds ^.99(3,7) =8.45, the decision is reached that a serious lack of fit
exists. That is, the hypothesis of no lack of fit is rejected. Another way
of stating this conclusion is as follows: The assumed linear model in
adequately describes the data.
8.18 NONLINEAR MODELS
If the /''-ratio in Table 8.16 had turned out to be nonsignificant, it
would have been concluded that the linear fit was probably adequate.
However, since the linear model was judged to be inadequate, the re
searcher is obliged to consider fitting some nonlinear model. That is, he
should attempt to discover a different mathematical model which bet
ter describes (or represents) the observations.
There are many alternatives to be considered at this stage. For
example, should a higher degree polynomial be investigated or should
some other functional form be considered? Pox-haps some exponential
function might be the appropriate model for the problem under investi
gation. A few mathematical models, other than polynomials, which are
frequently encountered in applied work are
77 = o>{3 ;
In 97 In-y + (lnc*)/3- T ;
1/77 - T + <*/3 x ;
77 -
oj > 0, ft > (8,78)
a > 0, ft > 0, y > (8.79)
a > 0, > 0, r > (8 . 80)
> 0, y > 0. (8.81)
These are xisually referred to as: (1) the simple exponential or com
pound interest function, (2) the Gompertz function, (3) the logistic
function, and (4) the Mitscherlich function, respectively.
The selection of an alternative to the linear model, i.e. to the first
degree polynomial, is not 'easy. The choice should be made only after
careful consideration of the basic mechanism of the system. If this is
not feasible, scatter plots should be examined. When it is evident that
8.19 SECOND ORDER MODELS
191
some degree of curvature is present in the data but no clear-cut choice
of mathematical model is possible, a reasonable approach is to syste
matically examine polynomials of increasing order (i.e., of higher
degree) .
8.19 SECOND ORDER MODELS
Since the first order model (a straight line) was an inadequate repre
sentation of the data in Table 8.14, it is now proposed that a second
order (quadratic) model be investigated. That is, the model
f) /^ ~y~ \ /? ~y" i- /? Y~ f Q fto^
in which X is identically 1, will be considered. Writing the data in
matrix form, namely,
F =
6.4'
5.6
6.0
7.5
6.5
8.3
7.7
11.7
10.3
17.6
18.0
L18.4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
4
2
4
3
9
3
9
4
16
4
16
5
25
5
25
5
25 _
and using the Abbreviated Doolittle Method (see Table 8.17), the
regression equation is found to be
1t == 8.43 3.14-XTi + l.OOXi. (8.83)
The accompanying analysis of variance is given in Table 8.18.
A comparison of Tables 8.16 and 8.18 indicates: (1) Fitting a quad
ratic term led to a significant reduction in the lack of fit sum of squares,
and (2) there is still a significant lack of fit. In other words, although
the quadratic equation was a better fit than the linear equation, the
second degree polynomial does not adequately describe the data.
What, then, should be the next step? Should higher degree polynomials
be investigated or should attention be directed toward some other
functional form? As indicated in Section 8.18, the answer to such a
question is not easily obtained. In fact, a "best' 7 answer may not exist.
192
CHAPTER 8, REGRESSION ANALYSIS
TABLE 8.17-The Fitting of F = 6 +&i^i+Z>u^i to the Data of Table 8.14
by the Abbreviated Doolittle Method
Front
Half
Row
bo bi
611
Terms
Back Half
Check
(0)
12 36
136
124
1
309
(1) . . .
136
576
452
1
o
1201
(2> : . : : :
2584
1920
1
5217
(3)
12 36
136
124
1
309
(4)
1 3
11 333333
10.333333
0.083333
o
o
25.75
(5)
28
168 000012
80.000012
2.999988
1
o
274
(6) .
1
6
2 857143
0. 107142
0.035714
o
9.785714
(7) ....
34 666640
34 666654
6,666569
5.999952
1
70.999931
(8)
1
1
192305
173076
028846
2.048077
TABLE 8.18-Third Analysis of Variance for the Data of Table 8.14
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
jF-Ratio
Due to &o
1
1281 3333
1281 3333
Due to bi 5o
1
228.5715
228.5715
695.59**
Due to bn 60, b\
1
34.6667
34.6667
105.50**
Lack of fit
2
3.4285
1.7142
5.22*
Experimental error ....
7
2 . 3000
0.3286
Total
12
1550.3000
Significant at the 5 per cent level.
'* Significant at the 1 per cent level.
Thus, rather than say what should be done, it seems politic to suggest
a procedure which can be modified at the discretion of the analyst. The
suggested procedure is: Rather than seek a better fit in terms of a
higher degree polynomial (i.e., a degree greater than 2), it is probably
wiser to cast about for some other functional form to represent the
data. In the case under consideration (i.e., the data of Table 8.14), an
examination of a scattergram would suggest that an exponential func
tion be given serious consideration. The suggested procedure does not,
of course, preclude the fitting of higher degree polynomials if that ap
pears to be the proper approach. For example, if we consider the data
analysed in Section 8.9, a third-degree polynomial will give an excel
lent fit* Of course, some form of exponential, perhaps of the logistic
variety, might also be appropriate.
8-20 ORTHOGONAL POLYNOMIALS
When the values of X (the independent variable) are equally spaced,
there is another method of fitting polynomial regression equations
which has much to recommend it. This is the method of orthogonal
polynomials. You will have noticed, when fitting polynomials by the
8.20 ORTHOGONAL POLYNOMIALS 193
method described earlier, that each time it was necessary to start the
solution from the beginning and solve a new set of normal equations.
For example, when fitting a second order model we were unable to use
the results of fitting the first order model. However, when the data
permit the use of orthogonal polynomial techniques, we can salvage
the previous results and simply perform the calculations required to
add a new term to the polynomial (of one less degree) determined at the
preceding stage.
The method of orthogonal polynomials will be illustrated only for
the case where the values of X are equally spaced at unit intervals and
where each X has but one Y value associated with it. If the X values
are equally spaced at intervals not equal to unity, we may code the X
values by dividing through by the length of the common interval and
then proceed in the manner to be developed below. If there is more than
one Y value associated with each X, the method is not applicable un
less we have an equal number of Y values associated with each X value*
In the latter case, the complete solution may be obtained by introduc
ing the proper divisor into the calculations. If the X values are un
equally spaced, a solution may be otained [see Kendall (13)], but the
operation is so cumbersome that it will not be presented in this text.
Thus, in all but the simpler cases, it is usually better for the research
worker to use the method described earlier in this chapter. However, if
the experimental data are amenable to simple treatment by the method
of orthogonal polynomials, the research worker is advised to use that
method, for it saves time and also allows him to calculate and evaluate
readily, step by step, the contribution made by fitting each additional
term in the regression function.
What, then, is the ^method of orthogonal polynomials? It may be
shown that any polynomial, for example,
Y = 6 + biX + - - - + b k X k (8.84)
may be rewritten as
? - A, + AI&+ - + A*& (8.85)
in which the f (z = l, --,&) are orthogonal polynomials 5 and the
A.i (i = 0, - , k) are constants defined by
Ao = $2 Y/n - Y (8.86)
and
'
t'\* ' * 1, -,* (8-87)
siv
For the case we are considering, that is, where X takes on the values
* Two polynomials are said to be orthogonal if, when X takes on a specified set
of values, S&-jbaO for i^k, where the summation means that we first compute
the product k'ikL f r each value of X and then obtain the sum of these products.
194
1,2,
CHAPTER 8, REGRESSION ANALYSIS
, n, the first three orthogonal polynomials may be expressed as
~ Z), (8.88)
- (x -
7n
J,
(8.89)
(8.90)
where the X* are constants (depending on ri) chosen so that the values
of the "s are integers reduced to their lowest terms. An abbreviated
table of values is given in Table 8.19; a more complete table may be
found in Anderson and Houseman (1),
TABLE 8.19-Partial Table of % Values
Degree of Polynomial
A-l
& =5 2
= 3
&4
n
! &
3 fc'
tf
2 la
i
_!
i +1
3 +1 1
2
+ 2 1
+ 1
2
+ 1
2
1 1 +3
1 +2
4
3
-fl +1
2
4 . .
+3 +1 +1
1 L _ Q
"^^ JL ~ ^
^
5
+ 2
+ 2 4-1
+ 1
Before considering a numerical example, the equations necessary for
calculation of the various sums of squares will be given. They are as
follows :
(8.91)
r~
(8.92)
S.S, due to b Q
S.S, due to fitting the ith degree term ===== .4t-(5D
Example 8.7
Consider the data of Table 8.3 and rewrite the. values in the form
shown in Table 8.20. Equations (8. 80) and (8.87) then yield f^ 56.3
+7.478i .0365& .6801&. If Equations (8.88) through (8.90) are
evaluated as far as possible by using the known values of n, 3Tand the X's,
and then substituted in the regression equation just found, we obtain
f = 21.7277 5.8458^+2.3449X^ 0. 1134AX The reduction in sum of
squares due to fitting the various terms in the regression function may,
of course, be calculated using Equations (8.91) and (8.92).
8.21 SIMPLE EXPONENTIAL REGRESSION
The regression function specified by Equation (8.78) is frequently
encountered in experimental work* Thus, it is appropriate that some
8.21 SIMPLE EXPONENTIAL REGRESSION
Table 8.20-Table for Calculating the A's and Corresponding
Suras of Squares
195
F
5i
8
ni
F
F
17
6
22
11
102
374
187
21
5
11
o
105
231
o
22
4
2
6
88
44
132
27
3
8
81
135
216
36. ..
2
10
7
72
360
252
49 .
H
13
4
49
637
196
56
o
14
o
784
64
1
13
4
64
832
256
80. ..
2
10
7
160
800
560
86
3
5
8
258
430
688
88
4
2
6
352
176
528
92
5
11
460
1012
94
6
22
11
564
2068
1034
732. . .
o
1361
73
389
X
1
1
1/6
S (0 2 --
182
2002
572
discussion of the associated methods of analysis be given. If the method
of least squares is used, the resulting normal equations are not amen
able to easy solution. Consequently, some other (approximate) ap
proach is necessary. The usual approximate procedure adopted is to
take logarithms (logarithms to the base 10 are most convenient) which
results in
log 17 = log a + (log ff)X.
Redesignating the quantities as f olio ws : Z = log
and W X, Equation (8.93) appears as
Z = A + BW
j, A ==log <x,
(8.93)
' = log/?,
(8.94)
and we immediately recognize this as being of the sameJForm as Equa
tion (8.12). Estimates of A and B, denoted by A and B, may then be
found following the methods described earlier. This solution, which is
equivalent to fitting a straight line by least squares to the data when
plotted on semi-logarithmic paper, is not identical with a least squares
solution of the original problem using Equation (8.78) and ordinary
graph paper. However, the approximation is sufficiently accurate for
most problems.
Example 8.8
Consider the data of Table 8.21. Using either the method of Section
196 CHAPTER 8, REGRESSION ANALYSIS
8.7 or the Abbreviated Doolittle Method, it is determined that
A =0.9469 and J3 = 0.002576.
TABLE 8.21-Protein. Content and Proportion of Vitreous Kernels in
Samples of Wheat
Sample Number
Proportion of
Vitreous Kernels
X(-WO
Protein Content F
Z = log F
1
6
10.3
1.013
2
75
12.2
1,086
3
87
14.5
1.161
4
55
11.1
1.045
5 ...
34
10.9
1.037
6
98
18.1
1.258
7
91
14.0
1.146
8
45
10.8
1.033
9
51
11.4
1.057
10
17
11.0
1.041
11
36
10.2
1.009
12
97
17.0
1.230
13
74
13.8
1.140
14
24
10.1
1.004
15
85
14.4
1,158
16
96
15.8
1.199
17 ....
92
15.6
1 . 193
18
94
15,0
1.176
19
84
13.3
1.124
20
99
19.0
1.279
Reproduced from M. TCzektel, Methods of Correlation Analysis (New York: John Wiley
and Sons, Inc., 1941), p. 82. By permission of the author and publishers.
8-22 THE SPECIAL CASE:77=/3X
In some instances, it is reasonable to assume that the true regression
line passes through the origin. That IB, if simple linear regression is be
ing considered, /5o in
& __L_ o v f8 OS"i
^ = PQ + plJ\ V.O.^.*V
is assumed to be and Equation (8.95) is rewritten as
(8.96)
where, of course, |S=i. It is clear that such an assumption, if justified,
will simplify the calculations! procedures. It can be verified that for
this special ease
j^^ga- ^ XY/ 23 X*- (8.97)
Please do not make the mistake of adopting this simpler form just
because it is easier to handle. Further, even if the assumption is justi-
8.23 WEIGHTED REGRESSIONS 197
fied (such as when X = height and 3^ weight of men), it may be better
to forego the simplifying assumption and consider *7 = /3o+/3i.-X" as being
more appropriate for the range of X values being studied in the experi
ment.
In summary, the mathematical model should only be chosen after
proper consideration has been given to all the factors involved.
8,23 WEIGHTED REGRESSIONS
Suppose the data to be considered are of such a nature that the as
sumption of homoscedasticity (i.e.,, homogeneous variances) is no longer
justified. That is, suppose we can not assume that o^-\^ is the same for
all X, but that we must assume
where the w^ are known constants. If -we restrict ourselves to the case
where 77 = /3v+-@iX, it may be shown that the resulting normal equa
tions are
/ k \ / k \ k n i
I T^ n*Wi J -f- ( 23 niW+Xi J &! = ]>3 23 Wi Y^
\ i=*\ / \ i^\ / ii J--.1
/ k X / k 2 \ k n-t
( 53 ntWiXt ) 6 + ( 23 n^WiXi ) 61 = 23 23 WiXiY+j
\ * i / \ i*=i / t=i j i
(8.99)
where
F<, = /3 + ^Si^i^ H- 6, v ; i 1, , fe (8,100)
j = 1, - - , f.
It is of particular interest to consider the case where erf is propor
tional to -XT;, that is, where we may write
v\ = <r 2 A0* = o- 2 -X"i, (8.101)
since this is a fairly common occurrence in certain areas of experimen
tation. Under these conditions, the normal equations reduce to
z^l
()o+(i: <**)&!- 2:2:
where
fl :===i ^ ^ fl<i.
198 CHAPTER 8, REGRESSION ANALYSIS
In this special case, a 2 is estimated by
- 2). (8. 103)
ti y-i
8.24 SAMPLING FROM A B1VAR1ATE NORMAL POPULA
TION
Let us consider, as far as practicable, the consequences of obtaining
a random sample of values of both X and Y from some bivariate popu
lation rather than first choosing values of X and then observing ran
dom F values associated with these chosen X values. What effect will
such a procedure have on our estimates? As we have stated the prob
lem, it is much too general to permit a satisfactory answer in this book.
However, if we make the assumption that our bivariate population is a
bivariate normal population, then we may examine the effect of obtain
ing random pairs of X and Y rather than choosing X values and then
observing the random values of Y associated with the selected values
of X.
In this case, two approaches are possible: (1) Obtain the best regres
sion equation for estimating a value of Y associated with a specified
value of -XT; (2) obtain the best regression equation for estimating a
value of X associated with a specified value of Y. That is, we can obtain
$ = 60 + &1-3T (8.104)
as in Section 8.7, or we can obtain
= co + CiF (8.105)
where
c*=*X tf,T (8.106)
and
ci - Z>3>/ 52 y*. (8.107)
It is to be noted that the above relations assume no "errors of meas
urement" in X and Y. If, however, our variables arc subject to errors
in measurement, so that we really observe Z X + * and W F+S,
where and 5 arc independently and normally distributed with moan
and variances <r and cr*> respectively, what estimation procedure may
we use? If, in the future, we measure % and wish to estimate Y, the
regression of W on % should be used ; if we measure W and wish to esti
mate .XT, the regression of Z on W should be calculated and used.
A reasonable question to ask at this point is: What effects do the
above-mentioned errors of measurement have on the accuracy and pre
cision of our estimates? Some answers arc:
(1) Tf the random, errors of measurement are associated only with
the dependent variable, and are not related to the true values,
8.25 ADJUSTED Y VALUES 199
they will not affect our estimate of the true slope but will
cause s% to overestimate <T E .
(2) If the random errors of measurement are associated only with
the independent variable, and are not related to the true
values, they not only cause S E to overestimate o- E but also
tend to produce underestimates of the true slope.
(3) If both variables are subject to error, the consequences are not
so easily determined, and much care should be taken when
making predictions based on such data.
Suppose, however, that we want to estimate the true relationship
between X and Y. To accomplish this we need further information
about v\ and cr 2 8 . Such information (i.e., estimates s and sf) can
sometimes be obtained by making duplicate measurements. However,
such a procedure is not always possible. When duplicate measure
ments are not available, other approaches must be explored. Many
scholars have considered the problems associated with regression anal
yses in which both variables are subject to error, and several solutions
have been proposed. However, because no general optimal solution has
yet been obtained and because the subject may rightly be considered
to be beyond the scope of this text, no attempt will be made to illus
trate any of the proposed methods of analysis.
8.25 ADJUSTED Y VALUES
Closely related to the reduction in sum of squares, mentioned sev
eral times in this chapter, is the technique of adjusting values of the
dependent variable to take account of differences among the associ
ated values of the independent variable. For example, if we are con
cerned with measurements on the gains in weights of certain animals, a
valid comparison among the gains does not seem possible unless we
adjust for such a value as the initial weight of the animals or the feed
consumed. That is, if one animal gains 60 pounds while consuming 300
pounds of feed, and another animal gains 40 pounds while consuming
200 pounds of feed, we do not feel justified in making a direct compari
son between 60 pounds and 40 pounds. We should first attempt to
make some adjustment, or correction, for the different amounts of feed
consumed. One way to make such an adjustment is through regression.
If, from the present or other data, we have an estimated regression
function Y = bo-{-biX, where F = gain in weight and JT = feed con
sumed, we can adjust the observed gains in weight to some common
value for feed consumed. The value of X most commonly selected is
the sample mean (X) but any value will do. The reason why the mean
is usually adopted as the point of comparison is that, in general, it is
near the center of the range of values of the independent variable.
What, then, is the procedure for determining adjusted Y values?
The formula defining adjusted Y values (adjusted to X, that is) is as
follows :
adj. Y = Y - bi(X - X) (8.108)
200
CHAPTER 8, REGRESSION ANALYSIS
and the nature of the adjustment is illustrated in Figure 8.8. Here only
three sample points have been plotted since these are sufficient to illus
trate the technique. It is seen that all the adjusted Y values (repre
sented by circles) appear on the line erected vertically at X because
we adjusted to X = "X. Note that it is possible to have adjusted Y\
> ad justed F 2 even when F x < F 2 . Thus, it is readily apparent that the
adjustment of a set of measurements based on a concomitant variable
may completely change the entire picture of an experiment. As a con
sequence, we might reach much different conclusions based on an anal
ysis of adjusted values than would have been reached if no account
were taken of the functional relation existing between the dependent
and independent variables.
X
FIG. 8.8 Illustration of adjusted Y values.
It should be evident that adjusted Y values may also be determined
when dealing with other than simple linear regression. For example, if a
multiple linear regression analysis has been performed and the regres
sion cqiiation determined to be
f> ~ fto + b,X l + . . . + b k x k , (8. 109)
then adjusted Y values are defined by
adj. F =* F - b l (X l - Z*) - b^X* - Z 2 ) -
- b*(X* - 3"*). (8.110)
Equation (8.110) would, of course, be evaluated using the appropriate
sample values (Y^ Xu, - - * , XM) and the calculated mean values.
Rather than dwell on the topic of adjxasted values at this time, we
shall defer further discussion until later in the book where a more effi
cient method of analysis, namely, covariance analysis, will be intro
duced.
8.26 THE PROBLEM OF SEVERAL SAMPLES OR GROUPS 2O1
8.26 THE PROBLEM OF SEVERAL SAMPLES OR GROUPS
In this section, a topic of considerable importance will be discussed.
It is : Given several samples or groups of observations, may all the data
be pooled into one large sample? This sort of problem has arisen earlier
in this book and it is not surprising that it also arises when dealing
with regression analyses.
Although the problem can arise regardless of the form of the regres
sion function, the discussion here will be limited to the case of simple
linear regression. If other functional forms are pertinent, a statistician
should be consulted.
When several sets of sample data are available, the question most
frequently asked is: Can one regression line be used for all the data?
More specialized questions are :
(1) Taking liberties with the system of notation adhered to up to
this point and letting 6 t - represent the estimate of fit, where 6*
is the estimated slope for the ith group and /? is the true slope
of the regression function in the population from which the
ith group is a sample, does /3i ==/3 2 = - - =/3 k ? In other words,
are all the sample slopes estimates of the same true slope?
(2) Assuming /3i = /32= Pk, would a regression fitted to the
group means be linear?
(3) Assuming /3i = /32= - - j8* and that the regression of the
group means is linear, is {3w = l3M, where &M is the true regres
sion coefficient for the means and j3w is the true pooled within
groups regression coefficient?
To mention one case where it is necessary to know the answers to the
questions stated above, we cite the technique known as analysis of co-
variance which we shall study in detail in a later chapter. This tech
nique has as one of its basic assumptions the requirement that the same
regression coefficient, /3, apply to all groups. Hence, the need for an
appropriate test is clear. Let us now outline the general procedure to be
followed. Suppose we have k groups and n* observations (on both X
and Y) in the zth group. We may present most of the necessary calcu
lations in Table 8.22, where
(8.111)
"* *< V J I / \ 7-.1
5*= S (Xf J?i)(F^-~- 7*)= y^jy^-Ft (8.112)
J-l y-1
2O2 CHAPTER 8, REGRESSION ANALYSIS
2
/ nt \
( 2 t F )
_ -. .
: (F<, - F,) 2 = Z) F* -- - > (8- 113)
i y-i
and
(& Tli
2:2:^
i*-l J=l
joi i^l ^=0=1 ~_^
> . Wi
t 1
f: (-YO- -
k
n . ^
z;^) 2:2: F
= 2:2: ^-F,- - ^ ^^i , (8 . 115)
/ k Hi
( Ti S 7 "
If we designate by >S 2 the sum of squares among the k group regres
sion coefficients, that is, if $2 is a sum of squares expressing the amount
of variation among 61, 62, ,&*, where bj is the regression coefficient
in the jfth group, it may be shown that
2
02 == ^w
A
(8.117)
y; J!i _ J
Similarly, we may designate the sum of squares of the deviations of the
F-means from the regression of y-means on X-means by AS f 3 , where this
is calculated as shown in Table 8.22. The square of the difference be
tween the regression coefficient computed from the "pooled within"
values (6^-) and the regression coefficient for the regression of the
means (6^) is given by (6^ &jir) 2 - If we multiply this by a suitable fac
tor, it becomes another estimate of cr^, assuming, of course, a constant
variance of Y for all X. This may be expressed as
(8.118)
CD
O
rt
o
CD
>-l
bJD
w
O
'5
(U
H
1
3
00
7
cr
CO
7 -w.l
1
-W3 \ w
> i S
Degrees of
Freedom
<N CS CS
1 1 1
s S
rH
^ |
I 1
-W3 W3 I
-W3
w
i 4
w
L ^ Pq
4
J
^^ >
*W3 i ^3
i
w
L L L
Cj 0? O
I ~h I
^0 ^O ^-O
ii
6- f
w
<3J 3
*W3 ^
II II
^
^ ^5
w
* <
-W3 p
ii ii
fe ^
pq pq^
EH
^ f
w
^ X ^
| |
^
Degrees of
Freedom
rH r-H -rH
L ' L
i_
-W3 I
tH
I
*wr
CL.
o
M
I
^ M - t *
giji
204 CHAPTER 3, REGRESSION ANALYSIS
and noting that in Table 8.22 we defined Si as the pooled sum of
squares of deviations from regression, we can show that
S T == .Si + S* + ^ 3 + 6V (8. 119)
Furthermore, the degrees of freedom associated with ST may be sub
divided in the following fashion :
" + (k - 1) + (k - 2) + 1, (8. 120)
JT) n . _ 2 = ( 32 < -
ml \ il
and these are associated with S, S%, /S 3 , and S 4 , respectively.
Now we are in a position to answer the questions posed at the begin
ning of this section. Let us consider these in order and indicate the
proper test procedures.
1. Can one regression line be used for all the observations?
(8.121)
/ '( -
2. To test 11:01 =/3&: This test and the succeeding ones are
usually cozisidered if F in Equation (8.121) turns out to be significant.
That is, we are curious as to the reason for the significance.
S*/(k 1)
3. To test whether regression of means is linear (assuming
S*/(k 2)
(8.123)
(Si
' \ * i
4. To test H\$W $M (assuming regression of means is linear and
It should bo clear that the order in which those tests arc performed is
very important since the assumptions necessary for the later tests are
tested as hypotheses in the earlier tests. Note also that if a sequence of
tests is applied, the critical level (true probability of Type I error) of
the sequence is not known though it is needed for proper interpretation
of the results.
8.27 SOME USES OF REGRESSION ANALYSIS
205
Example 8.9
Consider Table 8.23. To test H:/3i =/3 2 =/3 3 using a: = 0.01, we calculate
F== 150/15. 667 = 9. 5 with j>i=2 and 7^ = 300 degrees of freedom. Since
F 9.5 > 7^0.99(2,300), the stated hypothesis is rejected. The performance
of the remaining tests is left as an exercise for the reader.
TABLE 8.23-Hypothetical Data to Illustrate the Procedure
for Testing the Hypothesis
Group
Degrees
of Free
dom
Z* 2
!!C *y
Z^ 2
Z:y 2 -(I>}0 2 /:* 2
Degrees
of Free
dom
Mean
Square
A
101
101
101
400
200
400
800
600
600
4000
3000
2000
2400
1200
1100
100
100
100
B
C
Total
303
1000
2000
9000
4700
5000
300
302
15.667
Difference for testing Hi /3i=/3 2 =/33
300
2
150
Granting the assumption that two populations have a common vari
ance, the hypothesis HifBi pz may be tested against the alternative
by examining
*=(fti- 6aOA^^ (8.125)
(8.126)
where
(-ar -
y i
and
y i
n\
4
(8.127)
The value of i specified by Equation (8.125) is, of course, distributed as
"Student's" t with *> = ni+n2 4: degrees of freedom. Confidence limits
for 01* 02 may also be obtained by use of the foregoing equations. It is
to be noted that economies in calculation may be achieved by select
ing, whenever possible, the same -X" values for both samples.
8.27 SOME USES OF REGRESSION ANALYSIS
The uses to which regression techniques may be put are numerous. A
few of the more important are :
206
CHAPTER 8, REGRESSION ANALYSIS
(1) To reduce the length of a confidence interval when estimating
some population mean (or total) by considering the effect of
concomitant variables.
(2) To eliminate certain "environmental" effects from our esti
mates of treatment effects; that is, we may wish to examine
adjusted Y- values.
(3) To predict Y knowing values of X^ , X k (our auxiliary
variables) whether or not a causal relationship exists.
(4) To influence the outcome of the dependent variables assum
ing, of course, that we have a causal relationship.
There are many other uses for regression methods which might have
been listed. We have not attempted to exhaust the possibilities, nor
have we attempted to give our examples in any order of importance.
The relative importance of the different uses will vary depending on
the subject matter being discussed.
Problems
8.1 Derive the normal equations specified by Equation (8.14).
8.2
8.3
8.4
8.5
8.6
8.7
Given the following values, find:
>:* = 121 Z;,Y 20
Derive Equation (8.21).
Derive Equation (8.22) from Equation (8.25).
iX, (b) SB, (c) s bi .
- 82
n 10
Find the linear regression of F on X given the values;
X: 3 8 4 11 9
F: 5~"3~4"""l~2
Given that
n 277, 5* - 65, F - 72, ]>' 1600, y - 3600, Z^ - 2000,
compute: (a) SB, (b) s bii (c) sp for -Y==45.
Given the abbreviated analysis of variance shown below, perform the
following:
(a) Test /7:j5i0 using a = 0.01.
(6) Compute the standard error of estimate, $$.
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Due to regression
1
40
40
Deviations about regression
SO
200
4
Total
51
240
8.8 Given that
n - 38, 3? 5, "F 40,
answer the following:
(a) Determine ?*==(
- 100,
* 10,000,
- - 800
PROBLEMS 207
(6) Test # :/3i = using a = 0.05.
(c) Partition ^y 2 into two parts, one associated with the slope of the
linear regression and the other associated with the deviations
about regression.
(d) For the observation (X = S, F = 36), compute the adjusted
value of Y.
(e) Interpret both 61 and /3i.
8.9 Given that
n = 62, X = 10, Y = 20, >* = 40, ^y* -= 250, >;y = 80,
solve the following:
(a) Determine y = 60 + &i-X^.
(6) Compute a 99 per cent confidence interval for /3i. State all as
sumptions.
(c) Estimate the gain in information from using X as a statistical
control (the regression of Y on -XT) in estimating the population
mean of Y. (NOTE: Information is here used as a synonym for
"the reciprocal of the variance/ 7 )
8.10 Given that 61 = 0.2 grams of gain per gram of feed eaten, find the net
difference between the gains of two rats where one animal consumed
200 grams of feed and gained 60 grams while the other animal con
sumed 300 grams and gained 90 grams.
8.11 The data in the table given below represent the heights (Jf) and the
weights (F) of several men. We selected the heights in advance and
then observed the weights of a random group of men having the
selected heights.
X
60 in. HOlbs.
61 135
60 120
61 126
62 140
60 130
62 135
65 158
64 145
70 170
72 185
70 180
(a) Plot a scatter diagram.
(6) Obtain the estimated regression line F" =
(c) Calculate and interpret a 90 per cent confidence interval for /3i.
(d) Calculate and interpret a 98 per cent confidence interval for /5 -
(e) Calculate and interpret a 95 per cent confidence interval for
*~ 66*
(/) Test the hypothesis T:/3i = 0.
(g) Test the hypothesis H:0i = G.
W Test the hypothesis H:(3<>= 30.
2O8 CHAPTER 8, REGRESSION ANALYSIS
(i) Predict the weight of an individual who is 66 inches in height.
Give a "prediction interval."
(j) Estimate the height of a man whose recorded weight is 170 pounds.
Give both point and interval estimates.
(&) Test for "linearity of regression."
(Z) What proportion of the variation in Y is "explained" by the re
gression of weight on height?
(NOTE: Give all assumptions and use a probability of Type I
error equal to .05 in each test.)
8.12 Assuming the data given in Problem 4.3 to be a random sample from a
bivariate normal population, (a) calculate the regression for esti
mating weight from height, (b) calculate the regression for estimating
height from weight, (c) plot a scatter diagram and show both regres
sion lines thereon.
8.13 The Consumer Market Data Handbook, 1939 edition, U.S. Department
of Commerce, lists consumer market data by states, counties, and
cities. Among the types of information listed are "Population and
Dwellings/' "Volume and Type of Business and Industry, 1935,"
"Employment and Payrolls, 1935/' "Retail Distribution by Kinds of
Business, 1935/' and "Related Indicators of Consumer Purchasing
Power. " Among the latter are numbers of income tax returns, auto
mobile registrations, radios, telephones, electric meters, and magazine
subscribers.
Such information as listed above might be used by national ad
vertising agencies, large sales organizations, and by individual retail
or manufacturing agencies for various purposes in planning their
business activities. The numbers and kinds of analyses which might
bo considered for such data arc largo. We have selected only a small
portion for study in this problem.
The data given here present the filling station sales per capita (F)
and the automobile registrations per 1000 persons (-XT) for a group of
Iowa counties.
PROBLEMS
209
IOWA CONSUMER DATA
County
Per Capita
Sales of Fill
ing Stations
(yearly'}
Automobile
Registrations
per 1OOO
Persons
Adair
$17
206
Adams ...
25
233
Allamakee .
16
237
Appanoose
13
183
Audubon
28
243
Benton
27
230
Black Hawk
20
272
Boone
21
214
Bremer
22
314
Buchanan . ...
16
263
Buena Vista . . .
32
314
Butler
27
295
Calhoun
27
273
Carroll . . .
21
279
Cass .
30
283
Cedar
21
276
Cerro Gordo
23
265
Cherokee
43
254
Chickasaw
23
264
Clarke
32
194
Clay
23
285
Clayton
14
255
Clinton ... .
21
232
Crawford
19
238
Dallas
24
271
Davis
18
224
Decatur
12
203
Delaware
22
23O
(a) Plot these data on an 8X11 sheet of graph paper. On the abscissa
or -X"-axis place automobile registrations anc^or^the ordinate or Y-
axis plot sales per capita. Plot the point (X, "F) from the results
to be obtained below.
(6) Calculate the means, 'X and T, and the standard deviations for
X and Y.
(c) Fit a straight line to the plotted points by obtaining the regression
of Y on X as a least squares fit. What is the model in this case
i.e., for a single observation, county per capita sales by filling
stations? What parameters do the statistics feo and 61 estimate?
Explain in words the meaning of 6 and 61,, that is, interpret the
results of your analysis.
(d) Plot f* = bQ + biX on the scattergram.
(e) Calculate & and Y & f or each X,
(/) Calculate (F ]P") 2 for each X and thus obtain S(F F) 2 - Com-
210
CHAPTER 8, REGRESSION ANALYSIS
pare this value with X)?/ 2 ""^i5D^2/- Then obtain Sg and explain its
relationship to the values of Y ?.
(0) Determine 95 per cent confidence limits for fti.
8.14 On the basis of the following tabulations comparing years of service
with ratings, the management seeks to discover whether or not
there is a distinct tendency to rate old employees higher than
more recent additions to the working force.
Employee*
Service
(in years)
Rating
Employee
Service
(in years)
Rating
A
1
5
K
6
9
B
9
6
L . .
7
4
C
8
8
M
1
2
D . ...
3
8
N
1
3
E
3
6
O
3
8
F . . .
2
7
P. . .
1
6
G
4
5
O
2
5
H
5
6
R .
2
3
I
5
4
S. . ,
4
4
J
6
5
T
2
7
* Source: G. R. Da vies, Business Statistics, p. 338.
(a) Plot a scatter diagram ( X = service, F = rating).
(6) Obtain the regression line J^ fto + friAT.
(c) Compute s/^
(d) Compute s^.
(tf) Set your results out in an analysis of variance table.
(/) Test //:/3i = using (1) a S-tcst and (2) an latest.
(0) Estimate the average rating which might be given an employee
with (1) 4 years' service, (2) 15 years' service. Give both point
and interval estimates. Discuss the validity of these estimates.
(/&) Kstimate, hy interval, what rating an individual employee with
4 years' service might be expected to receive.
(NOTE: Whenever necessary, state all the assumptions made in order
to use the techniques involved.)
8,15 Assume you are an investment counselor for a large insxirancc com
pany. As one of yoxir duties, you woxild need to have some idea of the
amount of policy loans per year, i.e., loans to policyholders, using
their life insurance policies as collateral. Suppose you wish to estimate
the total amount of policy loans your company will make during the
coming year. Assume the date to be January I, 1948. You are given
the data sheet shown below. (I) What methods of estimation might
you use and what would your estimates be? (2) What further informa
tion might you request hi order to do a better job? (3) Give reasons for
the answers you make to (2).
PROBLEMS
21 1
Year
National In
come* (in mil
lions of dollars)
Estimated
Population
of ILS.A.t
(in thousands)
Policy Loans
Made by U.S.
Life Insurance
CompaniesJ
(in millions
of dollars)
1929. . .
87,355
121,770
2,379
1930
75 003
123,077
2,807
1. ...
58,873
124,040
3,369
2
41,690
124,840
3,806
3.
39,584
125,579
3,769
4
48,613
126,374
3,658
5. . .
56,789
127,250
3,540
6
64,719
128,053
3,411
7
73,627
128,825
3,399
8
67,375
129,825
3,389
9. .
72,532
130,880
3,248
1940
81,347
131,970
3,091
1
103,834
133,203
2,919
2. .
136,486
134,665
2,683
3 .
168,262
136,497
2,373
4. . .
182,407
138,083
2,134
5
181,731
139,586
1,962
6. . .
179,289
141,235
1,891
7
202,500
144,034
1,937
8
224, 400
146,571
* Statistical Abstract of the United States, 1949, p. 281.
t Op. cit., p. 7.
j Life Insurance Fact Book, 1949, p. 67.
Estimated.
8.16 Let us assume that one of your duties is that of preparing reports for
the managing director of the firm. They are engaged in manufacturing
and are, of course, interested in the average cost per unit of production.
Obviously, units of production are easily measured, but average cost
requires lengthy and difficult computations. If some relationship
between these two quantities can be determined empirically, an
estimation procedure may be employed. From past records, the
following data are available:
212 CHAPTER 8, REGRESSION ANALYSIS
F = Average Cost X = Units Produced
(in cents') (in thousands*)
1.1
9
1.9
13
3.5
5
5.9
17
7.4
18
1.4
8
2.6
14
1.4
12
1.9
7
3.5
15
1.0
10
1.1
11
4.6
4
17.9
23
(a) What methods would you employ to have available a means of
estimating values of Y if X were known?
(6) Describe briefly what devices you would use to determine the type
of curve that would best fit a given set of data.
8.17 An advertising concern is interested in prorating sales by counties
for Maryland. In hopes of using magazine circulation per 1000 popu
lation to aid them, they obtained the following data:
PROBLEMS
213
Magazine Circulation per
1000 Population
Per Capita
Sales (F)
159 279
114 184
67 137
79 126
112 213
124 184
129 181
58 133
85 161
127 228
64 129
131 182
75 142
116 199
141 268
133 189
76 161
48 105
68 102
127 235
150 259
136 232
114 216
= 4245
= 841 , 133 ]T .X" F = 482 , 786
(a) Determine the regression equation.
(6) If the circulation in County A were 90, what would you estimate
the per capita sales to be? What is the standard error of f"?
(c) Is the regression coefficient significant?
(d) What are your assumptions? Are they justified?
8.18 Given the following data satisfying the normality and homogeneous
variance assumptions, do you believe that the true regression is
actually linear?
X
4
3
6
7
18
19
18
13
26
25
24
21
38
35
28
31
44
43
39
38
214
CHAPTER S, REGRESSION ANALYSIS
8.19 For the following data, test the hypothesis that /3i=/3 2 = ^3 = /34,
where we assume normality, etc., as required for simple linear re
gression.
Samples
Degrees of
Freedom
Z* 2
S xy
Z:v 2
A
67
300
312
550
B
75
500
515
758
C .
115
200
216
375
D
34
200
300
500
8.20 Using the data given below, fit a second degree polynomial (parabola)
for gross profits per farm against months of labor to obtain a curve
for Iowa farms.
Farm No.*
Gross
Profits
Y
Months
of Labor
X
Farm No.*
Gross
Profits
Y
Months
of Labor
X
1. .
16.7
20
18
11.2
14
2. . .
17.9
19
19
9.4
15
3
17.4
24
20
8.7
12
4
14.9
15
21
12.2
17
5
16.2
24
22
7.7
14
6. . . .
14.0
15
23
11.5
14
7
15.1
24
24
7.3
13
8
18.3
24
25
11.8
16
9 .
11.3
16
26. .
15.1
23
10. . .
18,3
26
27
10.5
33
11
17.1
24
28
17.0
29
12
12.0
16
29
15.6
30
13
15.2
25
30
13.2
31
14, . . .
16.2
28
31
17.2
22
15
14.9
24
32
14.6
32
16
10.5
15
33
12.2
34
17
16.5
27
34
9.8
36
* Source: Selected values from Iowa farm records plus some supplementary hypothetical
observations.
8.21
with
= 244,
8.22
Given the linear regression:
Z^^" 2 = 58,000, and n = 100:
(a) What is the standard error of ^ = 254?
(&) For what ]?" value is its variance a minimum?
(c) Given that information is the reciprocal of the variance, how may
we maximize our information about the unknown parameter /?i
in estimating a linear regression similar to the above?
In a regression study the following preliminary calculations were
made: 3? = 20; T = 22; 23 (X JT) 2
(a) What is the estimate for the population regression coefficient?
PROBLEMS 215
(6) How do you interpret the population regression coefficient /3i?
(c) Obtain the regression equation in the form Y = b^ + b^X.
(d) Test the hypothesis H:@i = l.
8.23 An economist from the University of Hawaii and an economist from
the University of Chicago were comparing their studies of income and
the consumption of various goods. Among the items studied was
gasoline for use in private automobiles. Each had used a sample of
about 100 university employees chosen to cover the range of salaries
and wages. The Chicago economist reported that he had observed an
increase in gas consumption of 10 gallons per $100 increase in income
while the Hawaii economist noted an increase of only 4 gallons per
$100 increase in income. They then looked at the variances of their
regression coefficients and gave these figures, V(b c ) = 2.41 and
(a) Could the observed difference between the regression coefficients
be expected to occur more than once in 20 times by chance if we
consider the necessary assumptions for such a test to be fulfilled?
(6) Would your conclusion be changed if the change in gas consump
tion had been reported as 0.1 gallon per $1 increase for Chicago
and .04 gallon per $1 increase for Hawaii? Or what would the
variance of b H be, if the regression coefficients had been reported
in the latter unit, per dollar increase in income?
(c) When the Hawaii economist tested the hypothesis (/S^ 0), he
obtained a tf- value of 2.828= (4 0)/-\/2. Approximately what F-
value would he have obtained if he had examined the reduction in
sum of squares due to linear regression by preparing an analysis
of variance?
[NOTE: V() is another way of expressing s|, for example, t^(6 c )
=<j
8.24 The performance of a tensile strength test on a specific metal yielded
the following results:
Brinell Hardness Tensile Strength
Number (1OOO psi)
X Y
104 38.9
106 40.4
106 39.9
106 40.8
102 33.7
104 39.5
102 33.0
104 37.0
102 33.2
102 33.9
101 29.9
105 39.5
106 40.6
103 35.1
216 CHAPTER 8, REGRESSION ANALYSIS
(a) Determine the best linear regression equation by least squares
and obtain confidence limits for estimating the mean tensile
strength associated with a specified Brinell number.
(6) Is any functional form other than a linear equation indicated by
these data? Make the appropriate test and discuss your results.
8.25 Using the data of Table 8.10, obtain the following regression equa
tions :
(a) Y = b Q 4- biXi
(Z>) y & 4. b 2 x*
<V) Y = b 4- 3^3
(d) Y = 6 + &4^4-
For each regression equation, perform a complete analysis. Comment
on the four different values of 6 . Also, compare the results of this
problem with the multiple regression analysis obtained in Example 8.5.
8.26 The solubility of nitrous oxide in nitrogen dioxide was investigated
with the following results:
Reciprocal Temperature
(= 10OO/ degrees absolute)
3.801 3.731
3.662 3.593 3.533
Solubility
1.28 1.21
1.11 0.81 0.65
(per cent by weight)
1.33 1.27
1.04 0.82 0.59
1.52
0.63
Perform a complete regression analysis and interpret your results.
8.27 A Rockwell hardness test is fairly simple to perform. However, the
determination of abrasion loss is difficult. In an attempt to find a
way of predicting abrasion loss from a measurement of hardness, an
experiment was run and data collected on 30 samples. The following
results were obtained:
JT - 70.27, 7 = 175.4, I> 2 - 4300, **T,y* = 225,011,
>;y = 22,946, s% 3663, and f =- 550.4 5.336X.
Estimate the abrasion loss when hardness is 70. Discuss the usefulness
of the prediction equation.
8.28 A gauge is to be calibrated using dead weights. If X represents the
standard and Y the gauge reading, perform a linear regression analysis
based on the following results from 10 observations:
2* = 230, 7 = 226, 2Zy = 1532, > 2 1561, Z^y 2 = 1539.
Test H: #L=1 using = 0.01
PROBLEMS 217
8.29 Elongation of steel plate (F) is related to the applied force in psi
Given the data
X
1.33
26
2.68
51
3.50
66
4.40
84
5.35
101
6.27
117
7.11
133
8.93
150
9.76
182
10.81
202
perform a complete regression analysis and interpret your results.
8.30 It is desired to determine the relationship of a twisting movement to
the amount of strain imposed on a piece of test metal. Eight samples
were obtained and the following data observed:
Twisting Movement (X) Strain (F)
100 112
300 330
500 546
700 770
900 1010
1000 1100
1200 1323
1300 1515
Determine the "best" relationship between X and F. Interpret your
results.
8.31 The data given below and identified as F, Xi, and X* represent annual
figures for 1919 to 1943, a 25-year period, for three adjacent counties
in the semiarid central area of South Dakota. F* is the average yield
of oats in the ith year. XM is preseason precipitation in inches, e.g.,
9.82 for JSTii is the rainfall from August, 1918, to March 31, 1919, etc.
X%{ is the growing season precipitation in inches. This rainfall covers
the period April 1 to July 31 for each crop year listed. Due to the
nature of weather and yield data, we may assume that these data
fulfil our necessary assumptions for multiple linear regression. Do
a complete analysis and interpretation, of the data. The reader should
note, though, that these are time series data, and thus an ordinary
multiple linear regression analysis may be of doubtful validity.
21 8
CHAPTER 8, REGRESSION ANALYSIS
Year
F
^Ti
X 2
1919
30.8
9.82
14.85
1920
34.2
9.12
17.30
1 . .
14.3
6.24
9.92
2
34.5
14.06
9.33
3
32.7
5.29
12,01
4. . . .
36.0
7.74
10.87
5 .
33.8
9.40
11.78
6
3.7
4.22
7.14
7
26.1
8.11
14,44
8
18.6
6.30
8.95
9. . .
15.0
10.58
6.15
1930
23.8
8.62
8.63
1
4.4
10.53
6.19
2
23.5
7.05
8.86
3
0.1
7.75
7.97
4
0.0
4.41
4.93
5
19.7
7.05
11.27
6
0.0
6.90
5.37
7
4.5
7.97
8.78
8
14.4
5.41
10.37
9
13.4
7.30
8.78
1940
11.8
5.94
7.06
1
22.2
6.77
10.44
2
42.9
11.23
14.58
3
24.6
8.55
9,57
8.32
A study of 18 regions gives the following data on suicide rate, age,
per cent male, and business failures. Fit an equation for the linear
regression of Y on JSTi, X%, and X 3 , where
F = suicide rate
Xi age
X% * per cent male
X$ = business failures
and analyze completely. The summary of the data follows:
53 F 285.3 ^YXi * 8536.6165
531.09 53^-Ya 14500.1161
= 911.95 23FA r 3 - 29644.847
1800 53-^1^2 26913,822
-< 4905.6904 2^Yi-Y 3 53614.575
=^ 15731,2223 23^Xa = 91U31 .630
46218.4473
199843.52
8.33 Do a complete multiple linear regression analywin of the following data.
Interpret yoxir results.
PROBLEMS
219
T> o KHif-
Choles
terol
Dosage
(gm. per
day)
Average
Blood
Total
Choles
terol
(m#.)
Initial
Weight
(&00
Ratio of
Final
Weight to
Initial
Weight
Average
Food In
take per
kg. Initial
Weight
(gm. per
day)
Degree
of Athero
sclerosis
No.
Xi
X*
X*
X 4
X 5
Y
1
30
424
2.46
0.90
18
2
2
30
313
2.39
0.91
10
3. ...
35
243
2. 75
0.95
30
2
4
35
365
2.19
0.95
21
2
5
43
396
2.67
1.00
39
3
6
43
356
2.74
0.79
19
2
7
44
346
2.55
1,26
56
3
8
44
156
2 58
0,95
28
9. . .
44
278
2.49
1, 10
42
4
10. . .
44
349
2,52
0.88
21
1
11. . .
44
141
2.36
1.29
56
1
12
44
245
2.36
0.97
24
1
13
45
297
2.56
1.11
45
3
14
45
310
2.62
0.94
20
2
15
45
151
3.39
0.96
35
3
16
45
370
3.57
0.88
15
4
17
45
379
1.98
1.47
64
4
18 .
45
463
2.06
1.05
31
3
19 .
45
316
2.45
1.32
60
4
20 .
45
280
2.25
1.08
36
4
21 . .
44
395
2.15
1.01
27
1
22. ..
49
139
2.20
1.36
59
23
49
245
2.05
1.13
37
4
24
49
373
2.15
0.88
25
1
25
51
224
2,15
1.18
54
3
26 .
51
677
2.10
1.16
33
4
27..
51
424
2.10
1.40
59
4
28
51
150
2.10
1 .05
30
8.34 You arc presented with farm records for one year for a sample of 89
dairy farms located in a fairly homogeneous area in the same milk shed.
The records contain the following information:
Y milk sold per cow (Ibs.)
Xi = amount of concentrates fed per cow
X% = silage fed per cow
X 3 = pasture cost per cow
Xi, amount of other roughage fed.
You first decide to fit a multiple linear regression of F, milk sold, on the
four independent variatcs, the X's given above. Thus, the regression
equation is of the form
Y = & + biXi + bzX* +
220
CHAPTER 8, REGRESSION ANALYSIS
8.35
(a) List the numerical quantities and statistics you would compute to
obtain this regression equation for Y. You need not give detailed
formulas. In particular, you will wish to compare b 2 and b 4 , or
silage with other roughage fed in effect on milk production. Also,,
pasture is quite homogeneous in the area, so you suspect /3 3 may
not be different from zero. Include in your list such items as
needed for examination of the indicated regression coefficients.
(6) Supposing you obtain 61 =+0.30, what interpretation would you
make of this statistic?
(c) Can you suggest any other form for this regression function, using
only the given J^T's? If so, write it out.
Using the data given below, obtain a multiple linear regression equa
tion. (Do a complete analysis.) Then, consider other possible analyses
and comment on the "best" functional relationship.
DATA FROM 25 IOWA COUNTIES*
Corn
Yield
Percent
No.
No.
Percent
Value
Ob
serva
tion
per
Acre
1910-
1919
age Farm
Land in
Small
Grain
Improved
Acres
per
Farm
Brood
Sows per
1,000
Acres
age Far in
Land
in
Corn
per Acre
of Land
Jan. 1,
1920
Sum
T> um
ber
County
A r !
X 4
A"a
-Y 4
AT 5
F
W
I
Allamakee
40
11
103
42
14
$ 87
297
2
Bremer
36
13
102
58
30
133
372
3
Butler
34
19
137
53
30
174
447
4
Calhoun
41
33
160
49
39
285
607
5
Carroll
39
25
157
74
33
263
591
6
Cherokee
42
23
166
85
34
274
624
7
Dallas
40
22
130
52
37
235
516
8
Davis
31
9
119
20
20
104
303
9
10
Fayette
Fremont
36
34
13
17
106
137
53
59
27
40
141
208
376
495
11
Howard
30
18
136
40
19
115
358
12
Ida
40
23
185
95
31
271
645
13
14
15
Jefferson
Johnson
Kossuth
37
41
38
14
13
24
98
122
173
41
80
52
25
28
31
163
193
203
378
477
521
16
17
Lyon
Madison
38
34
31
16
182
124
71
43
35
26
279
179
636
422
18
Marshall
45
19
138
60
34
244
540
19
Monona
34
20
148
52
30
165
449
20
Pocahontas
40
30
164
49
38
257
578
21
Polk
41
22
96
39
35
252
485
22
23
24
Story
Wapello
Warren
42
35
33
21
16
18
132
96
118
54
41
38
41
23
24
280
167
168
570
378
399
25
Winnesluek
36
18
113
61
21
115
364
Sums
937
488
3342
1361
745
4955
11828
!Means
37.48
19.52
133.68
54.44
29.80
198 . 20
473.12
* Reproduced from; H. A. Wallace and G. W. Snedecor, Correlation and Machine Calcu
lation (revised ad,; Ames, Iowa; The Iowa State College Press, 1931). By permission of the
authors and publisher.
REFERENCES AND FURTHER READING 221
References and Further Reading
1. Anderson, R. L., and Houseman, E. E. Tables of orthogonal polynomial
values extended to A r = 104. Res. Bui. 297, Agr. Exp. Sta., Iowa State Univ.,
April, 1942.
2. Bowker, A. H., and Lieberman, G. J. Engineering Statistics. Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959.
3. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
4. Chew, V. (editor) Experimental Designs in Industry. John Wiley and Sons,
Inc., New York, 1958.
5. Davies, O. L. (editor) Statistical Methods in Research and Production. Oliver
and Boyd, Edinburgh, 1949.
6. Dixon, W. J., and Massey, F. J. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1958.
7. Eisenhart, C. The interpretation of certain regression methods and their use
in biological and industrial research. Ann. Math. Stat., 10 (No. 2):162 86,
1939.
8. Ezekiel, M. Methods of Correlation Analysis. John Wiley and Sons, Inc.,
New York, 1941.
9. Gray bill, F. A. An Introduction to Linear Statistical Models. McGraw-Hill
Book Company, Inc., New York, 1961.
10. Hader, R. J., and Grandage, A. H. E. Simple and multiple regression
analyses. In Experimental Designs in Industry (edited by V. Chew). John
Wiley and Sons, Inc., New York, 1958.
11. Hunter, J. S. Determination of optimum operating conditions by experimen
tal methods, Part II-l, Models and Methods. Industrial Quality Control,
15 (No. 6): 16-24, Dec., 1958.
12. Kempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
13. Kendall, M. G. The Advanced Theory of Statistics. Vols. I and II. Charles
Griffin and Co., Ltd., London, 1946.
14. Kramer, C. Y. Simplified computations for multiple regression. Industrial
Quality Control, 13 (No. 8):8-ll, Feb., 1957.
15. Prater, N. H. Estimate gasoline yields from crudes. Petroleum Refiner, 35
(No. 5):236-38, May, 1956.
16. Snedecor, G. W. Statistical Methods, Fifth Ed. The Iowa State University
Press, Ames, 1956.
C H APT E R 9
CORRELATION ANALYSIS
IN CHAPTER 8, methods of estimating functional relationships among
variables were presented. Such methods have many uses in experi
mental work. However, there is a related matter which also deserves
attention when discussing the joint variation of two or more variables*
It is: How closely are the variables associated? Or, in other words,
what is the degree (or intensity) of association among the variables?
9.1 MEASURES OF ASSOCIATION
The techniques that have been developed to provide measures of the
degree of association between variables are known as correlation meth
ods. This name reflects the universal practice of speaking about "meas
ures of correlation' * rather than about "measures of the degree (or in
tensity) of association." Consequently, when an analysis is performed
to determine the amount of correlation, it is referred to as a correla
tion analysis. The resulting measure of correlation is usually called a
correlation coefficient.
In, this chapter some of the more frequently used measures of corre
lation will be presented. However, because of the close ties between this
chapter and some of the preceding chapters (particularly Chapter 8),
it will be sufficient to give only a minimum of discussion.
9.2 AN INTUITIVE APPROACH TO CORRELATION
Because of the nature of the concept of correlation, it is clear that
(in most cases) it is closely related to the concept of regression. In fact,
for a given regression equation, it seems reasonable to expect that a
correlation coefficient will measure how well the regression equation
fits the data or, stating this in reverse fashion, how closely the sample
points baig the regression curve. Thus, a correlation coefficient will
undoubtedly be related to the standard error of estimate ($#) which
measures the dispersion of the points about the regression curve.
Pursuing this idea and denoting the correlation coefficient by the
symbol R, we express R as a function of s^ } for example,
*-/(*,). (9.D
If R is to perform satisfactorily as a measure of correlation, it is desir
able that it exhibit two characteristics:
(1) It should be large when the degree of association is high and
small when the degree of association is low.
(2) It should be independent of the units in which the variables
are measured.
C2223
9.4 CORRELATION IN SIMPLE LINEAR REGRESSION 223
One way to achieve the desired properties is to (approximately)
define R by
JV^l -4/4 (9.2)
where
^E
2 ^ST^ / -rr T>\ 9 // ~\ /Q '2\
4 = 13 (F ~ F)V( - 1), (9-4)
and g is the number of parameters in the true regression function that
were estimated by the regression equation symbolized by F. If n is
large relative to q, another approximation is
R*G* 1 - ]C (F - f)V ]C (F - F) 2 . (9.5)
Since 53(F- I^) 2 < 22 (F- F) 2 , it is clear that <B 2 <1. Further, if
the sample points hug the regression curve closely (i.e., the correlation
is high), R 2 will be close to 1. Similarly, if the regression curve is a poor
fit, the sample points "will be widely dispersed about the estimated
regression and R 2 will be close to 0, reflecting a low correlation,
Having given the foregoing intuitive approach to correlation, it is
necessary that a more precise approach be formulated. This will now
be done. It is hoped that the remarks given earlier in this section will
aid the reader in appreciating the discussions to follow.
9.3 THE CORRELATION INDEX
Rewriting Equation (9.5) as
(F - F) 2
and referring to Sections 8.8, 8.15, and 8.16, it is seen that
sum of squares due to regression
R* = - - - ~ --
corrected sum of squares
(9.7)
Since the ratio defined by Equation (9.7) may be calculated for any
estimated regression equation, it is a most general and useful measure
of correlation. It is referred to as the correlation index. In succeeding
sections, special cases will be examined in detail.
9.4 CORRELATION IN SIMPLE LINEAR REGRESSION
In Section 8.8, the partitioning of the sum of squares of the depend-
ent variable was discussed and the results presented in Table 8.2.
Referring to Table 8.2 and invoking Equation (9.6), we obtain
, _ { i: F g - CE
224 CHAPTER 9, CORRELATION ANALYSIS
(9 ' 8)
where r 2 is used instead of R 2 to conform with standard practice. It is
customary to talk about r rather than r 2 . Thus, we have
= (9,9)
which assumes the same sign as ^2xy, an d hence the same sign as bi.
It is readily seen that the correlation coefficient associated with
simple linear regression is easily obtained once a regression analysis has
been performed. Further, it is clear that
l<r<l (9.10)
where 1 represents perfect negative linear association in the sample
and +1 represents perfect positive linear association in the sample. A
value of is interpreted to mean that no linear association between X
and Y exists in the sample. Since r is only a sample value, any infer
ence to the sampled population must be carefully stated. More will be
said concerning this a little later.
Example 9.1
Referring to Tables 8.3 and 8.4, the coefficient of linear correlation
between X and Y for the Schopper-Riegler data is determined as
follows:
r* ** 10,177.59/(51,712.00 41,217.23) 0.9698
r - V0.9698 = 0.98.
Example 9.2
For the following data,
X
-2 4
1 1
1 1
2 4
it may be verified that r = 0, indicating no linear association. Please
note carefully the word "linear/ 7 for a moment's reflection will reveal
that X and Y are perfectly associated, the relationship being Y**X*.
What we calculated was a measure of linear correlation when the
indicated relationship is actually quadratic. This simple example should
call to your attention one of the greatest potential trouble spots in
9.5 SAMPLING FROM A BIVARIATE NORMAL POPULATION 225
correlation analysis, namely, the use of an inappropriate measure of
correlation.
The preceding discussion and interpretation of r, or perhaps we
should say of r 2 , is most valuable in regression analyses. Examination
of Equations (9.7) and (9.8) reminds us that lOOr 2 is the percentage of
the corrected sum of squares that is "explained by 7 ' the fitting of the
simple linear regression Y=b Q +biX. If this percentage is not large
enough to satisfy us, a better fitting regression equation should be
found. .
Some terms associated with the coefficient of correlation that are
sometimes encountered are:
r 2 = coefficient of determination, (9.11)
1 r 2 = coefficient of nondetermination, (9.12)
and
r 2 == coefficient of alienation. (9.13)
Example 9.3
For the Schopper-Riegler data, 7* 2 = 0.9698 and 1 -r 2 = 0.0302. Thus,
96.98 per cent of the variation in Y (Schopper-Riegler rating) is "ex
plained by" the linear regression of Y on X (hours of beating).
9.5 SAMPLING FROM A BIVARIATE NORMAL POPULA
TION
The interpretation of r given in the preceding section is valid for any
simple linear regression regardless of what assumptions are made con
cerning the variables X and Y. However, if a random sample is drawn
from a bivariate normal population, then r [defined by Equation (9.9) J
is a sample estimate of the population parameter
p
(9.14)
The reader should note that this is the same correlation coefficient
specified by Definition 3.31, and thus it is not surprising that r is some
times referred to as the sample product-moment correlation coeffic^ent.
When sampling from a bivariate normal population, it is natural to
want to test hypotheses about the true value of p. Since such tests are
simply further examples of the general techniques introduced in Chap
ter 7, only a brief explanation will be given.
To test H:p = Q versus the alternative A:p?*Q, we calculate
(9.15)
226 CHAPTER 9, CORRELATION ANALYSIS
and reject H if t>t a~./2) c n 2) or if < Z ( i_ a / 2 ) 0-2). However, a mini
mum amount of simple algebra will show that
t = = > (9.16)
*r * 6l
and thus the test just detailed is identically equivalent to the test of
.ff:/3i = versus A:/3i^O as given in Section 8.13. A review of that
section will remind you that the hypothesis might also be tested using
an J^-ratio [see Equation (8.43)]. Consequently, three equivalent
methods of testing are available, the choice being determined by the
form of the analysis.
Example 9.4
Given the sample observations
X
11
F
it is easily verified that r 0.98. Using Equation (9.15), we obtain
=( 0.98) V3/VO. 0413= 8.49. Since t= 8.49 < .995(3) = 5.841,
the hypothesis 7/:p is rejected in favor of the alternative A :p ^0.
Clearly, a 1 per cent significance level was used. It is suggested that
the reader consider //:/3i==0 versus A:/3i -p^O and compare the resulting
test statistic with that computed above.
If the hypothesis to be tested is //:p = p versus A:p^p^ y where
p r^0, the test procedure is more complicated. The complication arises
because (r p )/5 r is not distributed as "Student's" t unless p = CK
When p ?^0 7 an approximate test is provided by
*r - G)[log. (l + r) - log, (1 - r)]
- (1.1513) [io glo (l + r) - lo glo (l - r)]. (9.17)
Fisher (4) has shown that z r is approximately normally distributed
with mean # PO and variance 0% l/(n 3). The approximate test pro
cedure is to calculate
z - (* r - * Pa )/cr, (9.18)
and compare this quantity with fractiles of the standard normal dis
tribution. The hypothesis //:p = p () would be rejected if
or if
*< -*(i-*, a >- (9-20)
The research worker may also be interested in obtaining a confidence
interval estimate of p. This may be obtained by calculating
9.6 CORRELATION IN MULTIPLE LINEAR REGRESSION
227
and then using Equation (9.17) to solve for r L and r u ,
Quite frequently the research worker has several independent
samples, each randomly selected from a bivariate normal population,
from which estimates r x , - - - , r k are obtained. If the research worker
can accept the hypothesis Hip^ - - = p A , it is permissible to obtain
a pooled estimate of the common population correlation coefficient,
and this pooled estimate should, of course, be more reliable than any
of the individual estimates. If calculations are carried out as in Table
9.1 and the observed chi-square is not judged significant at the lOOa
per cent significance level, a pooled estimate of p (corresponding to the
''average z"} may be found.
TABLE 9. 1-Cal dilations for Testing the Hypothesis p = - - = : P A (^= = 3)
Sample
Size of
Sample
n 3
r
z
(v, a'Nrr
\'U ^^ O J At
(n 3)s 2
A . . .
102
102
102
99
99
99
. 63245
. 77459
. 67082
.74551
1.03168
.81223
73 . 80549
102.13632
80.41077
55.02273
105,37200
65.31204
B. .
C
Total
306
297
256.35258
.86314
225.70677
Jc / k
Average z = 23 (^ 3)#; / S (w 3)
t 1 ' i 1
(Averages) 23 (^ -~ 3)z t -
221.26817
v 2 for testinc: //:p, == p,.
4.43860
9.6 CORRELATION IN MULTIPLE LINEAR REGRESSION
When a multiple linear regression equation has been fitted to a set of
data, as in Section 8.15, it is natural to seek a measure of correlation
which reflects the "goodness of the fit." The correlation index defined
in Section 9.3 may be used to give us what we desire. Referring to Equa
tion (8.68), it is seen that
sum of squares due to regression
corrected sum of squares
(9.22)
This may also be expressed as
x *>y
Sy'
(9.23)
228 CHAPTER 9, CORRELATION ANALYSIS
which is analogous to the expression
as given, in Equation (9.8). If we calculate R = \/R 2 , where R* is defined
by Equation (9.22) or Equation (9.23), then R is known as the mul
tiple correlation coefficient. The significance of R may be assessed by
the F-test specified in Equation (8.76). No example will be given at
this time since nothing new and different is involved. However, some
of the problems at the end of the chapter will require the calculation
and interpretation of the coefficient of multiple correlation.
It is also worth noting that R 3 as defined by Equation (9.22), may
be thought of as a simple linear correlation between Y and Y where
Closely allied to the topic of multiple correlation is that of partial
correlation. By partial correlation is meant the correlation between two
variables in a multivariable problem under the restriction that any
common association with the remaining variables (or some of them)
has been "eliminated." Clearly, many partial correlation coefficients
may be calculated. For example, a first order partial correlation coeffi
cient is one which measures the degree of linear association between
two variables after taking into account their common association with
a third variable. Symbolically,
(9 ' 25)
,
Vl
*
where the subscripts refer to the three variables JTi, X%, and X$. Here,
of course, r^.z is attempting to measure the correlation between -XTi
and -XT 2 independent of JT 3 . It should also be clear that r*v (i, j = 1, 2, 3)
are simple linear correlation coefficients measuring the correlation be
tween Xi and Xj. A second order partial correlation coefficient may be
illustrated by
^12,3 ^14.3^24.3
=== (9.26)
. a
Vl - rl
which measures the correlation between .X\ and X% independent of X$
and X&.
Before proceeding to another topic, it will be worth digressing for a
moment to discuss a related matter (related to partial correlation,
that is) in regression. In Section 8.15, the equation
& - 60 + b l X l + - + b k X k (9,27)
9.7 THE CORRELATION RATIO 229
was discussed for the case k = 4. At that time, had we so desired, it
would have been appropriate to call attention to a different system of
notation which is sometimes encountered. For fc = 4, Equation (9.27)
would appear as
Y = b Q + biXi + b 2 X 2 + b z X* + 4X4. (9.28)
An alternative notation is
Y == 5 bo + byi, 234^1 + &F2. 134^2 + #^3. 124^3 + &r4. 123-^4, (9.29)
and in this form the analogy with partial correlation is evident. Strictly
speaking, the coefficients should be called partial regression coefficients
where, for example, &rx.234 represents how Y would vary per unit change
in XT. if X%, Xz, and X were all held fixed. Thus, 6 y 1.234 (or, as we usually
denote it, 61) gives only a partial picture of what happens to Y as Xi
changes. Hence the adjective "partial." It should be clear that the less
cumbersome notation was used (at the risk of not clearly defining the
meaning) solely to simplify the writing of the equations.
9.7 THE CORRELATION RATIO
Closely related to the correlation index is a quantity known as the
correlation ratio. Denoted by E*, it is defined by
(9.30)
where T\- is the mean of the ith group consisting of n* observations and
7 is the mean of all observations. Expressing Equation (9.30) in words,
among groups sum of squares
2 ^ - ?_^ - - ^ - (9.31)
corrected sum of squares
where the quantity labeled "among groups sum of squares" is most
easily found using the identity
(9.32)
z 1 , i*l
where
(9.33)
= total of the observations in the ith group
230 CHAPTER 9, CORRELATION ANALYSIS
and
k
T = ]T a* = nY
~1 (9.34)
= total of all observations.
It should be clear, of course, that
(9.35)
= total number of observations.
A moment's reflection will indicate that the value of E 2 is highly
dependent on the choice of groups. For example, if there is only one
observation in each group, the value of ffi is unity; if all the observa
tions are in one group, the value of E* is 0. Great care, then, must be
exercised when grouping the observations.
Another point of interest is the following : Once the observations are
assembled in groups, the value of E 2 is determined solely from the
values of the "dependent" variable. Consequently, the "independent"
variable need not be a quantitative variable. It can be a qualitative
variable. That is, subject to the dangers implicit in the groxiping, the
correlation ratio may be used to measure the correlation between a
quantitative variable and a qualitative variable.
Since grouping is so important, some guidance is necessary. One rule
of thumb is to have three to five groups, each containing a large num
ber of observations (say 100). Strict rules of procedure are hard to de
fine, but the preceding rule may prove helpful. Denoting the popula
tion correlation ratio by ?7 2 , Woo (13) gives tables for use in testing the
hypothesis J/:?7 = when we are willing to assume that the Ky are
normally and independently distributed (with common variance) in
each group.
Because the analysis of variance form of presenting results is so often
encountered, it should not be surprising to find it helpful in the present
situation. Referring to Table 9.2, it is seen that the sums of squares
needed in Equation (9.31) arc easily accessible. (NOTE: Now that the
opportunity has presented itself, we shall take a moment to review
the symbolism introduced in Section 7.20. It seems almost unnecessary
to remark that the letters M, (7, and W in the symbols M vv , G yv , and
W^j, were chosen to stand for the words "Mean, Groups, and Within,"
respectively. However, since this abbreviated method of representing
various sums of squares will be used extensively in later chapters, it is
a good idea to become well acquainted with the notation as early as
possible.)
9.8 BISERIAL CORRELATION
231
TABLE 9.2-Analysis of Variance Associated With the
Calculation of a Correlation Ratio
Source of
Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Mean .
1
jj//_ =: T^/M.
Af,/l
Among groups . .
Within groups . . .
k 1
k
/ \W>i 1)
G yy = 23 G?M - T*/n
W vv = Z ^ 2 - M yy - G vv
<?,/(* - 1)
/ *
Total
M
y- F 2
^'
9.8 BISERIAL CORRELATION
A measure of correlation encountered frequently in such areas of
specialization as education, psychology, and public health is the bi-
serial correlation coefficient. Only a brief discussion will be given in this
text. Those persons interested in more detail are referred to McNemar
(7), Pearson (10), and Treloar (12).
The biserial correlation coefficient, usually denoted by r b , is used
where one variable, Y, is quantitatively measured while the second
variable, X, is dichotomized, that is, defined by two groups. The as
sumptions necessary for a meaningful interpretation, of r b are :
(1) Y is normally distributed and suffers little due to broad group
ing (if grouping is necessary) .
(2) The true distribution underlying the dichotomized variable X
should be of normal form.
(3) The regression of Y on X is linear.
(4) The mean value of Y in the minor, or smaller, category as
specified by X, denoted by Fi, is to be on the regression line.
This assumption implies a large number of observations in the
minor segment.
If we define:
p = proportion of observations in the major category
q = proportion of observations in the minor category
z = ordinate of the standard normal curve at the point cutting off a
tail of that distribution with area equal to q
T 2 = mean of the Y values in the major category
S F = standard deviation of all the Y values,
then
(9.36)
232 CHAPTER 9, CORRELATION ANALYSIS
and this gives us a measure of the degree of linear association between
X and F.
It should be mentioned that, in a manner analogous to the way in
which we developed the correlation ratio, Pearson (10) introduced the
concept of a biserial correlation ratio, denoted by E b} which extends the
biserial correlation concept to cover any postulated regression function.
We shall not go into detail here, but the reader is referred to Pearson
(10) and Treloar (12) if he is interested in such problems.
9.9 TETRACHORIC CORRELATION
Another measure frequently encountered in some areas of research is
the tetrachoric correlation coefficient. This is generally denoted by r t and
is used to measure the degree of linear association between two vari
ables, X and Y, where both are dichotomized and the true underlying
distributions are assumed to be normal. That is, if we have samples
from a bivariate normal population but the measurements are not
available (we know only to which cell of a 2X2 contingency table each
observation belongs), we can obtain a measure of the correlation be
tween X and F. It is not feasible to present a formula for r t , but refer
ence to McNemar (7), Treloar (12), and other works will indicate cal-
culational methods for those interested in this particular statistic.
9.10 COEFFICIENT OF CONTINGENCY
Of some interest, also, is a measure of the degree of association be
tween two characteristics where our observational data are classified in
an rXc contingency table. In Chapter 7 we gave a method for testing
the hypothesis that these two characteristics, or classifications, were
independent of one another. Suppose, however, that we are more inter
ested in estimating the degree of association between them than test
ing the hypothesis of independence. How may we do thin? Pearson (9)
proposed for this purpose a measure known as the coefficient of con
tingency defined by
C= ^/T-1' C^-37)
where x 2 is the usual
as given in Chapter 7, In the case of a 2X2 table, this may seem to be
analogous to a tetrachoric correlation coefficient, but the coefficient of
contingency is of wider generality because wo no longer require the
assumption, of normality of the underlying distributions. Any distri
bution, discrete or continuous, is acceptable. However, there is a dis
advantage to this measxire of association; its maximum possible value
9.11 RANK CORRELATION 233
varies with the number of rows and columns, and thus two different
values of C are not directly comparable unless computed from tables
of the same size. For further remarks on this measure, the reader is re
ferred to McNemar (7) and Treloar (12).
9.11 RANK CORRELATION
Let us now consider a slightly different problem but one that arises
quite frequently in certain areas of research. The problem is as follows:
n individuals are ranked from 1 to n according to some specified char
acteristic by m observers., and we "wish to know if the m rankings are
substantially in agreement with one another. How may we answer such
a query? Kendall and Smith (6) have proposed a measure known as the
coefficient of concordance, W, for answering this question which is
defined by
W = - , (9.38)
m?(w? n)
where S equals the sum of the squares of the deviations of the total of
the ranks assigned to each individual from m(n + l)/2. The quantity
m(n + l)/2 is, of course, the average value of the totals of the ranks,
and hence 3 is the usual sum of squares of deviations from the mean.
W varies from to 1, representing no community of preference, while
unity represents perfect agreement. The hypothesis that the observers
have no community of preference may be tested using tables given in
Kendall (5) or, more simply (for n>7), by calculating
(9.39)
<mn(n -J- 1)
which is approximately distributed as chi-square with v = n 1 degrees
of freedom. If there are "ties" in some of the rankings, it may be neces
sary to modify our formulas somewhat; if such a case is encountered,
the researcher is referred to Kendall (5).
If we find W to be significant, the next step is to estimate the true
ranking of the n individuals. This is done by ranking them according to
the sum of the ranks assigned to each, the one with the smallest sum
being ranked first, the one with the next smallest sum being ranked
second, and so on. If two sums are equal, we rank these two individuals
by the sum of the squares of the ranks assigned to them, the one with
the smaller sum of squares obviously being ranked ahead of the other.
If W is not significant, we are not justified in attempting to find an
"average/' or "pooled," estimate of a true ranking, for we are not at
all certain that such a true ranking even exists.
When m = 2, that is, when only two rankings are available, a slightly
different approach is often used. In this case, a measure known as
Spearman's rank correlation coefficient is computed. Spearman's rank
correlation coefficient, denoted by r 89 is defined by
234
CHAPTER 9, CORRELATION ANALYSIS
!_
n
3
n
(9.40)
where c^ equals the difference between the two ranks assigned to the ith
individual. It can easily be seen that r s varies from 1 to +1, whereas
W varied only from to 1, 1 signifying perfect disagreement and
+ 1 signifying perfect agreement between the two rankings. A test of
the null hypothesis H : p s may be made using tables provided by
Olds (8). We must remember, however, that the same conclusion,
namely, to accept or reject H, could be reached by computing W and
comparing with the tabulated values for m = 2. Incidentally, we should
remark that Kendall (5) does not tabulate W itself but only the associ
ated value of S. This, of course, cuts down the amount of arithmetic
required since it is not necessary actually to compute the value of W
in order to perform our statistical test. Similarly, Olds (8) only tabu
lates
4.
t1
Example 9.5
Consider the data of Table 9.3. Calculations yield
0.771 with ) d
t 1
8.
Using 01 = 0.05, the hypothesis H:p s = Q is rejected. [NOTE: This con
clusion was reached after consulting the tables provided by Olds (8).]
Thus, it is concluded that the two judges are in. quite good agreement.
TABLE 9.3~Preferences for Six Lemonades as Expressed by Two Judges
Lemonade
Ranking Given
by Judge No. 1
Ranking Given
by Judge No. 2
Difference in
Ranks = d
A
4
4
B
1
2
1
c
6
5
1
D
5
6
1
JS
3
1
2
F
2
3
. i
Example 9.6
Consider the data of Table 9.4. It may be verified that m(n+l)/2
= 10.5, S ==25, 5, and W = 0.162. Examination of the tables in Kendall
(5) leads us to accept the hypothesis of no community of preference
9.12 INTRACLASS CORRELATION 235
TABLE 9.4 Preferences for Six Lemonades as Expressed by Three Judges
Lemonade
Ranking
Given by
Judge No. 1
Ranking
Given by
Judge No. 2
Ranking
Given by
Judge No. 3
Sum of Ranks
A
5
2
4
11
B
4
3
1
8
C
1
1
6
8
D
6
5
3
14
E
3
6
2
11
F
2
4
5
11
among our three judges, and thus we shall not attempt to estimate
any "true order of preference."
9.12 INTRACLASS CORRELATION
a>
The measure of correlation to be discussed in this section was devised
to assess the degree of association (or similarity) among individuals
within classes or groups. For this reason, the measure is known as the
intraclass correlation coefficient. (NOTE : Some authors have referred to
the intraclass correlation coefficient as the coefficient of homotypic corre
lation but the former term is more common.)
As an example of a situation in which the intraclass correlation co
efficient is the proper measure, consider the problem of measuring the
correlation between heights of brothers. Because all that is desired is a
measure of similarity between heights of brothers, any attempt to
label one as X and the other Y (for example, by age) would introduce
TABLE 9.5-Symbolic Representation of Data To Be Used in Calculating
the Intraclass Correlation Coefficient
Groups
1
2
k
F u
F 12
F 21
F 22
Y kl
F fc2
Observations*
F ln
F 2n
Y kn
Total
Gf* S~**
\ ^-'"2 ^fc
* Each observation is assumed to be of the form F# ^u+ i+ t -j where M is a constant, g^
is a random variable with mean and variance <r#, and e if is a random variable with
mean and variance <r 2 . That is, a linear model has been postulated which states that any
observation is a linear combination of three contributing factors: an over-all mean effect,
an effect due to the particular group to which the observation belongs, and an "error"
effect representing all extraneous sources of variation.
236
CHAPTER 9, CORRELATION ANALYSIS
TABLE 9.6 General Analysis of Variance for Calculating the Intraclass
Correlation Coefficient Using the Data of Table 9,5
Source
of
Variation
Degrees
of
Freedom
Sum
of
Squares
Mean
Square
Expected
Mean
Square
Mean
1
M-uu
Among groups
k1
Cr, rt /
s^-^-yiSQ
(T 2Ji rncr ^ f
Within groups
&O 1)
Ww
$*
<r 2
Total
kn
Y\ Y 2
a spurious element into the correlation. The spurious element, of
course, would be that an ordinary (simple linear) correlation would
measure the correlation between the heights of older brothers and the
heights of younger brothers rather than simply assess the "sameness"
of heights of brothers.
The intraclass correlation coefficient, denoted by r I} is most easily
calculated using analysis of variance techniques. Given the data of
Table 9.5, the variation among the kn observations may be summarized
as in Table 9.6, where
T -
+ G 2 +
G/n
(9.41)
(9.42)
(9.43)
and
R - M vv G uv . (9.44)
Since the population intraclass correlation coefficient is defined by
'G
(9.45)
a sample estimate is provided by
2
+ 4
MS a MS V
where
MS a + (n
MS a = mean square among groups
(9.46)
9.12 INTRACLASS CORRELATION 237
= s* + ns% (9.47)
= G vv /(]k - 1)
and
MS W = mean square within groups
= s* (9.48)
= W v3f /k(n 1).
It will be seen that if n = 2 ; the analysis would fit the situation described
earlier, namely, the correlation between the heights of brothers.
(NOTE: Once again we have availed ourselves of the opportunity to
introduce some new notation. This time the concept of components of
variance, denoted by s* and g 2 ^, has been used as an alternative way of
expressing mean squares. The relationship between "expected mean
squares' 7 and "mean squares" is, of course, simply the familiar rela
tionship between "population parameters" and "sample statistics."
The determination of the form of the various expected mean squares
will be examined in detail in succeeding chapters, where linear models
will be the main topic of discussion. Those who desire more informa
tion on this topic may jump ahead to the appropriate sections.)
If one is willing to assume that the individuals within groups are
random samples from normal populations (one population per group)
and that each population has the same variance, then the hypothesis
H:p r = is equivalent to the hypothesis .ff: 0-^ = 0, and this may be
tested using
F = MS a / MS w (9.49)
with degrees of freedom i>i = fc 1 and V2 = k(n 1).
Example 9.7
Given the data in Table 9.7, calculations will lead to the analysis of
variance shown in Table 9.8. From this we obtain r 7 = 0.6974. To test
/frpj^O, we calculate F = 30.857/5. 500 = 5. 61 with z>i = 7 and i>2=*8
TABLE 9.7-Heights of Eight Pairs of Brothers
rfeights
Pair (inches')
A 71; 71
B 69; 72
C 59; 65
D 65; 64
E 66; 60
P 73; 72
G 68; 67
H 70; 68
238 CHAPTER 9, CORRELATION ANALYSIS
TABLE 9.8-Analysis of Variance for Data of Table 9.7
Expected
Degrees of
Sum of
Mean
Mean
Source of Variation
Freedom
Squares
Square
Square
Mean. .
1
67.5
67.5
Among groups
7
216.0
30.857
o- 2 +2<r5
(Among pairs of
brothers)
Within groups
8
44.0
5.500
<r 2
(Between brothers
within pairs)
Total
16
327.5
degrees of freedom. Since F = 5.61 >F Q . 95^8) = 3.5, the hypothesis
Hip z = Q is rejected.
9.13 CORRELATIONS OF SUMS AND DIFFERENCES
Reference to Section 5.14 reminds us that, for any constants a and
any variables Xt, the linear combination specified by
U = 22 a^Xi (9.50)
has
v v = E[U] = jb <*tf** (9.51)
* i
and
where /*, is the mean of .XT*-, of IKS the variance of X*, arid &*$ is the
co variance of X* and JSTj. .Thus, if C7 JSTiJX r 2,
Mt/ = MI ^ (9.53)
and
2 .- _2 j_ -.2 4. 7<r ^Q ^4^1
" rr V i [ C/ o -.1 **\J -i o V, ^ * OTPy
Utilizing Definition (3.31), it is easily verified that Equation (9.54)
may be rewritten as
^ ^2 ^ ^.2 2p 12< r I o- 2 . (9,55)
Rearranging terms, we obtain
PROBLEMS 239
0-2 0.2 0-2
P 12 = -^ ~ I U = Xi + X* (9.56)
or
2 l 2 _-2
^ 1 2 U . T-T- -y yr XQ c^\
P 12 , U -A.1 ^-2- ^ y --> / >'
This leads to an alternative method of obtaining r 12 (the sample esti
mate of P 12 );r namely:
TT -\r _i V* /^O c:Q^
I c/ == j\. i } -A. 2 y^y . ooj
or
o2 _|_ ? 2 _ C 2
*t + *2 ^ . j, _ ^ _ ^ (Q 59)
Before terminating our discussion of the correlation of sums and dif
ferences, attention must be directed to the relationship between the
contents of this section and the "method of paired observations"
examined in Sections 6.9 and 7.9. Noting thatZ) = JX~ Y is analogous
to U = X X^ we recognize that a legitimate pairing of related ob
servations will yield a smaller standard error of the mean difference if
a positive correlation exists. Such a reduction in the standard error
represents a gain in efficiency (relative to nonpairing) which will be
reflected in a shorter confidence interval, an easier establishment of
statistical significance, or a smaller sample size. Clearly, the success of
pairing in any situation depends upon the extent to which the re
searcher can introduce positive correlation into an experiment.
Problems
9.1 Using the data of Example 9.4, test H:/3i = versus A :/3i ^0 using:
(a) a Z-tcst, (b) an ^-test. In both tests, let <* = 0.01.
9.2 If U = a + bX and V = c + dY, show that r uv r XY -
9.3 Verify Equation (9.16).
9.4 Interpret a simple linear correlation coefficient of 0.8.
9.5 If the simple linear (product-moment) correlation coefficient between
X and F is r jsrr = 0.8, what are the values of:
(a) r xv , (6) r x f , and (c) r y $l
9.6 Using the data of Problem 4.3 and the results of Problem 8.12, com
pute and interpret the appropriate measure of correlation.
9.7 Using the data of the problem indicated, compute and interpret the
appropriate measure of correlation:
(a) 8.4 (e) 8.8 00 8.14 (m)8.20
(6) 8.5 (/) 8.9 (J) 8.15 (n) 8.21
(c) 8.6 (g) 8.11 (/e) 8.16 (o) 8.22
(d) 8.7 (A) 8.13 (0 8.17 (p) 8.24
24O
CHAPTER 9, CORRELATION ANALYSIS
9.8
(?)
(r)
0)
8.26
8.27
8.28
(0 8.29 O) 8.32
O) 8.30 <X> 8.33
(v) 8.31 (?) 8.35
The following
F, selected at
table gives hypothetical data for
random from a bivariate normal
the covariates X and
distribution.
X
F
X
Y
12
74
18
149
20
170
16
142
17
147
13
144
11
75
18
173
8
46
11
101
8
59
16
140
4
20
15
132
12
90
5
35
9
74
14
96
12
77
6
50
16
144
3
24
11
110
5
26
10
99
8
95
13
109
6
73
15
109
17
159
(a)
(6)
(c)
(d)
Compute the means, the standard deviations, and the standard
errors of the means of X and F.
Make a scatter diagram to show the relation between these two
series. Also, draw one line through the plotted data showing the
mean of X and another showing the mean of F,
Fit a straight line to the points on the scatter diagram in order to
express mathematically the average relationship between these
two variables. The required equation is f^ bo + hiX. This calls for
the computation of:
/ x*v
(1) the regression coefficient bi -- >
3T). Find
(2) the F-intercept 6 7
The regression equation may be written
6 Q and 61 geometrically from the graph.
Calculate the estimated value of F for each of the 30 values of X
from the equation P"~&o + &i-Y. Also, compute the errors of esti
mate (F-F) for each X.
Interpret the constants 5 and &i obtained for !?"=*= bo+hiX.
Compute and interpret the standard error of estimate from the
formula
/j\ (y -
-y^
(00 Compute the sum of squares of the errors of estimate (deviations
from regression) with the formula
PROBLEMS 241
(70 Test the regression coefficient, 61, for significance.
(i) Compute the correlation coefficient using the formula
(f) Compute and interpret the coefficient of determination, r 2 .
(A?) Partition 53 2/ 2 into two parts: that associated with regression, and
that attributed to errors of estimate.
(Z) Compute the correlation coefficient between X and F.
(m) Compute the correlation coefficient between Y and Y.
(ri) Compute the correlation coefficient between x and y.
(o) Compute and interpret the 95 per cent confidence limits of /3i.
(p) Compute the standard errors for the estimated values of Y for
each of the following:
(1) the mean of all F's whose X value is equal to 10.
(2) particular Y's whose X value is equal to 10.
(q) Compute the sum of squares attributed to regression using thefor-
mula 52 (F F) 2 . The short-cut formula is (2>2/W2> 2 > or r^y*.
Show computationally that the three formulas give the same sum
of squares.
(r) Show computationally that (1 r^^^ = 2Z(Y F) 2 .
(s) Compute the regression of X on F; that is, compute the constants
in the equation X = b' + biY, where
and ft'o - 3T
Plot the regression on the same sheet on which the regression
^-^^Q+biX was plotted.
(0 Show that r 2 = Z>i&I, where 61 and bi are the two regression coeffi
cients.
(u*) Compute
(v) Show logically, algebraically, or geometrically that | r \ cannot be
less than nor greater than 1.
9.9 We have this sample of X and Y values:
9
4
11
2
7
5
10
1
8
3
242
CHAPTER 9, CORRELATION ANALYSIS
9.10
9,11
9.12
9.13
(a) Compute the product-moment correlation between Y and X for
this sample.
(&) What assumptions are required for testing the significance of a
sample value of rl What parameter is estimated by the sample
correlation?
Indicate or describe three methods for testing the hypothesis that
the true value of the correlation is in the bivariate population
from which the above sample was taken. (Exact formulas are not
required.)
Management seeks to discover a measure of correlation between length
of service on the part of a certain type of machine and the annual re
pair bills on such machines. From the following data:
(c)
Machine
Years of
Service
Annual Repair
Cost
A
1
$2.00
B
3
1 .50
C ., .
4
2,50
D
2
2.00
&
5
3.00
F. .
8
4.00
o.
9
4.00
H
10
5.00
/. . .
13
8.00
J
15
8.00
(a) Make a scatter diagram, designating years of service as the X
scries and annual repair costs as the Y series.
(6)
(c)
Find the correlation coefficient r.
Is the measure of correlation significant?
What are your assumptions?
Given that
22 yj = 1000
and that the Rum of squares due to regression is 640, compute the value
of r showing all your steps. What assumptions are necessary if r is to be
interpreted as a sample estimate of a population correlation coefficient?
The correlation coefficient between the C.A.V.D. Vocabulary and the
Graduate Record Verbal tests was 0.60 for a sample of 67 men students
and 0.50 for a sample of 39 women students. With a risk of Type 1 error
of 5 per cent, is this evidence that the two groups are random samples
from bivariate normal populations of the same correlation?
Given the following data and statistics for a random sample from a
bivariate normal distribution;
3?
7
6
20
22
100
2500
-400
-4
0.8
44
6.708
0.6708
(a) Give a detailed interpretation of the linear regression of Y on X.
REFERENCES AND FURTHER READING
243
Include all inferences that can be made about the population
regression. Also, interpret all inferences made.
(&) Interpret the above correlation coefficient.
(c) What assumptions are implicit in the use of the regression in (a) ?
9.14 Using the results below, test the hypothesis H:pi=p(i = lj - , 7).
Also run through the series of tests outlined in Section 8.26. State the
assumptions made in each case.
REGRESSION AND CORRELATION DATA IN SEVEN TYPES or SHEETING
Fabrics
Degrees
of
Free
dom
Z*'
Sary
S;y2
Correla
tion
Coefficient
Regression
Coefficient
Degrees
of
Free
dom
Sum of
Squares
Mean
Square
1
139
139
139
139
139
139
139
60357.14
60357.14
60357.14
60357.14
60357.14
60357.14
60357.14
989 . 64
1970.43
1647.50
192.86
5482.14
7605.00
12458.50
1965 . 89
2351.43
3190,85
3258.61
2804.04
2276.79
4375.60
0.0909
0.1654
0.1186
0.0138
0.4214
0.6487
0.7666
0.0164
0.0326
0.0273
0.0032
-0.0908
0.1260
0.2064
138
138
138
138
138
138
138
1949.66
2287.10
3145.88
3257.99
2306.11
1318.56
1804.00
2
3
4
7
Total
973
422499.98
30346.07
20223.21
-0.5028
966
16069,30
16.63
972
20201.41
Difference for testing among regression coefficients
^ = 688.68/16.63=341.41
6
4132.11
688.68
References and Further Reading
1. Bowker, A. H., and Lieberman, G. J. Engineering Statistics. Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959.
2. Brownlee, K. A. Statistical Theory and Methodology in Science and Engi
neering. John Wiley and Sons, Inc., New York, 1960.
3. Dixon, W. J., and Massey, F. J. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
4. Fisher, R. A. On the probable error of a coefficient of correlation deduced
from a small sample. Metron, 1 (No. 4) :3, 1921.
5. Kendall, M. G. Rank Correlation Methods. Charles Griffin and Co., Ltd.,
London, 1948.
6 y an d Smith, B. Babington. The problem of m rankings. Ann. Math.
Stat., 10:275, 1939.
7. McNemar, Q. Psychological Statistics. John Wiley and Sons, Inc., New York,
1949.
8. Olds, E. G. Distributions of sums of squares of rank differences for small
numbers of individuals. Ann. Math. Stat., 9:133, 1938.
9. Pearson, K. Mathematical contributions to the theory of evolution. XIII.
On the theory of contingency and its relation to association and normal
correlation. Drapers' Co., Res. Me?n., Biometric Series I. Cambridge Uni
versity Press, London, 1904.
IQ m On a new method of determining correlation when one variable is
given by alternative and the other by multiple categories. Biometrika,
7:248, 1910.
11. Snedecor, G. W. Statistical Methods. Fifth Ed. Iowa State University Press,
Ames, 1956.
12. Treloar, A. E. Correlation Analysis. Burgess Publishing Co., Minneapolis,
1942. .
13. Woo, T. L. Tables for ascertaining the significance or nonsignincance of
association measured by the correlation ratio. Biometrika, 21:1, 1929.
CHAPTER 10
DESIGN OF EXPERIMENTAL
INVESTIGATIONS
BEFORE PROCEEDING to the introduction and discussion of further
techniques of statistical analysis, time will be taken to examine certain
aspects of data acquisition. Such a digression, if it really is a digres
sion, is justified because the analysis of any set of data is dictated (to a
large extent) by the manner in which the data were obtained. The truth of
the foregoing statement will be illustrated many times throughout the
remainder of this book.
10.1 SOME GENERAL REMARKS
It has been well demonstrated in the preceding chapters that sta
tistics (as a science) deals with the development and application of
methods and techniques for the collection, tabulation, analysis, and
interpretation of data so that the uncertainty of conclusions based
upon the data may be evaluated by means of the mathematics of
probability. However, it should also be evident that there is some
thing more to statistics than the routine analysis of data using stand
ard techniques, For example, the reader should realize that the anal
yses are exact only if all the underlying assumptions are satisfied. Since
this is rarely true, much depends on the skill of the researcher in select
ing the method of analysis which best fits the circumstances of the
experimental situation being studied. Thus, it seems safe to say that
statistics is an art as well as a science.
10.2 WHAT IS MEANT BY "THE DESIGN OF AN EXPERI
MENT"?
Designing an experiment simply means planning an experiment so
that information will be collected which is relevant to the problem
under investigation. All too often data are collected which turn out to
be of little or no value in any attempted solution of the problem. The
design of an experiment is, then, the complete sequence of steps taken
ahead of time to insure that the appropriate data will be obtained in a
way which permits an objective analysis leading to valid inferences
with respect to the stated problem. Such a definition of designing an
experiment implies, of course, that the person formulating the design
clearly understands the objectives of the proposed investigation.
10.3 THE NEED FOR AN EXPERIMENTAL DESIGN
That some sort of design is necessary before any experiment is per
formed may be demonstrated by considering an example.
[244]
1O.4 THE PURPOSE OF AN EXPERIMENTAL DESIGN 245
Example 10.1
It is desired to determine the effect of gasoline and oil additives on
carbon and gum formation of engines. 1 Twenty additives are to be
tested in combination with a "control" gasoline and oil mixture. Eighty
similar engines are available for use in the experimental program.
As the problem is now stated it is far too general to permit the selec
tion, of a particular design. Many questions must be asked (and answers
obtained) before the statistician can propose a suitable design. Typical
questions are:
(1) How is the effect to be measured? That is, what are the char
acteristics to be analyzed?
(2) What factors influence the characteristics to be analyzed?
(3) Which of these factors will be studied in this investigation?
(4) How many times should the basic experiment be performed?
(5) What should be the form of the analysis?
(6) How large an effect will be considered important?
When we recognize that the foregoing questions are only a small sample
of those that might be asked, it is evident that much thought should be
given to the planning stage in any experimental investigation. In fact,
the importance of thus recommendation cannot be overemphasized.
10.4 THE PURPOSE OF AN EXPERIMENTAL DESIGN
The purpose of any experimental design is to provide a maximum
amount of information relevant to the problem under investigation.
However, it is also important that the design, or plan, or test program,
be kept as simple as possible. Further, the investigation should be con
ducted as efficiently as possible. That is, every effort should be made
to conserve time, money, personnel, and experimental material. For
tunately, most of the simple statistical designs are not only easy to
analyze but also are efficient in both, the economic and statistical
senses. For this reason, a statistician should be consulted in the early
stages of any proposed research project. He can often recommend a
simple design which is both economical and efficient.
Having said that the purpose of any experimental design is to pro
vide a maximum amount of information at minimum cost, it is evident
that the design of experiments is a subject which involves both sta
tistical methodology and economic analysis. A person planning an ex
periment should incorporate both of these features into his design.
That is, he should strive for statistical efficiency and resource economy.
However, an examination of books on statistical methods and the
design of experiments will seldom reveal many explicit references to
the cost aspects of the problem. This is unfortunate. On the other hand,
the subject of cost is implicit in most discussions of experimental design.
We have only to note the continual attempts to plan experiments using
1 Projects and Publications of the National Applied Mathematics Laboratories,
April through June, 1949, p. 79.
246 CHAPTER 1O, DESIGN OF EXPERIMENTAL INVESTIGATIONS
the smallest size sample possible, to realize that the cost aspect has not
been overlooked. Fortunately, as we have already observed, most
simple designs are both economical and efficient, and thus the statis
tician's efforts to achieve statistical efficiency usually also lead to
economy of experimentation.
10.5 BASIC PRINCIPLES OF EXPERIMENTAL DESIGN
It has been stated many times that there are three basic principles
of experimental design: replication, randomization, and local control.
Because of the fundamental nature of these concepts, each will be dis
cussed separately. Further, it is recommended that the reader strive for
as complete an understanding and appreciation of these ideas as pos
sible, for they will play a very important role in much of the remainder
of this book.
10.6 REPLICATION
By replication we mean the repetition of the basic experiment. The
reasons why replication is desirable are: (1) It provides an estimate of
experimental error which acts as a "basic unit of measurement" for
assessing the significance of observed differences or for determining the
length of a confidence interval. (2) Since, under certain, assumptions,
experimental error may be estimated in the absence of replication, it is
also fair to state that replication sometimes provides a more accurate
estimate of experimental error. (3) It enables us to obtain a more pre
cise estimate of the mean effect of any factor since cr-? = cr*/n. (In the
formula just quoted, o- 2 represents the true experimental error and n
the number of replications.)
It must be emphasized that multiple readings do not necessarily
represent true replication. This statement may best be substantiated
by an example.
Example 10.2
Two manufacturing processes are used to produce thermal batteries"
Sample batteries are obtained from each of two production lots, one
lot being produced by process A and the other by process B. The
batteries are then tested and the activated life of each battery is
recorded.
If an analysis of the above experiment were attempted, it would be
discovered that no valid estimate of error is available for testing the
difference between processes. The variation among batteries within
lots yields a valid estimate of error for assessing only the lot-to-lot
variability. True replication would require that batteries be tested
from each of several lots manufactured by each process. (NOTE: In
the example just given, the effects of lots and processes are said to be
confounded. This term will be discussed more fully a little later.)
Sometimes the absence of true replication is more easily recognized
than in Example 10.2. For instance, if multiple measurements of acti
vated life had been obtained by connecting several clocks to a single
10.7 EXPERIMENTAL ERROR AND EXPERIMENTAL UNITS 247
battery, the researcher would easily have recognized that the observed
data were not true replications but only repeated measurements on the
same experimental unit. Another example of the same type of spurious
replication (i.e., multiple measurements rather than true replication)
would be multiple determinations of the silicon content of a particular
batch of pig iron where the variability among processes was to be as
sessed.
10.7 EXPERIMENTAL ERROR AND EXPERIMENTAL
UNITS
In the preceding discussion of replication, the terms experimental
error and experimental unit were used. Because of their wide usage, it
is necessary to have a clear understanding of their meanings. An experi
mental unit is that unit to which a single treatment (which may be a
combination of many factors) is applied in one replication of the basic
experiment. The term experimental error describes the failure of two
identically treated experimental units to yield identical results.
At the risk of saying too much and thus confusing the reader, it is
my belief that some discussion of the preceding definitions is in order.
In one respect, the term "experimental error" is unfortunate, especially
the word ' 'error. " This word is probably a legacy from the physical sci
ences, particularly astronomy, where the investigators (observers) were
concerned with errors in both measurement and observation. However,
the influence of experimenters in both the biological and physical sci
ences should not be discounted entirely. The adoption of the word
"error" could just as easily be attributed to them, for they clearly
recognized the existence of errors of technique in the performance of
their experiments. But whatever the history of the word "error/ ' a
thoughtful examination of the definition of the terra "experimental
error" will reveal that its meaning to the statistician is much more
general. In each particular situation, it reflects: (1) errors of experi
mentation, (2) errors of observation, (3) errors of measurement,
(4) the variation of the experimental material (i.e., among experi
mental units), and (5) the combined effects of all extraneous factors
which could influence the characteristics under study but which have
not been singled out for attention in the current investigation.
There is another item related to the term experimental error which is
sometimes confusing to the statistical novice. This is the practice of the
professional statistician of referring to "the experimental error for
testing a particular effect." Such a phrase suggests that, in a given ex
periment, there may be more than one experimental error even though
examination of the assumed statistical model will reveal only one such
term. As confusing as this practice may be to the uninitiated, it serves
a useful purpose. As the reader progresses through the remainder of this
book, he will become more familiar with the way in which the expres
sion is used and thus, I hope, become more tolerant of what seems at
the moment to be an unwise use of words that have been carefully
248 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
defined. In an attempt to give a somewhat more specific defence at this
time, let me say that all the statistician is really doing is reminding you
of the fact that every statistic has its own standard error. Perhaps his
choice of w r ords is not the best, but it is a firmly entrenched part of the
language of experimental design. Thus, I strongly recommend that you
forgive the statistician his choice of words and that you concentrate on
the more important task of learning how and when to use statistical
methods.
Before terminating this discussion of experimental error, ways of
reducing its magnitude should be indicated. The following statements
are, of course, only general recommendations, for specific recommenda
tions can be made only "when a particular design problem is being con
sidered. Experimental error may usually be reduced by adoption of
one or more of the following techniques: (1) using more homogeneous
experimental material or by careful stratification of available material,
(2) utilizing information provided by related variates, (3) using more
care in conducting the experiment, (4) using a more efficient experi
mental design.
10.8 CONFOUNDING
In Section 10.6,, the word "confounded" was introduced to describe
a certain phenomenon which is fairly common in experimentation.
Since this phenomenon is so important in the design of experiments, it
is appropriate that time be taken to investigate and describe it more
thoroughly. This will best be done through the use of examples.
Example 10.3
A chemist has developed a new synthetic fertilizer and wishes to
compare it with an established product. He contacts a nearby university
and they agree to run an experiment on two available experimental
plots. The established product will be applied to one plot of ground and
the experimental product to the other. The characteristic to be meas
ured and used as the index of performance will be the yield (converted
to bushels per acre) of a specified cereal crop. However, when the two
yields are compared, we are unable to say how much of the difference
is due to fertilizers and how much is due to inherent differences (in fer
tility, soil type, etc.) between the two plots. That is, any comparison of
fertilizers is said to be confounded with a comparison of plots or, in
slightly different words, the effects of fertilizers and plots are con
founded.
Example 10.4
An analyst is engaged in determining the percentage of iron in chemi
cal compounds. Two different procedures are to be compared. The
analyst takes a sample of the first chemical compound and makes a
determination of the iron content using procedure A. Then he makes a
determination using procedure J5. This sequence (that is, first A and
then jB) of steps is repeated several times, each time on a new sample
from a different compound. But here again, as in Example 10.3, we are
1O.9 RANDOMIZATION 249
troubled by the existence of confounding. Any comparison of the two
procedures (A and B) will be confounded with a comparison of the first
and second determinations made (on each compound) by the analyst.
That is, if there is any improvement in technique (due to a learning
process) from the first to the second determination, this effect will be
confounded with the difference between procedures.
Examination of the preceding examples will show that the word "con
founded^ is simply a synonym for "mixed together." That is, two (or
more) effects are said to be confounded in an experiment if it is impos
sible to separate the effects when the subsequent statistical analysis is
performed.
Since one of the purposes of experimental design is to provide unam
biguous results, it would seem almost obvious that a good design should
avoid confounding. It is, therefore, disconcerting to the uninitiated to
learn that the statistician frequently deliberately introduces confound
ing into a design. However, as you will see later, such a procedure is
not followed indiscriminately. When confounding is introduced into a
design it is done so for a good reason, and the reason, is, as often as not,
to achieve economy through reduction of the size of the experiment.
10.9 RANDOMIZATION
It was noted in Section 10.6 that replication provides an estimate of
experimental error which can be used for assessing the significance of
observed differences. That is, replication makes a test of significance
possible. But what makes such a test valid? We have seen that every
test procedure has certain underlying assumptions which must be satis
fied if the test is to be valid. Perhaps the most frequently invoked as
sumption is the one which states that the observations (or the errors
therein) are independently distributed. How can we be certain that this
assumption is true? We cannot, but by insisting on a random sample
from a population or on a random assignment of treatments to the ex
perimental units, we can proceed as though the assumption is true.
That is, randomization makes the test valid by making it appropriate to
analyze the data as though the assumption of independent errors is
true. Note that we have not said randomization guarantees independ
ence, bxit only that randomization permits us to proceed as though
independence is a fact. The reason for this distinction should be clear;
Errors associated with experimental units that are adjacent in space
or time will tend to be correlated, and all that randomization does is to
assure us that the effect of this correlation on any comparison among
treatments will be made as small as possible. Some degree of correlation
will still remain, for no amount of randomization can ever eliminate it
entirely. That is, in any experiment, true and complete independence of
errors is an ideal that can never be achieved. However, such independ
ence should be sought, and randomization is the best technique de
vised so far to attain the desired end.
Sometimes the concept of randomization is introduced as a device
for "eliminating" bias. To illustrate the thinking back of this approach,
250 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
consider again Example 10.4. There, any comparison of procedures A
and B would be biased in favor of B if a learning effect existed. How
ever, if each time a new compound "was to be investigated the analyst
had decided at random which procedure to use first, the bias would
have been reduced, perhaps even eliminated. But even more would
have been accomplished. If there were other biases operating, these
would also have had their effects eliminated (or at least reduced) by
the randomization. That is, by randomly assigning treatments to the
experimental units, we try to make certain that treatments will not be
continually favored or handicapped by extraneous sources of variation
over which the experimenter has no control or over which he chooses
not to exercise control. In other words, randomization is like insurance;
it is always a good idea, and sometimes it is even better than we expect.
Regardless of the foregoing arguments in favor of randomization,
there have been (in the past) persons who have spoken out in favor of
systematic (nonrandom) designs. "Can we not," they ask, "obtain a
more accurate measurement of differences among treatments if such
treatments are applied to the experimental units in a systematic man
ner?" The only honest answer to this query is, "Possibly." Why, then,
does the statistician insist on randomization? The reason is, of course,
the same as expressed earlier: It is because the statistician wishes to
make certain inferences from the observed data and he desires to at
tach a measure of reliability to these inferences. If randomization is
not employed, the quoted measure of reliability may be biased. Fur
ther, any inference would be unsupported by a meaningful probability
statement. (NOTE: The reader is reminded of the discussion of judg
ment versus random samples presented in Section 4.2.)
There are, of course, situations in which complete randomization is
either impossible or uneconomical. The statistician should not, there
fore, adopt the unrelenting position of insisting on complete randomiza
tion in every case. On the other hand, neither should he agree to the
use of a completely systematic design, for the experimenter must
reconcile himself to the fact that some degree of randomization is re
quired for the valid application of most statistical analyses. Clearly,
some intermediate position between the two extremes 2 of complete
randomization or a strictly systematic design is often most realistic.
Once the experimenter and the statistician recognize one another's
problems, a compromise plan can usually be found which is mutually
satisfactory.
10.10 LOCAL CONTROL
In Section 10.5, it was stated that the three basic principles of ex
perimental design are replication, randomization, and local control.
2 The question of which is better, a systematic or a randomized design, has
never been completely settled. Most likely it never will be settled. Most designs
in common use today involve both systematic and random elements, and this
seems a reasonable state of affairs. For the person who wishes to pursue this
point farther, the literature offers many papers discussing the argument, both
pro and con. See references (2, 27, 35, 36, 44).
1O.11 BALANCING, BLOCKING, AND GROUPING 251
The first two of these basic principles have already been discussed and
it is now appropriate that time be devoted to the third.
In one sense, local control is synonymous with experimental design.
However, this interpretation of experimental design is very narrow, and
not consistent with our earlier definition. If we agree, then, that experi
mental design is as defined in Section 10.2, then local control is only a
part of the total complex. In this sense, local control refers to the
amount of balancing, blocking, and grouping of the experimental units
that is employed in the adopted statistical design. It was observed
earlier (Section 10.9) that replication and randomization make a valid
test of significance possible. What, then, is the function of local con
trol? The function, or purpose, of local control is to make the experi
mental design more efficient. That is, local control makes any test of
significance more sensitive or, in the language of Section 7.1, it makes
the test procedure more powerful. This increase in efficiency (or sensi
tivity or power) results because a proper use of local control will reduce
the magnitude of the estimate of experimental error. (NOTE: The
reader should recognize that local control can be exerted in several
ways. The more common methods have been suggested above and in
the last paragraph of Section 10.7.)
10.11 BALANCING, BLOCKING, AND GROUPING
In the preceding section, the terms balancing, blocking, and grouping
were introduced in connection with the principle of local control.
Rather than leave these words undefined, a few sentences of explana
tion will be given so that the researcher will understand what is im
plied. Actually, it is possible to say that the three terms are synony
mous. However, in this text we shall use them to describe different
aspects of design philosophy. It is hoped that this will not lead to con
fusion when other references are consulted.
By grouping will be meant the placing of a set of homogeneous exper
imental units into groups in order that the different groups may be
subjected to different treatments. These groups may, of course, con
sist of different numbers of experimental units.
Example 10.5
A pharmaceutical company is investigating the comparative effects
of three proposed compounds. The experiment will consist of injecting
rats with the compounds and recording the pertinent reaction. A litter
consisting of 11 rats (experimental units) is available. Each of the 11
rats is assigned at random to one of three groups subject only to the
restriction that the three groups contain 4, 4, and 3 rats, respectively.
The animals in the first group are then injected with compound A,
those in the second group with compound B, and those in the third
group with compound C.
By blocking will be meant the allocation of the experimental units to
blocks in such a manner that the units within a block are relatively
homogeneous while the greater part of the predictable variation among
252 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
units has been confounded with the effect of blocks. That is, using the
researcher's prior knowledge concerning the nature of the experimental
units, the statistician can design the experiment in such a way that
much of the anticipated variation will not be a part of experimental
error. In this way, a more efficient design is provided.
Example 10.6
Consider again the problem outlined in Example 10.5. This time,
however, let us assume that 12 rats are available and that the pedigrees
show 6 of them are from litter X, 3 are from litter F, and 3 from litter Z.
Since it may well be expected that rats in the same litter will perform
more nearly alike than rats from different litters (due to inherited
characteristics), it would seem natural to form three blocks. The first
block would contain the 6 rats from litter X, the second block would
contain the 3 rats from litter Y, and the third block would contain the 3
rats from litter Z. The three treatments (A, B 7 and C) would then be
assigned at random to the rats within blocks. Since each rat is subjected
to only one treatment, the block containing 6 rats would undoubtedly
end up with 2 rats seeing treatment A, 2 seeing treatment B, and 2
seeing treatment C. The other two blocks would have single rats seeing
each treatment.
By 'balancing will be meant the obtaining of the experimental units,
the grouping, the blocking, and the assignment of the treatments to the
experimental units in such a way that a balanced configuration results.
(Circular though the preceding definition is, I feel it projects the
thought I wish to impart. Consequently, I hope you will forgive the
poor logic.) It should be clear that we can have little or no balance,
partial balance, approximate balance, or complete balance in any par
ticular design. For instance, Example 10.5 illustrates a case of approxi
mate balance, while Example 10.6 might be construed as an illustra
tion of partial balancing. Rather than go on to manufacture further
examples at this time, let us defer the matter until later. As you pro
gress through the chapters on various designs which follow, it will
become abundantly clear that the statistician continually strives for
balanced designs. Thus, examples of completely balanced designs will
be available in excess.
10.12 TREATMENTS AND TREATMENT COMBINATIONS
Several times in the preceding sections, the word "treatments" has
been used with little or no explanation. Just what is meant by this
word? Like so many other terms in statistics, the word "treatments"
entered the literature because of its use in agronomic experimentation.
However, the word "treatments" (like "blocks" and "plots") has long
since lost its strict agronomic connotation. In fact, the three phrases
mentioned in the preceding sentence are now an accepted part of the
language of statistics, regardless of the area of application.
To the statistician, the word treatment (or treatment combination)
implies the particular set of experimental conditions which will be im-
10.13 FACTORS, FACTOR LEVELS, AND FACTORIALS 253
posed on an experimental unit within the confines of the chosen design.
By way of explanation, several illustrations will now- be given :
(1) In agronomic experimentation, a treatment might refer to:
(a) a brand of fertilizer, (b) an amount of fertilizer, (c) a depth
of seeding, or (d) a combination of (b) and (c). The latter
example would more properly be termed a treatment combi
nation.
(2) In animal nutrition experimentation, a treatment might refer
to: (a) the breed of sheep, (b) the sex of the animals, (c) the
sire of the experimental animal, or (d) the particular ration
fed to an animal.
(3) In psychological and sociological studies, a treatment might
refer to: (a) age, (b) sex, or (c) amount of education.
(4) In an investigation of the effects of various factors on the
efficiency of washing clothes in the home, the treatments were
various combinations of: (a) the type of water (hard or soft),
(b) temperature of water, (c) length of wash time, (d) type of
washing machine, and (e) kind of cleansing agent.
(5) In an experiment to study the yield of a certain chemical proc
ess, the treatments might be all combinations of: (a) the tem
perature at which the process was operated and (b) the
amount of catalyst used.
(6) In a research and development study concerned with batteries,
the treatments could be various combinations of: (a) the
amount of electrolyte and (b) the temperature at which the
battery was activated.
Many more examples could be cited from every field in which experi
mentation is performed. However, later chapters will abound with
such examples. Thus, it seems best that we move on to other matters.
10-13 FACTORS, FACTOR LEVELS, AND FACTORIALS
In any discussion of experimental design, the word "factorial" is
almost certain to be heard. Frequently, the reference is to a "factorial
design." However, this is actually a misnomer. There is no such thing
as a factorial design. The adjective "factorial" refers to a special way
in which treatment combinations are formed and not to any basic type
of design. Thus, if a randomized complete block design 3 has been
selected and the treatment combinations are of a factorial nature, a
more correct expression would be "a randomized complete block design
involving a factorial treatment arrangement." Some writers, such as
Yates (46), have recognized this situation and they speak of factorial
experiments rather than factorial designs. This shift in terminology,
while in the proper direction, does not completely resolve the difficulty
since the word "experiment" seems to imply that survey data are to be
excluded. To avoid any such implication, we shall speak not of factorial
* See Chapter 12 for a definition of this type of design.
254 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
designs nor of factorial experiments, but simply of factorials. It is to be
understood, of course, that this is only an abbreviation for a more
lengthy expression describing the nature of the treatments.
Having introduced the subject of factorials, it is desirable that spe
cific terms be defined in an explicit manner. This will now be done.
In most investigations, the researcher is concerned with more than one
independent variable and in the changes that occur in the dependent
variable as one or more of the independent variables are permitted to
vary. In the language of experimental design, an independent variable
is referred to as a factor. Referring to the illustrations in the preceding
section, it is noted that five factors were listed for the home washing
study, while the battery study involved only two factors. The reader
can easily find many more examples of investigations involving several
factors by consulting various technical journals.
Before proceeding to the definition of the next term arising in con
nection with factorials, it will be wise to indicate the generally accepted
notation used to represent factors. Most writers use lower case Latin
letters to represent factors. As an illustration, the five factors in the
home washing experiment might be represented by
OT = type of washing machine
a = kind of cleansing agent
b type of water
c = temperature of water
d = length of wash time.
A second illustration is provided by an investigation conducted by
Ratner (40). His experiment involved a study of how long it took to per
form a certain move, and the factors investigated were
d = distance
w = weight
o = operator-pair
It was mentioned earlier that the researcher is generally interested
in experimental results (observations on the dependent variables) as
one or more factors are allowed to vary. It will be seen in Ratner's study
that he considered 3 distances (d^, d 2 , <3 3 ), 10 weights (w i} - - - , 1^10),
and 4 operator-pairs (o x , o 2 , o 3 , o 4 ). In the home washing experiment,
the investigator used 2 types of machine, 2 kinds of cleansing agent, 2
types of water, 2 temperatures of water, and 2 lengths of wash time.
These various values, or classifications of the factors, are known as the
levels of the factors. That is, there were 10 levels of weight, 3 levels of
distance, and 4 levels of operator-pairs in Ratner's experiment. In the
home washing study, each factor appeared at 2 levels. These two ex
amples should indicate that the word "level" is a very general term
which may be applied in many varied situations. Ratner's investiga-
10.13 FACTORS, FACTOR LEVELS, AND FACTORIALS
255
tion of move times provides an excellent example of this diversity, for
tlie 3 levels of distance (6, 12, and 18 inches) are values of a continuous
variable, while the 4 levels of operator-pairs (i.e., 4 distinct pairs of
operators formed from 8 individuals) are classifications of a qualitative
variable.
Since so many experiments involve factorial treatment arrange
ments, it is necessary that some notation be adopted to represent the
various treatment combinations. Unfortunately, several systems of
notation appear in the literature. These are summarized in Table 10.1
TABLE 10.1-niustrations of Notations Used To
Represent Factorial Treatment Combinations
HF^f/ir* 4-rvt m-*i -f-
Method
j_ teatment
Combination
I
II
III
IV
V*
1
Q>\O\G\
111
daboCQ
000
(1)
2
Qf\b\Ci
112
dob^c^
001
c
3
a\b\cz
113
ao&o2
002
c*
4
CLib^Ci
121
dobiCQ
010
b
5
CLlbzCz
122
G^biCi
Oil
be
6
ciibzCs
123
aobiCs,
012
be 2
7
CL^b^Ci
211
ciibQCQ
100
a
8
dJ}\C2
212
CLlboCi
101
ac
9
aJb^Cz
213
dibaCz
102
ac 2
10
dob^Ci
221
a\b\CQ
110
ab
11 .
CLzbtCz
222
a^biCi
111
abc
12
CLzb^Cz
223
GlbiCz
112
abc*
* In this representation, the absence of a letter implies that the factor which it represents
is at the lowest level. In general, the exponents on the letters agree with the subscripts
used in Method III. Thus, ao&tffc becomes ab l c* = bc*. The symbol (1) is used to signify that
each factor is at its lowest level, that is, oo&o^o is equivalent to a6 c=* (1).
for a case involving 12 treatment combinations where the 12 combina
tions were formed from 2 levels of factor a, 2 levels of factor 6, and 3
levels of factor c. In this rep resent at ion, using Method I as an ex
ample, the symbol a.-^-c* (i=l, 2; j=l, 2; &=1, 2, 3) represents the
treatment combination formed by using the iih level of factor a, the
jth level of factor b, and the fcth level of factor c.
There is another item of terminology that should be mentioned in
the present context. This item is best explained by example. The fac
torial arrangement of the treatments used in Table 10.1 would be re
ferred to by the statistician as a 2X2X3 factorial. Similarly, Ratner's
investigation, would be termed a 3X10X4 factorial, while the home
washing study was a 2X2X2X2X2 = 2 5 factorial.
Before leaving (for the time being) the subject of factorials, it is only
fair that the reader be warned of a double use of certain symbols which
could (but should not) lead to confusion. The situation is as follows:
256 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
It is common practice to use the letters a, 6, c, to denote not only
the various factors but also the number of levels of the factors. For
example, a statistical model might be written as
Y# = M + <*i + ftj + aj; i = 1, - - - , a (10. 1)
j = 1, , *
where
fj, = mean effect
<xt = effect of the ith level of factor a
$3 = effect of thejth level of factor b
eij = experimental error
and
!>;= Z& =
=l y=i
while the 6/ are NID (0, cr) . In this and similar situations, the decision
to use a and 6 to denote not only the factors but also the number of
levels of each factor should not lead to any confusion. The sense in
which a letter is being used in any particular instance should always
be perfectly clear from the context.
10.14 EFFECTS AND INTERACTIONS
Whenever a statistician undertakes the design of an experiment, he
must first ascertain the objectives of the researcher. Frequently, the
objectives may be very simple. For example, the researcher may wish
to determine the effect on the yield of a chemical reaction of changing
the operating temperature while all other factors (variables) are held
constant at predetermined levels. On the other hand, he may have no
interest whatsoever in temperature; his concern might be only with
pH. In this case, an experiment would be planned to determine the
effect of pH under the restriction that all other factors (including tem
perature) are held constant.
Experiments such as those referred to in the preceding paragraph are
fine if the effects of pH and temperature (on the response variable) are
independent. However, if we know that the factors are interdependent,
or if we are doubtful of the validity of an assumption of independence,
then an experiment which estimates both main effects and interactions
should be recommended. Such an experiment would, of course, utilize
a factorial arrangement of the treatments.
Example 10.7
It is suggested that the effects of pTEL and temperature on the yield of
a certain chemical reaction are not independent. It is, therefore, recom
mended that a design be adopted which utilizes treatment combinations
10.14 EFFECTS AND INTERACTIONS 257
formed by combining different levels of the two factors involved. It is
decided that two levels of each factor will be investigated. Denoting
plS. by a and temperature by b, the four treatment combinations might
be:
= pEL of 4.0 and a temperature of 30C.
= pTEL of 4.0 and a temperature of 40C.
= pH of 4.4 and a temperature of 30C
of 4.4 and a temperature of 40C.
Before we can say how the performance of an experiment involving a
factorial set of treatment combinations will help answer our questions
concerning independence of the factors, it will be necessary to define
certain terms. These terms (effect, main effect, and interaction) have
already been used without explanation. The time has now arrived when
specific definitions must be given.
We shall consider first a 2 2 factorial such as the one used in Example
10.7. If we agree that the symbols a^bj (i = Q, l;y = 0, 1) can represent
not only the treatment combinations but also the average yields from
all experimental units subjected to the similarly designated treatment
combinations, it is possible to define effect, main effect, and interaction
as noted below. (NOTE : To avoid complicating the discussion, it has
been assumed that each average yield was obtained from the same
number of experimental units.)
Effect of a at level b of 6 = a-Lbo a &o (10.2)
Effect of a at level &i of b = a^bi a 6i (10.3)
Main effect of a [(ai&o #o&o) + (#i&i a &i)]/2
= G*i - ao)(Si + 6o)/2 (10.4)
= A.
Similarly,
Effect of b sit level a Q of a = ao&i a Q b (10.5)
Effect of 6 at level a of a = aj>^ #160 (10.6)
Main effect of 6 = [(a 5i # &o) + (^1^1 ~~ ^160) 3/2
= (ai + a )(6i - 6o)/2 (10.7)
= B.
If a and 6 were acting independently, the effect of a at 6 and the effect
of a at 61 should be the same. (A similar statement holds for the
effects of b at ao and ai.) Thus, any difference in these two effects is a
measure of the degree of interdependence between the factors, that is,
of the extent to which a and b interact. Accordingly, we define the
interaction between a and b by
258
CHAPTER 1O, DESIGN OF EXPERIMENTAL INVESTIGATIONS
AB =
(10.8)
If the symbols used in the preceding definitions are simplified by re
placing ao and 60 by unity, and a\ and 61 by a and 6, the effects and in
teractions may be defined by
4M = (a + 1)(6 + 1) (10.9)
2,4 = (a - 1)(5 + 1) (10.10)
2J5 = (a + 1)(6 - 1) (10.11)
2^LJ3 = (a !)(& 1) (10.12)
where M represents the mean effect (i.e., the mean yield of all experi
mental units).
Example 10.8
Let us assume that an experiment has been performed involving
treatments such as described in Example 10.7. To illustrate the compu
tation of main effects and interactions; three hypothetical cases will be
examined.
II
III
O-Q
bo
61
63
67
69
73
61
63
67
69
78
Z>0
Jl
63
67
69
70
Case I: 71^ = 68, A =4, B = 6, and AB = 0.
Case II: M = 69.25, A = 6.5, B = 8.5, and
Case II I: M = 67.25, A = 2.5, B = 4.5, and A B 1.5.
Having defined and illustrated (for a 2 2 factorial) the concepts of
effects, main effects, and interactions, it is appropriate that an attempt
be made to put these ideas into words rather than symbols. However,
the reader is reminded (again) that the understanding of a concept is
much more important than the memorization of any definition, whether
it be in words or in mathematical symbolism. With that reminder, let
us now attempt definitions of the two terms, "interaction" and "main
effect." Utilizing the earlier definitions and the illustrations in Example
10.8, we may say that:
(1) Interaction is the differential response to one factor in combination
with varying levels of a second factor applied simultaneously. That
is, interaction is an additional effect due to the combined influence
of two (or more) factors.
(2) The main effect of a factor is a measure of the change in the response
7 O.I 4 EFFECTS AND INTERACTIONS 259
variable to changes in the level of the factor averaged over all levels of
all the other factors.
It should be clear that the concepts described as effects and interac
tions will also be present in situations involving more than two factors.
For example, in a case involving four factors, there would be four main
effects, six two-factor interactions involving the combined effect of two
factors averaged over the other two factors, four three-factor inter
actions involving the combined effect of three factors averaged over
the one remaining factor, and one four-factor interaction involving the
combined effect of all four factors. Extensive discussion of these ideas
will be deferred until a later chapter.
Before terminating the discussion of effects and interactions, how
ever, two additional topics will be mentioned* One is a convenient
method of determining the effects in 2 n factorials; the other is the
definition of effects and interactions for 3 n factorials.
To illustrate the method of calculating effects in 2 n factorials, let us
consider a 2 3 factorial. Using the abbreviated notation for treatment
combinations given in Table 10.1, and letting these symbols also repre
sent the average yields of experimental units subjected to the similarly
designated treatment combinations, the main effects and interactions
may be found by adding and subtracting yields according to the signs
given in Table 10.2. It can easily be verified that this procedure is
simply a tabular device for calculating the effects and interactions
defined by
X = (a 1)(6 l)(c l)/2 2 (10.13)
where the sign in each set of parentheses is plus if the corresponding
capital letter is not contained in X and negative if it is contained in X,
and the right-hand side is to be expanded and the yields substituted for
the appropriate treatment combination symbols. Equation (10.13) may
be extended to the 2 n factorial case by simply adding more multipli-
TABLE 10.2-Schematic Representation of Effects and
Interactions in a 2 3 Factorial
Treatment Combination
Effect
C\T
(1) a b ab c ac be abc
Interaction
+ + + + + + + +
SM
_|_ _|_ _|_ -|-
4:A
____^_|__-_-|-~|_
4B
_|_ _(__(__ _]_
4AB
__[__^_^-_|_
4C
_|_ _)- + _|-
4 AC
^_1____ + +
4BC
- + + - + -- +
4ABC
260 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
cative factors as shown in Equation (10.14),
X = [O 1)(6 l)(c 1)0* 1) j/2"- 1 . (1O.14)
Wlien factors are investigated at only two levels, the best the re
searcher can do (apart from a simple test of significance) is to deter
mine: (1) whether the effect of a factor is positive or negative and
(2) whether the factors are independent. However, when factors are
investigated at more than two levels, the researcher can probe more
deeply. He now has the opportunity to see if the effect of a factor is
linear or nonlinear. In most experimental work, this is a very impor
tant item of information, and thus the researcher should give serious
consideration to factorials involving more than two levels of the fac
tors when planning an investigation.
If an experiment is designed involving two factors, each at three
levels, the main effects and interactions may be used to study the non-
linearity of the response variable. Rather than go into excessive detail
at this time, only the pertinent formulas will be presented. In these for
mulas we have again used the symbols a^bj (i = 0, 1, 2; j = Q, 1, 2) to
represent both the treatment combinations and the yields from the
treatment combinations.
Linear effect of a = A L = (a 2 a )(6o + 61 + 2 )/3 (10.15)
Quadratic effect of a = A Q = (a 2 2#i + # )(o + &i +- t>z)/6 (10.16)
Linear effect of b = B L = (<z + #1 + # 2 )(&2 o)/3 (10.17)
Quadratic effect of 6 = B Q = (a + ai + a 2 )(Z> 2 2i L + J )/6 (1O.18)
Linear X Linear interaction = A^B^ = (a 2 # )(&2 ~~ *o)/2 (10.19)
Linear X Quadratic interaction
(10.20)
= A L B Q = (a* a )( 2 26i + 6 )/4
Quadratic X Linear interaction
(10.21)
Quadratic X Quadratic interaction
(10.22)
= A Q B Q = (a* 2ai + a )(6 2 26i + 6 )/8
Example 10.9
Consider an experiment similar to that described in Example 10.7
but involving three levels of pH and three levels of temperature. As in
Example 10.8, three cases will be considered.
I
7o <Zl
10.15 TREATMENT COMPARISONS
II III
10
13
16
13
16
19
16
19
22
60
61
62
Case I: ^, = 6,
and j4.QjBQ = 0.
, o, -^IQ
= 3 and
r T T . j . Q
J. J. -L . -tTL L O ,
22
10
14
25
13
17
30
18
22
10
12
11
14
17
21
19
25
35
J and
1/8.
It should IOQ noted that the "no interaction" result in cases I and II
could have been predicted by observing that the pattern of differences
between yields at varying levels of b is the same for each level of a.
(NOTE: We could just as easily have examined the differences between
yields at varying levels of a for each level of &).
From the preceding discussion, it should be evident that there is a
great deal to be said about effects and interactions. As a matter of fact,
what started out to be a short section exposing the reader to general
concepts has grown (necessarily, I believe) into a rather detailed dis
cussion of the topic. On tlie other hand, the surface has only been
scratched. There is much more that can be said. Some of this additional
material will be discussed in later chapters, while the remainder will be
left to books devoted to experimental design. For those who wish, to
read further on these topics, the following references are recommend
ed: Cochran and Cox (13), Cox (14), Davies (16), Federer (20), Finney
(21 and 22), Kempt home (28), Quenouille (39), and Yates (46).
10.15 TREATMENT COMPARISONS
In most experiments involving several treatments, the researcher will
be interested in certain specific comparisons among the treatment
means. To aid in making such, comparisons, the statistician finds it
convenient to talk in terms of "contrasts," Algebraically, a contrast
among the quantities TI, - - - , T& (where 2\- is the sum of nt observa
tions) is defined by
c k5 T k
(10.23)
where
(10.24)
If each ni = n, that is, if each Tt is the sum of the same number of ob-
262 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
servations, then the necessary condition for a contrast reduces to
, = 0. (10.25)
Example 10.10
Consider an experiment involving batteries in which four treatments
are to be investigated. The four treatments happen to be four different
electrolytes. However, it is noted that electrolytes No. 1 and No. 2 are
quite similar in composition, that No. 3 and No. 4 are also similar, but
that Nos. 1 and 2 differ considerably from Nos. 3 and 4. It would, then,
be reasonable to plan comparisons of: (1) treatments 1 and 2 versus treat
ments 3 and 4, (2) treatment 1 versus treatment 2, and (3) treatment
3 versus treatment 4. Assuming that 20 batteries (experimental units)
are used and that they are allocated to the treatments in the ratio
4:2:5:9, what would be the form of the contrasts for the selected treat
ment comparisons? Denoting the treatment totals by T^i = l y 2, 3 7 4),
the desired contrasts are:
= 7r z + 7T 2 - 3T 3 3T*
C 2 - (l)Ti + (-2)2-2 + (0)2*8
C 3 - (0)2-1 + (0)2*a + (9)T, + (-5) 2V
One might ask how we obtained the coefficients c*j (i = 1, 2, 3, 4; j" =1,2,
3) used in the above comparisons. A short explanation at this moment
should serve to clear up any difficulties. Consider the case of compari
son Ci: What we are actually attempting to do is to compare the mean
of 6 observations (4 + 2) with the mean of 14 observations (5+9). It is,
of course, necessary to adjust for the spurious weighting given by our
comparison of treatment totals based on unequal numbers of ob
servations. Since the smallest integer which may be divided evenly by
both 6 and 14 is 42, we see that 7 and 3 are the indicated weights to
be used if our comparison is to be unaffected by the differing numbers
of observations associated with the various treatments. The remaining
coefficients are found in a like manner.
Example 10.11
Consider a research situation similar to that described in Example
10.10, but involving five treatments. Suppose that four batteries are
allocated to each treatment. If treatment No. 2 represents a commonly
used electrolyte, while Nos. 1, 3, 4, and 5 are newly developed electro
lytes in which Nos. 1 and 3 are of type A and Nos. 4 and 5 are of type
B, the contrasts specified in Table 10.3 are appropriate for the obvious
treatment comparisons.
Let us now take note of another item of importance. If two con
trasts,
(10.26)
10.15 TREATMENT COMPARISONS 263
TABLE 10.3-Symholic Representation of the Contrasts for the
Treatment Comparisons Specified in Example 10.11
Electrolyte
Contrast
1 2
3 4
5
Ci
1 +4
i i
1
C 2 -
+ 1
-4-1 ~1
1
C 3
+ 1
1
C 4
+1
1
and
C_ ^T"* I ,- T"' F I ^- HT^ / "1 f~\ O *7\
q Clq**- 1 ~T~ ^2^-t 2 ~T~ * * " ~T~ Ckq^- ky \1-\J . I )
are such that
]C **pc f = (p^ g), (10.28)
then the contrast C P is orthogonal to the contrast C ff . (NOTE: It is
common practice to speak of orthogonal contrasts or orthogonal treat
ment comparisons.) If Ui = n (for all i), the orthogonality condition
reduces to
23<^<^=0, pr*q. (10,29)
The reader can easily verify that the contrasts specified in Examples
10.10 and 10.11 are, in each case, orthogonal. In addition, the percep
tive student will have noted that the effects and interactions discussed
in Section 10.14 were also orthogonal contrasts.
At this time ? the following question might well be asked, namely,
"Are orthogonal contrasts better than nonorthogonal contrasts?" In
tuitively, orthogonal contrasts seem to be preferable. (NOTE : Actually,
they are preferable if one wishes the estimates derived from the dif
ferent contrasts to be uncorrelated.) However, occasionally it is desir
able to design an experiment with the expressed intent of analyzing a
set of nonorthogonal contrasts. In such cases, the probability state
ments accompanying the associated tests of significance are of an am
biguous nature (due to the correlation between the contrasts), and
much care should be exercised in interpreting the experimental results.
One final remark needs to be made and then we may move on to
another topic. The remark is the following: Regardless of the desira
bility of orthogonal contrasts, the statistician should not let his prefer
ence for such a state of affairs override the needs of the researcher. By
this is meant that, as nice as it is to have a set of orthogonal contrasts,
264 CHAPTER 1O, DESIGN OF EXPERIMENTAL INVESTIGATIONS
only those contrasts which are meaningful to the researcher should be
analyzed.
10.16 STEPS IN DESIGNING AN EXPERIMENT
Each statistician has his own list of steps which he follows when
designing an experiment. However, a comparison of various lists
reveals that they all cover essentially the same points.
According to Kempthorne (28), a statistically designed experi
ment consists of the following steps :
(1) Statement of the problem.
(2) Formulation of hypotheses,
(3) Devising of experimental technique and design.
(4) Examination of possible outcomes and reference back to the reasons
for the inquiry to be sure the experiment provides the required in
formation to an adequate extent.
(5) Consideration of the possible results from the point of view of the
statistical procedures which will be applied to them, to ensure that
the conditions necessary for these procedures to be valid are satis
fied.
(6) Performance of experiment.
(7) Application of statistical techniques to the experimental results.
(8) Drawing conclusions with measures of the reliability of estimates of
any quantities that are evaluated, careful consideration being
given to the validity of the conclusions for the population of objects
or events to which they are to apply.
(9) Evaluation of the whole investigation, particularly with other in
vestigations on the same or similar problems. 4
In a later section, these steps will be illustrated through the considera
tion of some design problems.
Since the designing of an experiment or the planning of a test pro
gram is such an important part of any investigation, the statistician
must make every effort to obtain all the relevant information. This
will usually require one or more conferences with the researcher, and
the asking of many questions. It has been my experience that the
amount of time consumed in this phase can be materially reduced if, at
the preliminary meeting between the researcher (e.g., a development
engineer) and the statistician, time is taken to explore the relationship
between research and/or development experimentation and the sta
tistical design of experiments. (NOTE: Frequently, there is a formid
able communications barrier which must be overcome.) One of the
best ways to convince the researcher of the need for the multitude of
questions posed by the statistician is to give him (in the first meeting)
a "check list" which specifies various stages in the planning of a test
program. (An even more efficient arrangement if you are the statisti
cian in an industrial organization is to distribute copies of such a list to
all persons who may at some time have need of your services.) One
4 O. Kempthorne, The Design and Analysis of Experiments, John Wiley and
Sons, Inc., New York, 1952, p. 10.
10.16 STEPS IN DESIGNING AN EXPERIMENT 265
such list, prepared by Bicking (3), is reproduced below for your con
sideration.
Check List for Planning Test Programs
A. Obtain a clear statement of the problem
1. Identify the new and important problem area
2. Outline the specific problem within current limitations
3. Define exact scope of the test program
4. Determine relationship of the particular problem to the whole re
search or development program
B. Collect available background information
1. Investigate all available sources of information
2. Tabulate data pertinent to planning new program
C. Design the test program
1. Hold a conference of all parties concerned
a. State the propositions to be proved
b. Agree on magnitude of differences considered worthwhile
c. Outline the possible alternative outcomes
d. Choose the factors to be studied
e. Determine the practical range of these factors and the specific
levels at which tests will be made
f . Choose the end measurements which are to be made
g. Consider the effect of sampling variability and of precision of test
methods
h. Consider possible inter-relationships (or "interactions") of the
factors
i. Determine limitations of time, cost, materials, manpower, instru
mentation and other facilities and of extraneous conditions, such,
as weather
j. Consider human relation angles of the program
2. Design the program in preliminary form
a. Prepare a systematic and inclusive schedule
b. Provide for step-wise performance or adaptation of schedule if
necessary
c. Eliminate effect of variables not under study by controlling,
balancing, or randomizing them
d. Minimize the number of experimental runs
e. Choose the method of statistical analysis
f. Arrange for orderly accumulation of data
3. Review the design with all concerned
a. Adjust the program in line with comments
b. Spell out the steps to be followed in unmistakable terms
D. Plan and carry out the experimental work
1. Develop methods, materials, and equipment
2. Apply the methods or techniques
3. Attend to and check details; modify methods if necessary
4. Record any modifications of program design
5. Take precautions in collection of data
6. Record progress of the program
266 CHAPTER 10, DESIGN OF EXPERIMENTAL INVESTIGATIONS
E, Analyze the data
1. Reduce recorded data, if necessary, to numerical form
2. Apply proper mathematical statistical techniques
F. Interpret the results
1. Consider all the observed data
2. Confine conclusions to strict deductions from the evidence at hand
3. Test questions suggested by the data by independent experiments
4. Arrive at conclusions as to the technical meaning of results as well
as their statistical significance
5. Point out implications of the findings for application and for further
work
6. Account for any limitations imposed by the methods used
7. State results in terms of verifiable probabilities
G* Prepare the report
1. Describe work clearly giving background, pertinence of the problems
and meaning of results
2. Use tabular and graphic methods of presenting data in good form for
future use
3. Supply^ sufficient information to permit reader to verify results and
draw his own conclusions
4. Limit conclusions to objective summary of evidence so that the work
recommends itself for prompt consideration and decisive action. 5
The reader should realize, of course, that the two lists (of steps in
designing experiments) presented in this section are only guides. Very
seldom will the various steps be tackled and settled in the particular
order given.^The statistician does not operate in such a mechanical and
routine fashion. Questions will be asked and answers received which will
trigger new lines of thought, and thus the planning conference will find
itself jumping from one step to another in a seemingly haphazard man
ner. Furthermore, it is not surprising to find, as the conference pro
gresses and new information is brought forth, the same step being con
sidered several times. Regardless of the repetition inherent in such a
procedure, it is a good procedure.
In summary, then, the designing of an experiment can be a time-
consuming and, occasionally, a painful process. Thus, the use of check
lists such, as those presented earlier can be most helpful (as a supple
ment to common sense) in making relatively certain that nothing has
been overlooked.
10.17 ILLUSTRATIONS OF THE STATISTICIAN'S AP
PROACH TO DESIGN PROBLEMS
To illustrate the manner in which a statistician approaches a design
problem, a series of examples will be considered. The first of these will
demonstrate the application of Kempthorne's nine steps, while the
6 Charles A. Bicking, "Some uses of statistics in the planning of experiments "
Industrial Quality Control, Vol. 10, No. 4, Jan., 1954, p. 23.
10.17 APPROACH TO DESIGN PROBLEMS 267
remainder will illustrate various topics discussed in Sections 10.1
through 10.16.
Example 10.12
Suppose a machine is constructed for the purpose of generating a
random series of 0*s and 1's. If the machine is truly a generator of
random binary elements, it should, among other things, yield 3 s 50 per
cent of the time and l*s 50 per cent of the time* It is proposed that an
experiment be devised to check on this particular aspect of the random
ness of the machine.
The preceding paragraph illustrates Kempthorne's Step 1, the
statement of the problem. If we formulate H:p = % (where p stands
for the probability of a 0) and A :p Q =?*%, we have taken care of Step 2.
The devising of an experimental technique and design (Step 3) is fairly
simple. In this case we shall operate the device a certain number of
times, say n, record the proportion of J s (po), an d !see if this is in close
enough agreement with the hypothesis H. If the agreement is good,
we accept H] if the agreement is poor, we reject H and accept A, the
alternative hypothesis. The only remaining part of Step 4 to be taken
care of is the determination of the number of operations of the device
that are required before we feel safe in making a decision. Suppose it is
desired that the probability of rejecting H :po = i (when it is really true)
should be no greater than a: = 0.05. This implies n>6, as can easily be
shown. Note carefully the concept of rejecting a true hypothesis. The
value of n would also be influenced by fixing the probability of accept
ing a false hypothesis, but we choose to ignore this in the present
example. Step 5 consists, in this case, of recognizing that the results
will be analyzed using the binomial distribution, and thus we should
make certain that the repeated events (operations of the device) are
statistically independent. Step 6 is evident, though sometimes trouble
some. When discussing Steps 3 and 4, the content of Step 7 was alluded
to, and all that remains is the formalizing of the analysis. Step 8 implies
that we should produce a confidence interval estimate of the true
probability of producing a with our device; that is, a point estimate, p Q>
is not sufficient. We must also be very careful to state that our conclu
sions only hold for the particular device operated, unless this device was
randomly selected from a larger group (or population) of devices.
Had other devices of a similar nature been investigated, the results of
our experiment should be evaluated along with all pertinent informa
tion from the allied studies (Step 9).
The reader will probably have recognized the similarity of this illus
tration to Example 7.4. It is, of course, the same. All we have done here
is "dress up" the problem and use it to illustrate the various steps in
the design of an experiment.
Example 10.13
Consider the problem of an engineer who wishes to assess the relative
effects of eight treatments (for the moment undefined) on the activated
life of a particular type of thermal battery. Assume that 64 relatively
homogeneous batteries are available for experimentation. With only this
much information, the most efficient design would be to randomly
assign the batteries to the eight treatments (groups) subject to the
268 CHAPTER 1 0, DESIGN OF EXPERIMENTAL INVESTIGATIONS
restriction that 8 batteries be allocated to each treatment. Such an
assignment is illustrated in Table 10.4. The reader should note that the
major design decisions reached in this example were concerned with
balancing and grouping. (NOTE: The type of design described above is
known as a completely randomized design.)^
TABLE 10.4-Random Assignment of Batteries to
Treatments as Described in Example 10.13
Treatments
ABCDEFGH
9
58
37
18
14
21
48
43
22
53
36
38
1
15
63
56
64
26
30
33
50
3
60
41
34
11
5
29
27
45
57
23
17
52
6
61
16
47
25
10
4
51
13
40
49
32
59
12
31
8
2
35
46
19
7
20
28
14
54
39
44
62
55
42
Numbers in the table represent serial numbers of units; a random order of testing would
also be determined.
Example 1O.14
As a second illustration of a completely randomized design, consider
the agronomist who has 28 homogeneous experimental plots available
for testing the relative effects of 4 different fertilizers on the^ yield of a
particular variety of oats. A reasonable design would be to impose, at
random, a different fertilizer on each plot. If the restriction is imposed
that 7 of the experimental plots be allocated to each fertilizer (treat
ment), complete balance will have been achieved.
Example 10.15
Referring to Example 10.13, suppose you are now advised that the
64 batteries consist of 8 batteries from each of 8 different production
lots. How will this additional information affect the design? If it is
suspected that there are real differences among the lots, the precision
of the experiment can be improved by removing the lot-to-lot variation
from the estimate of experimental error. Such an improvement in de
sign may be accomplished by assigning the treatments to the batteries
at random within each lot. Such a restricted randomization is illustrated
in Table 10.5. (NOTE: The type of design described above is known as
a randomized complete block design.) 7
The major benefit resulting from this type of blocking is a gain in
efficiency in analysis. That is, more sensitive tests of significance for
treatment differences can be made and shorter confidence interval
estimates of treatment effects can be obtained.
6 See Chapter 11 for further discussion of completely randomized designs.
7 See Chapter 12 for further discussion of randomized complete block designs.
1O.T7 APPROACH TO DESIGN PROBLEMS 269
TABLE 10.5 Random Assignment of Treatments to Batteries
Within Lots as Described in Example 10.15
Lots
1-ff
9-H
17 -R
25-C
33-J5
41-G
49-3
57-Z>
2-C
1CKE
18- A
26-D
34-F
42-H
50-.E
58- A
3-F
ll-D
19-B
27 -E
33-D
43-B
51-6=
59-F
4-B
12-F
2Q-G
2S-B
36-G
44-C
52-F
60-G
5-&
13-G
21-C
29-H
37-C
45-JS
53-H
61-C
6-G
14-C
22-F
30-G
3S-A
46-A
54-A
62-H
7~D
15-B
23-D
31-F
39-J3
47-F
55~D
63-B
S-A
16- A
24r-H
32-^1
4Q-H
48->
56-C
64-JS
Numbers in the table represent serial numbers of units; the letters represent treatments.
It will be observed that we have assumed Lot ISTo. 1 contains batteries 1 to 8, Lot No. 2
contains batteries 9 to 16, etc.
Example 10.16
Another illustration of a randomized complete block design is pro
vided by the following problem in nutrition research.. A nutritionist
wishes to assess the relative effects of four newly developed rations on
the weight-gaining ability of rats. He has 20 rats available for experi
mentation. Examination of the pedigrees of the experimental animals
indicates that the 20 rats consist of 4 rats from each of 5 litters. The
statistician would, under these circumstances, recommend that the
rations be assigned to the rats at random within each litter (block).
Example 10.17
Consider next a somewhat more complex problem. Assume that we
are again concerned with testing batteries., but this time the problem
arises during the development phase. The development engineer has to
reach a decision about three things: (1) how much electrolyte should
be incorporated in this particular model, (2) what weight of heat paper
should be used in the construction of the batteries, and (3) what effect
will the temperature at which the batteries are activated have on the
activated life of the batteries?
Denoting electrolyte by a, heat paper by 6, and temperature by c,
and assuming that two levels of each factor are to be investigated, the
eight treatment combinations might be as shown in Table 10.6.
It is decided that 16 batteries will be built to each of the four "elec
trolyte-heat paper' 7 specifications, providing a total of 64 batteries for
testing. As a precaution against bias being introduced because the last
batteries built might be better than the first batteries built, the 64 bat
teries will be built in a random order. Next, in each set of 16 batteries,
8 will be randomly selected for testing at low temperature, and the
remaining 8 will be reserved for testing at high temperature.
When this stage is reached, that is, once each battery has been built
and assigned a test temperature, the 64 batteries will be arranged in
random order for individual testing. As you can probably anticipate,
27O
CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
TABLE 10.6-Factors, Factor Levels, and Treatment Combinations for
the Experiment Described in Example 10,17
Fa
ctors and Factor Le
vels
Treatment
Combination
Amount of
Electrolyte
(gm/cell)
Weight of Heat
Paper (gm/cell*)
Test
Temperature (F)
(1)
1
4
50
<z
2
4
50
b
1
6
50
ab. ....
2
6
50
c
1
4
100
ac
2
4
100
be
1
6
100
abc
2
6
100
this last restriction frequently proves to be unpopular, especially if only
one temperature chamber is available. (NOTE: The design that has
been formulated is a completely randomized design involving a 2 3 fac
torial with 8 experimental units per treatment.)
Example 10.18
Suppose that we now consider a slightly different problem. Like many
of us engaged in research, the development engineer is often hard pressed
for funds. If this were the case in the situation described in Example
10.17, the development engineer might place a preliminary order for 8
batteries. Of the 8, 2 would be assembled to each of the 4 "electrolyte-
heat paper" combinations. His plan, of course, would be to test 1 bat
tery in each pair at low temperature and 1 at high temperature. This
testing would, naturally, take place in a random order.
Next, assume that, after the first 8 batteries are built and tested,
funds are made available for the building and testing of 8 additional
batteries. These would be ordered without delay in order to provide
some replication of the experiment. However, due to the way in which
the batteries were produced and tested, that is, first 8 and then 8 more,
it is clear that the combined analysis of all 16 batteries must take into
account the blocking which is implicit in the data. (NOTE: In this
example we have a randomized complete block design consisting of two
blocks and involving a 2 3 factorial set of treatment combinations.)
Example 10.19
Referring again to the problem described in Example 10.17, suppose
that two additional complications arise: (1) only 8 batteries can be
tested in a normal work day, and (2) in the interests of economy, the
test engineer wishes to place 4 batteries in the temperature chamber at
the same time. Under these restrictions, we have a natural set of blocks,
namely, days. Further, within each block it would be desirable to test
1 battery corresponding to each of the 8 treatment combinations.
Because of the temperature chamber restriction, we would decide, ran-
TO. 18 ADVANTAGES AND DISADVANTAGES OF DESIGNED EXPERIMENTS 271
domly for each day, whether to first test batteries at high temperature
and then test batteries at low temperature, or vice versa. Once this
decision is made, the random order of testing batteries within tempera
tures must be specified. As might be expected, the eventual analysis of
the data will take due cognizance of all restrictions placed on the test
program. (NOTE: The type of design illustrated in this example is
known as a split plot design*)
10.18 ADVANTAGES AND DISADVANTAGES OF STATIS
TICALLY DESIGNED EXPERIMENTS
Having spent considerable time discussing various aspects of, and
techniques in, experimental design, It is appropriate that the advan
tages and disadvantages of statistically designed experiments be con
sidered. These will, of course, be expressed In different ways by differ
ent people. However, as "was true for the steps involved in designing
experiments, an examination of various lists of advantages and disad
vantages will show that all the lists cover essentially the same points.
Advantages of Statistically Designed Experiments
Bicking (3) has listed the advantages of statistical designs over old
kinds of designs (nonstatistlcal) as follows :
(1) Close teamwork is required between the statisticians and the re
search or development scientists with consequent advantages in the
analysis and interpretation stages of the program
(2) Emphasis Is placed on anticipating alternatives and on systematic
pre-planning, yet permitting step-wise performance and producing
only data useful for analysis in later combinations
(3) Attention is focused on inter-relationships and on identifying and
measuring sources of variability in results
(4) The required number of tests is determined reliably and often may
be reduced
(5) Comparison of effects of changes is more precise because of group
ing of results
(6) The correctness of conclusions is known with definite mathematical
preciseness 9
If these advantages truly exist, and I believe they do, the value of
statistical aid in planning experiments is evident and should always be
sought.
Disadvantages of Statistically Designed Experiments
Happily, there are more advantages than disadvantages associated
with statistically designed experiments. In fact, I found it somewhat
difficult to formulate a list of disadvantages. However, a careful read
ing of Mandelson. (30), together with a realistic appraisal of the imple-
8 See Chapter 13 for further discussion of split plot designs.
9 Charles A. Bicking, "Some uses of statistics in the planning of experiments,"
Industrial Quality Control, Vol. 10, No. 4, Jan., 1954, p. 22.
272 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
mentation of certain statistically designed experiments, did yield the
following possible disadvantages:
(1) Such designs and their analyses are usually accompanied by
statements couched in the technical language of statistics. It
would be much better if the statistician would translate such
statements into terms that are meaningful to the nonstatis-
tician. In addition, the statistician should not overlook the
value of presenting the results in graphical form. As a matter
of fact, he should always consider plotting the data as a pre
liminary step to a more analytical approach.
(2) Many statistical designs, especially when first formulated, are
criticized as being too expensive, complicated, or time-con
suming. Such criticisms, when valid, must be accepted in good
grace and an honest attempt made to improve the situation,
provided that the solution of the problem is not compromised.
Before terminating our discussion of the advantages and disadvan
tages of statistically designed experiments, some mention should be
made of particular advantages and disadvantages associated with fac
torials. This is deemed necessary because of the important role that
factorials play in the design and analysis of experiments. (NOTE:
There will undoubtedly be some overlap between the advantages and
disadvantages given for statistically designed experiments in general
and those about to be given for factorials. However, a small amount
of repetition will not be harmful.)
Advantages of Factorials
(1) Greater efficiency in the use of available experimental re
sources is achieved.
(2) Information is obtained about the various interactions.
(3) The experimental results are applicable over a wider range of
conditions; that is 7 due to the combining of the various fac
tors in one experiment, the results are of a more comprehen
sive nature.
(4) There is a gain due to the hidden replication arising from the
factorial arrangement.
Disadvantages of Factorials
(1) The experimental setup and the resulting statistical analysis
are more complex.
(2) With a large number of treatment combinations the selection
of homogeneous experimental units becomes more difficult.
(3) Certain of the treatment combinations may be of little or no
interest; consequently, some of the experimental resources
may be wasted.
PROBLEMS 273
10.19 Summary
In this chapter several extremely important topics have been dis
cussed. It is recommended that they be re-examined from time to time
as the reader progresses through the succeeding chapters. Such a peri
odic reappraisal "will prove beneficial for a number of reasons, for ex
ample: (1) a thorough understanding of the concepts, principles, and
techniques involved is essential for a fruitful study of the experi
mental designs which are the subjects of the next three chapters, and
(2) an appreciation of these important principles should manifest itself
in improved experimentation.
Problems
10.1 Choosing practical situations from your special field of interest,
describe three problems whose solutions must be determined experi
mentally.
10.2 With reference to Problem 10.1, discuss the need for an experimental
design in each of the three illustrations.
10.3 It is sometimes said that experimental design is a subject which con
sists of two (almost distinct) parts: (a) the choice of treatments,
experimental units, and characteristics to be observed; (b) the choice
of the number of experimental units and the method of assigning the
treatments to the experimental units. Discuss this classification from
the points of view of the researcher and the statistician.
10.4 Define "systematic error" and discuss the relationship between this
factor and the statistical design of experiments.
10.5 Some terms that occur rather frequently in the literature are:
(a) accuracy, (b) precision, (c) validity, (d) reliability, and (e) bias.
Restricting your remarks to the theory of statistics or to applica
tions of statistical methods, define and discuss each of these terms.
10.6 Cox (14) uses "Designs for the Reduction of Error" as the title of
one of his chapters. What does this title suggest to you?
10.7 With reference to factorials, what is meant by the phrase "hidden
replication"?
10.8 Discuss the use of concomitant information in experimental design.
10.9 Choosing practical situations from your own special field of interest,
illustrate the concept of confounding. Give examples of: (a) unavoid
able confounding, (b) unintentional confounding, and (c) intentional
confounding.
10.10 Choosing practical situations from your own special field of interest,
illustrate the concept of randomization,
10.11 With reference to Problem 10.10, discuss the difficulties (if any)
associated with the randomization process.
10.12 Give your interpretation of the phrase "restricted randomization."
10.13 What would you do if, in the planning of a randomized complete
block design, the same order of treatments occurred (randomly) in
each block?
10.14 Discuss the following ways in which treatments can be assigned to
experimental units: (a) randomly, (b) subjectively, and (c) system
atically. Give illustrations which show the benefits, dangers, and
difficulties involved in each of the three approaches.
274 CHAPTER TO, DESIGN OF EXPERIMENTAL INVESTIGATIONS
10.15 Cox (14) uses the phrase "Randomization as a Device for Conceal
ment" as the heading of one of the sections in his book. Without
referring to his discussion, what do you believe he has in mind?
10.16 Cox (14) makes a distinction between factors that represent a treat
ment applied to the experimental units (treatment factors) and fac
tors that correspond to a classification of the experimental units into
two or more types (classification factors). Give illustrations of each
of these from situations in your own special field of interest.
10.17 The statement has been made that an uncontrolled and unmeasured
variable may be of sufficient importance to lead to the conclusion
that two controlled factors interact to a significant degree. Discuss
this idea, including all possible implications. What safeguards do
we have against such a result occurring?
10.18 Cox (14) also states that it is sometimes convenient to classify factors
as follows: (a) specific qualitative factors, (b) quantitative factors,
(c) ranked qualitative factors, and (d) sampled qualitative factors.
How would you define each of these? Compare your ideas with those
expressed by Cox.
10.19 Show graphically what is meant by an interaction. Illustrate your
ideas using the data of Examples 10.8 and 10.9.
10.20 Explain the relationship, if any, between regression functions (i.e.,
response functions) and the concepts of effects and interactions.
10.21 How would you go about selecting the factors to be investigated in
an experiment? Illustrate with examples from your own specific field
of interest.
10.22 Assuming the factors have been decided upon, how would you go
about selecting the factor levels? Illustrate with examples from your
own specific field of interest.
10.23 Choosing practical situations from your own special field of interest,
illustrate completely randomized, randomized complete block, and
split plot designs.
10.24 Indicate how the examples provided in answer to Problem 10.23
attempted to "control error/'
10.25 What is meant by the precision of an experiment? of a contrast?
10.26 What is meant by a sequential experiment? Is there any other kind?
Please discuss.
10.27 In Problem 10.3, reference was made to i . . . the choice of treat
ments, experimental units, and characteristics to be observed." Illus
trate each of these with examples from your own special field of
interest.
10.28 Discuss the following items relative to the selection of experimental
units: (a) number of units, (b) size of units, (c) shape of units,
(d) independence of units.
10.29 What is meant by a "control" treatment?
10.30 Cox (14) classifies observations into six groups: (a) primary observa
tions, (b) substitute primary observations, (c) explanatory observa
tions, (d) supplementary observations for increasing precision,
(e) supplementary observations for detecting interactions, and
(f) observations for checking the application of the treatments.
Please try to define and illustrate each of these. Then compare your
ideas with those expressed by Cox.
REFERENCES AND FURTHER READING 275
10.31 Building on the samples given in Section 10.16, construct your own
list of " steps in designing an experiment."
10.32 Contrast the one-factor-at-a-time method of experimentation with
the factorial approach. Construct a table which shows and compares
the advantages and disadvantages of each.
10.33 Define: (a) absolute experiments and (b) comparative experiments.
Give examples of each. With which type is this book mainly con
cerned?
10.34 Consider the following ''elements" of experimental method:
(a) control, or the elimination of the effects of extraneous variables
(6) accuracy of instruments and data acquisition
(c) reduction of the number of variables to be investigated
(d) planning of the test sequence in advance of the start of experi
mentation
(e) detection of malfunctions
(/) testing for reasonableness of results
{g} analysis and interpretation of results
Evaluate the foregoing list by comparing it with the ideas expressed
in this chapter.
10.35 Choosing practical situations from your own special field of interest,
give three examples of statistically designed experiments. For each
of these, point out where and how the concepts of this chapter were
employed.
10.36 Choose a practical problem in your own area of specialization. Fol
lowing, where practicable, the philosophy expressed in this chapter,
design an experiment to provide data relevant to the problem. Justify
all of your decisions and relate them to the discussion in the text. If
feasible, perform the experiment and then analyze and interpret the
results.
References and Further Reading
1. Anscombe, F. J. Quick analysis methods for random balance experimenta
tion. Technometrics, 1 (No. 2) :195-209, May, 1959.
2. Barbacki, S., and Fisher, R. A. A test of the supposed precision of systematic
arrangements. Ann. Eugen., 7:189, 1936.
3. Bicking, C. A. Some uses of statistics in the planning of experiments. In
dustrial Quality Control, 10 (No. 4) :20-24, Jan., 1954.
4. Bingham, R. S., Jr. Design of experiments from a statistical viewpoint,
Parts I and II. Industrial Quality Control, 15 (No's. 11 and 12) :29-34 and
12-15, May and June, 1959.
5. Bross, I. D. J. Design for Decision. The Macmillan Company, New York,
1953.
6. Brownlee, K. A. The principles of experimental design. Industrial Quality
Control, 13 (No. 8):12-20, Feb., 1957.
7 _ Statistical Theory and Methodology in Science and Engineering,
John Wiley and Sons, Inc., New York, 1960.
8. Budne, T. A. Random balance: Part I The missing statistical link in fact
finding techniques, Part II The techniques of analysis, Part III Case
histories. Industrial Quality Control, 15 (No's. 10-11-12) :5-10, 11-16, 16-19,
April, May, and June, 1959.
g a _ The application of random balance designs. Technometrics } 1 (No.
2): 139-55, May, 1959.
276 CHAPTER 10, DESIGN OF EXPERIMENTAL INVESTIGATIONS
10. Caplan, F. Statistical design in electronics production-line experimentation.
Industrial Quality Control, 12 (No. 5):12-13, Nov., 1955.
11. Chapin, F. S. Experimental Designs in Sociological Research. Harper and
Brothers Publishers, New York, 1947.
12. Chew, V., (editor) Experimental Designs in Industry. John Wiley and Sons,
Inc., New York, 1958.
13. Cochran, W. G., and Cox, G, M. Experimental Designs. Second Ed. John
Wiley and Sons, Inc., New York, 1957.
14. Cox, D. R. Planning of Experiments. John Wiley and Sons, Inc., New York,
1958.
15. Crump, S. L. Some aspects of experimental design. Industrial Quality Con
trol, 10 (No. 4):14-16, Jan., 1954.
16. Davies, O. !L. (editor) The Design and Analysis of Industrial Experiments.
Second Ed. Oliver and Boyd, Edinburgh, 1956.
17. DeLury, 33. B. On the design of experiments. Industrial Quality Control,
10 (No. 4):24-29, Jan., 1954.
18. . Designing experiments to isolate sources of variation. Industrial
Quality Control, 11 (No. 2) :22-24, Sept., 1954.
19. Duffett, J. R. Some experience with the design of experiments. Industrial
Quality Control, 11 (No. 3): 36-40, Nov., 1954.
20. Federer, W. T. Experimental Design. Macmillan Co., New York, 1955.
21. Finney, D. J. Experimental Design and Its Statistical Basis. The University
of Chicago Press, Chicago, 1955.
22. . An Introduction to the Theory of Experimental Design. The Univer
sity of Chicago Press, Chicago, 1960.
23. Fisher, R. A. Statistical Methods for Research Workers. Tenth Ed. Oliver and
Boyd, Edinburgh, 1946.
24. . The Design of Experiments. Fourth Ed. Oliver and Boyd, Edin
burgh, 1947.
25. Gilbert, S. Statistical design of experiments in metallurgical research. In
dustrial Quality Control, 12 (No. 5):13-18, Nov., 1955.
26. Hunter, J. S. Determination of optimum operating conditions by experi
mental methods, Part II 1-2-3, Models and methods. Industrial Quality
Control, 15 (No's. 6-7-8) :16-24, 7-15, and 6-14, Dec., 1958, Jan. and Feb.,
1959.
27. Jeffreys, H. Random and systematic arrangements. Biometrika, 31:1, 1939.
28. Kempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
29. Leone, F. C., Nottingham, R. B., and Zucker, J. Significance tests and the
dollar sign. Industrial Quality Control, 13 (No. 12) :5-20, June, 1957.
30. Mandelson, J. The relation between the engineer and the statistician. In
dustrial Quality Control, 13 (No. ll):31-34, May, 1957.
31. Mood, A. M., The heart of a reliability program. IRE Transaction on
Reliability and Quality Control, PGRQC-16:16-23, June, 1959.
32. National Bureau of Standards. Projects and Publications of the National
Applied Mathematics Laboratories. April through June, 1949.
33^ 9 Economy in the planning of experiments. Industrial Quality Control,
14 (No. 7):5-6, Jan., 1958.
34. Peach, P. The use of statistics in the design of experiments. Industrial
Quality Control, 3 (No. 3):15-17, Nov., 1946.
35. Pearson, E. S. Some aspects of the problem of randomization. Biometrika,
29:53, 1938.
35 _ ^ n illustration of "Student's" inquiry into the effect of balancing
in agricultural experiments. Biometrika, 30:159, 1938.
37. , and Wishart, J. (editors). "Student's" Collected Papers. Biometrika
Office, University College, London, 1942.
38. Purcell, W. R. Balancing and randomizing in experiments. Industrial
Quality Control, 7 (No. 4):7-14, Jan., 1951.
REFERENCES AND FURTHER READING 277
39. Quenouille, M. H. The Design and Analysis of Experiment. Charles Griffin
and Co., Ltd.,, London, 1953.
40. Ratner, R. A. Effect of variations in weight upon move times. Master of
Science Thesis, Iowa State University, Ames, 1951.
41. Satterthwaite, F. E. Random balance experimentation. T echnometrics y 1
(No. 2) :1 11-37, May, 1959.
42. Shainin, D. The statistically designed experiment. Harvard Business Rev.,
July-Aug., 1957.
43. Snedecor, G. W. Statistical Methods. Fifth Ed. The Iowa State University
Press, Ames, 1956.
44. "Student" (W. S. Gosset). Comparison between balanced and random
arrangements of field plots. Biometrika, 29:363, 1938.
45. Wilson, E. B., Jr, An Introduction to Scientific Research. McGraw-Hill Book
Company, Inc., New York, 1952,
46. Yates, F. The design and analysis of factorial experiments. Techn. Comm.
No. 85, Imperial Bureau of Soil Science, 95 pp., 1937.
47. Youden, W. J. Statistical design. A collection by the editors of Industrial
and Engineering Chemistry of a series of bimonthly articles by Dr. W. J.
Youden, National Bureau of Standards, during his six years (19541959)
as a Contributing Editor. American Chemical Society, Washington, D.C.
48. . Problems of the experimenter. National Convention Transactions,
American Society for Quality Control, pp. 41-47, 1959.
49. , Kempthorne, O., Tukey, J. W., Box, G. E. P, and Hunter, J. S.
Discussion of the papers of Messrs, Satterthwaite and Budne (including au
thors' responses to discussion). T echnometrics, 1 (No. 2):157 93, May, 1959.
50. Zelen, M., and Connor, W. S. Multi-factor experiments. Industrial Quality
Control, 15 (No. 9):14-17, Mar., 1959.
CH APTE R 11
COMPLETELY RANDOMIZED DESIGN
CHAPTER 10, several experimental designs were illustrated. In the
present chapter, we propose to discuss the simplest of these designs,
namely, the completely randomized design, in considerable detail.
Much attention will, of course, be given to methods of analyzing data
arising from such a design, and it will be observed that analysis of vari
ance (frequently abbreviated as AOV or ANOVA) is the method most
widely used.
11.1 DEFINITION OF A COMPLETELY RANDOMIZED
DESIGN
A completely randomized (CR) design is a design in which the treat
ments are assigned completely at random to the experimental units, or
vice versa. That is, it is a design that imposes no restrictions, such as
blocking, on the allocation of the treatments to the experimental units.
Of course, as in Examples 10.13 and 10.14, some degree of balance may
be sought.
Because of its simplicity, the completely randomized design is widely
used. However, the researcher is cautioned that its use should be
restricted to those cases in which homogeneous experimental units are
available* If such units cannot be obtained, some blocking should be
utilized to increase the efficiency of the design.
Example 11.1
Given four fertilizers, we wish to test the null hypothesis that there
are no differences among the effects of these fertilizers on the yield of
corn, We shall assume there are 20 experimental plots available to the
research worker* A sound procedure would be to place each fertilizer on
an equal number of experimental plots so that our estimates of the
mean effect of each fertilizer will have equal weight. Then, we insist
that the fertilizers be assigned to the plots at random. This may be
accomplished by numbering our plots from 1 through 20 and then
drawing tickets at random from a hat, 5 tickets being identified by
coloring or code mark with each of the 4 fertilizers. The first one drawn
specifies the treatment for plot No. 1, the second for plot No. 2, and
so on,
Example 11.2
If, in the preceding example, only 17 plots were available, some lack
of balance would be inevitable. Assuming that more precise information
is desired on fertilizer No. 1, the randomization procedure could be
modified so that, for example, 8 plots would be treated with fertilizer
No. 1, 3 plots with No. 2, 3 with No. 3, and 3 with No. 4.
t2781
11.2 ONE OBSERVATION PER EXPERIMENTAL UNIT
279
11.2 COMPLETELY RANDOMIZED DESIGN WITH ONE
OBSERVATION PER EXPERIMENTAL UNIT
If, in a completely randomized design, rit experimental units were
subjected to the th treatment (^=1, - - } t) and only one observation
per experimental unit was obtained, the data would appear as in Table
11.1.
TABLE 11.1 Symbolic Representation of Data in a Completely Random
ized Design (Unequal Numbers of Observations for Each Treatment)
Treatment
Total
1
2
t
Observations
F
F 21
? :
Totals
r,
r,
r*
t
i i
Numbers of observations
Means
T.
*
n*
7,
1-1
F - T / iZ n t
I -1
Using the equations:
\ Y 2 = total sum of squares
= sum of the squares of all the observations
= i: y: F
S j / j *- 131
= sum of squares due to the mean
= T 2 / i: nt,
' 1=1
= among treatments sum of squares
(11.1)
(11-2)
(11-3)
and
= experimental error sum of squares
28O
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE 11.2-ANOVA for Data of Table 11.1
Source of Variation
Degrees of Freedom
Sum of Squares
Mean Square*
IVEean
1
J\ftsu
M
Among treatments . .
t 1
T
T
Experimental error (within
treatments)
i: (*-
i*i
EVV
E
Total
t
~y ] m
y; r*
i 1
* The mean squares are found by dividing each sum of squares by the corresponding
degrees of freedom. To avoid confusion with symbols for effects and interactions (see
Chapter 10), the symbols for mean squares will always be set in boldface type. This pro
cedure will be adhered to throughout the remainder of this book.
(11.4)
TUT
-LYLyjf
the ANOVA shown in Table 11.2 is obtained. If each n = r&, that is,
if the number of experimental units per treatment is the same for all
treatments, Table 11.1 would be modified as shown in Table 11.3.
Equations (11.1) through (11.4) would be rewritten as:
t n
YU, (11-5)
(11.6)
TABLE 11.3 Symbolic Representation of Data in a Completely Random
ized Design (Equal Numbers of Observations for Each Treatment)
Treatment
1
2
. . .
t
Total
F "
F 2 i
Y tl
Observations
F 12
f 22
f' 2
h.
h.
Y tn
t
Totals
TI
Tz
T t
^> * T-
/ - * *
i i
Numbers of observations
n
n
n
tn
Means
^
F 2
Y t
7--T/*
11.2 ONE OBSERVATION PER EXPERIMENTAL UNIT
281
and
The resulting ANOVA is shown in Table 11.4.
TABLE 11.4-ANOVA for Data of Table 11.3
(11-7)
(11-8)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
JMean
1
Mint
M
Among treatments .
t 1
T
T
Experimental error (within treat
ments)
t(n 1)
J&tnj
E
Total
tn
T! Y 2
Up to this point, our discussion of a completely randomized design
with one observation per experimental unit has concentrated on the
calculation of the various sums of squares and mean squares, and on
the specification of the associated degrees of freedom. While the calcu
lation of the sums of squares and mean squares has been explained in
detail, no explanation has been given as to why the degrees of freedom
are as stated. However, the way in which the degrees of freedom are
found seems reasonably clear. Since the procedure will be illustrated
many times in this and succeeding chapters, no attempt will be made
to formulate and state general rules.
Before the preceding analyses of variance can be used for purposes
of statistical inference, certain assumptions must be made about the
observations. The nature of these assumptions will now be examined.
(NOTE: In general, the assumptions underlying analyses of variance
are the same as those usually associated with regression analyses.
These are additivity, linearity, normality, independence, and homo
geneous variances. That is, the statistical model most frequently assumed
in analysis of variance applications is a linear model to which has "been
appended certain restrictions about independent observations from normal
distributions.)
The basic assumption for a completely randomized design with one
observation per experimental unit is that the observations may be
represented by the linear statistical model
= JUL
y = 1, , n* (unequal numbers)
or
(11.9)
j = 1, - - , n (equal numbers)
282
CHAPTER 11 , COMPLETELY RANDOMIZED DESIGN
where /x is the true mean effect, n is the true effect of the ith. treatment,
and tj is the true effect of the yth experimental unit subjected to the
ith treatment. (NOTE : e^ also includes the effects of all other extra
neous factors. However, we rely on the process of randomization to
prevent these effects from contaminating our results.) In addition, it
is customarily assumed that M is a constant while the e/ are NID (0,, 00.
However, the specification of the model is still incomplete, for
nothing has been said about the T,-. The researcher has two choices as
to what he can say about the r t -, namely: (1) 2^i=i r t = 0, which reflects
the researcher's decision that he is concerned only with the t treat
ments present in his experiment, or (2) the T* are NID (0, o- T ), which
reflects the researcher's decision that he is concerned with a population
of treatments of which only a random sample (the t treatments) are
present in his experiment. These two choices lead to what the statis
tician refers to as Model I and Model II, respectively. Incidentally,
Model I is sometimes referred to as the analysis of variance (fixed
effects) model, while Model II is known as the component of variance
(random effects) model.
Once the foregoing assumptions have been made, the theory out
lined in Chapter 3 may be invoked to obtain "expected mean squares."
These expected mean squares can be of valuable assistance to the re
searcher, for they indicate the proper procedure to be followed in esti
mating parameters and/or testing hypotheses about parameters within
the framework of the assumed model. It is customary to exhibit these
expected mean squares in an additional column in ANOVA tables. So,
without further discussion at this time, we re-exhibit Table 11.2 as
Table 11.5 (Model I) and Table 11.6 (Model II) with certain expected
mean squares included. Similarly, Table 11.4 is re-exhibited as Table
11.7 (Model I) and Table 11.8 (Model II). (NOTE: While formal deri-
TABLE 11.5-ANOVA for Data of Table 11.1 Showing Certain Expected
Mean Squares (Unequal Number of Observations per Treatment: Model I)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
Mean
1
M
M
Among treatments ....
t 1
1 yy
T
-+vV(-
Experimental error. . .
Z frt - 1)
i 1
Eyy
E
"'
Total
t
ZF2
i 1
11.2 ONE OBSERVATION PER EXPERIMENTAL UNIT
283
TABLE 11.6-ANOVA for Data of Table 11.1 Showing Certain Expected Mean
Squares (Unequal Numbers of Observations per Treatment: Model II)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square*
IVIean
1
JWi/t/
M
Among treatments
t 1
T
T
0.2 _J_ nQ0 3
Experimental error
] ( ni -. i)
J~L*f~j
E
<r*
Tl
Total
t
23 ***
S F 2
1
* The constant no is a sort of an average nt, and it Is defined by
wo = f X>* - i ?/ Z^l/Cf - 1).
l_ x_l il t=l -J
TABLE 11.7-ANOVA for Data of Table 11.3 Showing Certain Expected Mean
Squares (Equal Numbers of Observations per Treatment: Model I)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
Mean
1
M
M
Among treatments. . . .
Experimental error. . .
t 1
t(n - 1)
Ryy
T
E
t
i
<r 2
Total
in
S F 2
TABLE 11.8-ANOVA for Data of Table 11.3 Showing Certain Expected Mean
Squares (Equal Numbers of Observations per Treatment: Model II)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
IS/Iean
1
M m
M
Among treatments . . .
tl
T
w
T
<r 2 -t-^cr?
Experimental error
t(nV)
Eiyy
E
o- 2
Total
tw>
TZ F 2
284 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
vations of expected mean squares will not be given in this book, certain
rules will be given to aid the researcher in finding these valuable quan-
titles. Until these rules are expounded, the reader is asked to accept
the results as given.)
Having performed the previously indicated calculations and having
determined certain expected mean squares (based on the specified as
sumptions), we are now ready to proceed to the making of statistical
inferences. Just what types of inference will be made will, of course,
depend on the purpose for which the experiment was conducted. Three
common inferences concern themselves with the following problems:
(1) hypotheses about the relative effects of treatments, (2) estimation
of the magnitude of components of variance, and (3) estimation of the
mean effects of individual treatments. Each of these will now be con
sidered.
Consider first the hypothesis of "no differences among the effects of
the t treatments in the experiment. 77 The way in w-hich this hypothesis
was phrased indicates that Model I has been assumed. Thus, the hy
pothesis may be expressed as H:T^ = O (i=l, , f). Examination of
the expected mean squares in Tables 11.5 and 11.7 indicates (in each
case) that, if H is true, both the experimental error mean square and the
among treatments mean square are estimates of <r 2 . Thus, if H is true,
the ratio
mean square for treatments
T/E = - - -- (11.10)
experimental error mean square
is distributed as F with
and i>2
degrees of freedom because of the assumption that the e/ are
NID (0, <r). If the value of F specified by Equation (11.10) exceeds
jFa-aOOiJ K 2 >, where lOOo: per cent is the chosen significance level, H will
be rejected and the conclusion reached that there are significant dif
ferences among the t treatments.
Had Model II been assumed, that is, had the hypothesis been
phrased as follows: "There are no differences among the effects of all
the treatments in the population from which the t treatments in the
experiment are a sample," the same test procedure would have evolved.
That is, under Model II, the hypothesis H:o* = Q would also be tested
by forming the ratio F = T/E. Why, then, have we been so concerned
over the distinction between the two models? There are two reasons
for our concern over the differences in assumptions between Models I
and II. These are: (1) the inferences in the two cases are about entirely
different populations; and (2) in more complex analyses, quite dif
ferent test procedures may be indicated. Many illustrations of these
11.2 ONE OBSERVATION PER EXPERIMENTAL UNIT 285
differences will be forthcoming in later sections of this and succeeding
chapters.
Consider next the estimation of components of variance. Regardless
of which model is assumed (that is, Model I or Model II), it is clear
that
s 2 the experimental error mean square == E (11.11)
is an estimate of o- 2 . However, if Model II is assumed, it is also possible
to estimate ov by calculating
2 (mean square for treatments) (experimental error mean square)
coefficient of <r* in the expected mean square for treatments
(T E)/no, for unequal numbers
(T E)/n, for equal numbers.
Finally, let us consider the estimation of the mean effects of indi
vidual treatments. It should be obvious that a point estimate of the
true mean effect of the ith treatment (Mt = M+ri) is given by T\-. How
ever, since confidence interval estimates are desired, it is necessary that
we determine the standard error of the treatment mean. In Section 6.5,
the estimated variance of a sample mean,
was used to define the standard error of the mean
. (11 . 14)
Consequently, the estimated variance of the mean of the ith treatment in
a completely randomized design with one observation per experi
mental unit is given by
^ _ experimental error mean square
number of observations in ith. treatment (11. 15)
= E/m = s*/m
and the standard error of the mean of the ith treatment is given by
(11.16)
Of course, if each ?i v = n, the same standard error would be attached to
each sample mean. A lOOy per cent confidence interval estimate of M*
would then be determined by calculating
_
= Y* =F
where v is the number of degrees of freedom associated with E.
286
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
Considerable time has been spent in discussing the analysis of a
completely randomized design with one observation per experimental
unit. For example, calculations were explained in detail, assumptions
were carefully stated, expected mean squares were introduced, and
test and estimation methods were developed with care. Such attention
to detail was deemed appropriate for an orderly development of the
methods involved. Further, the discussion given here will greatly expe
dite the future presentation of similar methods for more complex situa
tions. Some examples will now be given.
Example 11 .3
Consider that an experiment similar to that described in Example
10.13 has been performed. However, only four treatments were investi
gated and only 20 batteries were available for testing. The data in
Table 11.9 resulted. Following Equations (11.5) through (11.8), the
appropriate calculations are:
]T y* = 104,352
M yv = (1444) 2 /20 = 104,256,8
T yy =- [(369) 2 4- (371) 2 + (345) 2 4- (359) 2 ]/5 104,256.8
= 84.8
E yy = 104,352 104,256.8 84.8 = 10.4.
These lead to the ANOVA shown in Table 11.10. The expected mean
square for treatments has been given for both Model I and Model II.
Since F = 43.49 >-P T .9cs t ie> = 5,29, the hypothesis ffrr^O (i=l, 2, 3, 4)
or Hia^ Q, whichever applies, is rejected. Since the number of obser
vations per treatment is the same for each treatment, the standard error
of a treatment mean is -\/Q. 65/5 = -\/0- 13 = 0.36 second.
TABLE 1 1 .9-Activated Lives of Twenty Thermal Batteries Resulting From
Experiment Described in Example 11.3
Treatment
1
2
3
4
Total
73
74
68
71
73
74
69
71
Observations (in seconds)
73
74
69
72
75
74
69
72
75
75
70
73
Totals
369
371
345
359
1,444
Numbers of observations
5
5
5
5
20
Means
73.8
74.2
69.0
71.8
72.2
1 1 .2 ONE OBSERVATION PER EXPERIMENTAL UNIT
TABLE 11.10-ANOVA for Data of Table 11.9
287
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
F-
Ratio
Mean
1
104,256.8
104,256.8
Treatments .
3
84.8
28.27
U + (s/3) i: A
i 1
43.49
Experimental error
16
10.4
0.65
or
* + Sa-r
cr 2
Total
20
104,352.0
Example 11 .4
Consider an experiment to study the effect of storage condition on
the moisture content of white pine lumber. Five storage methods were
investigated, with varying numbers of experimental units (sample
boards) being stored under each condition. The data in Table 11.11
were obtained. Following Equations (11.1) through (11.4), the appropri
ate calculations are :
= 863.36
= (108.8) 2 /14
845.53
+
M * L 5 ' 3 ' 2 l 3
E yy = 863.36 - 845.53 10.66 = 7.17,
t (27.4)' t (7.1)'
845.53 = 10.66
TABLE 11.11-Moisture Contents of Fourteen White Pine Boards Stored
Under Different Conditions
Storage Conditions
Total
1
2
3
4
5
Observations (in per
cent)
7.3
8.3
7.6
8.4
8.3
5.4
7.4
7.1
8.1
6.4
7.9
9.5
10.0
7.1
Totals
Number of observa
tions
39.9
5
8.0
0.4
19.9
3
6.6
0.5
14,5
2
7,3
0.6
27.4
3
9.1
0.5
7.1
1
7.1
0.9
108.8
14
7.8
Means
Standard errors
288
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
This leads to the ANOVA shown in Table 11.12. Again, the expected
mean square for treatments is given for both Models I and II. Since
^ = 3.34 <jP.95C4,9) = 3.63, we are unable to reject the hypothesis
HtTi 0( = 1, - - - , 5) or fl r :cr? = 0. The standard errors of the treat
ment means, presented in Table 11.11 for convenience, were calculated
using Equation (11.16) where a 2 = 0.80.
TABLE 11.12-ANOVA for Data of Table 11.11
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
F-
Ratio
Mean ... ...
1
845.53
845.53
Storage conditions . .
Experimental error .
4
9
10.66
7.17
2.67
0.80
o
5
er 2 -f- ^ ^ WiTi/4
i 1
i?r
3.34
Total
14
863.36
THE RELATION BETWEEN A COMPLETELY RAN
DOMIZED DESIGN AND "STUDENT'S" *-TEST OF
H:jui = M2 VERSUS
11.3
In Section 7.20 it was mentioned that the analysis of variance tech
nique could be used as an alternative to "Student's" -test when
examining the hypothesis fl r :^i = /x 2 . Clearly, this same relationship
exists when we have a completely randomized design involving only
two treatments. In this instance, the hypothesis (under Model I) of
H :ri = T 3 = is equivalent to T:^i = ^2 where MI
11.4 SUBSAMPLING IN A COMPLETELY RANDOMIZED
DESIGN
In many experimental situations, several observations may be ob
tained on each experimental unit. If these observations are all on the
same characteristic (i.e.., on the same variable), the process of obtain
ing the observations is often referred to as subsampling. Some examples
of subsampling are:
(1) In the battery experiment of Example 11.3, several observa
tions per battery might have been obtained by connecting
several clocks to each battery. These several observations per
battery would be referred to as "samples within experimental
units."
(2) In a field experiment, the researcher may not have time to
harvest (totally) each experimental plot. Thus, he might ran-
11.4 SUBSAMPLING 289
domly select several quadrats per plot and harvest tlie grain
in each selected quadrat. Again, we would describe these ob
servations as "samples within experimental units. "
(3) In a food technology experiment involving the storage of
frozen strawberries, 10 pints (experimental units) were stored
at each of 5 lengths of storage time (treatments). When
ascorbic acid determinations were made after storage, two
determinations were made on each pint (samples within exper
imental units).
As you can well imagine, the addition of subsampling to the experi
mental program will have an effect on the eventual analysis. First, let
us see what changes are required in the assumed statistical model.
Under conditions such as have been described above, the appropriate
model is
Ytjk ~ p. + Tt + ij + 17 *y*; i = 1, - * , t
/=!,-,< (11.18)
where /z is the true mean effect, T%- is the true effect of the ith treat
ment, tij is the true effect of the jth experimental unit subjected to the
ith treatment, and rj^ Jk is the true effect of the &th sample taken from
the jth experimental unit subjected to the zth treatment. Proceeding
as before, we assume that /z is a constant, that the </ are NID (0, <r),
and that the 17^ are NID (0, o-,,). Of course, this still leaves the nature
of the r< unspecified. That is, do we assume Model I or II? This deci
sion will depend on the manner in which the treatments involved in the
experiment were selected. (NOTE: In most experimental situations,
Model I is appropriate because the researcher generally selects his
treatments in a nonrandom fashion and is only interested in making
inferences about the treatments actually present in the experiment.
However, since Model II better fits some situations, we will con
tinue to give it consideration.)
In order to simplify as much as possible the presentation of the
method of calculating the various sums of squares, let us adopt the
following notation:
jy = total number of observations in the whole experiment
t m
JEy = total of the n t j observations on the^'th experimental unit sub
jected to the ith treatment
-2
fc=*l
290 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
nj
Ti = total of all the ^2/ n i5 observations on the ith treatment
"> *
= z 2: YW =
= total of all N observations
t nj njj t nj
= z: r*.
i=i yi jfc=i t^i j=i i=i
Using the preceding notation, the various sums of squares are found as
shown below.
= total sum of squares
-34"i/j/ = sum of squares due to the mean
H (11.20)
= T*/N 9
T yy = among treatments sum of squares
-^
(11.21;
E yy = experimental error sum of squares
g / o / n *
^) - z: ( r? / 2: ^
*=i V ' y=i
r w , (11.22)
and
^y = pooled sum of squares among samples on the same
experimental unit (11.23)
- -2WW - T W - Eyy
11.4 SUBSAMPLING
291
"where
Y ' ij
average of the n^ observations on the /th experimental unit
subjected to the ith treatment
Y i = average of all observations on the fth treatment
and
7 average of all observations in the whole experiment
= T/N.
TABLE 11.13-Generalized ANOVA for a Completely Randomized Design
With Subsampling (Unequal Numbers: Model I)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
JVtean
1
J^flJ-H
M
Treatments . . .
Experimental
error
* 1
i: (**-D
ti
Tyy
E-vv
T
R
'l + C.<r*+ I^Ci: nyVrV(*-l)
i 1 V j-1 /
vK*
Sampling error
ib (D
i y i
Syy
S
2
0-,,
Total
2V
J2 Y*
These results would then be presented as in Table 11.13 in which the
constants Ci and c 2 are defined by
(11.24)
z: c* -
and
t / rti f n< \ t nf
23 ( S) ^?j / 2D ^*y ) X) 2D
1=1 \ y=i ' / i / i=i j i
N
(11.25)
292
CHAPTER II, COMPLETELY RANDOMIZED DESIGN
Had Model II been assumed, the ANOVA presented in Table 11.13
would be exactly the same except for the "expected mean square for
treatments/' which would appear as 0^+c 2 <r 2 +c 3 ov where
N
N
(11.26)
Example 11.5
Consider an experiment to investigate the fermentative conversion of
sugar to lactic acid. We wish to compare the abilities of two micro
organisms to carry out this conversion. A quantity of substrate is pre
pared and divided into two unequal portions. Each portion is then
divided into a number of 100 ml. subportions (experimental units) as
follows: No. 1, 4 units; No. 2, 3 units. Each of the 100 ml. units is
Inoculated with one or the other of the two microorganisms, the 4 units
being Inoculated with microorganism No. 1 and the 3 units with
microorganism No. 2. The fermentation is allowed to proceed for 24
hours, and then each experimental unit (100 ml. subportion) is ex
amined for the amount of residual sugar, expressed as mg. per 5 cc.,
to determine the amount of change produced by each microorganism,
the converted sugar having been shown previously to occur as lactic
acid. Varying numbers of determinations are made on each sample.
The data are recorded in Table 11.14.
TABLE 11. 14- Amount of Unconverted Sugar in the Substrate Following a
24-Hour Fermentation Due to Two Different Microorganisms
(Coded data for easy calculation)
Determi
nations
Microorganism No. 1
Microorganism No. 2
Sample number
Sample number
1
2
3
4
1
2
3
1
5.6
5.7
5.0
5.0
5.1
5.4
5.4
5.4
5.5
5.4
5.3
5.5
7.6
7.6
7.8
7.4
7.0
7.2
7.5
7.6
7.5
7.4
2
3.. . .
4
5
Sums
>
11.3
2
15.1
3
27.1
5
10.8
2
23.0
3
21.6
3
30.0
4
Following the calculational procedure outlined, we obtain:
3
= 5
i= 4
2= 3
= 22
= 2
11.4 SUBSAMPLING
293
1= 3
=== 3
.#22=21.6
r x = 64.3
r 2 = 74.6
T=138.9
and hence
]T F 2 = 902.07
Myy = (138.9) 2 /22 = 876.9641
, (74.6)2
876.9641
24.0927
(27.1) 2 t (10.8) 2 (23.0)2 (21.6) 3
. - j j _ j _
(30.0)
yy L 12 ' 10
= 901.0568 - 876.9641
= r(11.3) 2 (15. 1) 2
yy ~~ L 2 3
901.0568
= 901.9036 901.0568 = 0.8468
S vv = 902.07 876.9641 24.0927 0.8468 = 0.1664.
These results are presented in ANOVA form in Table 11.15.
TABLE 11.15-ANOVA for Fermentation Data of Table 11.14
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
Mean
1
876.9641
876.9641
Microorganisms ....
Experimental error.
Sampling error
1
5
15
24.0927
0.8468
0.1664
24.0927
0.1694
0.0111
2 j- i. N 2
17
Total
22
902.0700
On examination of the expected mean squares in Tables 11.13
and 11.15, it is seen that an exact test of the hypothesis ff:r l -==0
(i=l, -,) is impossible. This unfortunate circumstance results
from the fact that Ci^c 2 , and this is so because of the unequal num
bers of samples per experimental unit and the unequal numbers of ex
perimental units per treatment. This result clearly attests to the desir
ability of equal numbers of observations in the various subclasses, and
for this reason the statistician always recommends "equal frequencies"
when he is consulted at the design stage of any research project.
What, then, can be done in a situation such as described above?
That is, since unequal frequencies are sometimes inevitable, is there
any approximate test procedure that can be used? There is. However,
discussion of this approximation will be deferred until Section 11.7.
Before terminating the present discussion of subsampling in a com
pletely randomized design, the simplifications associated with equal
294 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
frequencies will be demonstrated. If, in a completely randomized
design, there are t treatments, n experimental units per treatment, and
m samples per experimental unit, the appropriate statistical model is
y ijk = p> + r + etj + -rjijk} i = 1, , t (11 . 27)
j = 1, . . . , n
k = 1, - - , m
where all terms are defined as before. The calculations are now speci
fied by
F 2 = total sum of squares
nm
*^~^ rr 1 / *. KJ-
y / - / wwi -^-~ /I//
X y -^ t / A/'AAZ' JKt yy y
i=l
experimental error sum of squares
t n
t 1 J=l
f 13 23
L i=i j=i
sum of squares due to the mean
(11.29)
T*/tnm,
treatment sum of squares
7) 2
(11.30)
*- F;) 2
i/m - 23 r!
- T m , (11.31)
and
5 1 y^F 2 Af T E (11*32)
where
771
f? X"^ V /-* 1 00\
-CSij ' X ^ * ijki \LL.3O)
23 23 Y^ = 23 Js, (H.34)
J-l *=-! 3=1
t n rn
11.4 SUBSAMPLING
t n
and
i = Ti/nm,
Y = T/tnm.
295
(11.35)
(11.36)
(11.37)
(11.38)
TABLE 11.16-Generalized ANOVA for a Completely Randomized
Design with Subsamplmg (Equal Numbers: Model I
and Model II)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
IVIean
1
M vv
M
Treatments
t 1
T
T
for* +in< r * + nmj* f i*/(f - 1)
Experimenal error . . .
Sampling error
t(n - 1)
tn(m 1)
Eyy
g
E
s
or
[a^ + m<r 2 + nmer^
a* + mcr z
<?
TJ
Total
tnm
y~L Y Z
These sums of squares would then be presented in ANOVA form as in
Table 11.16. Examination of the expected mean squares in Table 11.16
indicates that, because of the equal frequencies, there will be no diffi
culty in testing H:n = (i=l, - - , t) or H:<r* = Q. In addition, the
components of variance are easily estimated by
s* = S (11.39)
and
= CE S)/m.
(11.40)
And, finally, the standard error of a treatment mean is given by
(11.41)
(NOTE : Although not explicitly stated, it should be clear that a- 2 and
<rL could also have been estimated when unequal frequencies occur.
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
This can be seen by studying the expected mean squares in Table
11.13.)
Example 11.6
An agronomist conducted a field trial to compare the relative effects
of 5 particular fertilizers on the yield of Trebi barley. Thirty homo
geneous experimental plots were available and 6 were assigned at
random to each fertilizer treatment. At harvest time, 3 sample quadrats
TABLE 11.17-Coded Values of Yields from Ninety Sample Quadrats
Fertilizer Treatments
1
2
3
4
5
57
67
95
102
123
46
72
90
88
101
28
66
89
109
113
26
44
92
96
93
38
68
89
89
110
20
64
106
106
115
39
57
91
102
112
39
61
82
93
104
43
61
98
98
112
23
74
105
103
120
36
47
85
90
101
18
69
85
105
111
48
61
78
99
113
35
60
89
87
109
48
75
95
113
111
50
68
85
117
124
37
65
74
93
102
19
61
80
107
118
Under each treatment, the 18 observations are arranged in six groups of three. Each
group consists of the observed yields on the three quadrats taken from a single experi
mental plot.
were taken (at random) from each experimental plot and the yield
was obtained for each of the 90 quadrats. The data, in coded form, are
given in Table 11.17. Using Equations (11.28) through (11.32), we
obtain :
J^,Y* = 646,285
M yy = (7187)V0 = 573,921.88
TM = [(650) 2 -f- (1140)* + (1608) 2 + (1797)2 + (1992)*]/18 - 573,921.88
= 639,168.72 573,921.88 = 65,246,84
11.4 SUBSAMPLING
297
E vv = L(131) 2 + - . - + (344)]/3 - 639,168.72
= 641,001.67 - 639,168.72 = 1,832.95
S uv = 5,283.33 (by subtraction).
These are summarized in Table 11 .18. It is easily verified that F = 222.47,
with z/i = 4 and v^ = 25 degrees of freedom, is highly significant, and thus the
hypothesis H:r^ = Q(i = l, - - , 5) is rejected. (NOTE: An experienced
analyst could probably have predicted this result on examination of the
data, but the analysis and the statistical test make the conclusion an
objective one rather than a subjective one.) In case a confidence interval
estimate of a treatment mean is desired, the standard error of a treat-
TABLE 11.18-ANOVA for data of Table 11.17
Source of
Variation
Degrees
of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
F-
Ratio
Mean
1
573,921.88
573,921.88
Fertilizers. . .
Experimental
error
4
25
65,246.84
1,832.95
16,311.71
73.32
4 + 3<r* + (18/4) i: rf
I 1
c? _!_ 3^-2
222 .47
SamplinfiT error
60
5 283 33
88 O6
2
O~-n
v n
Total
90
646,285.00
ment mean is calculated. Its value is -\/E/nm -\/(73.32)/18 =
= 2.02. It is also clear that components of variance may be estimated in
a simple manner. For example, s% = 88.06. However, when an estimate
of <r 2 is sought, the calculations yield s 2 = (73.32-88.06) /3 <0. Since
cr 2 , by definition, is positive, it is unreasonable to quote a negative
estimate. Thus, in the present situation, the "best" estimate of cr 2 will
be taken to be zero, even though this is a biased estimate. More will
be said about the implications of this in a later section. For the moment,
we shall be content with observing that apparently the variation among
the true effects of different experimental units is small, and thus the
researcher might consider less replication (fewer experimental units per
treatment) in a future experiment of this type.
The reader will, no doubt, have realized that the concept of sub-
sampling may be extended to many stages. That is, we can have
"samples within samples within samples . . . , " and the resulting
ANO VA would reflect such multi-stage subsampling by partitioning the
total sum of squares into many more parts. Rather than continue the
discussion in general terms, we shall rely on problems at the end of the
298 CHAPTER IT, COMPLETELY RANDOMIZED DESIGN
chapter to illustrate not only the principles involved, but also the
mechanics of the appropriate calculations.
11.5 EXPECTED MEAN SQUARES, COMPONENTS OF
VARIANCE, VARIANCES OF TREATMENT MEANS,
AND RELATIVE EFFICIENCV
In Sections 11.2 and 11.4, the reader was introduced to the concepts
of components of variance, expected mean squares, and variances of
treatment means. In those sections, no reasons were given as to why
the expected mean squares contained the indicated components of
variance nor why the coefficients of the components of variance were
as given. We now propose to remove this deficiency. In addition, a
scheme will be proposed that permits the estimation of the relative
efficiency of different proposed designs involving various degrees of
subsampling. The discussion will be conducted with reference to Tables
11.16 and 11.18.
Reference to Tables 11.16 and 11.18 shows that the expected mean
square for sampling error contains only one component of variance.
This is so because the only factor which affects (or causes or produces)
the variation "among samples within experimental units" is the 77^
factor. However, the expected mean square for experimental error con
tains two components of variance since this source of variation reflects
the variation among the means of the samples taken from each experi
mental unit, and these means will vary not only because of the varia
tion from experimental unit to experimental unit, but also because of
the variation among the samples taken from each experimental unit.
To discuss the expected mean square for treatments, it is appropriate
to consider first the sum of squares. The treatment sum of squares
reflects the variation among the means of all the observations (on
samples) recorded for each treatment. Now, these means will vary
because of three contributing factors: (1) variation among treatments
(fertilizers), (2) variation among experimental units (plots) within
treatments, and (3) variation among samples (quadrats) within experi
mental units. Thus, the expected mean square involves three compo
nents of variance if Model II is assumed, or two components of vari
ance and one sum of squares if Model I is assumed. (NOTE: The
reader may verify the reasonableness of the foregoing remarks by sub
stituting the assumed linear statistical model for Y i3 ^ in the expres
sions for the various sums of squares.)
How were the various coefficients in the expected mean squares de
termined? The coefficient of <r* is 1 (and thus not shown) because this
reflects the variation among individual samples. The coefficient of <r z
is m (m = 3 in Table 11.18) because there were m observations (samples)
per experimental unit. The coefficient of o> when Model II is assumed,
or f s^t-i T^/(t 1) when Model I is assumed, is nm because there
were nm observations (m samples on each of n experimental units) per
treatment. In Table 11.18, n & and m = 3. We might note that
another way of expressing the justification of the coefficients described
11 ,5 EXPECTED MEAN SQUARES 299
above is to say that each treatment mean Is the average of nm observa
tions, while each experimental unit mean is the average of m observa
tions.
The estimation of the various components of variance has been well
illustrated in the preceding sections. However, a recapitulation will be
made to summarize the procedure. Since S is an unbiased estimator
of o\j, it is reasonable to write
si = S. (11.42)
Similarly, E is an unbiased estimator of cr^+mo- 2 , and thus we write
s* + ms* = E. (11.43)
If, then, we combine Equations (11.42) and (11.43) as shown in Equa
tion (11.44), an unbiased estimator of cr* is obtained:
m in
Now that the preceding estimates are available, it is possible to
determine (subject to sampling variation, of course) which factor is
contributing the most to the observed variation. Then, perhaps, an
improvement can be made in experimental technique, or the design
layout (configuration) can be changed, to better control the variation
in future experiments of the same type. To pursue this aspect of anal
ysis, the concept of "relative efficiency 7 ' of one design compared to
another design of the same type but involving different numbers of
experimental units and/or samples will be investigated.
Before such a comparison can be made, a criterion for measuring
efficiency must be established. The criterion adopted in this book will
involve the estimated variance of a treatment mean. We will say that
a design which provides a smaller estimated variance of a treatment
mean than does some other design is the more efficient of the two.
With reference to Table 11.16, and in agreement with the definition
given earlier, the estimated variance of a treatment mean is
estimated variance of the individual items contributing
to the mean
number of items (observations) averaged to get the mean
(11.45)
nm
4
n
Examination of Equation (11.45) leads to the following conclusions:
(1) If the estimates of the components of variance, s 2 and &*,
remairt relatively constant, an increase in n or m (or both)
3OO CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
will result in a smaller estimated variance of a treatment mean.
(2) An increase in n (the number of experimental units per treat
ment) will have more of an effect than an increase in m (the
number of samples per experimental unit) in reducing F(F Z -).
This supports the statement made in Section 10.16 to the
effect that "It (replication) enables us to obtain a more pre
cise estimate of the mean effect of any factor. . ^ . ^_
(3) If either s 2 or s* (or both) can be made smaller, F( F*-) can be
made smaller. This could be accomplished by choosing more
homogeneous experimental units or by improving the experi
mental technique.
Let us now return to the problem of estimating the efficiency of a
proposed design relative to the design used. To do this, we must first
estimate what the variance of a treatment mean would be if the pro
posed design were used. Assuming that: (1) the proposed design would
involve n' experimental units per treatment and m' samples per experi
mental unit and (2) the estimates of <r 2 and a* would remain un
changed, the new estimated variance of a treatment mean would be
2 I / 2
y-'(F,) = * m / (11.46)
nm
If F'(F T ) < F(F), the proposed^design is said to be more efficient than
the present design; if ^ 7 (F) > F(F t -), the proposed design is said to be
less efficient than the present design. Thus, as a measure of relative
efficiency i we use the ratio of F(F) and F'(FV). If the efficiency of the
proposed (new} design relative to the present (old) design is desired, one
calculates (in per cent)
R.E. of new to old = 100[F( 7<)/?'CF*)], (11.47)
while if the efficiency of the present (old") design relative to the proposed
(new} design is desired, one calculates (in per cent)
R.E. of old to new = 100 [F^F^/t^F,)]. (11.48)
Some texts use the concept of "relative information" and it would be
wise for us to see what relationship this bears to relative efficiency. If
information is defined as the reciprocal of the variance, then it is only a
matter of simple algebra to show that relative information is the same
as relative efficiency. For example,
R.I. of old to new = -y ^ _ X 100 = R.E. of old to new. (11 . 49)
Li/ ^ \ Y i) j
Similarly,
R.I. of new to old = R.E!. of new to old. (11.50)
11 .6 F-RAT1OS THAT ARE LESS THAN UNITY 3O1
It should be noted that there are other definitions of relative informa
tion to be found in the literature (e.g., Yates: Design and Analysis of
Factonal Experiments) which differ from relative efficiency. However,
if we define our terms as above, the two concepts may be used inter
changeably.
Example 11.7
The experiment on frozen strawberries discussed in (3) in the first
paragraph of Section 11.4 was performed. However, all that is available
is the abbreviated AISTOVA of Table 11.19. The estimates of the com
ponents of variance are 4 = 5 and s 2 = (20 5) /2 = 7.5 where the symbol
8 is used to denote determinations (rather than 77 to denote samples).
The estimated variance of a treatment mean is
10(2)
5 -f- 2(7.5)
20
= 1.
The question is then asked, "Is the present design more or less efficient
than a similar design employing 6 pints per storage time and 3 determi
nations per pint?" Calculating
5-1-3(7.5)
6(3)
1.53,
the answer is, "The present design is more efficient than the proposed
design." In fact, the efficiency of the present design relative to the
proposed design is: R.E. of old to new = 100(1.53/1) = 153 per cent.
TABLE 11.19-Abbreviated ANOVA of Ascorbic Acid
Content of Frozen Strawberries
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
Among storage times
4
4OO
100
<r* + 2^-*- 2 y-
Among pints treated alike . .
Between determinations on
pints treated alike. .
45
5O
9OO
250
20
s 4 j *'
*
F-RATIOS THAT
11.6 SOME REMARKS CONCERNING
ARE LESS THAN UNITY
In all the examples considered so far, tlie calculated F- values have
been greater than unity. Thus, in each of these cases, the only decision
to be made by the analyst was whether the calculated value should be
termed statistically significant or nonsignificant. If significant, the
hypothesis H:<n = Q (i=l, - - , Q or H:o* = was rejected; if not sig
nificant, the appropriate hypothesis was not rejected (perhaps even
3O2 CHAPTER I 1 , COMPLETELY RANDOMIZED DESIGN
accepted). However, it is possible (and quite probable) that a calcu
lated F-value will turn out to be less than unity. What should our con
clusion be in such a situation?
We can, of course, simply say that F was not significant and thus the
hypothesis cannot be rejected. However, such an easy dismissal of the
question is not wise, for it could cause us to ignore a valuable warning
sign. Suppose, as might happen, that F, with v\ and f 2 degrees of
freedom, is so small that F r = l/^P, with z> 2 and v degrees of freedom, is
significant. What should our conclusion be in this case? It appears as
though something should be rejected; but what is it? In this situation,
it seems reasonable to reject the postulated statistical model.
If the statistical model is rejected because of a significant F' value,
what are the steps that should then be taken? Some of these are :
(1) The experimental procedure should be reviewed to see if the
various assumptions are satisfied. For example, if the proper
randomization was not employed, the validity of the inde
pendence assumption is doubtful.
(2) If sufficient observations are available, the assumption of
normality could be checked by plotting the data either on
regular graph paper or on normal probability paper.
(3) The assumption of homogeneous variances might be checked,
but this would require a large number of observations within
subclasses.
(4) The underlying phenomenon should be restudied to see if the
assumed linear model is a good approximation to the true state
of affairs. If, as a result, the assumed model is rejected, a
search should be made for a new model which better describes
the observed data and the phenomenon under investigation.
11.7 SATTERTHWAITE'S APPROXIMATE TEST PRO-
CEDURE
When discussing the analysis of a completely randomized design
involving subsampling, it was noted that no exact test of
J2 r :ri = 0(i= 5 =l, * , f) was possible when the experiment involved
unequal frequencies at the various stages of subsampling. At that time,
it was promised that an approximate test procedure would be explained
later. We are now ready to fulfill that promise.
The proposed approximation, due to Satterthwaite (29), proceeds as
follows : Using estimates of the components of variance, mean squares will
be synthesized which will have the same expected value if the hypothesis
to be tested is true. These synthetic mean squares will then be used to form
a ratio which is approximately distributed as F.
How are the synthetic mean squares formed? If we denote the actual
mean squares existing in an ANOVA by MSi, MS%, , MSk, then a
synthetic mean square may be obtained by forming a linear combina
tion such as
L = aiMSi + a 2 MS 2 + - - - 4- a k MS k (11.51)
11. 8 SELECTED TREATMENT COMPARISONS 3O3
where the a t - are constants. The degrees of freedom associated with L
are then estimated by
. . . *
*/v h
where, of course, v represents the degrees of freedom associated with
MSi(i~L, - * , fc). Sometimes both the numerator and denominator
mean squares (in the approximate /^-ratio) will be synthesized. How
ever, it is more likely that only one synthetic mean square will be used
in any given situation.
Because of the lack of uniqueness of the approximate .F-ratio (dif
ferent ^-ratios could result from the use of different synthetic mean
squares) and because of the necessity of approximating the degrees of
freedom, the procedure is of limited usefulness. However, if used with
care, it can be of value to the researcher and/or statistician. The reader
is referred to Cochran (10) for a further discussion of this problem.
Example 11.8
Referring to Example 11.5, we recall that an exact test of H":ri = r 2 =
was impossible. This was so because c\ ?^C2 in Table 11.15. It is decided
to form a "synthetic experimental error mean square" that will have an
expected value of cr^ + czo-*. This could be done by calculating
-j #2S>
[i -
O (0.1694) + [1 -
The approximate jFVratio would then be F = 24.0927/Z/ with degrees of
freedom v = 1 and v<2. = v, where
[ai(0.1694)]V5 + [a 2 (0.0111)] 2 /15
The details of the numerical calculations are left as an exercise for the
reader,
11.8 SELECTED TREATMENT COMPARISONS: GENERAL
DISCUSSION
In Section 10.15, the idea of making specific comparisons among
treatment means was introduced. At that time, also, the concept of an
orthogonal contrast was presented, and it was suggested that orthogo
nal contrasts were to be preferred over nonorthogonal contrasts. How
ever, the researcher was warned not to let the statistician's desire for
orthogonality override his (the researcher's) needs.
In this section some general comparisons among treatments will be
examined, not to illustrate the concept of a contrast, but to demon
strate the manner in which the ANOVA is modified to provide the
proper analysis. Because of the infinitely many possibilities, this will
best bejione by discussing a few illustrative cases.
3O4
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
For example, consider an experiment involving t treatments and n
experimental units per treatment in which no subsampling occurred.
If treatment No. 1 were a "control" treatment, it would be of interest
to make tlie following specific comparisons among the treatments:
(1) treatment No. 1 versus the rest and (2) among the rest. The sums of
squares for these two comparisons would be determined as follows:
- T*/tn
SS(l versus rest)
\Tlfn + (r 2 + - -
xSVSXamong the rest)
These results, when coupled with the basic ANOVA, "would be pre
sented as in Table 1 1 .20, where the sums of squares (degrees of freedom)
for the selected comparisons are offset to indicate that they are portions
of the treatment sum of squares (degrees of freedom). (NOTE: In this
example, the sums of squares for the two comparisons add up to the
treatment sum of squares. The reader is "warned that this will not
always be the case.)
TABLE 11.20-Generalized ANOVA Showing Two Selected Treatment
Comparisons
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
Mean
1
Jl4T,_.,
M
Treatments
t 1
T
T
T/E
1 vs. rest
1
(CV)
Ci
Ci/JB
Among the rest. .,.,,...
t 2
( O2 lint
C 2
Co/E
Experimental error
t( n l)
Ew
E
Total
in
51 ^ 2
A second illustration based on the same type of design would be the
case in which the t treatments segregate into k groups containing
t\, t%, - * , tk treatments, respectively, where
In such a case, the natural comparisons would be: (1) among groups
and (2) among treatments within the ith group; i= 1, - - , fc. The sum
of squares for the first of these fc + 1 comparisons would be calctdated
as f olio ws :
G yy =
groups) =
T*/tn.
11.8 SELECTED TREATMENT COMPARISONS 3O5
The sum of squares among treatments in the first group is given by
The sums of squares among treatments in each of the remaining /b 1
groups would be found in a similar manner. The results would then be
presented in ANOVA form as in Table 11.21. (NOTE: Once again the
TABLE 11.21-Generalized ANOVA Showing k+1
Selected Treatment Comparisons
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
1
-tkt r/7/
AT
^lean
Treatments
t1
-t j/j/
r
T/E
Among groups
k 1
Gyy
G
G/E
'Within group 1 ....
hl
(WOw
Wi
W,/E
^^ithin group 2
2 2 1
(TiT 2 ) w
W*
Wz/E
\jyithin group k ....
te 1
(TF*)w
w k
W k /E
j f i >
77
E
Experimental error
t(ni)
-&yv
~n_ *- 1
f/tsr
y^ y2
lotal
tn
^1^ ^
sums of squares for the various comparisons add up to the treatment
sum of squares.) , . .
One more general illustration will be given. In this instance, assume
(again) that one treatment is a "control." However, the researcher
wishes to do more than compare: (1) control versus rest and (2) among
the rest He also wishes to compare, separately, each noncontrol treat
ment versus the control. Thus, in addition to the sums of squares indi
cated in the first illustration, he would also compute:
(C,) w = -55(1 vs 2) = (Tl
s 3) = (r*
vs = (Tl + T t }/n - (Tx + T t )*/2n.
These results would then be presented as in Table 11.22. (NOTE : This
306
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE 11.22-Generalized ANOVA Showing t+1
Selected Treatment Comparisons
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
Mean
1
Mm/
M
Treatments
t1
T
T
T/E
Control vs. rest . . .
1
(C^yy
Ci
Ci/jE
Among rest
t 2
\\-s i &)yy
C 2
C 2 /
Control vs. 2
1
\\^%)yy
C 3
CS/JE
Control vs. 3
1
(CO,*,
C 4
c 4 /
Control vs t .
1
\\-"t I .1/7/rr
v 1 1
Cn-i/E
T^XTDCr intent" ?1 prror
t ( n ^\\
Total
tn
T! F 2
time neither the degrees of freedom nor the sums of squares for the
comparisons will add up to the treatment sum of squares.)
Example 11 .9
The experiment described in Example 10.10 was performed and we
wish to investigate the specified comparisons. Assuming that the data
given in Table 7.20 were the results of this experiment, it iib seen that:
(C^ V y = [(184 + 68) 2 /6 + (170 -h 378) V14] - (800) 2 /20
= 34.3
(C^yy - [(184)V4 + (68) V2] - (252) V6 - 192.0
(C yy = [(170) 2 /5 + (378) 2 /9] - (548) 2 /14 = 205.7.
Combining these figures with those of Table 7.21, we get Table 11.23.
Examination of the F- values in Table 11.23 indicates that all the treat
ments (electrolytes) differ significantly in their effects on the charac
teristic (of the batteries) being studied. (NOTE: See Problein 11.30 for
the expected mean squares.)
11.9 SELECTED TREATMENT COMPARISONS: ORTHOG
ONAL AND NONORTHOGONAL CONTRASTS
Having spent considerable time discussing treatment comparisons in
general, let us now concentrate on the subject of contrasts, and par
ticularly on orthogonal contrasts.
It may be verified that the sum of squares associated with a particu
lar contrast is given by
= c
^j
1=1
-(
(11.53)
11.9 ORTHOGONAL AND NONORTHOGONAL CONTRASTS
307
where all symbols except t are defined as in Section 10.15. The symbol
t is used here, rather than k as in Section 10.15, to conform to the nota
tion being used in the present chapter. If each treatment total
TABLE 11.23-ANOVA for Experiment of Example 11.9
(Data in Table 7.20) Showing the Analysis of a
Specified Set of Treatment Comparisons
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
IVIean
1
32,000
32,000
Treatments (electro
lytes)
3
432
144
72
Ci (1 and 2 vs. 3
and 4)
1
34.3
34.3
17.15
C 2 (1 vs. 2)
1
192.0
192.0
96
C 3 (3 vs. 4)
1
205.7
205.7
102.85
1 ft
32
2
Experimental error . . .
nrv-vfoi
20
32 464
is the sum of the same number of observations (that is, if
i=l, - - - , 0> Equation (11.53) simplifies to
n
= n for
(11.54)
n
The results would then be presented in an AN OVA in agreement with
the format adopted in the preceding section. [NOTE: If a set pf^ 1
orthogonal contrasts among t treatments is investigated, the individual
sums of squares (one for each contrast) will add up to the treatment
sum of squares. ]
Example 11.10
Consider again the experiment described in Example 10.10 and
analyzed in Example 11.9. The sums of squares associated with the
three contrasts could also have been calculated as follows:
[(7) (184) + (7) (68) + (-3) (170) + (-3)(378)]'
[4(7) 2 + 2(7) 2 + 5(-3) 2 + 9(-3)]
[(1)(184) + (-2) (68) + (0X170) + (0)(378)] 2
+ 2(-2) + 5(0)2 + 9(0)2]
(0)(68) + (9) (170) + (-5)(378)] 2
(CO*
(COw j- 4(0)2 + 2(Q)2 + 5(9)2 + 9(_5)*]
The ANOVA will, of course, be the same as in Table 11.23.
308
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
Example 11.11
The experiment described in Example 10.11 was performed, and^the
data shown in Table 11.24 were recorded. The appropriate calculations
are:
^F 2 = 32,378
M yv = (800) 2 /20 = 32,000
T yy = [(ISO) 2 4- (160) 2 4- (160) 2 4- (164) 2 4- (136) 2 ]/4 - 32,000
EW = 32,378 32,000 248 = 130
[(-1)(180) + (4) (160) + (-1X160) + (-1X164) + (-1)(136)] 2
248
(CO
4[(-l) 2 + (4) 2 + (-1) 2 + (-1) 2 + (-1) 2 ]
(Q)(16Q) + (1)(160) + (-1X164) + (-
4[(1) 2 + (O) 2 + (I) 2 + (-1) 2 + (-1) 2 ]
+ (0)(16Q) + (-1)(16Q) + (0)(164) + (0)(136)] 2
4[(1) 2 4- (0) 2 4- (~1) 2 4- (O) 2
i- (0)(160) 4- (0)(160) 4
(O) 2 ]
= 100
50
98
^ * )y " 4[(0) 2 + (O) 2 + (O) 2 4- (I) 2 + (~1) 2 ]
These results are then summarized as in Table 11. 25. Using a: = 0.05,
all contrasts except Ci are judged to be statistically significant. (NOTE:
See Problem 11.31 for the expected mean squares.)
TABLE 11.24r-Data From Experiment Described in Example 10.11
and Discussed in Example 11.11
Electrolytes
1
2
3
4
5
40
38
44
41
34
45
40
42
43
35
46
38
40
40
34
49
44
34
40
33
TABLE 11. 25- AN OVA for Experiment Described in Example 10.11
(Data in Table 11,24; Discussion in Example 11.11)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
TV/T^ko-n
1
4
1
1
1
1
15
32,000
248
100
50
98
130
32,000
62
100
50
98
8.67
Elcr^T^lytes . ^ .
7.15
11.53
5.77
11.30
Ci
Co
C s
CA
L* > f\<a.T*1 TY1 d^T"! "f~ t 1 f^TTVIT
Total
20
32,378
IK9 ORTHOGONAL AND NONORTHOGONAL CONTRASTS 309
Up to this point, ttie discussion of contrasts has centered on: (1)
ANOVA techniques for isolating the sums of squares associated with
each contrast and (2) the use of the corresponding mean squares to test
the hypothesis that the true effects estimated by the contrasts are 0.
However, the problem of estimation should not be overlooked.
If the true effect estimated by a contrast C 3 - is denoted by the
symbol <f>j, it is desirable to construct a confidence interval estimate of
0y. That is, two numbers, L and U, are sought such that we can be
100-y per cent confident that <f> will be between L and U. To determine
Lf and U, the standard error of a contrast is needed. Defining the esti
mated variance of a contrast by
= v ( 2: CV
the standard error^of a contrast is given by VF(C/).
The nature of V(Tf) will, of course, depend on whatever assumptions
are made concerning the observations. If we are dealing with a com
pletely randomized design involving t treatments and n experimental
units per treatment in which no subsampling has been performed and
if the usual assumptions (see Section 11.2) have been made, then
;? y (11.56)
and
r\
l s~~t ~p- , \'\S "&(("* \ ^1 1 ^7^
U)
where v stands for the number of degrees of freedom associated with s 2
in Equation (11.56).
Example 11.12
Consider the experiment discussed in Examples 10.11 and 11.11. The
data were presented in Table 11.24 and the ANOVA in Table 11.25.
For this case, we have:
F(Ci) = 4(8.67)[(-l) 2 + (4) 2 -h (-1) 2 + (-1) 2 + (-1) 2 ]
F(C 2 ) = 4(8.67) [(I) 2 + (O) 2 + (I) 2 + (-1) 2 + C-l) 2 ]
F(C 3 ) = 4(8.67)[(1) 2 + (O) 2 + (-1) 2 + (O) 2 + (O) 2 ]
F(C 4 ) = 4(8.67)[(0) 2 + (O) 2 + (O) 2 + (I) 2 + (-1)'].
310 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
Since s 2 ==8.67 had 15 degrees of freedom, confidence intervals for
4> i (z = l, 2, 3, 4) may easily be constructed using Equations (10.23)
and (11.57).
11.10 ALL POSSIBLE COMPARISONS AMONG TREAT
MENT MEANS
In Sections 11.8 and 11.9, the usual method of analyzing comparisons
among treatment means was discussed in considerable detail. How
ever, one very important (statistical) restriction on the use of the de
scribed method was not mentioned. This restriction is as folio TVS: The
comparisons to be studied should be selected in advance of any analysis of
the data. That is, the method of analyzing contrasts described in the
preceding sections would not, in general, be valid if the comparisons
were decided upon after a perusal of the data and (perhaps) a pre
liminary ANOVA. In other words, the comparisons should have been
decided upon during the planning stage.
The restriction stated in the preceding paragraph can, however, work
a hardship on the researcher. Much experimentation is of a purely
exploratory nature and little, if any, idea of which comparisons might
be of interest is available prior to the collection and analysis of the
data. In such cases, the researcher would like to gain more from the
analysis than a simple statement that the treatment means are, or are
not, statistically significant. He would also like to know, for example,
if some of the treatments might be considered equivalent and which
treatment is "best."
How can the researcher attain the goals stated in the preceding
paragraph? This problem has received much attention from statisti
cians in recent years, and some of those who have made contributions
in the area are: Bechhofer (4), Duncan (15, 16), Dunnett (17), Hartley
(22), Keuls (24), Kramer (25, 26), Newman (27), Scheffe (30), and
Tukey (33, 34, 35, 36). Incidentally, the methods of Duncan, Scheff6,
and Tukey (the major protagonists) are discussed in detail in Federer
(19), and numerical illustrations are given for each method.
Before proceeding to discuss the method which I favor, time will be
taken to mention an associated technique which has been widely used
by researchers for many years. This technique involves what is known
as a least significant difference or LSD, which is defined by
LSD =
where v represents^the degrees of freedom associated with the variance
estimate used in F(Fi F y ). The LSD technique operates as follows:
If the absolute value of the difference between any two treatment
means exceeds the LSD, the effects of the two treatments are judged
to be significantly different ; if the absolute value of the difference does
11. TO COMPARISONS AMONG TREATMENT MEANS 311
not exceed the LSD, no such conclusion is reached. The reader is
warned that indiscriminate use of the LSD technique is dangerous for,
if we have enough treatments, the probability is high that at least one
of the t(t 1)/2 differences will, due to chance alone, be judged sig
nificantly different. Thus, the use of the LSD is to be discouraged.
(NOTE: When Z = 2, the LSD is a legitimate, but redundant, device.)
The method to be used in this book for making (when desirable) all
possible comparisons among treatment means is that proposed by
Scheff6 (30). While this method has not been the one most widely
adopted, it does have certain advantages. These advantages are:
(1) it is closely related to the concept of a contrast, (2) it uses tables
that are widely available (viz., .F-tables), and (3) it is easy to use. Let
us now see how the technique works.
Recalling that a contrast is defined by
(11.59)
the procedure is to calculate
' 2 (11.60)
where
A* = (t - l^ci^oc,,.,,), (11.61)
^(Ti), (11.62)
vi = t 1, (11.63)
and ?2 stands for the degrees of freedom associated with the denomi
nator mean square used in the 7^-test of H\T~T<L * =T*. Then, if
\Cj\ >A[F(C/)] 1/2 , the hypothesis H:<pj = will be rejected. (See
Section 11.9 for the definition of </.) That is, if the absolute value of
Cj exceeds A[F(C/)] 1/2 , the contrast <7/ will be said to differ signifi
cantly from 0. [NOTE: The original F-test rejects -ff:r; =
(i= 1, - - - , f) if and only if at least one <7/ is significantly different, by
Scheff^'s techjiique, from 0. The application of Scheffe's procedure per
mits us, then, to determine which of the C/ are significant. ]
Example 11.13
Consider the experiment described in Examples 10.10 and 11.9. The
data were presented in Table 7.20 and analyses in Tables 7.21 and 11.23.
The value of A to be used in making any desired comparison is found
to be 3.98 since
A* = - l)/^!-*)^.^) - (3)F.99ca.i6) = (3) (5.29) - 15.87.
312 CHAPTER 11 , COMPLETELY RANDOMIZED DESIGN
Let us examine contrast C% described in Example 10.10, namely,
c 2 = (I)T! + (-2)r a + (0)T 3 + (O)r 4 = (i)(i84) + (-2) (as) - 48.
The estimated variance of C% is given by
p-(C 2 ) = (l) 2 (4s 2 ) + (-2)*(2*2) = 12^ 2 = 12(2) =24.
Therefore,
2 = (3.98) V24 = (3.98) (4,899) 19.498.
Since j Cy| = 48 > 19.498, we conclude that the difference between the
effects of treatment No. 1 and treatment No. 2 is statistically signifi
cant. Incidentally, this agrees with the conclusion reached in Example
11.9. Other comparisons among the treatment effects could be made in
a like manner.
Example 11.14
Consider the experiment described in Examples 10.11 and 11.11. The
data were presented in. Table 11.24 and the analysis in Table 11.25. In
this illustration, A 2 = 4(4.89) = 19.56 and thus A =4.42. If we are inter
ested in Cz as defined in Table 10.3, it may be verified that (72 = 40,
F(C 2 ) = 16s 2 -16(8.67) =138.72, [F(Cy ] 1/2 = 1 1.78, and -A[F(C 2 )] 1/2
= 52.07. Since | C*\ =40 <A [F(C 2 ) ] 1/2 = 52.07, we conclude that C 2 is
not significantly different from 0.
It is noted, however, that this conclusion is the opposite of that
reached in Example 11.11. Why is this? The reason may be explained
as follows: Scheff^'s method will not lead to significant results (if the
appropriate null hypothesis is true) as frequently as will the classical
approach of orthogonal comparisons because we have been permitted
to examine the data before deciding on the analysis. This, obviously,
should lead to fewer cases of claiming significance when no real dif
ferences exist. This is as it should be, for, if we can look at the data
before deciding on the comparisons to be investigated, we should be
able to lessen our chances of making errors. From the point of view of
estimation, this decrease in the "frequency of errors" takes the form of
longer confidence intervals (i.e., our estimates are less precise) than
those provided by the classical approach.
11.11 RESPONSE CURVES: A REGRESSION ANALYSIS OF
TREATMENT MEANS WHEN THE VARIOUS TREAT
MENTS ARE DIFFERENT LEVELS OF ONE QUANTI
TATIVE FACTOR
The reader may be wondering why the subject of this section is
under discussion at this time. Did we not discuss regression analyses
completely enough in Chapter 8? Of course we did, but now we wish
to utilize the techniques of regression to make more complete and in
formative analyses of data arising from completely randomized designs
in which the treatments are different levels of a single quantitative
factor.
11.11 RESPONSE CURVES 313
How is this possible? Let us suppose th^at the treatments being
examined are: (1) different levels (or rates) of application of the same
fertilizer, (2) different weights of an object being moved in a time-and-
motion study project, or (3) different intensities of a given stimulus in
a psychological experiment. If situations such as these arise, it seems
reasonable to investigate how the measured characteristic varies with
changes in the level of the treatment. That is, we would like to know
if the change in the measured characteristic takes place in a linear,
quadratic, , . . fashion as the level of the treatment is increased or
decreased. In other words, we wish to gain some idea of the shape of
the response curve so that an estimate may be made of the optimum
level of the treatment.
Just how will the type of analysis indicated above be carried out?
The first step is to plot the treatment means, thus gaining some idea
as to the general shape of the response curve. Once this has been done,
the researcher will be ready to undertake a more rigorous analysis of
his sample data.
Equations for various possible response curves could, of course, be
determined using the techniques of Chapter 8. However, the deter
mination of the equation of the response curve is not the immediate
aim of our analysis. The immediate aim is to reach an objective de
cision (based on more than a simple plotting of the means) as to the
nature of the regression function that will best describe the effect of
the treatment on the response variable.
Perhaps the most convenient way of reaching the goal stated in the
preceding paragraph is to determine how much of the treatment sum
of squares would be associated with each of the terms (linear, quad
ratic, . . . ) in a polynomial regression. If the various levels of the
treatment being studied are equally spaced, this analysis can best be
carried out using the method of orthogonal polynomials introduced in
Section 8.20. (NOTE: The assumption of equal spacing will, in general,
present no problem, for both the researcher and the statistician will
ordinarily plan the experiment in such a way as to insure that the
assumption will be satisfied. That is, in most applications equal spacing
is the usual state of affairs.)
If each treatment total (T*) is the sum of n observations, the desired
sums of squares are found using
(t
g^
due to the kth degree term = - ; (11 . 64)
n
,-, 1;
where the ** are orthogonal polynomial coefficients. Extensive tables
of orthogonal polynomial coefficients are given in Anderson and
314 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE 11.26-Partial Table of Orthogonal Polynomial Coefficients
*-2
t = 3
*-4
t = 5
i
k=l
k=l k = 2
* = 1 ^ = 2
k = 3
k =l k = 2 k = 3
= 4
1
1
1 +1
3 +1
_ 1
2 +2 1
+ 1
2,
2
-^
+3
1 1 +2
4
3
+ 1 +1
+ 1 1
3
2
+ 6
4
+3 +1
+ 1
+1 1 2
4
5
_|_2 +2 +1
Houseman (1) ; for your convenience an abbreviated tabulation is
provided in Table 11.26. In agreement with the notation previously
adopted, the sums of squares associated with the linear, quadratic,
cubic, . , . terms will be denoted by (T ) yy , (T Q } yy , (T c ) vy , - - - . In
addition, since it is unlikely that the researcher will wish to isolate
more than a few terms when studying the treatment sum of squares,
the balance (if any) will be represented by (T^^yy. For example, if the
linear, quadratic, and cubic effects were isolated, the sum of the squares
of the deviations from regression would be given by
T yy
(11.65)
The results of the foregoing calculations may then be summarized as in
Table 11.27.
Example 11 .15
Consider the data in Table 11.28. Although an examination of the
treatment totals suggests that a linear response function may be ap
propriate, the quadratic effect will also be isolated for illustrative pur-
TABLE 11.27-Generalized ANOVA For a Completely Randomized
Design Showing the Isolation of the Linear, Quadratic, and Cubic
Components of the Treatment Sum of Squares
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
Mean
1
MW
M
Treatments
t 1
T
T
T/E
T L
1
( T T Iini
T L
TL/E
To
1
(To}w
TQ
To/E
T c
1
(Tc)
Tc
Tc/E
jTx>e t v
t 4
Tritr-n
Tr> ev /E
Experimental error. .
t(n 1)
JJtrtJ
E
Total
tn
T: Y*
11.11 RESPONSE CURVES
315
TABLE 11. 28- Yields (Converted to Bushels/Acre) of a Certain Grain
Crop in a Fertilizer Trial
Level of Fertilizer
No
Treatment
10 Ibs.
per Plot
20 Ibs.
per Plot
30 Ibs.
per Plot
40 Ibs.
per Plot
20
25
23
27
19
25
29
31
30
27
36
37
29
40
33
35
39
31
42
44
43
40
36
48
47
Totals
Means
114
22.8
142
28.4
175
35
191
38.2
214
42.8
poses. The following sums of squares were obtained:
52 F 2 = 29,560
M yy - (836) /25 = 27,955.84
TW = [(H4) 2 + (142) 2 + (175) 2 + (191) 2 + (214) 2 ]/5 - 27,955.84
= 1256.56
Eyy - 29,560 - 27,955.84 - 1256.56 347.60
h (0)(175) + (1)(191) + (2)(214)] 2
5[(-2) + (-1) 2 + (O) 2 + (I) 2 + (2) 2 ]
(249) 2
50
1240.02
[(2) (114) + (-1X142) + (-2)(175) + (-1X191) + (2)(214)] 2
(-27) 2
70
5[(2)
10.41
(-2)
(2) 2 ]
w = 1256.56 - 1240.02 - 10.41 = 6.13.
These are summarized in Table 11.29. Examination of the F-ratios
TABLE 11.29-ANOVA for Data of Table 11.28
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
.F-Ratio
!MLean . . .
1
27,955.84
27,955.84
Fertilizer levels ....
T L
4
1
1,256.56
1,240.02
314.14
1,240.02
18.07
71.35
TQ
1
10.41
10.41
0.60
TD e-o
2
' 6.13
3.07
0.18
Experimental error
20
347.60
17.38
Total
25
29,560.00
316 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
confirms our subjective judgment that the response of yield to rate of
application of the fertilizer is linear within the range of the levels of
fertilizer applied. This suggests that the rate of application of the
fertilizer might be increased even more, with an accompanying increase
in the yield. However, the reader is warned that extrapolation of the
linear relationship much beyond 40 Ibs./plot could (possibly) lead to
erroneous conclusions. Another way of putting this is to say that the
optimum level of fertilizer application has probably not yet been
reached, and further experimentation should be carried out along these
lines.
11.12 ANALYSIS OF A COMPLETELY RANDOMIZED DE
SIGN INVOLVING FACTORIAL TREATMENT COM
BINATIONS
By now the reader should be gaining some facility in the calculation
of sums of squares associated with various sources of variation. Thus,
the advent of another special situation, namely, factorial treatment
combinations, should present no new problems. In fact, once the reader
realizes that the factorial analysis is simply another way of partitioning
the treatment sum of squares, he is well on the way to a solution.
Let us now examine the details.
It has previously been noted that the usual statistical model asso
ciated with a completely randomized design involving t treatments and
n experimental units per treatment is
Ya = M + T* + e, y ; i = 1, - - , t (11.66)
j = l y . . . y n .
If we are now informed that the t treatments are actually all combina
tions of a levels of factor a and b levels of factor & (that is, t = a&) , the
statistical model may be rewritten as
Ftf* = fji + oLi + fr + (<*#)# + #*; i = 1, , a (11 . 67)
J = 1, ' , ft
k = 1, - - , n
where j^is^ the true mean effect, ca is the true effect of the ith leveLof
factor a, / is the true effect of the jth level of factor &, (<*)*/ is the
true effect "of the interaction of the ith level of factor a with the jth
level of factor 6, and e-^k is the true effect of the &th experimental unit
subjected to the (i?)th treatment combination. As usual, it is assumed
that M is a constant and that the e*/* are NID (0 ? a-} . Rather than discuss
assumptions concerning oa } /3j, and (cqS)*/ at this time, our attention
will be directed towards the calculation of the various sums of squares.
When tlie method of calculation has been explained, we shall return
to the assumptions and, as a consequence, to the expected mean
squares, estimation and test procedures, and other related topics.
A moment's reflection will confirm that the basic calculations are
unchanged. That is, ^,Y 2 , M yy , T vy , and E yy will all be calculated as
11.12 FACTORIAL TREATMENT COMBINATIONS
before. However, if we adopt the following notation:
Ai = total of all observations associated with the ith level
of factor a
b n (11.68)
= z; z; Y
y=i =1
BJ = total of all observations associated with the jth level
of factor 6
(11.69)
- ib i: Y*>
1=1 A^I
and
TV = total of all observations associated with both the ith
level of factor a and thejth level of factor b
= entry in the (i/)th cell of the a X 6 table (11.70)
= i:
it may be shown that
Ay V = sum of squares associated with the different levels of a
(F< - F) 2
(11.71)
= sum of squares associated with the different levels of b
= an
(11.72)
and
S^ab = among subclasses (cells) sum of squares for the #X6 table 1
(U.73)
a b
1 The reader \vill recognize that, in this particular situation, Sab = T yy . How
ever, the new notation and terminology were introduced at this time to acquaint
the reader with a system (of notation, terminology, and calculation) that will
prove most valuable when factorials involving more than two factors are analyzed.
318 CHAPTER T1 r COMPLETELY RANDOMIZED DESIGN
Using the preceding results, it may be verified that
= sum of squares associated with the interaction of
factors a and b
= n : ;
*=i y i
,, - F, - Y, +
(11.74)
These results are summarized in ANOVA form in Table 11.30.
TABLE 11.30-ANOVA for a Two-Factor Factorial in a Completely
Randomized Design
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Mean
1
MW
M
Treatments
A
aI
&-int
A
B
b 1
Bin,
B
AB
(a 1)(6 1)
(A&)yy
AB
Experimental error
ab(n 1)
JCLim
E
Total
abn
T\ F 2
Having explained the calculation of the various sums of squares,
your attention Is now directed to the assumptions associated with the
or*, j, and (a/8) v -. There are four possible sets of assumptions that can
be made with respect to the true treatment effects. These are discussed
below.
Model I: Analysis of Variance (Fixed Effects) Model
This model is assumed when the researcher is concerned only with the
a levels of factor a and the b levels of factor b present in the experiment.
Mathematically, these assumptions are summarized by:
=" 0.
y i
Model II: Component of Variance (Random Effects) Model
This model is assumed when the researcher is concerned with: (1) a
population of levels of factor a of which only a random sample (the a
levels) are present in the experiment and (2) a population of levels of
factor b of which only a random sample (the 6 levels) are present in
11.12 FACTORIAL TREATMENT COMBINATIONS 319
the experiment. Mathematically, these assumptions are summarized
as follows:
at are NID (0 ? <rj
ft- are NID (0, <rj
(aftis are NID (0,
Model III: Mixed Model (a Fixed, b Random)
This model is assumed when the researcher is concerned with: (1)
only the a levels of factor a present in the experiment and (2) a popu
lation of levels of factor 6 of which only a random sample (the 6 levels)
are present in the experiment. Mathematically, these assumptions are
summarized as follows :
i>;= 2b(/3)* =
t wi t -=i
ft- are NID (0, ^).
Please note that 231=1 ()# was not assumed to be 0.
Model III: Mixed Model (a Random, b Fixed)
This model is assumed when the researcher is concerned with: (1) a
population of levels of factor a of which only a random sample (the a
levels) are present in the experiment and (2) only the & levels of
factor & present in the experiment. Mathematically, these assumptions
are summarized as follows:
oa are NID (0 7 <r)
= 0.
Please note that X)?-! G*)v was not assumed to be 0.
While the logic underlying the preceding mathematical formulations
is beyond the scope of this text, it is hoped that the validity of the
expressions will be substantiated by the arguments which will accom
pany the specification of the several F-tests. Thus, it is requested that
the reader accept the expressions in good faith and concentrate on
learning the methods of analysis. In the long run, this will prove most
beneficial.
Based on the foregoing assumptions, the expected mean squares may
now be derived. As in the preceding examples, the derivations will be
omitted and only the results tabulated. The expected mean squares
for each of the four cases are shown in Table 11.31.
Examination of the expected mean squares in Table 11.31 will indi
cate the proper F-tests for such hypotheses as HI : a.* = (i = 1 , - , a) ,
#2:/3y = (j = l, - - , &), Jy 3 :(a/S) iy = (i=l, - - - , a; j = 1, - - , 6),
,
O
U
is
8 g
Ml
'
l
w j?
i ^
?
i i
J
O
a
-W!
1
S
i
M
"| "| "|
1
M M
b b b b
O
TJ
7
-S
>
03
-W3
8
I
tT
8
N ^ T M |
03
"O
1
"b Is % *b
03
03
w
W
1 1
"1 T
'03
"1 "1 "I
C* C4 C4 C4
b b b b
hH
1
iH
^
I I "1
w.i -wi tS
% *b "b "b
|
4-J
8
T3
S
s
03
Source of \
! 1
a .|
S S ^ ccj -^ cl
S S H
11.12 FACTORIAL TREATMENT COMBINATIONS
321
TABLE ll.Sa-.P-Ratios for Testing the Appropriate Hypotheses When Dealing
With a Two-Factor Factorial in a Completely Randomized Design (See
Table 11.30 for the ANOVA and Table 1 1.31 for the Expected
Mean Squares)
Source of Variation
F-Ratio
Model I
Model II
Model III
(a fixed, b
random)
Model III
(a random,
b fixed)
Mean
Treatments
A
A/E
B/E
AB/E
A/AB
B/AB
AB/E
A/AB
B/E
AB/E
A/E
B/AB
AB/E
B . . . . ...
AB
Experimental error. . .
Total
=0. For your convenience these are
H 4:0^ = 0, H 5 :o% = Qy and H&
specified in Table 11.32.
Before attempting a discussion of the reasons why the expected mean
squares (and thus the ^-tests) are as indicated, a three-factor factorial
will be considered. When this has been done, a general discussion of
test procedures will be undertaken and numerical examples presented.
When a three-factor factorial is associated with a completely
randomized design involving n experimental units per treatment com
bination, the appropriate statistical model is
i = 1, - - , a (11.75)
y = i, - ,
k = 1, - - - , c
*=!,-,
in which all terms are defined in a manner analogous to ttie definitions
accompanying Equation (11.67). The basic sums of squares, namely,
]F^F 2 , M yy , T yy , B^d E yy &*& calculated in the usual way. Then, if one
forms an aX&Xc table, an aX& table, an aXc table, and a &Xc table,
the remaining sums of squares may be found as follows:
among cells sum of squares for the
table
(11.76)
t=i y i fc i
322 CHAPTER II, COMPLETELY RANDOMIZED DESIGN
X b table
(11.77)
v y
= among cells sum of squares for the a X b table
Sac among cells sum of squares for the a X c table
\h 5T* 2 ,
= 2-., 2^ T ik /bn M yv , (11.78)
Sbc = among cells sum of squares for the b X c table
T\/an-M (11.79)
ben M m , (11.80)
= 22 JB,-/acn M yy , (11.81)
== X) C k /abn M yy , (11.82)
A=I
= S ab Ayy Byy, (11.83)
~= ^ac - Ayy ~ C VU , (11.84)
= Sic B VV Cyy, (11 . gS)
In the above expressions, the various totals are denned as shown
below :
T %jk total of all observations associated with the ith level
of factor a, the yth level of factor b, and the kth level
of factor c
== entry in the (ijtyth cell of the a X b X c table (11. 87)
and
total of all observations associated with the ith level
of factor a and the yth level of factor 6
entry in the (ij)th cell of the a X b table (11.88)
* 1 Z=l
11.12 FACTORIAL TREATMENT COMBINATIONS 323
total of all observations associated with the iih level
of factor a and the kth level of factor c
entry in the (ijfe)th cell of the a X c table (11.89)
b n b
/ / s j * ijkl ===: S _^ -t iyky
y-1 Z=i y_i
total of all observations associated with the/th level
of factor b and the kih level of factor c
entry in the (jfyih cell of the b X c table (11.90)
i=l Z=l 1=1
total of all observations associated with the ith level
of factor a
* (ii .91)
total of all observations associated with jth level of
factor b
</ = ib s r w (n.92)
and
Ck = total of all observations associated with the th level
of factor c
= z i: i: YM =
The pertinent sums of squares are summarized in ANOVA form in
Table 11.33.
As was the case with a two-factor factorial, the assumptions concern
ing the true treatment effects can take several forms. In fact,, for a
three-factor factorial, there are eight different situations. Rather than
discuss all of these, only four representative cases will be exhibited.
These are described below.
324 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE 11.33-ANOVA for a Three-Factor Factorial in a Completely
Randomized Design
Source of Variation
Degrees of Freedom
Sum of
Squares
Mean
Square
IVLean
1
-Ww
M
Treatments
A
a I
Ayy
A
B
b 1
Byy
B
C
c\
r*
^w
C
AB
(a 1)(6-1)
(AB~) m
A#
AC
(a l)(e-l)
(AC) m
AC
BC
(6 l)(e 1)
(.BQyy
BC
ABC
(o l)(6-l)(c-l)
(ABQ W
ABC
Experimental error. .
abc(n 1)
Eyy
E
TVktcil
/T hfyt
y F Z
-^^ *
Model I : Analysis of Variance (Fixed Effects) Model
This model is assumed when the researcher is concerned only with
the a levels of factor a, the b levels of factor 6, and the c levels of factor
c present in the experiment. Mathematically, these assumptions are
summarized by:
i=i
y=i
&
= 0.
Model II : Component of Variance (Random Effects) Model
This model is assumed when the researcher is concerned with: (1) a
population of levels of factor a of which only a random sample (the a
levels) are present in the experiment, (2) a population of levels of factor
b of which only a random sample (the b levels) are present in the experi
ment, and (3) a population of levels of factor c of which only a random
sample (the c levels) are present in the experiment. Mathematically,
these assumptions are summarized as follows:
<x z are NTD (0, <r a )
/3j are NID (0,
fc are NID (0, <r T )
tf are NID (0, cr a/3 )
* are NID (0, <r Y )
& are NID (0,
ik are NID (0,
11.12 FACTORIAL TREATMENT COMBINATIONS 325
Model III: Mixed Model (a and b Fixed, c Random)
This model is assumed when the researcher is concerned with: (1)
only the a levels of factor a present in the experiment, (2) only the &
levels of factor 6 present in the experiment, and (3) a population of
levels of factor c of which, only a random sample (the c levels) are
present in the experiment. Mathematically, these assumptions are
summarized as follows:
** =
i i y i
Y* are NID (0, <J T ).
Please note that
c c
]C Or)**, ]C (^y)y*, and
&=1 A=l
were ?zoi assumed to be 0.
Model III: Mixed Model (a Fixed, b and c Random)
This model is assumed when the researcher is concerned with: (1)
only the a levels of factor a present in the experiment, (2) a population
of levels of factor 6 of which only a random sample (the & levels) are
present in the experiment, and (3) a population of levels of factor c of
which only a random sample (the c levels) are present in the experi
ment. Mathematically, these assumptions are summarized as follows:
f are NID (0,
k are NID (0, <r r )
are NID (0,
Please note that
5 c &
k9 and
j 1
were ?^o^ assumed to be 0.
Based on the foregoing assumptions, the expected mean squares are
derived and the results presented in Table 11.34. The proper 7^-tests for
various hypotheses are shown in Table 11.35.
Having outlined the methods of calculation associated with two- and
three-factor factorials in a completely randomized design, we are now
fi
fl
I '
fr 9 ;
UJ o
s -
cu o3
ted
ial
e 1
to
ab
Ex
Fa
pec
4.
11
a ~,
I
s
3
S ^
o .^
1
g
S -n
$
"
-
+
t
b
a
5i
Wl
W3
+
b
I
^
4-
-1
-3
+ +
t
t
"I
I
OS ^
+
"I
t
Wl
O
H
11.12 FACTORIAL TREATMENT COMBINATIONS
327
ready to discuss the expected mean squares exhibited in Tables 11.31
and 11.34 and the ^-ratios exhibited in Tables 11.32 and 11.35. Perhaps
the best way to approach this topic is to talk about the types of in
ferences that the researcher wishes to make. You will recall that the
various Models (I, II, and III) reflect the researcher's desire to make
inferences about : (1) only the levels of the factors present in the experi
ment, (2) populations of levels of factors of which only a random
TABLE 11.35-F-Ratios for Testing the Appropriate Hypotheses When
Dealing With a Three-Factor Factorial in a Completely Randomized
Design (See Table 11.33 for the ANOVA and Table 11.34
for the Expected Mean Squares)
Source of Variation
.F-Ratio
Model I
Model II
Model III
(a and b
Fixed, c
Random)
Model III
(a Fixed,
b and c
Random)
Mean
Treatments
A/E
B/E
C/E
AB/E
AC/E
BC/E
ABC/E
no exact test
no exact test
no exact test
AB/ABC
AC/ABC
EC/ABC
ABC/E
A/AC
B/BC
C/E
AB/ABC
AC/E
BC/E
ABC/E
no exact test
B/BC
C/BC
AB/ABC
AC/ABC
BC/E
ABC/E
A
B
c
AB . . . .
AC
BC
ABC
T^yp^Tirnprital error
Total
sample (of levels from each population) is present in the experiment*
and (3) a mixture of the two preceding situations, respectively. In each
of these situations, the researcher may reason as follows :
(1) When dealing with a situation in which Model I applies, the
conclusions reached about any particular effect will be un-
contaminated by any other effect since, by proper definition
of the terms in the statistical model, the average contribution
of every other effect can be made equal to zero. Consequently,
all ^-values will be calculated by forming the ratio of the
mean square for the effect under scrutiny and the experi
mental error mean square. That is, all effects are tested against
experimental error.
(2) When dealing with a situation in which Model II applies, the
conclusions reached about any particular effect will be con
taminated by all those effects which represent interactions
328 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
between the effect under scrutiny and other effects present in
the experiment. This reflects the researcher's realization that
his conclusions (inferences) about the effect under scrutiny are
uncertain not only because of the e's but also because of the
chance contributions of the randomly selected levels of any
factor. That is, a different random sample (of levels of any
factor) might lead to different conclusions, and the researcher
attempts to incorporate this uncertainty into his conclusions
by testing a particular effect against an "error" which includes
an estimate of this additional variability. Thus, the expected
mean squares will be as shown in Tables 11.31 and 11.34 where
it is observed that each expected mean square contains all the
components of variance whose subscripts contain all the
letters representing the effect under scrutiny. This is the
mathematical way of expressing the ''contamination" dis
cussed above. Consequently, the -P-tests are as specified in
Tables 11.32 and 11.35. [NOTE: This illustrates the remark
made in Chapter 10, namely, " . . . the (proper) experimental
error for testing a particular effect/ 7 ]
(3) When dealing with a situation in which Model III applies, the
conclusions reached about any particular effect may or may
not be contaminated by other effects. That is, we have a
mixture of cases (1) and (2). To summarize what could be a
rather involved discussion, let us state the following rule:
The expected mean square for any effect will contain, in
addition to its own special term, all components of variance
which represent interactions between the effect under
scrutiny and other effects whose levels were randomly
selected. It will not contain components of variance repre
senting interactions between the effect under scrutiny and
other effects whose levels comprise the entire population
(of levels) to be investigated.
The expected mean squares specified in Tables 11.31 and 11.34
are, of course, simply results of the above reasoning and, as a
consequence, the F-tests are as shown in Tables 11.32 and
11.35. [NOTE: Again, this illustrates the remark made in
Chapter 10, namely, " . . . the (proper) experimental error
for testing a particular effect/']
To aid the researcher in writing out expected mean squares for different
situations, the following sequence of steps is recommended:
(1) Include, when applicable, a component of variance for each
subsampling stage.
(2) Include a component of variance representing experimental
error.
(3) Include every component of variance whose subscripts include
11.12 FACTORIAL TREATMENT COMBINATIONS 329
all the letters specifying the effect Math which the expected
mean square is associated.
(4) Insert coefficients in front of each component of variance in
accordance with the approach discussed in Section 11.5.
(5) Delete from the set specified in step (3), all terms representing
interactions between the effect associated with the expected
mean square and other effects whose levels were not randomly
selected.
(6) For a main effect, replace the component of variance for that
effect by a "sum of squares divided by the appropriate degrees
of freedom" if the effect is a "fixed effect,"
Before presenting illustrations of the methods discussed in this
section, some additional remarks need to be made. In the interest of
economy, these are presented here in the briefest form possible:
(1) It will have been noted that the degrees of freedom for inter
action effects were specified without any explanation. The
general rule is: For an interaction effect denoted by
ABCD - - - , the degrees of freedom are
v = (a !)(& l)(c 1)(<2 1) - - - .
(2) When, as in Table 11.35, no exact tests of certain hypotheses
are available, approximate tests can be made following
Satterthwaite's procedure. (See Section 11.7.) For example,
when Model II was assumed in Table 11.34, an approximate
test of H: o-l = Q is
p * A/[AB + AC - ABC}.
(3) Conclusions (inferences) about one factor in a factorial must
take due cognizance of all interactions of this factor with other
factors. That is, recommendations about one factor must give
consideration to the way in which its effect is influenced by
other factors.
Example 11.16
Consider a 4X3 factorial in a completely randomized design with
three experimental units per treatment combination. The data are
given in Table 11.36. Proceeding as indicated, the following sums of
squares were calculated:
53 F 2 = 564,389
Myy = (4023) 2 /36 = 449,570.2
T vy Sat, = [(306) 2 + * * 4- (268) 2 ]/3 449,570.2 = 67,160.8
E vv = 564,389 449,570.2 67,160.8 = 47,658,0
A yy [(726) 2 + (991) 2 + (1022) 2 + (1284) 2 ]/9 449,570.2 = 17,351.7
Byy = [(1624) 2 + (1500) 2 + (899) 2 ]/12 449,570.2 = 25,061.2
= 67,160.8 - 17,351.7 - 25,061.2 = 24,747.9.
33O
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
These results are summarized in ANOVA form in Table 11.37, where
the expected mean squares are shown for each of the four cases illus
trated in Table 11.31. Since the data were hypothetical, no F-tests will
be performed- Such tests and the resulting inferences will be illustrated
in succeeding examples in which actual experimental data will be
examined.
TABLE 11.36-Hypothetical Data for Illustrating the ANOVA for a
4X3 Factorial in a Completely Randomized Design
ox
#2
a 3
a 4
bi Z>2 63
b\ bz bz
bi b% b$
&i b% &s
128
34
16
152
40
118
76
102
132
180
220
60
42
134
18
128
88
80
158
96
60
90
220
48
136
172
46
216
76
93
168
162
68
150
156
160
Example 11 .17
Consider an agronomic experiment to assess the effects of date of
planting (early or late) and type of fertilizer (none, Aero, Na, or K) on
the yield of soybeans. Thirty-two homogeneous experimental plots were
available. The treatments were assigned to the plots at random, subject
only to the restriction that 4 plots be associated with each of the 8
treatment combinations. The data are given in Table 11.38 and the
ANOVA (assuming Model I) in Table 11.39.
TABLE 11. 38- Yields of Soybeans at the Agronomy Farm, Ames, Iowa, 1949
(In bushels per acre)
Date of
Planning
Fertilizer
Experimental Units Within Treatments
1
2
3
4
Early
Check
Aero
Na
K
Check
Aero
Na
K
28.6
29.1
28.4
29.2
30.3
32.7
30.3
32.7
36.8
29.2
27.4
28.2
32.3
30.8
32.7
31.7
32.7
30.6
26.0
27.7
31.6
31.0
33.0
31.8
32.6
29.1
29.3
32.0
30.9
33.8
33.9
29.4
Late
Assuming oi = 0.01, it is seen that the hypothesis "date of planting has
no effect' J must be rejected. Examination of the mean yields indicates
that the later date of planting is better (i.e., is associated with higher
yields). Of course, more information is needed concerning the distinc
tion between "early" and "late" before explicit recommendations can
be made. No statistically significant effects were noted for either
Q
M
o
*i
CO
PQ
"co:
H
-WI
j ( E
0?
^ ^
cs
11
.. A ^
b b b
^
? ? f
"* "
*b *b *b b
-W.I
o
H ?
5
s
cr
CO
T3
"..
cu
E
+ + + .
b *b *b t>
CJ
1*
H-t
> 1
^ 8 "cf
1
S
c,^ t ^
o b ^
*b *b b b
HH
"5 ?V c ^"^
r o>
**W" <0 ^"^ *W3
o
i" 1 s"
b -b "b "b
000*0^0
C^S OS vO ^O ^>*
53 *
O CO* O ^ V2
?. oo ro o> g
w
ON ^5 C*4 ^ -<
cq 4>- CM O\ C5
o
1-8 1
CO o^
CO
o ^ ^ ^ gg
S
CO
3 o
, ^ CO CM ^
NO
CO
Q
g
^ ^ "5
g ^ -S
1 *
-i
O
H
CO ^
g o ^3 BQ X g*
^5! r i ["J
KH CT^ * '
332
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
fertilizers or for the interaction between fertilizers and date of planting.
(NOTE: Had a. been chosen as 0.05, the interaction effect would have
been significant* This illustrates the dependence of the inferences upon
the choice of significance level, a fact which is sometimes overlooked,
or forgotten, by the analyst. That is, we must always remember that a
statement about significance or nonsignificance is a direct function of
the selected value of a..}
TABLE 11.39-ANOVA for Experiment Described in Example 11.17
(Data Given in Table 11.38)
Degrees
Source of
of
Sum of
Mean
Expected Mean
JF-
Variation
Freedom
Squares
Square
Square
Ratio
Mean
1
30,368.80
30,368.80
Treatments
Dates of planting
1
32.00
32.00
<r* + (16/1) i w
10.42
Fertilizers
3
16.40
5.47
2 +rs/,n^* 2
O" ~j {&/ *J ) f M i
1.78
Fertilizers X dates
y-i
sf 4 -v 2
of planting, . . .
3
38.40
12.80
4.17
t i y-i
Experimental error
24
73.74
3.07
cr 2
Total
32
30,529.34
Example 11.18
Consider a 3X4X3 factorial in a completely randomized design with.
6 experimental units per treatment combination. The data are given
in Table 11.40. Proceeding as directed earlier, Tables 11.41 through
11*44 were obtained and the following sums of squares calculated:
23 ^ 2 = 27,981
M yy = 19,703.56
T vy
E yy
$bc
A
flyy
Byy
Cyy
Sabc = 3283.27
4994.17
2913.27
1065.32
670.83
941.79
463.79
84.93
1507.69
38.60
122.11
124.36.
11.12 FACTORIAL TREATMENT COMBINATIONS
333
TABLE 11.4O-Hypothetical Data for Illustrating the ANOVA for a
3X4X3 Factorial in a Completely Randomized Design
ax
a 2
as
bl b% bs &4
61 b* b* b 4
6, 6 2 *3 &4
3
10
9
8
24
8
9
3
2
8 9
8
2
10
9
8
29
16
11
3
2
7 5
3
8
10
2
8
27
16
15
8
2
15 7
14
Ci
1
6
8
14
14
13
8
5
9
30 9
2
7
8
9
6
18
10
2
16
14
7 6
11
8
1
10
12
3
8
8
4
11
2 2
9
29
45
47
56
115
71
53
39
40
69 38
47
4
12
3
8
22
7
16
2
2
2 7
2
7
10
5
8
28
18
10
6
6
6 5
9
7
9
2
7
27
15
12
7
7
16 1
13
Ci
14
5
7
15
34
11
9
5
13
11 8
3
7
9
8
2
19
9
12
12
13
6 6
12
7
6
12
3
3
15
8
4
12
3 2
10
46
51
37
43
133
75
67
36
53
44 29
49
5
10
5
8
23
9
17
3
2
8 6
3
9
10
27
8
28
16
11
7
8
9 8
15
15
7
6
15
30
14
12
5
11
18 3
8
C3
8
6
4
18
16
12
13
15
17
8 7
16
7
17
3
10
17
10
20
9
9
8 6
17
3
2
10
5
3
7
8
6
11
7 3
14
47
52
55
64
117
68
81
45
58
58 33
73
These results are summarized in ANOVA form in Table 11.45. Since
the data were hypothetical, no expected mean squares are given.
Neither are any F-tests performed. The reader is referred to the prob
lems at the end of the chapter for illustrations of various tests and the
resulting inferences.
TABLE 11.41
Table Formed From the Data of Table 11.40
a
i
a>i
i
a
3
61
b*
* 3
64
61
b*
&3
64
61
b z
63 64
Cl
29
45
47
56
115
71
53
39
40
69
38 47
c% ....
46
51
37
43
133
75
67
36
53
44
29 49
47
.57,
.5,5
64
117
68
81
45
58
58
33 73
334
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE HA2-aXb Table Formed From the Data of Table 11.40
&! 122 365 151
148 214 171
139 201 100
163 120 169
TABLE 11.43-aXc Table Formed From the Data of Table 11.40
177 278 194
177 311 175
c 3 218 I 311 222
TABLE 11.44~6Xc Table Formed From the Data of Table 11.40
&i
184 185 138 142
232 170 133 128
222 178 169 182
TABLE 11.45-ANOVA for Data of Table 11.40
Source of Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Mean
Treatments
A
B
C
AB
AC
BC
ABC
Experimental error ,
2
3
2
6
4
6
12
180
19,703.56
941.79
463 . 79
84.93
1,507.69
38.60
122.11
124.36
4,994.17
19,703.56
470.90
154.60
42.46
251.28
9.65
20.35
10.36
27.75
Total
216
27,981.00
Even though this section is already quite long, there are several
items which need mentioning before we leave (for the time being) the
subject of factorials. These items are: (1) general computational pro
cedures for factorials involving four or more factors, (2) special compu
tational methods for 2 n and 3 n factorials, (3) subsampling in com
pletely randomized designs involving factorial treatment combinations,
11.12 FACTORIAL TREATMENT COMBINATIONS 335
and (4) analysis of response curves associated with the various main
effects and interactions. A brief discussion of each of these will be
given in the following paragraphs.
The general computational procedure for factorials proceeds as
follows. First compute ^Y*, M yy , T yy , E yy , and any sums of squares
required because of subsampling. Then, to subdivide T w in, say, a
four-factor factorial, form in succession the four-way table, all three-
way tables, and all two-way tables. As each table is formed, compute
the border totals as a check on the entries you have made in the cells
of the tables. Then, starting with the two-way tables, calculate the
sums of squares for each of the main effects and for each of the two-
factor interactions. Then, proceeding to the three-way tables, calculate
the sums of squares associated with each of the three-factor
interactions. And, finally, utilizing the four-way table, the sum of
squares associated with the four-factor interaction may be obtained.
The extension to 5, 6, - , AT factors is easy. After obtaining the
basic sums of squares, form the AT-way table, all possible (N l)-way
tables, all possible (N 2)-way tables, . . . , all possible three-way
tables, and all possible two-way tables in the order mentioned. Then
calculate, in the following order, all main effect sums of squares, all
two-factor interaction sums of squares, . . . , all (N l)-f actor inter
action sums of squares, and the A^-factor interaction sum of squares.
Whenever all the factors are at p levels, and there are n factors, the
statistician refers to such an arrangement as a p n factorial. Of particular
interest are those cases where p = 2 or 3. When such cases arise, there
are available to the research worker certain special computational
techniques. These are explained in considerable detail in such references
as Yates (39) and Kempthorne (23), and may be pursued by those
readers whose primary interest is in computation. Since the methods
outlined earlier in this section are valid for all cases, there seems little
reason to burden the reader with a specialized technique at this time.
Accordingly, we shall do no more than has already been done, that is,
point out the existence of the methods and give pertinent references
for the use of interested persons.
Wlten subsampling occurs in a completely randomized design in
volving factorial treatment combinations, the methods of analysis are
simply a combination of those given in this section and Section 11.4.
Thus, no detailed discussion of computational techniques will be pre
sented at this time. However, to illustrate the nature of the ANOVA's,
two cases will be mentioned. The first of these will involve only one
subsampling stage, while the second will involve two stages of sub-
sampling. If only one stage of subsampling is involved, the appropriate
statistical model (for a two-f actor factorial) is
Yij kl = M + on + ft- + () ^ + i/fc + IK**; *=!,-, (11 . 94)
J - 1, - , b
=!,,
/ = 1, - - , p,
336 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
TABLE 11 46-Abbreviated ANOVA for a Two-Factor Factorial in a
Completely Randomized Design Involving One Stage of Subsamplmg
(Model I)
Source of
Variation
Mean.
Treatments
Experimental error.
Sampling error
Total
Degrees of
Freedom
- 1
b -
(a !)(&
ab(n 1)
abn(p 1
Expected Mean Square
T 2 -4- pcr z + pnb y^ ca/(a 1)
17 i-i
b 2
CT 2 + 2?<T 2 + PW* X &'/(& ~ *)
" 3-1
^ + <r* + pn,f: &ftl-/(a
and the ANOVA would appear (in abbreviated form) as m Table 11.46.
(NOTE : If only one sample were obtained from each experimental unit ;
e K if one small sample is taken from a field plot to estimate the yield
of the entire plot, p in Table 11.46 is set equal to 1 and the line for
"sampling error" is deleted. However, if the whole plot is harvested,
the sampling error is and the ANOVA would be as shown m Table
11 30 ) In the second case to be examined, that is, a case involving
two stages of subsampling, the appropriate statistical model (for a two-
factor factorial) is
(11.95)
on
i = 1,
3
k
m =
i,
i,
i,
, a
, n
, P
,d,
and the ANOVA would appear (in abbreviated form) as in Table 11-47.
The extension to cases involving more than two stages of subsampling
should be obvious. -..-,-, ^ ^.v,
As indicated in Section 11.11, it is often advisable to examine the
response curve which summarizes the effects of the various levels of a
factor upon the characteristic being measured. When our data nt a
factorial arrangement, we may find it possible to examine response
11.12 FACTORIAL TREATMENT COMBINATIONS
337
TABLE 11.47-Abbreviated ANOVA for a Two-Factor Factorial in a
Completely Randomized Design Involving Two Stages of Subsampling
(Model I)
Source of
Variation i
Degrees of
Freedom
Expected Mean Square
jMCean.
1
Treatments
A
a 1
.2_j - j o .2 + ^ p<3 . 2 _j_^p w 5 ]T VO 1)
B .
bl
b
2_|_^.2_|_^ 2 _|_ j p y- /3i/(b 1)
AB
(a~ 1)(6 1)
a b
crl+d^-t-dp^-i-dpn^, ^Z (ctpYa/(a 1)(6 1)
Experimental error . .
First stage sampling
error
ab(n-l)
abn(p 1)
17 -l y-x
<T*-h<Z<T*+2><r 2
2_|_^.2
Second stage sam
pling error . .
abnv(d 1)
5 l if
o-*
5
Total
abnpd
curves associated with the levels of 2 or more factors. For example, if
we have 2 factors, a and &, we may subdivide the 2 sums of squares,
A yy and B yy , into parts designated as (A L ) VV} (Ao) y3/ , - - , and (BxJ)y V)
(Bo)y y , - , respectively. That is, we may obtain the linear, quad
ratic, - - , sums of squares associated with each of the factors a and b.
However, since we are now dealing with factorials, it is also possible
to subdivide the interaction sum of squares, (AB} yy . The parts into
which (AB}yy may be subdivided will be designated as (Ax,BrJ) V y,
(AxJEtQ^yy, (A Q BL) yi/ , (AgjBq)^, . If a, third factor, c, were present,
we would then have such quantities as (Ci^) yyy (Co) yy , f A - n -^
(A L B L Cz,)
yv ,
^ etc. The number of
possible subdivisions is, of course, limited by the number of levels of
the various factors involved. Because we have already devoted so
much time to the discussion of factorials in a completely randomized
design, the details of this technique (i.e., response curve analyses for
the various main effects and interactions) will not be discussed here.
However, the technique will be discussed in the following chapter in
connection with a randomized complete block design. Since the method
is the same regardless of the design (as long as the completely random
ized design has equal numbers of observations in each category), the
person desiring the details now can jump ahead and read Section 12.12.
The reader will appreciate, I am certain, that the foregoing discussion
338 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
has only scratched the surface of the subject of analyzing factorials.
However, I also feel that sufficient material has been given to enable
the researcher to handle the most commonly occurring situations.
Should more complex situations arise, reference to one or more of the
books listed at the end of the chapter should prove helpful. If not, a
professional statistician should be consulted.
11.13 NONCONFORMITY TO ASSUMED STATISTICAL
MODELS
By now the reader is well aware that the usual assumptions in analy
sis of variance involve the concepts of additivity, normality, homo
geneity of variances, and independence of the errors. However, up to
this point, little has been said about: (1) tests to assess the validity
of the assumptions, (2) the consequences if the assumptions are not
satisfied, and (3) transformations which, if applied to the original data,
may justify the use of the assumptions in connection with the trans
formed data (i.e., the data as they appear after the transformation has
been applied). In this section each of these topics will be discussed
briefly. For those who wish more details, several references are given.
In particular, three excellent expository articles are those by Bartlett
(3), Cochran (9), and Eisenhart (18).
First, let us consider various statistical tests that have been proposed
to check on the validity of the several assumptions.
Homogeneity of Variances
In Section 7.21, Bartlett's test was given for testing the hypothesis
fl":cr? = cr|= - - =<T| where a random sample of n^ observations had
been taken from the ith normal population (i= 1, - , fc) . Clearly, this
test is appropriate for checking on the homogeneity of variances. How-
eyer 5 __Bartlett^s^test has been shown to be quite sensitive to_non-
normality. Thus, if nonnormality is suspected or has been demon
strated, the test should be modified as suggested by Box and Anderson
(5). For a discussion of other tests, the reader is referred to Anscombe
and Tukey (2), Box and Anderson (5), David (12), and Dixon and
Massey (14).
Normality
To check on the assumption of normality, one can use the chi-square
test of goodness of fit given in Section 7.15. An alternative, and perhaps
preferred, method is the Kolniogorov-SniiiTLov^^tiesJL -discussed in
Chapter 15. For those who are satisfied with a less objective approach,
the data (or the residuals) may be plotted on normal probability paper
and a subjective judgment rendered.
Additivity
When the assumption of additivity is questioned, the problem is
somewhat more involved. This is so because there are three major
11.13 NONCONFORMITY TO ASSUMED STATISTICAL MODUS
causes of nonadditivity, namely, (1) the true effects may be multi
plicative, (2) interactions may exist but terms representing such effects
have not been included in the assumed model, and (3) aberrant obser
vations may be present. If the experimental design is such that inter
action effects may be isolated, the methods of the preceding section
may be used to check on (2). However, if this is not possible, the
researcher may use the more general tests suggested by Tukey (32, 37)
and by Ward and Dick (38) . Rather than give the details of these tests,
we refer the reader to the original publications. If access to these pub
lications is not possible, perhaps the illustrations in Snedecor (31) and
Hamaker (21) will suffice.
I ndepend ence
The assumption of independence or, granting normality, of un-
correlated errors is a crucial assumption and its importance should not
be overlooked. Of course, by utilization of the device of randomization,
the researcher can do his best to see that the correlation between errors
will not continually favor (or hinder) any particular treatment. If one
wishes to test for randomness, methods are available. However, since
these will be discussed in Chapters 15 and 16, no details will be given
at this time. The interested reader may jump ahead to the appropriate
sections. (NOTE: The procedure discussed in Section 11.6 may also
be helpful in this situation.)
In general, the__consequences are not serious when the assumptions
madeira connection, with analyses of variance are not strictly satisfied.
That is, moderate departures from the conditions specified by the
assumptions need not alarm us. For example, minor deviations from
normality and/or some degree of heteroschedasticity (lack of homo
geneity of variances) will have little effect on the usual tests and the
resulting inferences. In summary, the analysis of variance technique
i^jpiite^obust r ^|i(i,thus the researcher can rely on, its doing a good job.
und^r JDQSk JSJECIITQ qt *vn f*.^ However, since trouble can arise because of
failure of the data to conform to the assumptions, ways of handling
such situations must be examined.
When some action is needed to make tlxe data conf ormjto the jusual
approach is to transform the original data
inlsuchlTway that the transformed data will meet the conditions specie
fied by the assumptions. For example, if the true effects are multipli
cative instead of additive, it is customary to take logarithms and thus
change, for instance,
Y = jjLcttffrs (11.96)
into
Y' = log Y = log M + log en + log fy + log . (11.97)
Fortunately, in most cases, one transformation will suffice. That is, it
is usually not necessary to make a series of transformations, each to
346
CHAPTER 11, COMPLETELY RANDOMISED DESIGN
correct a separate "deficiency " in the original data. The reason for
this fortunate state of affairs is that, in general, the utilization of a
transformation to correct one particular deficiency (say, nonadditivity)
will also help with respect to another deficiency (say, nonnormality) .
With this in mind, the more common transformations are summarized
in Table 11.48. Further details may be found in Bartlett (3) and
Tukey (32).
TABLE 11.48-Some Common Transformations
Transformation
Conditions Leading to
Its Application
Name
Equation
Logarithmic . . . .
F' = log Y
1. The true effects are multiplicative (or
proportional) .
or
2. The standard deviation is proportion
al to the mean.
Square root ....
F'=VF
or
The variance is proportional to the
mean (e.g., when the original data are
samples from a Poisson distribution).
F'-VF+l
Arcsine . . .
F' = arcslne Vp
The variance is proportional to p,
(1 ju) as, for example, when the orig
inal data are samples (expressed as
proportions or relative frequencies)
from binomial populations.
Reciprocal
F'=1/F
The standard deviation is propor
tional to the square of the mean.
Before leaving the subject matter of this section, one other technique
for handling heterogeneous variances should be mentioned. This tech
nique is as follows: Partition the experimental error sum of squares in
correspondence with any partitioning of the treatment sum of squares.
However, this technique, valid though it may be, is seldom employed
because; (1) it is difficult and time-consuming to perform and (2) each
portion of E vy will usually possess a very small number of degrees of
freedom so that the subsequent F-tests will be of little value (i.e., they
will not be very powerful or discriminating tests). Because this tech
nique is used so rarely, no further discussion will be given at this time.
However, an example of subdividing the experimental error sum of
squares will be presented in the next chapter.
11.14 THE RELATION BETWEEN ANALYSIS OF VARI
ANCE AND REGRESSION ANALYSIS
Perhaps the most concise statement that can be made concerning the
relation between analysis of variance and regression analysis is the
11.15 PRESENTATION OF RESULTS 341
following: Analysis of variance and regression analysis are essentially
the same. Why, then, have we spent so much time (and we are not
through yet) discussing analysis of variance as a separate topic? The
answer is : Because there are many cases (based on specific conditions)
that are more easily explained using the methods of this and succeeding
chapters than those given in Chapter 8.
Because of the complexity of the topic, the general equivalence of the
two methods (i.e., analysis of variance and regression analysis) will not
be discussed in this book. The interested reader is referred to Graybill
(20) and Kempt home (23) for a general discussion of the basic theory,
and to Chew (8) for some illustrative examples.
11.15 PRESENTATION OF RESULTS
Even though an ANOVA table is very convenient for summarizing
certain aspects of the analysis of a set of data, it suffers from a rather
serious deficiency, namely, that it tends to overemphasize tests of
hypotheses and underemphasize estimation. Since estimation is the
more important of these two aspects of statistical inference, this could
be serious if steps are not taken to remedy the situation. Two steps that
can be taken to improve matters are: (1) always accompany an
ANOVA table with tables of means, together with their standard er
rors, and (2) "whenever possible, portray the results in graphical form.
If these two steps are taken and if a readable report is prepared, the
results of your research will be more easily understood and appreciated.
Example 11.19
Re-examination of Example 11.3 will show that the means were
given in Table 11.9, the ANOVA in Table 11.10, and the standard error
of the mean in the discussion. Actually, the standard error, which was
the same for each mean because of the equal sample sizes, might better
have been included in Table 11.9.
Example 11.20
Re-examination of Example 11.4 will show that the suggestion made
in Example 11.19 was adopted in that case. That is, the standard errors
were presented along with the means to which they applied.
Example 11.21
Re-examination of Example 11.6 will show that the ANOVA was
given in Table 11.18 and the standard error of a treatment mean was
included in the discussion. However, the treatment means were not
explicitly exhibited although they could easily have been obtained. Had
a complete report of the research been prepared, this deficiency would
have been noted and removed.
Example 11.22
Re-examination of Examples 11.11 and 11.12 will show that standard
errors were (implicitly) found for each of the selected contrasts. As
noted in the discussion of Example 11.12, the point and interval esti-
342
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
mates of the true effects of the contrasts could then be calculated. These
would, of course, be included in the research report.
Example 11.23
Re-examination of Example 11.15 will indicate that any research
report concerning this experiment would have benefited by a graph
showing the treatment means (average yields) as a function of the
amount of fertilizer applied to the experimental plots. It is suggested
that the reader plot these means and examine the graph in connection
with the recommendations made in Example 11.15.
Example 11.24
Re-examination of Example 11.17 will reveal that the treatment
means were not given. Since they are pertinent to the conclusions, we
give them in Table 11.49. The standard errors of the treatment means
TABLE 11.49 Treatment Means for the Experiment Discussed in Example
11.17 (Data in Table 11.38; ANOVA in Table 11.39)
Date of
Planting
Fertilizer
Early
Late
Average
Check
32.68 (0.88)
31.28 (0.88)
31.98 (0.62)
Aero
29.50 (0.88)
32.08 (0.88)
30.79 (0.62)
Na
27.78 (0.88)
32 . 48 (0 . 88)
30.12 (0.62)
K
29.28 (0.88)
31.40 (0.88)
30.34(0.62)
Average
29.81 (0.44)
31.81 (0.44)
30.81
The figures in parentheses in tlie table are the standard errors of the means to which
they are appended.
shown in Table 11.49 were calculated by taking the square roots of the
folio wing estimated variances:
V(Yi) = V (date of planting mean) = 3.07/16 = 0.1919
V(Y^ =- V (fertilizer mean) = 3.07/8 = 0.3838
V(Yii) = V (date of planting X fertilizer mean) =* 3.07/4 = 0.7675.
A graphical presentation of the means is given in Figure 11.1 where,
of course, the reader must realize that the slopes of the lines are a direct
reflection of the scales adopted. However, since our main use of the
graph will be in the interpretation of the interaction, this will not matter,
for we shall be concerned only with the slopes of the lines relative to
one another. A study of Figure 11.1 will confirm the conclusions
reached in Example 11.17, namely: (1) the late date of planting is
apparently better than the early date of planting, (2) there is little
difference among the main effects of the four fertilizers, and (3) there
is some indication of a possible interaction. (NOTE: This last conclu
sion is suggested by the lack of "parallelism" of the plotted lines.)
In addition to the remarks made in the first paragraph of this section
and illustrated in Examples 11.19 through 11.24, the reader should
11.15 PRESENTATION OF RESULTS
343
33
LJ
32
01
O
31
r^
30
CD
*~*
29
o
_J
UJ
28
>~
27
EARLY
CHECK AERO Na
TYPE OF FERTILIZER
K
FIG. 1 1 .1 Graphical representation of the mean
yields given in Table 1 1 .49.
realize that many experiments are conducted and analyses of variance
performed only to estimate components of variance. Important as this
topic is, it is felt that the discussion given earlier in the chapter will
prove sufficient for most applications. Should further details be desired,
it is suggested that a professional statistician be consulted.
One other topic should be mentioned in connection with the presen
tation of results. This topic is concerned with the general way in which
ANOVA's are commonly presented. Two customs have become quite
firmly established over the years and they are as follows:
1. (a) If an .P-ratio exceeds the 95 per cent point but does not
exceed the 99 per cent point, the F-ratio (or the mean
square for the effect being tested) is tagged with a single
asterisk (*).
(b) If an F-ratio exceeds the 99 per cent point, the F-ratio
(or the mean square for the effect being tested) is tagged
with a double asterisk (**) .
2. If space is at a premium, only an abbreviated A1STOVA will be
presented. When this is done, it is customary to include only
the columns for: (1) sources of variation, (2) degrees of freedom,
and (3) mean squares.
Incidentally, when the asterisk convention is used, it is good practice
to define the symbols at the bottom of every ANOVA table by use of
the following footnotes:
* Significant at a = 0.05.
** Significant at a. = 0.01.
344
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
The use of these customs will be illustrated in succeeding chapters.
If any words can be put together to summarize the implications of
this section, they are as follows: Do not forget the reader. Remember, you
are writing not for yourself but for others. Anything you can do to make
your assumptions, procedures, results, analyses, and conclusions more
understandable will add to the value of your research.
Problems
11.1 What are the proper objectives of analyses of variance (using experi
mental or survey data) ; that is, for what purposes may we properly
use analyses of variance?
11.2 Forty technicians were available to investigate 5 methods of deter
mining the iron content of a certain chemical mixture. Eight of the
technicians used method No. 1, 8 used No. 2, and so on. The assign
ment of technicians to methods was performed in a random manner.
Each technician made only one determination. Given that: (1) the
total of the 40 observations was 80, (2) the among methods mean
square was 6, and (3) the pooled variance among technicians within
methods was 8, fill in the following ANOVA table. (NOTE: omit the
spaces marked X.}
Source of Variation
Degrees
of
Freedom
Sum
of
Squares
Mean
Square
Expected
Mean
Square
Mean
X
Among methods , T , . .
Among technicians
within methods
Total
X
X
11.3 Given the following abbreviated ANOVA:
Source of Variation
Degrees
of Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
Among treatments
4
244
61
<r 2 + 7 23 -/4
Among experimental units
within treatments
30
270
9
t~i
cr 2
(ct) Write out the appropriate model.
(&) State the null hypothesis, both in words and symbolically, that
the experiment was probably designed to test,
(c) Test the hypothesis given in the answer to (&) using a probability
of Type I error equal to .05.
PROBLEMS 345
11.4 A process is designed to produce a fishline that will have a "15-lb.-
test" rating. The braided line may be treated with 4 different water-
proofings. The hypothesis is that the 4 treatments have the same,
if any, effect on the test rating of the cord. Twenty samples of each
type of treated cord are tested for breaking strength. Assuming that
analysis of variance is a valid technique to use in this case, set up the
appropriate table showing the proper subdivision of the degrees of
freedom. Discuss any further analyses that might be useful in investi
gating the treatments.
11.5 It is desired to test 10 different baking temperatures when we use a
standard cake mix. Fifty sample batches of mix are prepared, and 5
are assigned at random to each of the 10 temperatures. Six judges
score the cakes, and the average score is recorded for each cake. Give
the proper subdivision of the degrees of freedom, and write out the
mathematical model assumed. State the hypothesis to be tested. Dis
cuss and evaluate the method of analysis.
11.6 Four methods of performing a certain operation have been tried and
we have 10 observations for each method. The mean productivities
under each method are 60, 70, 80, and 90, respectively. Not having
the original data from which to calculate the sums of squares, we
assume that the coefficient of variation (square root of the pooled
estimate of a z divided by the average of all observations) is 0.1. On
this assumption, test the hypothesis that the "method population
means" are equal.
11.7 Given that the means of 10 individuals in each of 5 groups are 30, 32,
34, 36, and 38, and that the variance of a group mean is 8, compute
the analysis of variance.
11.8 An investigation to study the variation in average daily gains made
by pigs among and within litters when fed the same ration gave the
following results:
Source of Variation
Degrees of
Freedom
Mean
Square
Among litters
29
0.0576
Among pigs in the same litter
180
0.0144
How would you use this information to design experiments to test the
effects of different rations on average daily gains?
11.9 Community X and community Y are two neighboring small towns.
Community X is supplied with electricity by a private power com
pany, while community Y operates a municipally owned but ineffi
cient high-cost power plant. As a result, cost of electricity to home-
users is higher in community Y than in community X] for example,
the charge for the first 50 watts is $3.00 in X and $4.50 in Y. A ran
dom sample of household meter readings for the same month was
taken in each community. The following values were obtained:
346
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
11.10
X Sample
(16 Observations)
(Kw-hours used)
F Sample
(21 Observations)
(Kw-hours used)
28 16
6 12
12 22
36 18
14 28
24 18
4 4
58 16
16 22
60 22
28 4
6 14
30 34
14 16
76 30
54 26
22 44
16 58
18
Analyze these data in two ways :
(a) Compare home consumption of electricity in the two communities
by means of the comparison of two groups using "Student's"
(&) Prepare an analysis of variance of these two samples.
It is suspected that five filling machines in a certain plant are
filling cans to different levels. Random samples of the production
from each machine were taken, with the following results:
Machine
A
B
C
D
E
11.95
12.18
12.16
12.25
12.10
12.00
12.11
12.15
12.30
12.04
12.25
12.08
12.10
12.02
12.10
12.02
Analyze the data and state your conclusions.
11.11 The amount of carbon used in the manufacture of steel is assumed to
have an effect on the tensile strength of the steel. Given the following
data, perform the appropriate analysis and interpret your results.
The tensile strengths of six specimens of steel for each of three dif
ferent percentages of carbon are shown. (The data have been coded
for easy calculation.)
PROBLEMS
347
Percentage of Carbon
0.10
0.20
0.30
23
42
47
36
26
43
31
47
43
33
34
39
31
37
42
31
31
35
11.12 A public utility company has a stock of voltmeters which are used
interchangeably by the employees. The question arises as to whether
all the voltmeters are homogeneous. Since it would be too expensive
to check all the meters, a random sample of 6 meters is obtained and
all 6 are read three times while being subjected to a constant voltage.
The following data, expressed as deviations from the test voltage,
were recorded. Analyze and interpret.
Meter
0.95
0.33
2.15
1.20
1.80
1.05
1.06
1.46
1.70
0.62
0.88
0.65
1.96
0.20
0.48
1.50
0.20
0.80
11.13 An experiment had for its objective the evaluation of variance com
ponents for the variation in ascorbic acid concentration (mg. per
100 g.) in turnip greens. Two leaves were taken from near the center
of each of 5 plants. Ascorbic acid concentration was determined for
each leaf. This was repeated on each of 6 days, a new selection of
plants being obtained each day. The following data were collected:
Day
Leaf
Plant
1
2
3
4
5
1
A
9.1
7.3
7.3
10.7
7.7
B
7.3
9.0
8.9
12.7
9.4
2
A
12.6
9.1
10.9
8.0
8.9
B
14.5
10.8
12.8
9.8
10.7
3
A
7.3
6.6
5.2
5.3
6.7
B
9.0
8.4
6.9
6.8
8.3
4
A
6.0
8.0
6.8
9.1
8.4
B
7.4
9.7
8.6
11.2
10.3
5
A
10.8
9.3
7.3
9.3
10.4
B
12.5
11.0
8.9
11.2
12.0
6
A
10.6
10.9
10.4
13.1
7.7
B
12.3
12.8
12.1
14.6
9.4
348
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
11.14
Plants, days, and leaves are to be considered as random variables
(there might be some question about days). Calculate the analysis of
variance and evaluate the variance components for leaves of the same
plant, plants of the same day, and days.
Suppose we have the mathematical model
Yak M + n + e i3 + Si,* i = 1, 2, 3, 4
j - 1, 2
k 1, 2
where n is the true effect of the fth treatment, # is the effect of the
jth experimental unit subjected to the ith treatment, and 5 t -y& is the
Ath determination on the (ij) th experimental unit. We wish to test the
hypothesis H:ri = for all i. The following values are known:
3Ti = 8 .En 3 12 = 5
T 2 = 7 JS 2 i = 3 22 = 4
T 3 = 10 31 = 2 32 = 8
2% = 7 ^41 = 5 ^42 = 2
11.15
and ]Cy 2 = 18. Complete the appropriate analysis of variance, test
the hypothesis, and interpret your results.
Given the following abbreviated ANOVA of data collected from an
experiment involving 6 treatments, 10 experimental units per treat
ment, and 3 determinations per experimental unit :
Degrees of
Mean
Source of Variation
Freedom
Square
Expected
Mean Square
Treatments , , , .
5
12,489
O"g f~ 3o"
"+^r?
5 tZ
Exp. units within treatments. . . .
54
3,339
2 _!_ 1
<T$ -j- OCT
2
Det per experimental unit
120
627
2
(TK
11.16
(a) Write out the model assumed, stating explicitly what each term
represents.
(6) Test the hypothesis that the 6 treatments have the same popula
tion mean.
(c) Compute the variance of a treatment mean.
(d) Given that th.6 sample mean for treatment No. 3 is 193.7, com
pute and interpret the 95 per cent confidence interval for esti
mating the true population mean of treatment No. 3.
(e) Assuming that the estimates of the components of variance would
remain unchanged, would it be more or less efficient to use 9 ex
perimental units per treatment and 4 determinations per experi
mental unit? Show all calculations necessary to support your an
swer. What is the gain or loss in information?
We conducted a completely randomized experiment to study some
chemical characteristics of 5 varieties of oats. We assigned each variety
at random to 6 plots, making a total of 30 plots. Instead of harvesting
PROBLEMS
349
11.17
the entire plot, we selected at random 8 3-by-3-foot samples from
each plot. For each sample we made 3 chemical determinations.
Indicate the proper complete subdivision of the total degrees of free
dom. Give the expected mean square for each source of variation.
Indicate the proper F-test to test the hypothesis that the population
means for the 5 varieties are equal.
Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Groups
3
600
Experimental units within groups
36
120
[Determinations per experimental unit. . .
80
12
(a) Give the expected mean squares, assuming that we are interested
in just these groups but that experimental units and determina
tions are random variables.
(6) Test the hypothesis that the group population means are equal.
Interpret your result.
(c) Compute the variance of a group mean (per determination) .
11.18 Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Among treatments
4
20
Experimental units within treatments . . .
Determinations per experimental unit. . . .
15
20
15
4
Obtain estimates of all the components of variance, and interpret
each in terms of the model y;yfc==M+'7~i+;j+Siyfc stating explicitly
all assumptions that you make.
11.19 Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
Treatments
3
1800
600
0-!+3o- 2 +30ov
Experimental units
within treatments ....
Determinations per
experimental unit ....
36
80
3600
960
100
12
<r 2 s+3<r 2
<r!
(a) Compute the variance of a treatment mean.
(6) Test the null hypotheses jff:er? = 0, and interpret your answer.
(c) The sample mean of treatment No. 1 is given to be 80. Compute
a 95 per cent confidence interval for estimating the true popula
tion mean of -treatment No, 1.
350 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
11.20 Given, the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Treatments
4
960
Experimental units within treatments
determinations per experimental unit
35
40
320
20
11.21
(a) Test the hypothesis that the population treatment means are all
equal. Interpret your answer.
(6) Give the expected mean squares in the above analysis of variance.
(c) Compute the variance of a treatment mean,
(d) Estimate the gain or loss in information if the above experiment
were to be repeated with 10 experimental units per treatment and
a single determination per experimental unit. State all your
assumptions.
Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Among treatments
9
570
Experimental units within treatments
Among determinations on same experi
mental unit
90
200
190
10
(a)
(6)
GO
11.22
11.23
11.24
Compute F to test the hypothesis that the 10 treatments have
the same true effect.
Compute the variance of a treatment mean per determination in
the above experiment.
Assuming that the estimates of the components of variance
would not change, estimate the gain or loss in information in
estimating the treatment means if 20 experimental units per
treatment were selected with a single determination on each
experimental unit in repeating the experiment.
The cities and towns of Arizona have been allocated to 5 strata (or
groups) according to population. In each stratum we select at random
10 cities (or towns) ; in each of these cities we select at random 4
blocks; and in each of these blocks we select at random 2 households.
Indicate the proper subdivision of the degrees of freedom for a com
plete analysis of variance of some item such as the average income
of the head of each household.
Set up the analysis of variance table and show the degrees of freedom
for the following experiment: Six spray treatments are applied com
pletely at random in an orchard of 100 trees (all being used). Each
treatment is applied to sets of 2 trees; then the yield of each tree is
estimated by obtaining 4 samples around the perimeter. Note that
all but 2 treatments contain 8 sets; the remaining 2 treatments con
tain 9 sets. Show the expected mean squares.
The following abbreviated analysis of variance was prepared from
chemical determinations made on samples of a legume hay. The hay
PROBLEMS
351
samples were obtained from a completely randomized experiment
involving 16 treatments.
ANALYSIS OP VARIANCE OF CHEMICAL DETERMINATIONS
SAMPLES
Source of Variation
Degrees of
Freedom
Mean
Square
Treatments
15
550
Plots with, same treatment
220
Samples from plots treated alike. . .
256
20
11.25
11.26
(a)
State a suitable hypothesis about these treatments, and make the
proper test. The experimenter wished to reject the hypothesis
only under the condition of a 1 per cent chance of a Type I error.
What is your conclusion?
Indicate the function of replication in this experiment for study
ing the effects of various treatments on a legume hay.
How might the replication and sampling procedure for this ex
periment be changed in order to increase replication without
changing the total number of samples to be analyzed?
Estimate the possible gain in relative efficiency for your proposed
change in the experiment.
(e) What assumption is required for making this calculation?
In an effort to develop objective methods of estimating the yield of
corn, an experimental survey was conducted in a district of central
Iowa. A random selection of fields was made, and within those fields
2 sampling units (consisting of 10 hills each) were selected at random
and the grain yield determined by harvesting and weighing. The
analysis of variance (on a 10-hill s.u. basis) is as follows:
(6)
(c)
(d)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Fields
47
2098.7
44.65
S.u.'s within fields. . . .
48
554.5
11.55
How much information would have been lost if only one s.u. per field
had been taken?
IData from a sample survey of farms in the Midwest were to be sum
marized by means of the analysis of variance. Eight types of farming
areas were included in the study, and within each area 5 counties
were selected at random. Within each of the chosen counties, 20
farms were selected at random and farm management records taken
for each. A partial list of the summary calculations was as follows for
the item "farm income' 7 :
Total corrected sum of squares = 8,183,000
Sum of squares for among counties within areas = 352,000
Mean square for type of farming areas ==33, 000.
(a) Prepare and complete an analysis of variance for "farm income"
from the above information.
352
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
11.27
(6) What is the variance of a type-of-farming area mean as deter
mined from your analysis of variance?
(c) Suppose it had been decided to select only 2 counties in each
area and sample 50 farms in each county. What is the relative
efficiency of the plan used to the procedure suggested here?
(<2) The type-of-farming areas included in the study were arbitrarily
selected upon the basis of some known differences. Are the areas
different with respect to "farm income?"
(e) Write out the expected values of the mean squares used in
ans wering (d) above.
Given the following abbreviated analysis of variance of the data from
a completely randomized experiment with 4 treatments, 8 experi
mental units per treatment, 3 samples per experimental unit, and 2
determinations (of some chemical or physical characteristic) per
sample :
Source of Variation
Degrees of
Freedom
Mean
Square
Treatments
3
19,200
Among experimental units treated alike ....
28
4 800
Among samples per experimental unit
64
2,400
Hetwe^ri det^rrni -nations pfvr sample , ,
96
1,200
Estimate the gain or loss in efficiency in estimating the treatment
effects if we had used 12 experimental units per treatment, 2 samples
per experimental unit and 1 determination per sample.
11.28 Describe the assumptions underlying the application of the analysis
of variance technique.
(a) Which of these assumptions can the research worker check for
any particular analysis?
(&) Which assumption can be fulfilled by the research worker in an
experimental situation by appropriate procedures?
(c) For what purposes do we employ the analysis of variance tech
nique?
(c) What criterion should be applied for judging the validity of an
^-ratio obtained from an analysis of variance?
11.29 (a) Explain in your own words the meaning in the analysis of vari
ance of (1) a variance component and (2) a fixed effect?
(6) Consider the following abbreviated analysis of variance:
ANALYSIS OF VARIANCE or CALORIES CONSUMED IN ONE DAY FOR A
SAMPLE OF IOWA WOMEN OVER THE AGE OF 30
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Among zones
2
16,960,000
8 480 000
Among counties in zones . .
97
41 128 000
424 000
Between segments in counties in
zones
100
40 , 000 , 000
400 OOO
Among individuals in segments
in counties in. zones
600
180 000 000
300 000
PROBLEMS 353
For this problem we shall assume that 4 individuals (women over
30) were interviewed in each segment and that 2 segments were se
lected at^ random in each county. The zones are open country, rural
community, and urban. Counties appearing in the sample for the
zones were 50, 25, and 25, respectively, yielding the 97 degrees of
freedom for counties in zones.
(1) Estimate the variance components for individuals, segments, and
counties from this analysis of variance.
(2) Test the hypothesis: Calories consumed on this day are the same
for all zones. Show the mean squares used in forming the ^-ratio,
and indicate in general why these are the proper mean squares to
use for the test.
11.30 With reference to Example 11.9 and Table 11.23, show that the ex
pected mean squares for the three comparisons are:
Efca -h
n 4 )
n*
11.31 With reference to Example 11.11 and Table 11.25, show that the ex
pected mean squares for the four comparisons are:
11.32 Given the additional information that, in Problem 11.10, machine
A is a standard machine and machines JS, (7, D, and E are experi
mental models, modify the original analysis to assess the relative per
formance of the five machines.
11.33 Apply the technique of Section 11.10, to the following problems:
() 11-9 (d) 11.12
(&) 11.10 (e) 11.13
(c) 11.11 (/) 11.14
11.34 If you did not use the technique of Section 11.11 in the analysis of
Problem 11,11, please do so now.
11.35 It is suspected that the age of a furnace used in curing silicon wafers
influences the percentage of defective items produced. An experi
ment was conducted using four different furnaces and the data given
below were obtained. Analyze and interpret the data.
354 CHAPTER IT, COMPLETELY RANDOMIZED DESIGN
PERCENTAGE OP GOOD WAFERS IN 8 EXPERIMENTAL TRIALS
PER FURNACE (THE SAME NUMBER OF WAFERS WERE
USED IN EACH FURNACE IN EACH TRIAL)
Furnace
A (age 1 year)
B (age 2 years)
C (age 3 years)
D (age 4 years)
95
95
80
70
92
85
80
65
92
92
82
70
90
83
78
72
92
83
77
72
94
88
75
66
92
89
78
50
91
90
78
66
11.36 It is suspected that tlie environmental temperature in which batter
ies are activated affects their activated life. Thirty homogeneous
batteries were tested, 6 at each of five temperatures, and the data
shown below were obtained. Analyze and interpret the data.
ACTIVATED LIEE IN SECONDS
Temperature (C.)
25
50
75
100
55
60
70
72
65
55
61
72
72
66
57
60
73
72
60
54
60
68
70
64
54
60
77
68
65
56
60
77
69
65
11.37 It is suspected that both the machine on which bearings are produced
and the operator of the machine influence the critical dimension,
namely, the inside diameter. To check on this, the data given below
were obtained under normal production conditions. Analyze and
interpret the data.
PROBLEMS
INSIDE DIAMETERS OF BEARINGS (IN INCHES)
355
Machine
1
2
3
Operator
A
B
C
D J3
1.02
1.03
1.05
1.03
1.02
1.03
1.03
1.06
1.03
1.03
1,02
1.03
1.04
1.02
1.04
1.03
1.06
1.02
1.07
1.02
1.06
1.05
11.38 An experiment was conducted to assess the effects of temperature
and humidity on the effective resistance of a standard type of re
sistor. The following data were obtained. Analyze and interpret the
data.
CODED RESISTANCE VALUES
Temperature
20
F.
70F.
160F.
Humidity
10%
50%
10%
50%
10%
50%
23
24
26
24
25
27
24
24
25
25
26
26
25
25
26
26
26
28
24
26
26
26
28
28
11.39 Given the following abbreviated analysis of variance:
ANALYSIS OP VARIANCE or NET INCOME PER CROP ACRE
Source of Variation
Degrees of
Freedom
Mean
Square
Between soil areas
4
625
Soil conservation programs ....
SAXSCP
3
12
400
225
Between farms in subclasses . . .
80
100
(a) Assuming both of the main classifications are fixed effects, indi
cate the appropriate /^-ratios for tests of the hypotheses: (1) soil
areas do not differ in income; (2) the soil conservation programs
have no effect on income per crop acre.
(&) Assuming that soil areas were selected at random and, also, soil
conservation programs were selected at random from a larger
number (not entirely realistic, but possible), indicate the F-ratios
for the tests listed in (a) above.
356
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
11.40 In the accompanying table pasture-acres per farm for a sample of 36
farms in Audubon County, Iowa, for the year 1934 are presented.
The sample consists of 3 farms from each soil-tenure grouping; there
are 12 such groups. It is expected that there may be some interaction
effects among the soil and tenure classes. We shall consider the tenure
grouping as a fixed effect and the soil groupings as random sampling
from a larger population. Perform the necessary calculations, set up
the analysis of variance table, and discuss the results. Also examine
the homogeneity of variance in the tenure groups by the Bartlett test.
DATA ON PASTURE ACREAGE, AUDUBON COUNTY, IOWA, 1934
Tenure Group
Soil Group
I
II
III
IV
Farm
Owners 1
37.0
40.1
57.0
36.0
52.0
38.0
72.6
65,2
71.0
(Pasture acr
50.0
28.6
37.2
42.0
54.5
58.0
54.0
58.0
29.0
es per farm)
49.0
43.7
27.0
50.9
34.0
43.8
67.4
32.5
43.8
56.0
69.0
54.7
55.0
41.0
54.6
63.0
45.0
60.0
2
3
Tenants 1
2
3. .
Mixed 1
2
3
11.41 Five varieties and 4 fertilizers were tested. From each experimental
plot 3 quadrats were selected at random and their yields recorded as
follows :
Varieties
Fertilizers
1
2
3
4
5
57
26
39
23
48
1
46
38
39
36
35
28
20
43
18
48
67
44
57
74
61
2
72
68
61
47
60
66
64
61
69
75
95
92
91
98
78
3
90
89
82
85
89
89
99
98
85
95
92
96
98
99
99
4
88
95
93
90
98
99
99
98
98
99
PROBLEMS
35Z
11.42
(a) Construct an analysis of variance table.
(fe) On the basis of the appropriate model, write the expected mean
squares conforming to the following assumptions:
(1) varieties and fertilizers random selections;
(2) varieties and fertilizers both given sets;
(3) varieties a random selection fertilizers a given set.
(c) Test the hypothesis of equal variety means. Test the hypothesis
of equal fertilizer means.
(d) Construct a table showing the means and their standard errors.
(e) What conclusions do you reach as a result of this experiment?
A building superintendent wishes to compare the relative perform
ance ratings of various combinations of floor wax and length of
polishing time. Three waxes are to be investigated along with 3
polishing times. Eighteen homogeneous floor areas are selected and
2 are assigned at random to each of the 9 treatment combinations.
Analyze and evaluate the following data.
PERFORMANCE RATINGS
(HIGH is BETTER THAN Low)
Wax
A
B
C
Polishing time
(in minutes)
15
30
45
15
30
45
15
30
45
7
7.5
8.2
7
7.2
7.1
8
9.2
9.6
8
7.4
8.6
7
7.6
7
8
9.4
9.5
11.43 An experiment was performed to assess the effects of type of material
and heat treatment on the abrasive wear of bearings. Two bearings
were tested at each of 10 treatment combinations. Analyze and
interpret the following data.
AMOUNT OF WEAR (CODED DATA)
Material
A
B
C
D
R
Heat
treatment*
O M
O M
O M
O M
O M
23 30
25 31
42 45
44 50
37 39
38 39
41 44
42 49
20 24
25 30
* = oven dried; M moisture saturated.
11.44 From each of 5 lots of insulating material, 10 lengthwise specimens
and 10 crosswise specimens are cut. The following table gives the
impact strength in foot-pounds from tests on the specimens. Analyze
and interpret the data.
358
CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
Lot Number
Type of Cut
I
II
III
IV
V
1.15
1.16
0.79
0.96
0.49
0.84
0.85
0.68
0.82
0.61
0.88
1.00
0.64
0.98
0.59
0.91
1.08
0.72
0.93
0.51
Lengthwise
0.86
0.80
0.63
0.81
0.53
specimens
0.88
1.01
0.59
0.79
0.72
0.92
1.14
0.81
0.79
0.67
0.87
0.87
0.65
0.86
0.47
0.93
0.97
0.64
0.84
0.44
0.95
1.09
0.75
0.92
0.48
0.89
0.86
0.52
0.86
0.52
0.69
1.17
0.52
1.06
0.53
0.46
1.18
0.80
0.81
0.47
0.85
1.32
0.64
0.97
0.47
Crosswise
0.73
1,03
0.63
0.90
0.57
specimens
0.67
0.84
0.58
0.93
0.54
0.78
0.89
0.65
0.87
0.56
0.77
0.84
0.60
0.88
0.55
0.80
1.03
0.71
0.89
0.45
0.79
1.06
0.59
0.82
0.60
1 1 .45 Five batches of ground meat are charged consecutively into a rotary
filling machine for packing into cans. The machine has 6 filling
cylinders. Three filled cans are taken from each cylinder at random
while each batch is being run. The coded weights of the filled cans
are given below. Analyze and interpret the data.
PROBLEMS
359
Cylinder
Batch
1
2
3
4
5
1
1
4
6
3
1
1
3
3
1
3
2
5
7
3
3
2
1
2
3
2
1
3
1
1
1
5
1
1
3
1
2
2
1
3
1
4
3
3
1
1
3
3
3
4
2
2
3
3
3
1
1
4
2
1
5
1
2
1
2
1
1
1
3
*
5
2
_____ i
1
6
3
3
3
1
3
1
1
3
4
2
2
11.46 The following data on the density of small bricks resulted from an
experiment involving 3 different sizes of powder particles, 3 pres
sures, and 3 temperatures of firing. The 27 combinations of this
3X3X3 factorial were run in duplicate. Analyze and interpret the
the following coded data.
Size
Pressure
Temperature
1900
2000
2300
5-10
5.0
12.5
20.0
340 375
388 370
378 378
316 386
338 214
348 378
374 350
334 366
380 398
10-15
5.0
12.5
20.0
260 244
322 342
330 298
388 304
300 420
260 366
266 234
234 258
350 284
15-20
5.0
12,5
20.0
134 140
186 30
40 210
146 194
412 428
436 490
152 212
194 208
230 254
360
CHAPTER 11 r COMPLETELY RANDOMIZED DESIGN
11.47 During the manufacture of sheets of building material the perme
ability was determined for 3 sheets from each of 3 machines on each
day. The table below gives the logarithms of the permeability in
seconds for sheets selected from the 3 machines during a production
period of 9 days. The 3 machines received their raw materials from a
common store. Analyze and interpret the data.
Day
Machine
Log of Permeability
1
1
2
3
1,404
1.306
1.932
1.346
1.628
1.674
1.618
1.410
1.399
2
1
2
3
1.447
1.241
1.426
1.569
1.185
1.768
1.820
1.516
1.859
3
1
2
3
1.914
1.506
1.382
1.477
1.575
1.690
1.894
1.649
1.361
4
1
2
3
1.887
1.673
1.721
1.485
1.372
1.528
1,392
1.114
1.371
5
1
2
3
1.772
1.227
1.320
1.728
1.397
1.489
1.545
1.531
1.336
6
1
2
3
1.665
1.404
1.633
1.539
1.452
1.612
1.690
1.627
1.359
7
1
2
3
1.918
1.229
1.328
1.931
1.508
1.802
2.129
1.436
1.385
8
1
2
3
1.845
1.583
1.689
1.790
1.627
2.248
2.042
1.282
1.795
9
1
2
3
1.540
1.636
1.703
1.428
1.067
1.370
1.704
1.384
1.839
References and Further Reading
1. Anderson, H. L., and Houseman, E. E. Tables of orthogonal polynomial
values extended to 2NT= 104. Res. Bui. 297, Agr. Exp. Sta., Iowa State Univ.,
Ames, April, 1942.
2. Anscombe, F. J., and Tukey, J. W. The analysis of residuals. Unpublished
REFERENCES AND FURTHER READING 361
handout, Gordon Conference on Statistics in Chemistry and Chemical
Engineering, New Hampton, N. H., July, 1957,
3. Bartlett, M. S. The use of transformations. Biometrics, 3:39, 1947.
4. Bechhofer, R. E. A sequential multiple-decision procedure for selecting the
best one of several normal populations with a common unknown variance,
and its use with various experimental designs. Biometrics, 14:40829, 1958.
5. Box, G. E. P., and Andersen, S. L. Permutation theory in the derivation of
robust criteria and the study of departures from assumption. Jour. Roy.
Stat. Soc., Series B, 17:1-34, 1955.
6. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
7. Chew, V. (editor) Experimental Designs in Industry. John Wiley and Sons,
Inc., New York, 1958.
g^ f Basic experimental designs. Experimental Designs in Industry.
John Wiley and Sons, Inc., New York, 1958.
9. Cochran, W. G. Some consequences when the assumptions for the analysis
of variance are not satisfied. Biometrics, 3:22, 1947.
10. . Testing a linear relation among variances. Biometrics, 7:17, 1951.
11. , and Cox, G. M. Experimental Designs. Second Ed. John Wiley and
Sons, Inc., New York, 1957.
12. David, H. A. Upper 5 per cent and 1 percent points of the maximum F-ratio.
Biometrika, 39:422-24, 1952.
13. Davies, O. L. (editor) The Design and Analysis of Industrial Experiments.
Second Ed. Oliver and Boyd, Edinburgh, 1956.
14. Dixon, W. J., and Massey, F. J- Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
15. Duncan, D. B. Multiple range and multiple .F-tests. Biometrics, 11:142,
1955.
16. . Multiple range tests for correlated and heteroschedastic means.
Biometrics, 13:164-76, 1957.
17. Dunnett, C. W. A multiple comparison procedure for comparing several
treatments with a control. Jour. Amer. Stat. Assn., 50:10961121, 1955.
18. Eisenhart, C. The assumptions underlying the analysis of variance. Bio
metrics, 3:1-21, 1947.
19. Federer, W. T. Experimental Design. Macmillan Co., New York, 1955.
20. Graybill, F. A. An Introduction to Linear Statistical Models. McGraw-Hill
Book Company, Inc., New York, 1961.
21. Hamaker, H. C. Experimental design in industry. Biometrics, 11:25786,
1955.
22. Hartley, H. O. Some recent developments in analysis of variance. Communi
cations on Pure and Applied Mathematics, 8:4772, 1955.
23. EZempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
24. Keuls, M. The use of the "Studentized range" in connection with an analysis
of variance. Euphytica, 1:112-22, 1952.
25. Kramer, C. Y. Extension of multiple range tests to group means with un
equal numbers of replications. Biometrics, 12:30710, 1956.
26. . Extension of multiple range tests to group correlated adjusted
means. Biometrics, 13:13-18, 1957.
27. Newman, D. The distribution of the range in samples from a normal popula
tion expressed in terms of an independent estimate of standard deviation.
Biometrika, 31:20-30, 1939.
28. Quenouille, M. H. The Design and Analysis of Experiment. Charles Griffin
and Co., Ltd., London, 1953.
29. Satterthwaite, F. E. An approximate distribution of estimates of variance
components. Biometrics, 2:110, 1946.
30. Scheff<, H. A method for judging all contrasts in the analysis of variance.
Biometrika, 40:87-104, 1953.
362 CHAPTER 11, COMPLETELY RANDOMIZED DESIGN
31. Snedecor, G. W. Statistical Methods. Fifth Ed. The Iowa State University
Press, Ames, 1956.
32. Tukey, J. W. One degree of freedom for nonadditivity. Biometrics, 5:23242,
1949.
33 ^ Comparing individual means in the analysis of variance. Biometrics,
5:99, 1949.
34. . Quick and dirty methods in statistics. Part II: Simple analyses for
standard designs. Proc. Fifth Ann. Conv. A.S.Q.C., p. 189, 1951.
35. . Allowances for various types of error rates. Unpublished material
presented before the Institute of Mathematical Statistics and the Eastern
North American Region of the Biometric Society at Blacksburg, Va., Mar.
19, 1952.
36. . The problem of multiple comparisons. Unpublished dittoed notes,
Princeton Univ., Princeton, N. J., 1953.
37. . Reply to query number 113. Biometrics } 11:11113, 1955.
38. Ward, G. C., and Dick, I. D. Nonadditivity in randomized block designs
and balanced incomplete block designs. N. Z. Jour. Sci. and Tech., 33:430
35, 1952.
39. Yates, F. The design and analysis of factorial experiments. Tech. Comm.
No. 35 , Imperial Bureau of Soil Science, 95 pp., 1937.
CH APTE R 12
RANDOMIZED COMPLETE BLOCK
DESIGN
IN THIS CHAPTER, the most widely used of all experimental designs, the
randomized complete block design, will be discussed. The discussion
will follow closely the pattern adopted in Chapter 11 with,, once again,
the greatest attention being given to methods of analysis.
12.1 DEFINITION OF A RANDOMIZED COMPLETE BLOCK
DESIGN
A randomized complete block (RCB} design is a design in which: (1) the
experimental units are allocated to groups, or blocks, in such a way
that the experimental units within a block are relatively homogeneous
and that the number of experimental units within a block is equal to
the number of treatments being investigated, and (2) the treatments
are assigned at random to the experimental units within each block. In
the foregoing, the formation of the blocks reflects the researcher's
judgment as to potential differential responses from the various experi
mental units while the randomization procedure acts as a justification
for the assumption of independence. (See Chapters 10 and 11.)
Example 12.1
Six varieties of oats are to be compared with reference to their yields,
and 30 experimental plots are available for experimentation. However,
evidence is on file which indicates a fertility trend running from north
to south, the northernmost plots of ground being the most fertile. Thus,
it seems reasonable to group the plots into five blocks of six plots each
so that one block contains the most fertile plots, the next block contains
the next most fertile group of plots, and so on down to the fifth (south
ernmost) block which contains the least fertile plots. The six varieties
would then be assigned at random to the plots within each block, a new
randomization being made in each block.
Example 12.2
An experiment is to be designed to study the effect of environmental
temperature on the transfer time of a certain type of electrical gap.
Twelve different temperatures are to be investigated. A check of the
stockroom indicates that gaps are available from six different production
lots. Since it has previously been established that gaps from different
lots exhibit different characteristics, even when subjected to the same
conditions, some blocking is desirable. Accordingly, 12 gaps are selected
at random from each of the six production lots, and each such set of 12
gaps is hereafter referred to as a block. Then the 12 temperatures are
E3631
364
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
assigned at random to the gaps within each block, a new randomization
being made in each block.
Example 12.3
Ten rations are to be tested for differences in producing a gain in
weight for steers. Forty steers are available for experimentation and
they are allocated to four blocks (10 steers per block) on the basis of
their weights at the beginning of the feeding trial, with the heaviest
steers being in one block, the next heaviest steers being in the second
block, and so on. The 10 treatments (rations) were assigned at random
to the steers within each block, as shown in Figure 12.1.
Block 1
H
B
F
A
C
I
E
J
D
G
Block 2
A
I
G
H
J
D
F
E
C
B
Block 3
E
A
C
I
B
H
D
G
J
F
Block 4
J
F
D
B
H
I
A
C
G
E
FIG. 12.1 Random arrangement of treatments as
described in Example 12.3.
12.2 RANDOMIZED COMPLETE BLOCK DESIGN WITH
ONE OBSERVATION PER EXPERIMENTAL UNIT
The basic assumption for a randomized complete block design with
one observation per experimental unit is that the observations may be
represented by the linear statistical model
i = 1, - - - , b
j - 1, - - - ,t
(12.1)
where p, is the true mean effect, f3g is the true effect of the ith. block,
TJ is the true effect of the jth treatment, and e^ is the true effect of
the experimental unit in the ith block which is subjected to the y
treatment. In addition,
= and e# is NID(0, a).
As in Chapter 11, either Model I or Model II may be assumed with re
spect to the TJ.
Using the symbolism of Table 12.1 and the following equations:
F 2 = total sum of squares
& t
-z
(12.2)
ONE OBSERVATION PER EXPERIMENTAL UNIT
365
TABLE 12.1-Synabolic Representation of the Data in a Randomized
Complete Block Design With One Observation per Experimental Unit
Treatment
Block
1 ... j . . . t
Total
Mean
1.,
Fii Fi - Fi
B
y
i . , . ...
F- y v -
*j i
fj'h
3
Total
Mean
TI TJ Tt
Y.i Y.s Y. t
T
F..
TABLE 12.2-Generalized ANOVA for a Randomized Complete Block
Design With One Observation per Experimental Unit: Model I
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
IVtean
1
J\ft/*t
M
Blocks ...
b 1
K-.-r
B
o- 2 4- 1 2D /3i/0
>-D
Treatments
t 1
T
i 1
Experimental error
(6 !)(/! 1)
&VV
E
/-i
o- 2
Total
to
T:F*
=sum of squares due to the mean
= among blocks sum of squares
= among treatments sum of squares
S
y i
(12.3)
(12-4)
(12.5)
366 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
and
= experimental error sum of squares
H
(12.6)
- B yy - T yy ,
the ANOVA shown in Table 12.2 is obtained.
Following the line of reasoning developed in Section 11.2, it is easily
verified that the hypothesis H:rj = Q C? = l, - - ? may be tested by
computing
mean square for treatments
(12,7)
experimental error mean square
which, if jffistrue, is distributed as Fwithvi = t 1 and *> 2 = (6 !)( 1)
degrees of freedom. If the value of F specified by Equation (12.7) ex
ceeds j^ci-cooi, * 2 )> where lOOa per cent is the chosen significance level,
H will be rejected and the conclusion reached that there are significant
differences among the t treatments.
As before, it is also possible to estimate cr 2 by s 2 E. Then, too, if
Model II had been assumed, o> would be estimated by
sr = (T - E)/b. (12.8)
In either case, that is. Model I or Model II, the estimated variance of
a treatment mean is given by
F(F. y ) = E/b (12.9)
and the standard error of a treatment mean is given by
Vs 2 /b. (12.10)
A IQOy per cent confidence interval for estimating M/ M + T/ is then
found by calculating
= Y.J T *[ci+-y)/*iooVS7& (12.11)
where ^=(& !)(* 1)-
Example 12.4
The experiment described in Example 12.3 was performed and the
data of Table 12.3 were obtained. Use of Equations (12.2) through
(12.6) yields the ANOVA shown in Table 12.4. Because the F-ratio is
significant, we reject HIT, =0(./ = 1, , 10) and decide that in all
likelihood the 10 treatments (rations) are not equally effective in pro
ducing a gain in weight on steers. The treatment means and their
standard error, given in Table 12.3 for convenience, may then be used to
determine the best treatment (or treatments) and to indicate the
direction which future research should take.
12.2 ONE OBSERVATION PER EXPERIMENTAL UNIT
367
TABLE 12.3-Gains in Weight (in Lbs,) of Forty Steers Fed Different Rations
(Data coded for easy calculation)
TV**a t
Block
ment
1
2
3
4
Total
Mean
A
2
3
3
5
13
3,25
B . .
5
4
5
5
19
4.25
C
8
7
10
9
34
8.50
D
6
5
5
2
18
4.50
E,
1
2
1
2
6
1.50
F
3
5
7
8
23
5.75
G . .
8
8
7
8
31
7.75
H
6
12
2
5
25
6.25
I
4
5
6
3
18
4.50
J
4
4
2
3
13
3.25
Total
47
55
48
50
200
Mean
4 7
5.5
4.8
5.0
5.0
Standard error of a treatment mean =-\/(3 -43) /4 = 0.93
TABLE 12.4-ANOVA for Experiment Described in Example 12.3 and
Discussed in Example 12.4 (Data in Table 12.3)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
F-Ratio
IVTean
1
10OO
1000 00
Blocks
3
3 8
1 26
4
0-24. (10/3) Y\ Bi
Treatments
9
163.5
18.17
i a
O-S-l-^/Q) ]JT T *
5.29**
KxTD^riimpTi'tfil error
27
92 7
3 43
J-1
<r 2
Total
40
1260
** Significant at a =0.01.
Before moving along to the next topic connected with the analysis
of randomized complete block designs, there is one point that needs
discussion. That is, why do we not test P:i = (i=l, - , 6)?
Examination of Table 12.2 will show that the expected mean square
for blocks is of the same form as the expected mean square for treat
ments, and this suggests that a logical procedure would be to test H f
by calculating F = B/E. Why is it, then, that the statistician says this
should not be done? The answer may be found by noting the manner
368 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
in which the randomization was performed. You will recall that the
treatments 'were assigned at random to the experimental units within
each block but that the blocks were formed in a decidedly nonrandom
fashion. Because of this feature of the randomized complete block de
sign, a statistical test of the block effect should not be performed. [NOTE:
In some cases where "blocks" are replaced by "replications" and where
the replications may be considered as random samples of all possible
replications (i.e.. Model II with respect to replications), an ^P-test for
replications may be appropriate. However, even in such a case, the
jF-test would be of less importance than the estimation of the com
ponent of variance for replications, o> ]
12.3 THE RELATION BETWEEN A RANDOMIZED COM
PLETE BLOCK DESIGN AND "STUDENT'S" *-TEST
OF jHr: M z>==0 WHEN PAIRED OBSERVATIONS ARE
AVAILABLE
In Section 7.9, procedures were given for testing the hypothesis
H:fj,i = fj.2 for three different cases. In Section 11.3 it was stated that
the analysis of a completely randomized design was equivalent to one
of these, namely, Case I. In this section we wish to show the equiv
alence of the analysis of a randomized complete block design with two
blocks and "Student V t-test of If'.jAi juiz when paired observations
are available, that is, to Case III. The crucial step is to note the equiv
alence of "pairs" and "blocks." Once this association of terms is made,
the equivalence of the techniques may easily be demonstrated. Rather
than burden the reader with the details of the algebraic proof of the
equivalence, we will rely on the "power of an example" to convince him
of the truth of our claim. (NOTE : The reader should also reflect on the
obvious connection between the material of this section and the con
tents of Section 9.13.)
Example 12.5
Consider again the experiment described in Example 7.21 and the
data presented in Table 7.6. Denoting pairs (samples) by blocks and
utilizing Equations (12.2) through (12.6), the ANOVA of Table 12.5 is
obtained. It is seen that the F- value is significant at a. = 0.05, and this
permits us to reject the hypothesis that the two treatments (i.e., two
different steel balls) are doing an equivalent job. [NOTE: ^ = 7.89
= 2 = (2.81) 2 ; see Example 7.21.] It may be verified that the two treat
ment means are 54.2 and 46.2, respectively. Also, the standard error of
a treatment mean is determined to be V60. 8/15 = 2.01.
12.4 SUBSAMPLING IN A RANDOMIZED COMPLETE
BLOCK DESIGN
When subsampling is employed in a randomized complete block de
sign, the appropriate statistical model is
12.4 SUBSAMPLING 369
TABLE 12.5-ANOVA for Experiment Described in Examples 7.21 and
12.5 (Data in Table 7.6)
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
jF-Ratio
Mean
1
75 601 2
75 601 2
Blocks
14
2,649 8
189 3
15
<rM-(2/14} Y^ 8*
Treatments
1
480
480.0
i-i
2
cr 2 H-(15/l) T" Tf
7 89*
Exp^T~i~^nt2l e.rrnr ,
14
851,0
60.8
J-l
<r 2
Total
30
79,582.0
* Significant at =O.05.
J = 1, ' * ' > t
k = 1, - - , n
and the terms are defined in the usual manner. The various sums of
squares are found as follows :
]Y Z = total sum of squares
, (12.13)
= sum of squares due to the mean
H (12.14)
= T^/btn^
= among cells sum of squares for the block X treatment table
& t
XT^ x~^ T / M K/T M9 1 1 ^^
= / ; / ^ J. ij/n M-yyi {LZ. L3}
= sampling error sum of squares
B =
yy
block sum of squares
M vy ,
(12.16)
(12.17)
= treatment sum of squares
Tj/bn
(12.18)
37O
and
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
E yy = experimental error sum of squares
=== >bt -L$ity J- yy
where
T = grand total
b t n
- z: z 2: r*
t^l y^i &=!
7\7 = total of all observations in the iih block that were
subjected to thejth treatment
(12.19)
(12.20)
(12.21)
?i = total of all observations in the iih block
t n t
(12.22)
and
TV == total of all observations subjected to the^th treatment
== S -* 2 _^r * ijjc == S * J- tj -
(12.23)
Using the preceding results, the ANOVA shown in Table 12.6 is
obtained.
TABLE 12.6-Generalized ANOVA for a Randomized Complete Block
Design With n Samples per Experimental Unit: Model I
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
F-Ratio
M!ean
1
Ay. tii t
M
Blocks
61
B
2 ^A 2
0. _I_** .2_L./** > " Q.
Treatments
21
T
T
i 1
t
^^. no .^^ r f )n y^ X 2
T/E
Experim en tal error . .
(, !)(_!)
Jtyy
E
/ i
a 2_|_ 2
Sampling error
bt(n 1)
C-
s
V ]
o- 2
-n
Total
Un
y^r 2
12.5 PRELIMINARY TESTS
371
Example 12.6
An experiment was performed to assess the relative effects of five
fertilizers on the yield of a certain variety of oats. The location of the
30 experimental plots available for use in the experimentation was such
that it seemed advisable to group the plots into six blocks of five plots
each. The treatments were then randomly assigned to the plots within
each block. At the end of the growing season the researcher decided to
harvest (for purposes of analysis) only three sample quadrats from
each plot. The data of Table 12.7 were obtained and these led to the
ANOVA shown in Table 12.8. It is noted that the five fertilizer means
are significantly different and, thus, it is most important that the
proper fertilizer be recommended to the farmer. A tabulation of the
fertilizer means, together with the appropriate standard error, would
be of great help in reaching the correct decision.
TABLE 12.7-Coded Values of Yields From Ninety Sample Quadrats
Blocks
Fertilizer Treatments
1
2
3
4
5
1
57
67
95
102
123
46
72
90
88
101
28
66
89
109
113
2
26
44
92
96
93
38
68
89
89
110
20
64
106
106
115
3
39
57
91
102
112
39
61
82
93
104
43
61
98
98
112
4
23
74
105
103
120
36
47
85
90
101
18
69
85
105
111
5
48
61
78
99
113
35
60
89
87
109
48
75
95
113
111
6
50
68
85
117
124
37
65
74
93
102
19
61
80
107
118
12.5 PRELIMINARY TESTS OF SIGNIFICANCE
At this time I wish to digress for a few moments from the pattern
established in Chapter 11, and followed thus far in the present chapter,
to discuss a matter of considerable importance. This topic, namely,
372 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
TABLE 12.8-ANOVA for Data of Table 12.7; Discussion in Example 12.6
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
F-Ratio
Mean
1
573,921.88
573,921.88
Blocks
5
354.19
70.84
JU 2
<r 2 -f-3<7- 2 -K15/5) J^Qi
KeT'tilf ^ers . . ,
4
65 246 84
16311 71
s
O^+ScrM-ClS/i) y~\ Tj
220 61**
Experimental
error
20
1,478.76
73.94
* 3-1
o- 2 -i-3<r 2
Sampling error.
60
5,283 33
88 06
"17 i^
a*
G rj
Total
90
646,285.00
Significant at ex =0.01.
the use of preliminary tests of significance, could just as easily have
been discussed in Chapter 11 or it could, without difficulty, be deferred
until later. However., Example 12.6 at the end of Section 12.4 brought
it to my attention, and thus we shall consider it at this point.
Some practitioners suggest that when, as in Example 12.6, the experi
mental error mean square is less than the sampling error mean square,
the two sums of square and their degrees of freedom be pooled. (The
same suggestion; i.e., to pool, is also frequently made when the experi
mental error mean square exceeds, but not significantly, the sampling
error mean square.) That is, a pooled sum of squares (Ew+Sw) is
divided by the pooled degrees of freedom [(& 1) (2 l)+&^(n 1)], and
this new mean square is then used as the denominator in the .F-ratio
for testing H:r 3 - = Q (y= 1, - - , t). If such a procedure is followed, the
statistical test of jETrry O (j1, **,) will be based on a preceding,
or preliminary, test of significance, the hypothesis H':<r 2 = Q being
tested by the preliminary test. Because such procedures are sometimes
followed, we must make certain that we understand their advantages
and disadvantages.
Problems of the above type have been investigated by Paull (10),
and it will pay us to spend a few moments reviewing his conclusions
and recommendations. To make the exposition easier to follow, we
shall tie it in with Table 12.6 . Suppose the experimenter decides he will
always pool the two mean squares as indicated in the preceding para
graph, that is, he will never perform a preliminary test of significance
concerning H':<r 2 = Q. If, in fact, <r 2 does equal 0, this procedure is fine.
But suppose a- 2 > ; then the denominator in the final F-test (of HiTj =
for allj) tends to be too small. Thus, in such a situation, the final F-test
tends to produce too many significant results when the mill hypothesis
12.6 ESTIMATION OF COMPONENTS OF VARIANCE 373
H is really true. This is bad, for it implies that, quoting Paull, "a test
which, the research worker thinks is being made at the 5 per cent level
might actually be at, say, the 47 per cent level." 1
The use of a preliminary test of significance is clearly an attempt to
guard against such a possibility. It will not, of course, eliminate such
occurrences entirely. To be useful, however, it should keep the actual
(effective) significance level achieved by the final (or dependent) F-test
close to the value at which, the research worker desires to operate.
Another property which should be required of a preliminary test is that
it increase the power of the final -F-test relative to the power of a
"never pool" test. The recommendations for the performance of pre
liminary tests of significance for pooling mean squares in the analysis
of variance as formulated by Paull may be found in the reference
quoted. However, if the research worker follows the rule of "never
pooling," he will not go far wrong, and that is the rule we shall adopt
in this text.
12.6 ESTIMATION OF COMPONENTS OF VARIANCE AND
RELATIVE EFFICIENCY
The problems of estimating components of variance and predicting
relative efficiency are no different in a randomized complete block
design than they were in a completely randomized design. Thus, they
will not detain us long.
As before, <r% is estimated by s* = S ( see Table 12.6) and <r 2 is
estimated by
s* = (E - S)/n. (12.24)
If, as in Table 12.8, the algebraic solution leads to a negative value of
s 2 , it is customary to disregard the algebraic solution and use as the
estimate. Of course, this is a biased procedure, but it is aesthetically
more satisfying since no population variance can be negative.
The relative efficiencies of various allocations -within a randomized
complete block design will, as was the case when discussing a completely
randomized design, be determined by studying the variance of a treat
ment mean. It may be shown that
^ _ experimental error mean square
number of observations per treatment (12.25)
= E/bn = (4 + ns*)/bn.
Thus, if the estimates of the components of variance remain unchanged,
the efficiency of this design relative to one in which we might use &'
blocks and n' samples per experimental unit would be predicted by
R.R. of old to new - 100 [F'(F. y J/F(F.,.)] per cent (12.26)
1 A. E. Paull, "On a preliminary test for pooling mean squares in the analysis of
variance," Ann. Math. Stat., Vol. 21, 1950, p. 541.
374
where
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
(12.27)
Example 12.7
^ Consider Table 12.9. It is easily seen that s% = 10 and s 2 = 12, yielding
V(Y .j,} =58/24. To determine the efficiency of the design used relative
to one involving four blocks and six samples per experimental unit,
we first calculate F'(F. y .) = [10 + 6(12) ]/4(6) =82/24. Thus, the
estimated relative efficiency is 100(82/24)/(58/24) = 141 per cent.
TABLE 12. 9- Abbreviated ANOVA on Yields of Ten Varieties of Soybeans
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
Blocks
5
3,000
600
cr 2 -Mt<r 2 +(40/5)
rf
Varieties
9
4,500
500
o- 2 +4o- 2 -K24/4)
"F T Z -
Experimental error. . . ,
Sampling error
45
180
2,610
1,800
58
10
"
v
The ideas of this section may easily be extended to cases involving
many stages of subsampling. No examples will be given, but several
of the problems at the end of the chapter will provide the necessary
practice in the manipulations.
12.7 EFFICIENCY OF A RANDOMIZED COMPLETE
BLOCK DESIGN RELATIVE TO A COMPLETELY
RANDOMIZED DESIGN
In some instances, the investigator wishes to estimate the efficiency
of his use of a randomized complete block design relative to what
might have happened if the treatments had been completely random
ized over all the experimental units. That is, he wishes to know if he
gained or lost in efficiency by grouping the experimental units into
homogeneous groups (blocks) . One method of comparing the efficiency
of different designs is by use of uniformity data. Cochran (4) has dis
cussed this particular approach, and the reader is referred to his article
for further details. A second method of comparing efficiencies is to
consider algebraically what might have happened to the experimental
error mean square under complete randomization. To accomplish this,
it is convenient to proceed as though dummy treatments had been
applied to the experimental units. That is, we suppose that all experi
mental units were subjected to the same (viz., no) treatment and then
proceed to estimate what the experimental error mean square would
12.7 EFFICIENCY OF RANDOMIZED COMPLETE BLOCK DESIGN
375
have been under complete randomization. Following this line of reason
ing, and defining the efficiency of a randomized complete block design
relative to a completely randomized design by
estimated experimental error mean square for a CR design
experimental error mean square from the RCB design
it can be shown that
(b 1)J5 + b(t
R.E. =
(bt
(12.28)
(12.29)
where JB and E refer to the mean squares (in the randomized complete
block design) for blocks and experimental error, respectively.
Example 12.8
Consider the ANOVA presented in Table 12.4. In this case, the effi
ciency of the randomized complete block design relative to a completely
randomized design is estimated to be
R.E.
3(1.26) +4(9) (3.43)
39(3.43)
0.95.
It is seen that, because of the small magnitude of B relative to E, no
appreciable gain in efficiency resulted from the formation of the blocks
and the use of a randomized complete block analysis. That is, apart
from the "insurance" feature of the RCB design, the added effort was
not worthwhile.
Example 12.9
The data in Table 12.10 resulted from a particular manufacturing
operation, the operation being performed by one of four different ma
chines. The data were collected on five different days, hereafter referred
to as blocks. Calculations yielded the abbreviated ANOVA shown in
Table 12.11. Proceeding according to Equation (12.29), the randomized
complete block design is estimated to be 131 per cent as efficient as a
completely randomized design would have been.
TABLE 12,10-Output From Four Machines Producing Part No. Z-15
(Output = number of units produced In one day)
Machine
Day
A
B
C
D
1
293
308
323
333
2
298
353
343
363
3
280
323
350
368
4
288
358
365
345
5
260
343
340
330
376 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
TABLE 12, 11- Abbreviated ANOVA for Data of Table 12.10
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean Square
Blocks
4
2,146.2
536.55
Treatments
3
13,444.8
4,481.60**
Experimental error
12
2,626.2
218.85
** Significant at <*=0.01.
12.8 SELECTED TREATMENT COMPARISONS
It is often desirable to make certain specific comparisons involving a
selected number of the treatments. For a completely randomized de
sign, such comparisons were discussed in Sections 11.8 and 11.9. In this
section the same general topic will be examined in conjunction with a
randomized complete block design.
A moment's thought should be sufficient to convince the reader that
the calculations will be performed in the same manner as indicated
earlier. For example, if a randomized complete block design with one
observation per experimental unit is involved, the sum of squares for
a particular contrast, C*, would be given by
= ( i^ **r/Y / b i:
\ y i / / y i
2
Cjk,
(12.30)
where the Cjk are the coefficients specifying the contrast. As before, if
tI orthogonal contrasts are studied, the sum of the tI individual
sums of squares will equal the treatment sum of squares.
Example 12.10
Upon reading the complete description of the project referred to in
Example 12.9, certain additional information about the four machines
is brought to light. For example, machine A is the standard type of
machine now in use in the industry, while machines B, C, and D are
new designs which may be considered as possible substitutes. Further,
it is known that B and C contain moving parts made of some aluminum
alloy, while D does not have this feature. Also known from the manu
facturers* specifications is the fact that B is self-lubricating, while C is
not. Therefore, the comparisons represented symbolically in Table
12.12 seem to be indicated. Partitioning of the treatment sum of
squares is then carried out using Equation (12,30), and the abbreviated
ANOVA of Table 12.13 is obtained.
12.9 SUBDIVISION OF THE EXPERIMENTAL ERROR
SUM OF SQUARES WHEN CONSIDERING SELECTED
TREATMENT COMPARISONS
Before proceeding to the next general topic connected with random
ized complete block designs, another digression seems desirable. This
12.9 SUBDIVISION OF EXPERIMENTAL ERROR SUM OF SQUARES
377
TABLE 12.12-Symbolic Representation of the Selected Treatment
Comparisons Described in Example 12.10 (Data in Table 12.10)
Comparison
Machine
A
B
C
D
1
+ 3
1
+ 1
^
1
+ 1
+ 1
1
2
2
3
TABLE 12.13-Abbreviated ANOVA for Data of Table 12.10 Showing the
Subdivision of the Treatment Sum of Squares
(Discussion in Example 12.10)
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean Square
Blocks
4
2,146.2
536.55
Treatments
A vs. rest
1
13,142.4
13,142.4**
B and C vs. D
1
172.8
172.8
B vs. C
1
129.6
129.6
Experimental error
12
2,626.2
218 85
** Significant at a = 0.01.
time we will be concerned with the possibility of partitioning the experi
mental error sum of squares. (NOTE: You will recall that this topic
was mentioned in the last paragraph of Section 11.13.)
When is such a procedure in order? That is, when should the experi
mental error sum of squares be subdivided? The reason for sub
dividing E yy (if such a procedure is adopted) is that we are not satisfied
with our assumption of homogeneous variances of the e's. If such an
assumption is questioned its validity may, of course, be investigated
using Bartlett's test and if selected treatment comparisons are being
examined, it is desirable to subdivide E vy in a manner similar to the
subdivision of T yy . Such a procedure insures that any particular treat
ment comparison will be tested against the appropriate error. That is,
the expected value of the "error mean square for testing C k " will con
tain the same components of variance (other than the treatment effects)
as the expected value of the mean square associated with C^ In other
words, if we are faced with different variances o-^ (f = l, , Z>;
y=l, * - , ), the procedure of subdividing E yy will insure that the
expected mean squares for a particular comparison and its associated
error will each contain the same linear combination of the c%j. This, of
course, provides us with unbiased tests for the comparisons under in
vestigation.
Since the decision to implement the procedure (yet to be described)
378 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
for subdividing E w is one which everyone will have to make at some
time or other, some guiding rule is needed. The following appears to
be a reasonable rule : If there is any serious doubt as to the homogeneity
of the variances, subdivide the experimental error sum of squares as a
precautionary measure. It should be noted, though, that if the degrees
of freedom associated with the various parts of E yy are small, the re
sulting tests may be relatively insensitive (i.e., of low discriminatory
power). In practice both the following conditions are usually true:
(1) The degree of heterogeneity among the error variances is, as
a rule, not too great. Therefore, for most practical purposes,
the variances may be considered homogeneous.
(2) The numbers of degrees of freedom associated with the parts
into which the experimental error sum of squares is sub
divided are generally quite small.
Consequently, the rule stated above should be modified to read; Be
cause of the truth of statements (1} and (j) above, it is generally not wise
to subdivide the experimental error sum of squares. However, if the hetero
geneity of variances is such that a subdivision is necessary (regardless of
the fact that small numbers of degrees of freedom will result), the sub
division should be carried out in accordance with the procedure to be ex
plained in the next paragraph.
The method of subdividing the experimental error sum of squares in
agreement with a particular subdivision of the treatment sum of square
is as follows :
(1) Set up a table showing the values of the contrasts within each
block.
(2) Calculate the portion of the experimental error sum of squares
for a particular contrast using
(Ek) yy === experimental error sum of squares for C k
= f ib el* - ci/b\ / i: 4
L i=l J / /=!
where
C ki = C c&Yv (12.32)
-=i
and
C k == c jk Tj - T C ki . (12.33)
j i i=i
Example 12.11
Consider the experiment discussed in Examples 12.9 and 12.10. Using
Equations (12.32) and (12,33) in conjunction with Tables 12.10 and
12.12, we obtain Table 12,14. Then, using Equation (12.31), we get, for
12.9 SUBDIVISION OF EXPERIMENTAL ERROR SUM OF SQUARES
379
example,
experimental error sum of squares associated with Ca
experimental error sum of squares associated with the comparison
"B versus C"
4.
(_ 3)2 __ (36)V5]/2
426.4.
Similarly, (EJ vy = 1087.27 and (#2)^ = 1112.53. Thus, we finally obtain
the abbreviated ANOVA shown in Table 12.15.
TABLE 12.14-Sums for the Selected Treatment Comparisons in Each
Block (Data of Table 12.10)
Comparison
Block
Ci
C 2
C 5
1
85
35
+ 15
2
165
30
10
3
201
63
+ 27
4
204
+33
+ 7
5
233
+23
3
Total
888
72
+36
TABLE 12. 15- Abbreviated ANOVA for Data of Table 12.10 Showing
The Subdivision of the Experimental Error Sum of Squares
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
*
Blocks
4
2,146.20
536.55
<r'+(4/4) /*'
Treatments :
A vs rest
1
13,142.40
13,142.40
<r 2 + (25/60) (3-n r 2 T 3
r^ 3
B and C vs. D . . .
B vs C
1
1
172.80
129.60
172.80
129.60
<r 2 +(25/30)(r 2 +r3-2r 4 ) 2
0.2^ (25/10) (r 3 r 2 ) 2
Experimental error:
A vs rest
4
1,087.27
271 . 82^)
B and C vs. D . . .
^ vs. C
4
4
1,112.53
426.40
278. 13 >
106. 60J
cr*
* The symbol o- 2 was used in each expected mean square as a matter of convenience. If
the variances are homogeneous, it is correct; if the variances are not homogeneous, the
symbol <r 2 would be replaced by various linear combinations of the o\ y .
The procedure explained and illustrated in this section can sometimes
be used to advantage when analyzing a particular set of data. How
ever, the reader should realize that it is a special technique and will,
therefore, be used only rarely.
38O CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
12.10 ALL POSSIBLE COMPARISONS AMONG TREAT
MENT MEANS
Since the problem of making all possible comparisons among treat
ment means in a randomized complete block design is handled in
exactly the same manner as similar comparisons were handled in a
completely randomized design, no additional discussion is necessary.
The reader is, therefore, referred to Section 11.10 for the appropriate
details.
12.11 RESPONSE CURVES IN A RANDOMIZED
COMPLETE BLOCK DESIGN
Once again it is sufficient to state that the techniques explained in
the preceding chapter are directly applicable to the present situation.
Thus, the reader is referred to Section 11.11 for the computational
details. However, to emphasize the "sameness/* an illustrative example
will be presented. (NOTE: The reader will find it rewarding to com
pare the following example with Example 11.15.)
Example 12.12
Considering the data of Table 12.16 and using the methods described
in Section 11.11, the abbreviated ANOVA shown in Table 12.17 is
obtained.
TABLE 12. 16- Yields (Converted to Bushels/Acre) of a Certain Grain
Crop in a Fertilizer Trial
Block
Level of Fertilizer
No
Treat
ment
10 Ibs,
per Plot
20 Ibs.
per Plot
30 Ibs.
per Plot
40 Ibs.
per Plot
1
20
25
23
27
19
25
29
31
30
27
36
37
29
40
33
35
39
31
42
44
43
40
36
48
47
2
3
4
5
Treatment
totals
114
142
175
191
214
12.12
FACTORIAL TREATMENT COMBINATIONS IN A
RANDOMIZED COMPLETE BLOCK DESIGN
Because of tlie detail with which the analysis of factorial treatment
combinations was discussed in connection with a completely random
ized design (see Section 11,12), only a summary discussion seems
appropriate here. Accordingly, all that will be given are two linear
12.12 FACTORIAL TREATMENT COMBINATIONS
381
TABLE 12. 17- Abbreviated ANOVA for Data of Table 12.16 Showing
the Isolation of the Linear and Quadratic Portions of the Treatment
Sum of Squares
Source of Variation
Degrees of
Freedom
Sum of Squares
Mean
Square
Blocks
4
154 16
38 54
Treatmen ts
4
1256 56
314 14
Linear
1
1240 02
1240 02**
Quadratic
1
10 41
10 41
[Deviations from re
gression . .
2
6 13
3 07
TC-jrp^Hrnenfal errnr
16
193.44
12 09
** Significant at = 0.01.
statistical models, their associated ANOVA's, and some numerical
examples.
Two -Factor Case
ijk
Pi
(12.34)
Three- Factor Case
Yjjki = JJL + pi + 0.3 + fa +
j = 1,
k = 1, .
1>
= 1,
= 1,
= 1,
(12.35)
, a
, b
, c.
In the preceding equations, M is the true mean effect, p^ is the true
effect of the ith replicate (or block), the various terms involving a, (3
and y are the true effects of the several factors and their interactions,
and the e's are the true effects of the experimental units. The general
ized ANOVA's associated with Equations (12.34) and (12.35) are
shown in Tables 12.19 and 12.18, respectively.
Example 12.13
Consider the data in Table 12.20. Performing the usual calculations,
we obtain the abbreviated ANOVA shown in Table 12,21. Testing
#1:0^ = (.7 = 1, 2), we calculate ^ = 32.00/3.16 = 10.2, and this leads to
the rejection of Hi. Testing T 2 :/3;b = (& = 1, 2, 3, 4), we calculate
p = 5.47/3.16= 1.73, and this does not permit H% to be rejected. To test
# 3 :(o!0)/fc = (.7 = 1, 2; fc = l, 2, 3, 4), we calculate F = 12.80/3.16 = 4.05
and this leads to rejection of H$ at the 5 per cent significance level but
TABLE 12.18-Generalized ANOVA for a Three-Factor Factorial in a
Expected Mean Square
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Model I
M
Mean
Myy
T>
p
Replicates, . . .
Treatments
f 1
K-w
1
A
A
n-i+rbt: T!</fo-l)
6-1
AB.
(a- 1)0- 1
By
Cyy
B
a*+rab
1-1
AB
AC
( a _l)( c _l)
(AQyy
AC
(r z +rb > . > . (a7)yi/(0"" 1 -)w"' 1 ^
/-i 1-1
6 c^ j
EC
(d-l)fr-l)
(BC)yy
BC
<r*+ra^T, X) (fryOfci/C^ l)(c 1)
fc-i 1-1
a 6 c
ABC
(0-1) (6- !)(<?-!)
(ABQyy
ABC
;l jfe-1 i-1
Experimental
error
(r-lXafa-l)
Ryy
E
**
_T
EV2
Total
TdOC
*
**+rc i: :
,--1
[3821
Randomized Complete Block Design
Expected Mean Square
Model II
Model in
(a and b fixed, c random)
Model III
(a fixed, 5 and c random)
j-i
[3831
ft
-s
a
o
O
O
5
o
5-,
5
O
oS
O
I i -C>
'S I
d
.
1 *
S ,8
d
d
"2
I
1
4-
N b
b
"b
4-
b
o
b
"
4-
5
*b
S
4-
w b
^
-W2
a Z
c oi
(D
S
o
ck
J?
w
"-M rj
O O
I *
g a
e*
53
B
5
S
12.12 FACTORIAL TREATMENT COMBINATIONS
385
TABLE 12. 20- Yields of Soybeans at the Agronomy Farm, Ames, Iowa, 1949
{In bushels per acre)
Date of
Planting
Fertilizer
Replicate
1
2
3
4
Early
Check
Aero
Na
K
Check
Aero
Na
K
28.6
29.1
28.4
29.2
30.3
32.7
30.3
32.7
36.8
29.2
27.4
28.2
32.3
30.8
32.7
31.7
32.7
30.6
26.0
27.7
31.6
31.0
33.0
31.8
32.6
29.1
29.3
32.0
30.9
33.8
33.9
29.4
Late
TABLE 12.21-Abbreviated ANOVA for Data of Table 12,20
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean Square
Replicates. .
3
7.31
2.44
4
o-*+(8/3) 52 P*
Dates of planting
1
32.00
32.00
i-i
2
<r 2 +(16/l) 2Dj-
Fertilizers
3
16.40
5.47
31
o- 2 +(8/3) X) A
Fertilizers X dates of planting.
Experimental error
3
21
38.40
66.43
12.80
3.16
Jb-l
rH-cv3)i:]fc<0)i*
j-i jbi
Or*
not at the 1 per cent significance level. [NOTE : Depending on whether we
use <x = 0.05 or ex = 0.01, the recommendations will differ. If a. = 0.05, dif
ferent fertilizers would probably be suggested for each date of planting;
if a: = 0.01, it is possible that the same recommendation (concerning
fertilizers) would be made for each date of planting.]
Example 12.14
An experiment such as described in Example 10.18 was performed.
The resulting data are given in Table 12.22. The associated ANOVA is
presented in Table 12.23. It will be noted that none of the factors led to
significant results.
So far in this section we have summarized the cases involving two
and three factors with one observation per experimental unit. How
ever, there are two other topics associated with factorials in a random
ized complete block design which also deserve our attention at this
386
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
TABLE 12.22~Surge Voltages Resulting From the Experiment Described
in Example 10.18 and Discussed in Example 12.14
Heat
Tempera
Replicate
Electrolyte
Paper
ture
(a)
(*)
to
I
II
6.08
6.79
1
6.31
6.77
1
6.53
6.73
1
6.04
6.68
1
1
6.12
6.49
1
1
6.09
6.38
1
1
6.43
6.08
1
1
1
6.36
6.23
TABLE 12.23-ANOVA for Data of Table 12.22
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
IVIean . .
1
651 .6533
651.6533
Repli cates
1
0.2997
0.2997
Treatments
A
1
. 1462
0.1462
2.21
B
1
0.0018
0.0018
0.03
C
1
0.0232
0232
0.35
AB
1
0.0001
0.0001
0.00
AC
1
. 0047
0.0047
0.07
BC
1
0.0176
0.0176
0.27
ABC
1
0.0883
0.0883
1.33
Experimental error
7
0.4632
0.0662
Total
16
652.6981
time. These are: (1) subsampling and (2) the analysis of response
curves. Each of these will now be discussed.
Subsampling in a randomized complete block design which incor
porates factorial treatment combinations leads to analyses such as
shown in Tables 12.24 and 12.25. Since no new techniques are involved,
numerical examples will not be given. The reader is referred to Sections
11.12 and 12.4 for further details.
As was mentioned in Section 11.12, when factorial treatment combi
nations are involved, it is possible to subdivide the treatment sum of
squares into several parts such as (A.i^ yy) (Ao) vy , ; CB.L)IW*
(Bo)w* * ; (A-jJBi^yy, {A Q B L }y Vy (A iJB Q } y V y and so on. The pro
cedure to be followed will parallel that presented in Section 11.11, the
only difference being the refinements introduced to subdivide the
interaction sum of squares. Because of this, the technique will be pre
sented in terms of two numerical examples.
12.12 FACTORIAL TREATMENT COMBINATIONS
367
TABLE 12. 2 4- Abbreviated ANOVA for a Two-Factor Factorial in a
Randomized Complete Block Design With n Samples per
Experimental Unit
Source of Variation
Degrees of Freedom
Expected Mean Square
Replicates . .
Treatments:
A
r1
a\
61
AB.
ncr z -{-rn
I-IX&-D
Experimental error.
Sampling error.
TABLE 12.25-Abbreviated AISTOVA for a Two-Factor Factorial in a
Randomized Complete Block Design With n Samples per Experimental
Unit and d Determinations per Sampling Unit
Source of Variation
Degrees of
Freedom
Expected Mean Square
Replicates. . .
Treatments:
A
B
r1
a I
b-l
AB.
Experimental error .
Sampling error
Determinations. . . .
rab(nl*)
rabn(d 1)
Example 12.15
Consider the data of Table 12.26. The first step in the analysis is the
formation of the aX& table shown in Table 12.27. Remembering that
each entry in Table 12.27 is the sum of r = 2 observations and using the
polynomial coefficients given in Table 11.26, we find that
, , , [(-3)(35) + (-1X42) -4- (1)(59) + (3)(60)]
(2)(3)[(-3) 2 + (-1) 2 + (I) 2 + (3)*]
(-1X59) + (1X60)]*
(I) 2 ]
70,53
1.50
388
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
-1)(35) + (3) (42) + (-3) (59) H
(2) (3) [(-I)" + (3) 2 + (-3) 2 + (I) 2 ]
(0)(65) + (1)(67)]*
= .56
5.63
(I) 2 ]
(-2)(65)
(2)(4)[(1)*+ (-2)" +
where the divisors are:
.02
(1) for (.Ai^yy, (Ao)yy, and (^L c)w = (r6) X (sum of the squares of the
coefficients) ;
(2) for (Bz^yy and (Bo)yyi(ra) X(sum of the squares of the coefficients).
TABLE 12.26-Coded Data for Use in Illustrating the Calculation of the
Linear, Quadratic, . . . Effects in a Two-Factor Factorial Experiment
Conducted in a Randomized Complete Block Design
Replicate
<3l
<Z 2
a 3
a 4
1
f6i
l&a
7
5
8
6
9
11
7
10
2
i
u,
[ 6l
^2
4
7
6
6
9
6
10
9
10
12
8
11
1
U 3
6
7
10
12
TABLE 12.27-aX& Table Formed From the Data of Table 12.26
ai
a 2
a 3
a 4
Totals
b-L
14
17
18
15
64
6 2
11
12
21
21
65
&3
10
13
20
24
67
Totals
35
42
59
60
196
To illustrate the computation of the various sums of squares which
comprise (AB^yy, let us take the case of (AQBiJ)yy as an example. It
should be understood that any other of the sums of squares may be
found in a similar manner if the appropriate word-substitution, for
quadratic and linear, is made in the next few sentences. Obtain the sum
of the products of the a quadratic polynomial coefficients by the totals
in the cells of the aX& table for each level of &. Then apply the b linear
polynomial coefficients to these "sums" and obtain the usual sum of
products. Square this last sum, and divide the squared quantity by the
product of the sums of squares of the two sets of polynomial coefficients
(quadratic for a and linear for 6) used in the computation. Also divide
by r, the number of replicates, since each total in the a X b table was the
sum of r observations. The resulting value is the sum of squares due to
. For our numerical example we have
for 6 i: (1X14) + (-1X17) + (-1X18) + (1)(15) = - 6
12.12 FACTORIAL TREATMENT COMBINATIONS
389
for
for
and hence
(A Q B L ")yy =
4- (-1)(12) + (-1X21) -f- (1X21) - - 1
,: (1)(10) 4- (-1X13) -I- (~D(20) 4- (1)(24) = 1
2[(1) 2 4- (-1) 2 4- (-1) 2 4- (D 2 ][(-l) 2 + (O) 2 4- (I) 2 ]
3.06.
The reader should verify that the remaining sums of squares in Table
12.28 may be found in a like manner.
TABLE 12.28-Abbreviated ANOVA for Data of Table 12.26
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean Square
Replicates
1
1 50
1 50
Treatments :
A L
1
70.53
70.53
AQ
1
1.50
1 5O
Ac
1
5.63
5 63
B L
1
56
56
Bo
1
.02
,02
A L B L
1
25 31
25.31
A,BQ
1
2.60
2 6O
AQ&L
1
3 06
3 06
A.oBo
1
.20
20
AC&L
1
.32
.32
AC&Q
1
2.60
2.6O
Experimental error
11
3.5O
.32
Example 12.16
Consider next the data of Table 12.29. To save time, the calculation
of the sums of squares will be illustrated for only three effects : A&, AQB^
and AQB C CL. To check these results, the reader will find it advantageous
to form the aX&Xc and the aXb tables. Then,
(0)(1404) -f- (1)(925)]*
(2)(4)(6)[(-l) 2 + (O) 2 -f- (I) 2 ]
[(1X325) + (-2)(432)
330.04
996.67
(-2)*
1066.06
(-1) 2 4- (I) 2 4- (3)J
) 2 ][(-D 2 + (3)* +
(I) 2 + (3) 2 4- (5) 2 ]
-3) 2 4-
where
D =
4- (~3) 2 4- (-1
and the divisors are;
(1) for (^1 ,) yy ' (rbc) X(sum of the squares of the coefficients);
(2) for (.AQBj^yjfi (re) X (product of the sums of the squares of the
coefficients) ;
(3) (AQBcCi^vy* ( r ) X (product of the sums of the squares of the
coefficients) .
39O
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
TABLE 12.29-Hypothetical Data for Use in Illustrating the Computation
of Certain Sums of Squares in a Three-Factor Factorial, the Basic
Design Being a Randomized Complete Block
"O f>-rVli
<Z]
L
a
2
a
3
cate
&1
2
bz
&4
61
b*
6 3
?>4
61
b*
b s
&4
1
c\
C%
3
7
23
9
7
18
18
9
25
24
7
15
23
15
13
12
36
35
43
60
61
62
15
18
14
24
30
31
29
26
24
17
11
15
19
8
23
2
c
c*>
L C6
'^1
c*
c z
7
6
10
7
20
9
13
8
12
6
19
22
25
20
30
11
25
26
36
7
11
7
16
24
11
15
10
15
13
13
12
46
42
35
30
40
63
18
27
60
64
66
26
28
12
20
20
15
32
15
17
25
30
32
15
32
29
30
25
25
12
13
8
20
15
15
5
6
7
20
10
22
c
cs
^
8
8
9
15
10
12
26
20
28
30
8
11
13
17
8
10
40
45
66
20
30
25
30
15
34
18
19
15
35
30
15
15
10
4
5
8
The numerators for (<Az,)i/2/ and (AQBi^yy were found by the same
procedures as similar quantities in the preceding example. The numera
tor for {AoBcCi^yy was obtained by a simple extension of the same
principles. In this case the extension may be explained as follows:
Operate with the CL coefficients on the c entries in the aX&Xc table for
each level of b within each level of a and obtain a set of sums of products,
one for each level of 6 within each level of a. Next, use the Be coeffi
cients and operate on the sums just obtained. This will provide us with
some "BcCz totals" for each level of a. Then use the AQ coefficients to
give us the numerator for (AQBcCiJ)yy. The procedure should now be
clear, and the extension to any number of factors will be a simple, even
if a time-consuming, job.
12.13 MISSING DATA IN A RANDOMIZED COMPLETE
BLOCK DESIGN
Many times, even after considerable effort has been expended and
due diligence exercised in planning an experiment, there are things
which occur to the disadvantage of the research worker. One of the
most common of these "disturbances" is the problem of missing obser
vations. Missing observations arise for many reasons: An animal may
die, an experimental plot may be flooded out, a worker may be ill and
not turn up on the job, a jar of jelly may be dropped on the floor, or
the recorded data may be lost. What effect does this have on our
methods of analysis? Since most experiments are designed with at least
some degree of balance, or symmetry, any missing observations will
usually destroy this balance. Thus, we now expect our original planned
analysis to be complicated and some modifications in procedure to be
required. We could, of course, in many instances treat the data as a
case of disproportionate subclass numbers and use methods of analysis
appropriate to such situations (see Chapter 13). Ho wever, other
12.13 MISSING DATA 391
approaches are sometimes open to the statistician, and we shall exam
ine these in this section, pointing out the difficulties which arise and
indicating the computational procedures to be followed in each case.
First, let us mention two cases of missing data in a randomized com
plete block design which present no difficulties as to computational
procedures: (1) a complete block is missing, or (2) a treatment is
completely missing. When one or more complete blocks are missing
we simply proceed with the standard type of analysis, provided we
still have at least two blocks remaining; that is, we analyze the data
as though we had planned only on the number of blocks which are
actually available. For the case in which no data are available on one
or more treatments (assuming we still have at least two treatments
remaining), we may again proceed in the regular manner. However,
in this instance, the research worker should certainly inquire into the
reasons for the lack of data on certain treatments. It is apparent that
many things might have caused such a happening, each of which could
possibly lead to different decisions or recommendations on the part of
the experimenter. Without a specific example, further discussion on
such points can only be of a vague nature; rather than continue in
general terms, we shall postpone further remarks on this type of
situation until the need arises.
A more commonly occurring situation is the one in which one obser
vation is missing. Here we run into difficulty in our analysis. Either
we must treat the data by methods appropriate to disproportionate
frequencies, or we must find some other scheme which we hope will be
simpler to apply. One such device is to estimate a value to replace the
missing observation and then to proceed with the usual analysis for
randomized complete block designs. How does one obtain an estimate
of the missing observation? The estimation procedure currently favored
by statisticians is to assign that value for the missing observation
which will minimize the experimental error sum of squares when the
regular analysis is performed. To mathematics students this is another
familiar problem in differential calculus; calling the missing observa
tion M, the experimental error sum of squares is computed, or rather
the algebraic expression for the experimental error sum of squares is
formulated, and by differentiating this expression with respect to M
and equating to 0, a solution may be obtained. For the student not
proficient in mathematics, this procedure may be summarized by the
following formula which will provide an estimate of the missing obser
vation in accordance with the above principle:
tT bB S
' C12.36)
where
t== number of treatments
6 number of blocks
T = sum of observations with the same treatment as the missing
observation
B = sum of observations in the same block as the missing observation
of all the actual observations.
392
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
This value, that is, M, is then entered in the appropriate place in the
table of data and the augmented data are analyzed in the customary
manner.
We are now ready to construct our analysis of variance table and to
test the hypothesis T:ry = 0(j = 1, - , Z). However, certain changes
must be made in the form of our analysis of variance table if we are to
avoid biased results. The first change is easy to apply and proceeds as
follows: Reduce the degrees of freedom associated with both experi
mental error and total by 1. That is, the degrees of freedom for experi
mental error become (6 !)( 1) 1 and the degrees of freedom for
total become bt 1. The second change is a little more cumbersome to
apply. Before detailing this change, let us discuss what it is and why it
is necessary. It may be proved that, under the null hypothesis, the
expected value of T w /(tl'), the treatment mean square calculated
from the augmented data, is greater than <r 2 , the expected value of the
experimental error mean square. Thus any test of hypothesis which
does not correct for this fact will be a biased test and can only be
considered approximate. The correction for this bias, the second change
mentioned above, is to decrease the treatment sum of squares, T yy ,
by the amount
[B (t l)^f] 2
Correction for bias = Z = , (12 .37)
*(* - 1)
which gives us a new treatment sum of squares
T'yy = T yy Z, (12.38)
and the analysis of variance indicated in Table 12.30 is finally obtained.
TABLE 12. SO-Generalized ANOVA for a Randomized Complete Block
Design With One Missing Observation
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F-Ratio
]VIean
1
Mw
M
Blocks
61
JStnt
B
Treatments
t 1
T f
T f
T'/E
Experimental error
(_!)( __!)__ 1
EWJ
E
Total
btl
73 F 2 Z
Example 12.17
An experiment was conducted by Tinker (13) to investigate the
consistency of blink-rates during reading. Data were recorded for six
successive 5-minute periods of reading. As we have extracted only part
of the available data for our example, care should be exercised in
drawing conclusions from the analysis -which follows. The original paper
should be consulted by those desiring further information on the sub
ject matter. We will assume that the experiment was performed on
four individuals they will be our blocks and the six periods will
represent the treatments. Moreover, to illustrate the techniques of this
12,13 MISSING DATA
393
section, we will assume that the observation on Subject A for the
fourth period is missing. Our observed data are given in Table 12.31.
TABLE 12.31 Number of Blinks for Successive Five-Minute Periods of Reading
Periods
Sub
jects
1
2
3
4
5
6
A
24
23
28
30
41
B
18
17
17
19
19
18
C
41
41
49
39
19
27
D
46
69
74
58
54
50
Adapted from M. A. Tinker, "Reliability of Blinking Frequency Employed as a Measure
of Readability," Jour. E,xp. Psych., XXXV, 421.
Substituting ia our formula, we find our estimate of the missing
value to be
tT + bB S 6(116) -f- 4(146) 821
M
30.6.
(*- 1)(Z>-D 5(3)
The correction for bias in the treatment sum of squares is found to be
[JB - (f - l)Af] [146 - 5(30.6)] 2
Z = *=!) " 6(5) - ^
and so we arrive at the analysis of variance presented in Table 12.32.
TABLE 12. 3 2- Abbreviated ANOVA of Number of Blinks During Reading
Source of Variation
Degrees of
Freedom
Sum of Squares
Mean Square
Blocks ....
3
5233.82
1744.61
Treatments
5
339.90
67.98
Experimental error
14
1068.40
76.29
The test of the null hypothesis ^ r :ry = 0(y= 1, - - * , 6) gives rise to an
F-value less than unity. Noting that F'*=l/F is not significant, we
conclude that the blink-rate is consistent during reading when meas
ured over six successive 5-minute periods. It is evident that there are
wide differences among individuals, a fact which is not surprising and
which confirms our judgment in performing the experiment as we did,
that is, by removing the inter-individual differences which otherwise
would have appeared as part of the experimental error sum of squares.
(NOTE: Actually, the value we have assumed to be missing was re
corded as 27 in the original source of data. It will pay the reader to do
the analysis with the true value entered in the table for comparison with
the approximate solution presented above. This should give him an
indication, but only an indication, of how reliable the estimation pro
cedure is.)
If two or more values are missing, the same general procedure (using
the calculus) may be followed to provide estimates. For th.e person
not familiar with, the requisite mathematical tecliniques, equivalent
results may be obtained by use of the following iterative method.
Suppose that two values are missing: For one of these substitute the
394 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
mean of all recorded observations, and then estimate the second mis
sing value using Equation (12.36); next, place this estimate in its
proper place in the table, remove the general mean from its position
in place of the first missing observation, and then estimate a value
for the first missing observation using Equation (12.36). After about
two cycles you will find very little or no change in successive estimates
of the same missing value. When this point is reached, you have the
estimated values. This procedure may easily be extended to cases where
three or more observations are missing.
What changes are necessary before one proceeds with the usual F-
tests if we have been forced to estimate several missing values? First,
we must reduce the degrees of freedom associated with both experi
mental error and total by the number of observations estimated. Sec
ond, the treatment sum of squares must be reduced by a specified
quantity to avoid a biased test procedure. If we have only two missing
observations (not in the same block), the necessary correction for bias
is given by
- \_B> - (jt - 1)M']* + [B" -(jt- VM"\*
Z, = } ( 1 2 . o 9 )
t(t - i)
where
t number of treatments
B' = total of all the observations in the same block as the first
missing observation
J5" = total of all the observations in the same block as the second
missing observation
M' = estimate of the first missing observation.
M" = estimate of the second missing observation.
If more than two observations are missing, or if two observations are
missing in the same block, a formula giving the correction for the bias
in the treatment sum of squares may be found in Yates (14, 15).
Problems
12.1 The folio wing data are from an experiment involving a randomized
complete block design. Complete the appropriate analysis of vari
ance, and test the hypothesis that the true effects of the four treat
ments are equal. State all your assumptions.
PROBLEMS
395
Treat
ment
Block
1
2
3
4
1
20
18
16
17
2
18
18
16
2O
3 . .
20
18
17
18
4
20
16
20
17
5
19
16
16
2O
12.2 In a randomized complete block experiment with 5 treatments in 10
replications, the variance among the 5 treatment means was 100.
Complete the following abbreviated ANOVA, and test the hypothesis
that the 5 treatment effects are the same.
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Replicates
90
Treatments
Experimental error . . .
5
12.3 Upon calculating the analysis of variance of the yields of 6 varieties
planted in 8 randomized complete blocks, the 3 sums of squares, for
varieties, for blocks, and for experimental error (or remainder),
were each 245. Complete, as far as is possible, the appropriate
analysis of variance, and compute a value of F for testing the
significance of the differences among varieties. Interpret your result
in terms of the appropriate model, and give your conclusions.
12.4 Examine the results given below to learn about the effectiveness of
chalk and lime applications in neutralizing soil acidity and thus in
creasing the stand of beets.
Number of Beets per Plot
Block
Control
Chalk
Lime
1
49
135
147
2
37
151
131
3
114
143
103
4 ...
140
146
147
396 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
12.5 Analyze the data in the following table and interpret the results.
RATIO OF DRY TO WET WHEAT
Nitrogen
Applied
Block
None
Early
Middle
Late
1
0.718
0.732
0.734
0.792
2
0.725
0.781
0.725
0.716
3
0.704
1.035
0.763
0.758
4
0.726
0.765
0.738
0.781
12.6 To study the relative efficiencies of 5 different types of filter, an ex
periment is to be performed using a certain brand of oil. Fifteen
quarts of oil (in 1-qt. tins) are purchased and the same amount of
foreign material is added to each quart. Since only 5 tests can be
performed in any one day, we proceed as follows: (1) allocate, at
random, the 15 quarts into three groups of 5 each; (2) allocating the
groups to the days, assign the treatments at random to the quarts
within groups; (3) perform the experiment; and (4) collect, analyze
and interpret the data.
AMOUNT or FOREIGN MATERIAL CAUGHT BY FILTER
Type of Filter
"Rlnr^L-c
(Days)
A
B
C
D
E
1
16.9
18.2
17.0
15.1
18.3
2
16.5
19.2
18.1
16.0
18.3
3
17.5
17.1
17.3
17.8
19.8
12.7 In a paired experiment there were 10 pairs with the sum of the
squares of the deviations of the differences from their mean being
]d2 = 360. The totals for the 2 treatments were 7^ = 160 and
2^= 120. Complete the following abbreviated ANOVA.
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Pairs or replications
100
Treatments , , . .
Experimental error
PROBLEMS
397
12.8 Discuss the following statement: "If only one sample is obtained from
each, experimental unit, e.g., if one small sample is taken from a
field plot to estimate the effect on the whole plot, n is set equal to 1
in Table 12.6, and the line for sampling error is omitted. However,
if the whole plot in our field plot example is harvested, then the
sampling error is reduced to 0, and we have an analysis as in Table
12.2."
12.9 In a randomized complete block experiment on the accuracy of de
termination of ascorbic acid concentration in turnip greens (Heinze-
Kanapaugh method), 4 weights of sample were tried in 5 replications.
Two determinations, A. and B, were made on each sample. The results
(in micrograms per milliliter of filtrate) were as follows :
Sample
Weight
(Grams)
Replication
1
2
3
4
5
A
B
A
B
A
B
A
B
A
B
5
34.2
12.8
5.8
3.5
37.2
12.8
8.2
3.5
47.0
21.5
10.2
5.0
52.5
22.0
13.0
6.0
48.5
24.5
16.5
9.8
46.5
23.0
11.0
6.8
44.2
17.8
9.5
5.2
44.2
17.8
15.2
3.5
42.5
17.0
11.0
3.8
43.5
17.5
10.5
4.7
2
1
0.5
Complete the analysis of variance for these data.
12.10 Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Expected
Mean
Square
Replicates
3
176
Treatments
7
352
"KxTvcTimeTi'fcaJ. error . .
21
88
Sampling error
96
40
Determinations
256
10
(a) Give the experimental error mean square in the above analysis
for the following:
(1) if 10 had been added to each determination,
(2) if each determination had been multiplied by 10.
(6) Fill in the expected mean squares in the above table, assuming we
are interested in just these 8 treatments but that replicates,
samples, and determinations may be considered as random
variables.
398 CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
12,11 Given the following abbreviated ANOVA:
12.12
12.13
Source of Variation
Degrees of
Freedom
Mean
Square
Blocks
3
Treatments
8
Experimental error
Samples within plots. . . .
24
144
1084
381
if
(a) What is the construction of the experiment?
(b) What is the variance of a treatment mean?
(c) Give the answer to (6) if we have only 1 sample per plot.
(d) What is the maximum precision obtainable by sampling; i.e.,
we take k samples, k > co ?
We conducted a field experiment to estimate the effect of 9 fertilizers
on the yield of oats. Instead of harvesting each plot completely, we
took 12 samples, 3 by 3 feet, from each plot. The abbreviated
ANOVA is as follows:
Source of Variation
Degrees of
Freedom
Mean
Square
Replicates
3
384
Xreatrnents
8
960
Experimental error,
24
192
Among samples within plots
396
24
(a) Assuming that the components of variance do not change, esti
mate the gain or loss in information in the above experiment, had
6 replicates been used with 8 samples per plot.
(6) What would the above mean squares be if the analysis of variance
had been computed using the totals of the 12 samples in each
plot?
Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Expected
Mean
Square
Replicates
3
288
Treatments
7
432
Experimental error
21
144
Among samples within experimental units . .
Among detenriinatioris per sample. ........
96
256
72
6
(a) Compute the variance of a treatment mean.
(6) Give the expected mean squares.
PROBLEMS
399
12.14
(c) Compute the gain or loss In efficiency, or information, had 6 repli
cates been used with 8 samples from each experimental unit and
1 determination per sample.
(d) Give the experimental error mean square for the following:
(1) if 10 had been added to each determination,
(2) if each determination had been multiplied by 10.
(e) Test the hypothesis that there are no differences among the true
effects of the eight treatments.
A chemist is confronted with the problem of just where he should
expend his efforts in the following situation: A series of 8 soil treat
ments are applied in a randomized complete block design with 2 repli
cations, 3 soil samples from each plot are taken in the field, each
sample is divided into 2 portions in the laboratory, and duplicate de
terminations for each portion are analyzed for nitrate nitrogen. The
following mean squares are given:
Source of Variation
Degrees of
Freedom
Mean
Square
Treatments. . . . .....
7
11700
Experimental error .
7
1300
Samples within pints ,,,,,.
32
100
Portions within samples . . . ....
48
20
Determinations within portions
96
16
12.15
Find the expected mean squares and estimate the variance compo
nents. What might be his gain or loss in efficiency in future experi
ments if he used 6 replicates, but still continued to run only 24
analyses per treatment, e.g., 2 samples per plot, 2 portions per sample
and 1 determination per portion?
In an experiment to test the effect of 6 treatments on some soil
characteristic, we obtained the following abbreviated ANOVA. A
total of 6 soil samples was selected at random from each plot, and
2 chemical determinations were made of each sample.
Source of Variation
Degrees of
Freedom
Mean
Square
Replicates
4
240
Treatments . ,
5
360
Experimental error
20
120
Samples within plots
150
60
Determinations per sample
180
4
(a) Compute the variance of a treatment mean (per determination).
(6) Estimate the gain or loss in efficiency in the above experiment if
we had taken 8 samples per plot and had made only 1 determi
nation per sample.
4OO
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
12.16 Given the following abbreviated A1STOVA for a randomized complete
block design:
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Expected
Mean Square
Blocks
9
.4074
Treatments. .
3
1.1986
Experimental error. . .
27
,6249
(a) Complete the analysis; fill in expected mean squares.
(6) Estimate the efficiency of this design relative to a completely
randomized design.
(c) Compute the standard error for a treatment mean and for the
difference between 2 treatment means.
(d) The treatment means are 1.464, 1.195, 1.325, and 1.662. What
mean or means do you suspect might represent different popula
tions?
12.17 The following data give the gains in weight of pigs in a comparative
feeding trial. Analyze and interpret the data, paying attention to
the comparison of Rations I, II, and III with Rations IV and V.
GAINS 03? PIGS iisr A COMPARATIVE FEEDING TRIAL
Replicate
Ration I
Ration II
Ration III
Ration IV
Ration V
1
165
168
164
185
201
2
156
180
156
195
189
3
159
180
189
186
173
4
167
166
138
201
193
5
170
170
153
165
164
6
146
161
190
175
160
7
130
171
160
187
200
8
151
169
172
177
142
9
164
179
142
166
184
10 .
158
191
155
165
149
PROBLEMS
401
12.18
12.19
12.20
The following data are extracted from a larger experiment concerned
with oat-seed treatment. The following yields in grains were obtained
with 2 rates of the same compound over 7 replicates:
R
Lte
Replicate
1
2
Check
1
360
391
408
2
436
382
409
3
413
414
340
4.
353
416
324
5
328
375
304
6
269
422
268
7
220
227
290
What conclusion do you draw? In a separate column are the yields
for the untreated seed. Is seed treatment worth the added expense in
this instance?
Results similar to those in Problem 12.18 are also available for flax.
What advice would you give about the use of Ceresan M as against
224, and about the advisability of seed treatment?
Replicate
Ceresan M
224
Check
1
19.2
14.4
13.2
2
14.8
24.6
19.2
3
26.7
22.9
17.4
4
17.6
22.7
16.4
5
22.1
22.0
15.8
6
21.7
22.0
14.6
7
23.9
20.4
12.5
8
19.1
16.0
13.0
A project studying farm structures was concerned with the insulation
of poultry houses. The data obtained from a study of a set of model
structures (total number of eggs over 4 replicates of each treatment)
were as follows:
Standard house+laying mash 250
3" wall insulation+laying mash 280
3" wall insulation+laying mash+cod liver oil 350
6" wall insulation+laying mash 310
6" wall insulation+laying mash+cod liver oil 400
Construct a reasonable set of 4 orthogonal comparisons based on the
above treatments. Calculate the sum of squares for one of your com
parisons and test for significance. The following is part of the original
analysis:
4O2
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
Source of Variation
Degrees of
Freedom
Sum of
Squares
Treatments
4
3470
Experimental error
12
1728
12.21 Assume a randomized complete block experiment with 4 treatments
and 8 replicates. One of the treatments is a control or check and the
other 3 are different methods of treatment. Assume that the mean
effect of all 32 experimental units is 40, that the mean effect for the
control is 34, and that the mean effect for method B is 42. Also the
following abbreviated ANOVA is given:
Source of Variation
Degrees of
Freedom
Mean
Square
Replicates
7
32
Treatments
3
64
Experimental error
21
16
(a) What is the experimental error variance per experimental unit?
(6) Compute the coefficient of variation per experimental unit,
(c) Compute the variance of a treatment mean.
(d) Is the difference between the mean effects of control and method
B significant at the 1 per cent level?
(e) Compute and interpret the 95 per cent confidence interval esti
mate of the mean difference between the control and method B.
12.22 An experiment was conducted to assess the relative merits of 5 dif
ferent gasolines. Since vehicle to vehicle variations in performance
are inevitable, the test was run using 5 cars, hereafter called blocks.
The following descriptions of the 5 gasolines are available:
A: control
B: control+ additive X manufactured by company I
C: control + additive Y manufactured by company I
D: control -4-additive IT manufactured by company II
E: control + additive V manufactured by company II.
The data, in miles per gallon, are given below. Please analyze and
interpret the data.
Blocks (Cars)
Treatments
(Gasolines)
1
2
3
4
5
A
22
20
18
17
19
B
28
24
23
19
25
C
21
23
25
25
27
D
26
21
21
22
20
E
27
25
22
20
24
PROBLEMS
4O3
12.23
12.24
12.25
12.26
12.27
Subdivide the experimental error sum of squares in each, of the follow
ing problems in accordance with the principles given in Section 12.9:
(a) 12.4 (d) 12.19
(b) 12.17 (e) 12.20
(c) 12.18 (f) 12.22
Using the technique presented in Section 11.10, analyze further the
data given in the following problems:
(a) 12.1 (d) 12.6 (g) 12.19
(b) 12.4 (e) 12.17 (h) 12.20
(c) 12.5 (f) 12.18 (i) 12.22
In Table 12.31 we presented some data on blinking rates in successive
5-minute periods of reading. After substituting for a missing observa
tion, these data were analyzed using a randomized complete block
design. Ignoring the fact that we had to estimate a missing observa
tion, discuss critically the use of a randomized complete block design
in analyzing data of this type. If you feel that the use of a random
ized complete block design was unjustified, state reasons to support
your contention and give what you believe to be an appropriate
method of handling such data. Examine all your assumptions
carefully.
Five levels of fertilizer, 0, 10, 20, 30, and 40, were applied to corn in a
randomized complete block design. A preliminary analysis of vari
ance gave the following results:
d.f.
Replicates 4
Fertilizers 4
Experimental error 16
M.S.
2500
2800
1500
The sums of the yields in the 5 plots of each level were:
Level 10 20 30 40
Total yield
20
140
260
300
280
What additional computations would you make to interpret the
effect of treatments? Make these computations, and interpret the
results.
The strength index of cotton fibers was thought to be affected by the
application of potash to the soil. A randomized complete block experi
ment was conducted to get evidence. Here is a summary of the plot
strength indexes:
Treatment
/"|~> -J j-v-C "C?"" /*"%
Replications
^Jrounas ot &.2.(J
per Acre)
1
2
3
36
7.62
8.00
7.93
54
8.14
8.15
7.87
72
7.76
7.73
7.74
108
7.17
7.57
7.80
144
7.46
7.68
7.21
404
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
12.28
Analyze the data. Plot the mean strength index for each treatment,
Y, against the pounds of fertilizer per acre, X. The sum of squares
attributable to regression is 0.5662 with 1 degree of freedom (verify
this). Subtract this from your sum of squares for treatments (4
degrees of freedom). The remainder (3 degrees of freedom) is the
sum of squares of deviations from regression. Complete the analysis
of variance. Test the hypothesis of regression. What conclusions
do you draw?
The following are the yields (tons per acre) of sugar beets on plots
which, 2 years earlier, had been treated with lime:
TV*a.a frn pin f
Replications
(Tons per Acre)
1
2
3
4
5
1
13.7
13.3
12.6
14.7
10.8
2
16.9
17.1
14.7
15.7
15.4
3.. .
17.3
17.1
16.9
16.2
14.6
4
17.8
16.5
17.9
15.7
16.3
12.29
Analyze the data. Test the hypothesis that there is no effect of
treatment. Plot the treatment means, Y, against the rate of applica
tion of Erne, X. Do you think the regression is linear? As evidence,
divide the sum of squares for treatments into 2 parts : attributable to
regression, 35.52; and remainder. Test the null hypothesis that there
is no deviation from linear regression. Instead of thinking about re
gression, you might have divided the treatment sum of squares into
these 2 parts: (1) due to difference between mean of first treatment
and mean of the other 3 combined, 43.02; and (2) differences among
means of the last 3 treatments. What conclusions do you reach?
Consider an experiment to assess the relative effects of 4 different
treatments (i.e., packing pressures) on the function time of a certain
explosive actuator. Casings are available from 4 different production
lots. Four casings were randomly selected from each of the lots and
the treatments were assigned at random within each lot. Given the
data shown below (operation time in milliseconds), analyze and
interpret the results.
Packing Pressures (psi)
Blocks
(Lots)
10,000
20,000 30,000
40,000
1
12
17 10
12
2
11
16 9
11
3
10
15 8
11
4
9
15 8
10
PROBLEMS
405
12.30 Analyze and interpret the following data on yields of sweet potatoes
obtained with various combinations of fertilizer (n = N, p PzOs,
Replicate 1
Replicate 2
npk
Yield
npk
Yield
npk
Yield
npk
Yield
npk
Yield
npk
Yield
133
45
211
39
333
70
212
83
211
56
133
65
111
34
313
62
311
40
221
52
321
49
112
48
221
42
222
65
212
45
322
65
333
92
311
56
323
69
233
92
132
53
313
101
122
75
332
79
213
58
123
56
121
54
111
50
312
86
213
95
331
51
332
91
223
69
331
61
232
74
222
81
232
72
131
73
322
85
132
89
223
109
231
84
122
56
112
55
113
60
123
90
113
68
323
103
312
82
321
75
231
78
233
122
131
98
121
64
Totals
509
608
554
713
707
675
Grand
Totals
1671
2095
12.31 Given the following abbreviated ANOVA:
Source of Variation
Degrees of
Freedom
Mean
Square
Replicates
4
70
Treatments :
A
3
50
B
3
160
AB
9
40
Experimental error
60
10
Interpret the effects of a and & assuming that:
(1) the various levels of both a and 6 are fixed or selected;
(2) the various levels of both a and 6 are random variables;
(3) the levels of a are fixed, but the levels of & are random;
(4) the levels of a are random, but the levels of & are selected.
12.32 Mr. X sprayed apple leaves with different concentrations of a nitro
gen compound, then determined the amounts of nitrogen (ing. per
sq. dcm.) remaining on the leaves immediately and at two subsequent
times. The object was to learn the rate at which the nitrogen was ab
sorbed by the leaves. There were two replications of each treatment.
The first entry in each cell of the table is for the first replication.
4O6
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
Time
Levels of Nitrogen
n\
n*
ns
t
2.29
2.24
0.46
0.19
0.26
6.50
5.94
3.03
1.00
0.75
1.16
8.75
9.52
2.49
2.04
1.40
1.81
ti
t z
12.33
Obtain the analysis of variance which, subdivides the 8 degrees of
freedom for treatments into individual comparisons: N&, NQ, T L ,
T Q , N L T L , N L T Q , N Q T L , and N Q T Q .
2 3 FACTORIAL FIELD PLAN WITH YIELDS
Replicate 1
Replicate 2
Replicate 3
Replicate 4
(1) 7 b 24
abc 39 ac 31
a 30 c 21
be 27 ab 39
ab 36 be 31
(1) 19 ac 36
abc 41 b 30
c 30 a 33
a 28 ac 31
c 24 b 19
ab 35 (1) 13
be 26 abc 36
abc 66 (1) 11
a 31 be 29
c 21 ac 33
& 25 <zZ> 43
12.34
Complete the analysis of variance, computing the treatment sum of
squares for each of the individual treatment effects, and subdividing
the experimental error corresponding to the subdivision of the treat
ment sum of squares.
ANALYSIS or VARIANCE
Source of
Variation
Degrees
of Free
dom
Mean
Square
Replicates
A
3
1
192
100
B
1
2500
AB
1
900
Experimental
error
9
32
TABLE
<Z
#1
Sum
&o -
120
80
200
bl
160
240
400
Sum
280
320
600
(1) Interpret the effects of a and 6 assuming both are fixed variates.
(2) Compute and interpret the 95 per cent confidence interval esti
mate of the true mean difference between treatments
and
PROBLEMS
4O7
12.35 The following yields of grass -were reported for one year in dry matter
per 1/57-acre plots. This was a randomized complete block design.
El
(Ha
ephant Gr
rvests per y
LSS
ear)
Gu
(Ha
ate mala Gi
rvests per y
ass
ear)
Blocks
2
3
4
2
3
4
1
109
222
187
277
246
252
2
97
125
163
293
263
181
3
133
134
143
260
194
224
4
113
173
179
325
190
248
Discuss the complete 2X3 factorial experiment, displaying the perti
nent estimates; outline tentative conclusions before making the
analysis of variance and tests of hypotheses.
12.36 Analyze and interpret the following set of experimental data: crop
oats; location Flathus, Correctionville; year 1944; comment
yield in bushels per acre.
Replicate
' 1 *r^c* "f~TT"l f-T~l "f"
Treatment
1
2
3
Total
/ yt\ f p\k\ . .
32.2
33.9
34.6
100.7
7^27?ll ....
37.4
40.9
38.9
117.2
J^1??2&1 ....
30.6
39.4
33.8
103.8
fl% f p%k\ ... ...
52.4
48.0
43.9
144.3
> l\'P\k%
29.9
34.5
36.5
100.9
n%'f)~\ m fe%
42.2
29.9
34.1
106.2
^^l7^2^2
31.8
32.5
34.2
98.5
46.6
49.5
46.7
142.8
Total
303.1
308.6
302.7
914.4
12.37 An experiment was conducted to assess the effects of 3 raw material
sources (i.e., suppliers) and 4 mixtures (i.e., compositions) on the
crushing strength of concrete blocks. Twenty-four blocks were se
lected, 2 at random from those manufactured by each of the 12
treatments, and the experiment was conducted as a randomized com
plete block with 2 replicates. The resulting data are given below.
Analyze and interpret.
4O8
CHAPTER 12, RANDOMIZED COMPLETE BLOCK DESIGN
Suppliers
Replicate
Mixtures
^1
B
C
D
1
1
2
57
46
65
73
93
92
102
108
2
1
2
26
38
44
67
81
90
96
99
3
1
2
39
40
57
60
96
100
105
116
12.38 The following is a randomized complete block design with two missing
plots. Fill in estimates for the missing values, and complete the
analysis of the data.
Trea
tment
Block
1
2
3
4
Block Totals
1
43
35
37
42
157
2
45
39
40
47
171
3. ...
42
30
M"
43
115 +M"
4
M f
43
48
49
140+-34T
5
41
34
36
44
155
Treatment
Totals
1714-M'
181
161+lf"
225
73S+M'+M"
References and Further Reading
1. Anderson, R. L., and Bancroft, T. A. Statistical Theory in Research. McGraw-
Hill Book Company, Inc., New York, 1952.
2. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
3. Chew, V, (editor) Experimental Designs in Industry. John Wiley and Sons,
Inc., New York, 1958.
4. Cochran, W. G., Catalogue of uniformity trial data. Jour. Roy. Stat. Soc.
(SuppL), 4 (No. 2) 1937.
5 f and Cox, G. M. Experimental Designs. Second Ed. John Wiley and
Sons, Inc., New York, 1957.
6. Davies, O. L. (editor) The Design and Analysis of Industrial Experiments.
Second Ed. Oliver and Boyd, Edinburgh, 1956.
7. Dixon, W. J., and Massey, F. J. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
8. Federer, W. T. Experimental Design. Macmillan Co., New York, 1955.
9. Kempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
10. Paull, A. E. On a preliminary test for pooling mean squares in the analysis
of variance. Ann. Math. Stat., 21:539, 1950.
REFERENCES AND FURTHER READING 4O9
11. Quenouille, M. H. The Design and Analysis of Experiment. Charles Griffin
and Co. Ltd., London, 1953.
12. Snedecor, G. W. Statistical Methods. Fifth Ed. The Iowa State University
Press, Ames, 1956.
13. Tinker, M. A. ^Reliability of blinking frequency employed as a measure of
readability. Jour. Eocptl. Psych., 35:418, 1945.
14. Yates, F. The analysis of replicated experiments when the field results are
incomplete:. Emp. Jour. Exptl. Agr., 1:129, 1933.
15. . Incomplete randomized blocks. Ann. Eugen., 7:121, 1936.
CH APTE R 13
OTHER DESIGNS
THE COMPLETELY RANDOMIZED and randomized complete block designs
discussed in Chapters 11 and 12, respectively, are only two of the
many useful statistical designs that have been developed for special
situations. Unfortunately, not all of the available designs can be dis
cussed in this book. However, after considering such factors as fre
quency of use and potential contribution to more efficient experimen
tation, a select group of designs and analysis techniques has been
chosen for presentation in this chapter. Persons desiring information
on other designs should consult a professional statistician and/or refer
to the references at the end of this chapter.
13.1 LATIN AND GRAECO-LATIN SQUARES
The Latin square (LS} design is frequently used in agricultural and
industrial experimentation. It is a special design that permits the re
searcher to assess the relative effects of various treatments when a
double type of blocking restriction is imposed on the experimental
units. Viewed in this way, the Latin square design is a logical extension
of the randomized complete block design and two examples should be
sufficient to illustrate the ideas involved.
Example 13.1
Suppose we have 5 fertilizer treatments to be investigated and 25
plots available for experimentation. If the soil shows a fertility trend
in two directions (say N >S and E >W), it would seem reasonable to
set up blocks of (5) plots in bath directions. This is precisely what is
done under the names rows and columns. The treatments are then
applied at random, subject to the restriction that each treatment appear
but once in each row and each column.
Example 13.2
Consider the problem of testing 4 machines to see if they differ sig
nificantly in their ability to produce a certain manufactured part. It is
well known that different operators and different time periods in the
work day will have an effect on production. Thus, we set up 4 operators
as "columns" and 4 time periods as "rows" and then assign, at random,
the machines to the various cells in the square, subject to the restriction
that each machine be used only once by each operator and in each time
period.
These two examples should acquaint the reader with the basic concepts
involved in a Latin square design. The idea of a square is evident, of
[4101
13.1 LATIN AND GRAECO-LATIN SQUARES 411
course, since, if m, treatments are to be investigated, we need m z ex
perimental units,
The basic assumption for a Latin square design with one observation
per experimental unit is that the observations may be represented by
the linear statistical model
yV/Gfc) M + pi + ry + T k + e;y(fc) ; i = 1, - , nt (13 . 1)
1 ... w
K 1 3 , 77Z,
where
vn. tn m
-A =
and the #<*> are independently and normally distributed with mean
and common variance <r 2 . The subscript k is placed in parentheses to
indicate that it is not independent of i and j. The constants p t -, y^, and
Tk are, of course, the true effects associated with the ith row, Jth
column, and fcth treatment, respectively.
Because of the possible economies due to reduced sample sizes, the
Latin square design has great appeal to researchers in all fields. In par
ticular, the engineer has been a prolific user of the Latin square design,
but, unfortunately, he has not always used the design "wisely. An
examination of the postulated statistical model will show that the
interactions among rows, columns, and treatments have been assumed
to be 0. In many engineering or industrial experiments involving a
Latin square design (where the rows and columns usually refer to
real chemical, physical or other factors) , it is precisely this assumption
that appears to have been overlooked by the researcher. (NOTE:
When information about interactions is lacking or when the assumption
of interaction is of doubtful validity, a full factorial should be run.)
Having pointed out the advantages and limitations of a Latin square
design, let us now summarize the appropriate calculations. These are:
= total sum of squares
771 7M. fft, 7n tn T^fL _
- 2 _ y^ y^ Tr 2 _ y-
/- -^ .{ -* -*- iy (&) s >
M yy = sum of squares due to the mean
(13 . 3j
R yy = row sum of squares
2 (13.4)
C yy = column sum of squares
2 (13.5)
412
CHAPTER 13, OTHER DESIGNS
yy = treatment sum of squares
and
= experimental error sum of squares
* ~ X y -^ """"" JML yy ~~~~ JK-yy \-s yy ~~ JL yy
(13.6)
(13.7)
where Jfg,-, Cj, and T k represent the indicated row, column, and treat
ment totals, and T denotes the total of all the observations. The result
ing ANOVA is shown in Table 13.1.
TABLE 13.1-Generalized ANOVA for an mXm Latin Square
Design With. One Observation per Experimental Unit
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected
Mean Square
F-Ratio
Mean
1
MM
M
Rows
m I
JR
02 _j_ t m /(m l}] y^ P!-
0!nlvimTR . . . , -.,-,,
m1
c
i i
<r*+\ m /(in l')] T/
Treatments
T
T
y-i
o^+Iw/Cw 1)] 2
T/E
Experimental error.
(ml)(m 2}
E
i i
<r 2
Total
m*
TF*
Example 13.3
The data shown in Table 13.2 resulted from an experiment such as
described in Example 13.2. Assuming that time periods, operators, and
machines do not interact (either pairwise or as a complete set), the
ANOVA of Table 13.3 is obtained. This leads to the conclusion that
there are significant differences among the outputs of the 4 machines.
Further examination of the data should permit selection of the most
productive machine or machines.
Should a single observation be missing in an experiment conducted
according to an mXm Latin square design, its value may be estimated
using
M =
m(R
T}
(m 1) (m 2)
(13.8)
13.1 LATIN AND GRA ECO -LATIN SQUARES
413
TABLE 13.2-Number of Units Produced by Four Machines
in a Latin Square Design
(The random assignment of the machines is shown
by the letters in parentheses)
Time
Periods
Operators
1
2
3
4
1
31 (Q
39 (>)
57 ()
85 {A)
43 (Z>)
96 (A)
33 (C)
46 ()
67 01)
40 (J5)
40 (>)
48 (C)
36 (J5)
48 (C)
84 (A}
50 (Z>)
2.
3
4
TABLE 13. 3- Abbreviated ANOVA for Data of Table 13.2
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
Time periods ...
3
408.188
136.06
J^ 2
<r 2 -K4/3) 5Z Pi
Operators
3
88.688
29.56
* i
<T2-f-(4/3) ]T T *
Jv/Cacnines
3
4946.688
1648.90**
*-i
<7 2 -f-(4/3) 2^ rl
Experimental error .....
6
515.874
85.98
Jt-i
o- 2
** Significant at <* = 0.01.
where
R = sum of observations in the same row as the missing observation
C = sum of observations in the same column as the missing obser
vation
T = sum of observations with the same treatment as the missing
observation
> f==sum of all the actual observations.
After substituting the value of M in the table, the various sums of
squares are calculated as indicated above. However, it must be remem
bered that the treatment sum of squares so calculated (T vy ) will be
biased upwards, and a correction must be applied before we test the
hypothesis H:rk = Q (/b= 1, - , m). This correction is made by com
puting a new treatment sum of squares (T 7 ^) defined as
T f
JL <UU
Tyy
where
jr
[S R C (m 1) T\*
(m 1) 2 - 2) 2
(13.9)
(13.10)
414 CHAPTER 13 f OTHER DESIGNS
Remember that the degrees of freedom associated with experimental
error and total are each reduced by one (in view of the single missing
observation); that is, the degrees of freedom for experimental error
ar& now (m l)(m 2) 1, and the degrees of freedom for total are
now m 2 1. No example will be given for the above technique, but
one or two of the problems at the end of this chapter will illustrate the
principles involved.
By now the reader should be sufficiently adept at the calculations
involved in analyses of variance so that lengthy discussions of such
topics as subsampling, selected treatment comparisons, factorials,
analysis of response functions, estimation of components of variance,
and predictions of the relative efficiencies of various allocations of the
observations in terms of experimental and sampling units would be a
waste of time. Accordingly, we will do no more than state that the
techniques introduced in Chapters 11 and 12 may easily be extended
and adapted for use with Latin square designs. However, to make cer
tain that the previously mentioned extensions and adaptations are made
properly, a few problems requiring their use have been included in the
set at the end of this chapter.
Before terminating our discussion of the Latin square design, mention
must be made of its efficiency relative to completely randomized and
randomized complete block designs. (NOTE: This discussion will, of
course, be closely related to that of Section 12.7.) If we designate the
mean squares in the Latin square for rows, columns, and experimental
error by -R, C, and E, respectively, we may readily evaluate the effi
ciency of a Latin square design relative to either a completely ran
domized or randomized complete block design. For the efficiency of a
Latin square design relative to a completely randomized design, we
calculate
R + C + (m 1)JE
R.E. = - - - - (13.11)
If, however, we wish to compare a Latin square design with what
might have happened had a randomized complete block design been
utilized (assuming the rows were used as blocks), the following formula
is appropriate:
C + (m 1)JS
R.E. = - - - - (13.12)
mE
If columns were used as the blocks, we put R in place of C in Equation
(13.12).
The concept of a Latin square design can be extended rather easily
to that of a Graeco-Latin square (G~LS} design. Rather than go into the
details of a Graeco-Latin square, we shall only indicate, by example,
the nature of the design. Those persons interested in using such a de
sign are advised to consult a professional statistician.
13.2 SPLIT PLOTS 415
Example 13.4
Chew (21) describes an experiment which could be used to compare
five formulations (<x, /3, y, d, e) for making concrete bricks, using material
from 5 batches, prepared on each of 5 days, and tested on 5 different
machines (A, B } C, D, N). One possible randomization of a Graeco-
Latin square design for this situation is shown in Table 13.4, It will be
noted that: (1) each Latin letter appears exactly once in each row and
each column, (2) each Greek letter appears exactly once in each row
and each column, and (3) each Latin letter appears exactly once with
each Greek letter,
TABLE 13.4-Symbolic Representation of the Graeco-Latin Square Used
in Example 13.4
"O f\w5i
Columns (Days)
(Batches)
1
2
3
4
5
1. . . - .
Ace
By
Ce
D8
JES
2 .
BQ
C5
Da.
Ey
Ae
3
Cy
-De]
E8
Ad
BOL
4
D5
^ JL
Ea.
Ay
Be
C8
5. ...
JSe
A3
Bd
COL
Dy
As tempting as Graeco-Latin squares are to the industrial experi
menter (because of the potential savings in numbers of observations) ,
they should be used with caution. This recommendation sterns from the
same type of limitation that was emphasized for Latin square designs,
namely, no interactions are tolerated.
13.2 SPLIT PLOTS
A fairly simple design which, is frequently used in experimental work
is the split plot OSP) design. In this design we are concerned with two
factors, but we wish more precise information on one of them than on
the other. Let us assume that we have factors a and b and desire more
accurate information on b than on a. The usual scheme is to assign the
various levels of factor a at random to the whole plots (main plots) in
each replicate as in a randomized complete block design. Following
this, the levels of b are assigned at random to the split plots (sub-plots)
within each whole plot. Under such a scheme of randomization, which
may arise not only from the desire for more precise information on one
factor than on another but also because of the nature of the factors and
the way in which they must be applied to the experimental units, the
analysis of variance appears as in Table 13.5.
Example 13.5
An experiment similar to that described in Example 10.19 was per
formed. However, in this case, there were six replicates, three tem
peratures, and four levels of electrolyte. (NOTE: In contrast to Ex-
416 CHAPTER 13, OTHER DESIGNS
TABLE 13.5-Generalized ANOVA for a Split Plot Design
Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Expected Mean
Square
F- Ratio
1
r 1
a 1
(r-l)O-l)
-1
O_ !)(&-!)
(r _D * C a_D
M yv
R vv
Ayy
CEOw
J? VJ/
C^B) W
CEOiw
M
R
A
E a
B
AB
E b
Whole plots
Replicates.
A
o-i+bo-z+rb y^ ct.i/(a, 1)
a i
2 , 2
CT1-J-0CT2
<r!+rai;/sJ/(6-l)
AI
-?+r z; i (<*/?)**/(- 1) (&-1)
j_i fc i
2
CTl
^4/^a
Whole plot
error. . . .
Split plots
B
B/E b
AB/E b
AB
Split plot
error ....
Total
rob
]F*
ample 10.19, heat paper was not a factor in this experiment.) The data
are given in Table 13.6 and the resulting ANOVA in Table 13.7. No
calculational details are reported, since these are assumed to be straight
forward. Further interpretation of the data is impossible because of lack
of information regarding the exact nature of the treatments.
TABLE 13.6 Activated Lives (in Hours) of 72 Thermal Batteries Tested
in a Split Plot Design Which Used Temperatures as Whole Plots
and Electrolytes as Split Plots
Replic
ate
Temperature
Electrolyte
I
2
3
4
5
6
Low
A
2.17
1.88
1.62
2.34
1.58
1.66
M! edium
B
C
D
A
1.58
2.29
2.23
2.33
1.26
1.60
2.01
2.01
1.22
1.67
1.82
1.70
1.59
1.91
2.10
1.78
1.25
1.39
1.66
1.42
0.94
1.12
1.10
1.35
High
B
C
D
A
1.38
1.86
2.27
1.75
1.30
1.70
1.81
1.95
1.85
1.81
2.01
2.13
1.09
1.54
1.40
1.78
1.13
1.67
1.31
1.31
1.06
0.88
1.06
1.30
B
C
D
1.52
1.55
1.56
1.47
1.61
1.72
1.80
1.82
1.99
1.37
1.56
1.55
1.01
1.23
1.51
1.31
1.13
1.33
13.3 COMPLETE FACTORIALS WITHOUT REPLICATION
TABLE 13.7-Abbreviated ANOVA for Data of Table 13.6
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean Square
Whole plots
Replicates
5
4 1499
. 8300
Temperatures . .
2
1781
O 0890
\Vlaole plot error
10
1 . 3622
0.1362
Split plots
Electrolytes . .
3
1.9625
6542**
Temperature X
electrolyte
6
2105
0351
Split plot error
45
1 2586
0280
Significant at a: = 0.01.
Before leaving the subject of split plot designs, we must take note
tliat the principle of "splitting" may be carried on for several stages;
that is, we may employ split-split plot designs, etc. For more detailed
discussion of such designs and for some illustrative examples, one
should consult the references at the end of the chapter.
13.3 COMPLETE FACTORIALS WITHOUT REPLICATION,
FRACTIONAL FACTORIALS, AND INCOMPLETE
BLOCKS
Because most experimenters are interested in investigating the effects
on a response variable of the simultaneous variation of many factors,
a large number of designs incorporate factorial treatment combinations.
However, as the number of factors increases, the size of the experiment
becomes prohibitive. In addition, it becomes difficult to control the
magnitude of the experimental error within reasonable bounds.
In an attempt to reduce the experimental error to a reasonable mag
nitude, the principle of confounding (see Chapter 10) was utilized to
create a group of designs known as incomplete block designs. These
designs are so named because not all the treatment combinations are
present in each block, that is, the blocks are incomplete. With ade
quate replication, these designs proved very useful in agricultural
experimentation .
Since incomplete block designs are not usually included in a first
course in statistical methods, the decision has been made to omit dis
cussion of them from this book. However, several of the references
listed at the end of the chapter discuss at length the methods of
analysis appropriate to such designs.
When engineers and physical scientists became interested in statis
tically designed, multi-factor experiments, they decided that both
replicated complete factorials and incomplete block designs were un-
418
CHAPTER 13, OTHER DESIGNS
satisfactory in that they required too many experimental units.
Further, it was evident that, as a general rule, the experimental
errors in industrial experiments were much smaller than those en
countered in agricultural experiments. Because of the small experi
mental errors, one common approach has been to avoid replication
(i.e., subject only one experimental unit to each treatment combina
tion) and to estimate the experimental error by pooling the mean
squares associated with the higher order interactions. (NOTE: This is
equivalent to assuming that the true high order interaction effects
are 0.) This technique, referred to in the title of this section as complete
factorials without replication, is, as we have said, used quite often.
(NOTE: Actually, it is a completely randomized design involving
factorial treatment combinations and utilizing only one experimental
unit per treatment combination.)
Rather than devote a lot of space to the discussion of the models
and assumptions for the many possible situations, let us consider an
example. It is hoped that this will prove sufficient for a reasonable
understanding of the principles involved. For those persons who wish
to consider the matter more thoroughly, I again recommend the refer
ences listed at the end of the chapter.
Example 13.6
Davies (28) considered a laboratory experiment to investigate the
yield of an isatin derivative as a function of acid strength (a), time
of reaction (b), amount of acid (c), and temperature of reaction (d). Two
levels of each factor were used, namely:
a: 87 per cent, 93 per cent
b: 15 minutes, 30 minutes
c: 35 ml., 45 ml.
d: 60C., 70C.
The data shown in Table 13.8 led to the ANOVA of Table 13.9.
TABLE 13.8-Yield of Isatin Derivative
(g. per 10 g. of base material)
Acid
Strength
(a)
Reaction
Time
(*)
Temperature of Reaction (d)
601
70 1
Amount of acid (e)
Amount of acid (c)
35 ml. 45 ml.
35 ml. 45 ml.
87
93
15 rain.
30 rain.
15 min.
30 min.
6.08 (1) 6.31 (e)
6.53 (b) 6,12 (be)
6.04 (a) 6.09 (ae)
6.43 (ab) 6.36 (abc)
6.79 (d) 6.77 (ed)
6.73 (bd) 6.49 (bed)
6.68 (ad) 6.38 (aed)
6.08 (abd) 6.23 (abed)
Source: O. 31. Davies, (editor), Design and Analysis of Industrial Experiments. Second
Edition. Oliver and Boyd, Edinburgh, 1956, p. 275, Table 7.7. By permission of the author
and the publishers.
13,3 COMPLETE FACTORIALS WITHOUT REPLICATION
TABLE 13. 9- Abbreviated ANOVA for Data of Table 13.8
419
Source of Variation
Degrees of
Freedom
Mean Square
Main effects
A
i
0~f A /C "2
B
i
. 14oo
Onrn Q
C
i
.UUlo
Ono 1 1
D
i
-Uzoo
OOOOQ
Two-factor interactions
AB
i
. ZWo
Or\r\r\c\
AC
i
. uuuu
Of\f~\A /
AD
1
,UU4fcO
01 r\A r\
BC
1
- 1U4U
Or\-i *7j<
BD
1
.U17o
00 co c
CD
1
. ZDZ.S
Of\r\OQ
Experimental error
(pooled high order interactions) .
5
. UU^O
0.0385
Source: O. L. Davies, (editor), Design and Analysis of Industrial Experiments, Second
Edition, Oliver and Boyd, Edinburgh, 1956, p. 277, Table 7.72. By permission of the
author and the publishers.
As helpful as it was, the approach taken in the two preceding para
graphs and illustrated in Example 13.6 (i.e., the utilization of a com
plete factorial without replication) was not enough. Experiments were
still too large to suit the researcher. Some other way had to be found
to reduce the size and cost. One such attempt was the development
of fractional factorials in which only some (a fraction) of the treatment
combinations are actually investigated.
Once an experimenter has decided that some form of fractional fac
torial is appropriate for his needs, the question naturally arises: "Which
treatment combinations should be included in the experiment?" The
answer to this question depends, of course, on what assumptions the
experimenter is willing to make or, to phrase it differently, on what
information he is willing to forego. As we all know, you can seldom
get something for nothing, and the desired smaller experiment with
its associated savings can only be achieved at a cost, namely, the
cost of giving up part of the information usually derived from complete
factorials.
To illustrate the nature of a fractional factorial, let us consider a
one-half replicate of a 2 6 factorial. If we have 32 experimental units and
subject them to the treatment combinations shown in Table 13.10, the
principles introduced in Section 10.14 may be invoked to show' the
equivalences (i.e., confoundings) of effects listed in Table 13.11. If the
experimenter is willing to assume that all interaction effects involving
three or more factors are 0, this fractional factorial is adequate to
TABLE 13.10-Treatment Combinations To Be Used in a One-Half Replicate
of a 2 6 Factorial in Which the Defining Relation is I = ABCDEF*
Experimental
Unit
Treatment
Combination
Experimental
Unit
Treatment
Combination
1
(1)
17
/7/7
2
de
18
CLUr
3
ef
19
&&
nJ*>-F
?4
yj
df
20
(LQr&J
ft-f
15
ab
21
aj
-L.J
*6
abde
22
ud
Z,-
|7
dbef
23
o&
"kjj*-f
8
abdf
24
uaej
t/
19
ac
25
PJ
ffj
10
acde
26
CQf
11
acef
27
ce
ffj^-f
12
acdf
28
CCL6J
f-f
13
\J,V*MJ
be
29
C J
siT-.^J
14
bcde
30
duCCL
15
beef
31
cibcc
16
bcdf
32
dbcdef
sif^^f
^^/
Q>OCj
- The use of the symbol I rather than M (as in Chapter 10) is to agree with convention
The equality sign is used as an abbreviation for "is completely confounded with."
TABLE 13.11-Confounded Effects in a One-Half
Replicate of a 2 6 Factorial in Which the
Defining Relation is I^ABCDEP
I=ABCDEF
A=BCDEF
B = ACDEF
AB^CDEF
C=ABDEF
AC=BDEF
BC=ADEF
ABC=DEF
> = ACEF
ABD = CEF
CD=ABEF
ACD = BEF
ABCD=EF
E^ABCDF
BE=ACDF
A BE = CDF
CE=ABDF
ACE=BDF
BCE=ADF
ABCE^DF
DE=ABCF
BDE^ACF
ABDE=CF
CDE=ABF
ACDE=BF
BCDE=AF
ABCDE=*F
13.4 UNEQUAL BUT PROPORTIONATE SUBCLASS NUMBERS
421
TABLE 13.12-Abbreviated A1SFOVA for the Experiment
of Table 13.10
Source of Variation
Mean
A
B
C
D
R
F
AB
AC
AD
AR
AF
BC
BD
BE.
BF
CD
CR
CF
DR
DF
RF
Experimental error
(higher order interactions) .
Total
Degrees of
Freedom
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10
32
estimate all main effects, all two-factor interactions, and experimental
error. Under such an assumption, the appropriate ANOVA is as given
in Table 13.12.
Fractional factorials have wide application in industrial experimen
tation. Thus, it will pay research workers in both engineering and the
physical sciences to become better acquainted with these valuable aids
to efficient experimentation. As with other topics mentioned in this
section, it is felt that a detailed discussion is beyond the scope of this
book. For this reason, the interested reader is referred to the publica
tions listed at the end of the chapter.
13.4 UNEQUAL BUT PROPORTIONATE SUBCLASS
NUMBERS
The reader will have noticed that practically all the recommended
statistical designs require a balanced configuration, that is, an equal
number of observations in each group. The one exception was the
completely randomized design. However, even in that case, it was
422
CHAPTER 13, OTHER DESIGNS
noticed that unequal numbers of observations in the subgroups could
lead to difficulties in interpretation. (See Section 11.4.)
In this section we propose to examine one other case of unequal fre
quencies which presents little difficulty in the way of calculation. This
case involves a factorial set of treatment combinations in which the
cells of, say, the aX& table contain different numbers of observations
but these numbers happen to be proportional. That is, the number of ob
servations in the (ij)th cell are such that n^^u^Vj where u\\ u* * - :u a
are the proportions in the rows and #1:^2- - :v& are the proportions
in the columns. Rather than go into details, a numerical example will
be given and it is hoped that this will be sufficient to illustrate the
ideas involved. Persons desiring further details should consult the
references at the end of the chapter.
Example 13.7
Suppose we have 3 varieties of oats to be tested for yield differences and
that we also wish to investigate the effects of 3 fertilizers. There are 28
experimental plots available to the researcher. Further, we will assume
that from previous experiments we already know considerably more
about varieties B and C than about variety A; thus, we shall plant
variety A on twice as many plots as varieties B and C. It is also con
sidered desirable to assign the 3 fertilizers to the plots in the ratio
3:2:2; that is, we shall apply fertilizer No. 1 to 12 plots and each of
fertilizers No. 2 and No. 3 to 8 plots. The assignment of the treat
ment combinations to the plots was made completely at random,
and the resulting yields, in bushels per acre, are recorded in Table 13.13.
Calculating the various sums of squares in the usual manner, we arrive
at the abbreviated ANOVA of Table 13.14.
TABLE 13. 13- Yields of 3 Varieties of Oats Subjected to 3 Different
Fertilizer Treatments
(In bushels per acre)
c\<* +
Fertilizer
vjat
Variety
1
2
3
A
50, 51, 52, 56, 60 55
42, 40 38 38
55 56 56
58
B
65, 69, 67
50, 50
62, 62
C
67, 67, 69
48 50
65, 67
TABLE 13. 14- Abbreviated ANOVA for Data of Table 13.13
Source of Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
Treatments
Varieties
2
818.9
409 45
Fertilizers .
2
1455
727 50
Varieties X fertilizers
4
52.3
13 075
Among plots treated alike
19
100 5
5 289
13.5 UNEQUAL AND DISPROPORTIONATE SUBCLASS NUMBERS 423
13.5 UNEQUAL AND DISPROPORTIONATE SUBCLASS
NUMBERS
Let us now examine the case where our data may be represented by
tlie model
Y*Sh = /* + < + fo + ()* + * i = 1, - , a (13 . 13)
J = 1, ' , 6
where the various terras are defined as before but the n^ are not equal
for the various cells of the aX& table. Further, the n^ are not propor
tionate as they were in Section 13.4. What difficulties in analysis result
from this fact? Why is it that we refer to the case of "unequal and dis
proportionate subclass numbers" as an undesirable situation? The
answer is, of course, because we encounter complications in analyzing
such data. Let us now take note of some of the problems that arise.
Suppose that we went ahead, ignoring the fact that the subclass
numbers are disproportionate, and calculated the various sums of
squares in the usual fashion. If this procedure were followed, we
would find that the sums of squares so calculated (assuming that each
sum of squares was calculated directly; that is, no sum of squares was
obtained by subtraction) would not sum up to agree with the total
sum of squares. In other words, because of the disproportionality of the
subclass numbers, the different comparisons with which the sums of
squares are associated are -nonorthogonal. This, of course, would lead to
biased test procedures unless some adjustment were made. The other
major difficulty which arises when dealing with cases involving dis
proportionate subclass numbers is that the simple (unweighted) treat
ment means obtained from the data are biased estimates of the true
treatment effects. This could lead to serious errors if inferences were
made without attempting to correct for the above-mentioned bias.
What then, should be the method of analysis for such situations?
The usual approach is to utilize regression techniques and obtain a
general least squares solution. However, because of the many varia
tions which may be employed (e.g., different models and/or different
orders of estimating the unknown parameters), neither detailed ex
planations nor numerical illustrations of such solutions will be included
in this book. If you should encounter a situation in which a general
least squares solution is required, I would suggest that you do three
things: (1) review the contents of Chapter 8; (2) study the appropriate
sections of some of the references at the end of this chapter; and (3)
consult a professional statistician.
424 CHAPTER 13, OTHER DESIGNS
13.6 RESPONSE SURFACE TECHNIQUES
One of tlie most significant contributions to statistical methodology
in recent years has been the development of systematic procedures
for determining, experimentally, those levels of the factors under inves
tigation which produce an optimum response. These procedures, fre
quently referred to as response surface techniques, can be of value to
researchers in almost every field of specialization. Unfortunately, a
satisfactory description of the many ramifications of these techniques
is more than can be accomplished in this text. Therefore, we shall be
content with a few general observations on the topic and then refer the
reader to other sources where these ideas are discussed in greater detail.
Response surface techniques are, in essence, a blending of regression
analysis (Chapters) and experimental design (Chapters 10, 11, 12, and
13) to provide an economical means of locating a set of experimental
conditions (i.e., a combination of factor levels) which will yield a
maximum (or minimum) response. However, one very important fea
ture has been added. That feature is the sequential nature of the explora
tion of the response surface. While it is true that most research is of the
continuing variety (and therefore sequential), the majority of the
techniques discussed heretofore in this book have been of the nonse
quential type. Thus, the insertion of the sequential element into the
pattern of the investigation is, from one point of view, a long overdue
step.
In capsule form, the steps involved in the application of response
surface techniques are as follows :
(1) Choose base levels of the factors to be investigated. (Depend
ing on the judgment of the experimenter, these levels may be
close to or far removed from the optimum levels.)
(2) Since, at this stage, linear effects are thought to be dominant
over nonlinear effects, select one other level of each factor.
(3) Utilizing either a complete or fractional 2 n factorial, estimate
(by examining the effects; i.e., the linear regression coefficients)
the direction in which the greatest gain may be expected.
(4) Moving in this direction, that is, along the path of steepest
ascent, to the extent that the experimenter deems reasonable,
a second experiment (again utilizing a complete or fractional
2 n factorial) is performed.
(5) Repeat steps (3) and (4) until a near-stationary region is
found.
(6) Then, utilizing a complete or fractional 3 n factorial or a com
posite design 1 to estimate the second order effects, the nature
of the response surface may be explored in the near-stationary
region and the optimum conditions located.
It will be realized that the preceding steps are only an indication of
1 A composite design is essentially a complete 2 n factorial with sufficient points
added to permit estimation of the second order effects.
13.8 OTHER DESIGNS AND TECHNIQUES 425
the procedure. Depending on the problem and the assumptions that
the experimenter is willing to make, the "rules" may be modified. (For
example, 3 n factorials or composite designs might be used in step No. 3.)
However, regardless of the details, the philosophy of sequential experi
mentation has much to recommend it. In fact, the concept of exploring
a response surface in a sequential manner with the objective of locating
a maximum (or minimum) point on the surface is one with which all
experimenters should be familiar. For those who are interested in
pursuing this topic further, an excellent exposition is available in
Davies (28). Both the theory and the application of response surface
techniques are also discussed in a number of the other references listed
at the end of this chapter.
13.7 RANDOM BALANCE
Another recent contribution to experimental design is the concept of
random balance investigations in multi-multi-factor experiments.
As proposed by Satterthwaite (43), the random balance technique
permits the researcher to screen a large number of possible contributing
factors in an experiment involving a limited number of test runs. That
is, random balance is a device for considering (simultaneously) the many
factors involved and, as is always important, keeping the size of the
experiment within reasonable bounds. When the experiment has been
performed, examination of the results should permit isolation of the
more important factors for further investigation.
In a random balance experiment, all factors and levels are consid
ered by choosing at random the level of each factor to be used in forming
a particular treatment combination. (NOTE: From a practical point of
view, the following restriction on complete randomization has been
found desirable: Each level of a particular factor should be used an
equal, or nearly equal, number of times.) Since random balance experi
mentation was first proposed, there has been much discussion, both pro
and con, as to its worth. Personally, I believe that random balance has
much to recommend it and that we will see a rapid increase in its use,
especially in industrial experimentation. However, the theory on which
it is based has not been fully explored, and thus the controversy over
its merits continues. For those interested in the possibilities and/or
wisdom of using random balance in their own experimentation, I sug
gest a careful reading of the appropriate references at the end of this
chapter.
13.8 OTHER DESIGNS AND TECHNIQUES
As was stated in the opening paragraph of this chapter, the number
of designs and analysis techniques that have been developed for special
purposes are many. Thus, it has been possible to mention only a few
in this book. The two most common designs, the completely random
ized design and the randomized complete block design, were discussed
in detail in Chapters 11 and 12, respectively. In this chapter a few of
426
CHAPTER 13, OTHER DESIGNS
the more specialized designs and techniques have been described or
alluded to. An examination of some of the references which follow will
bring many other special designs to your attention. It is my hope that
the presentation thus far, brief through it has sometimes been, will
have whetted your appetite and that you will continue your readings
and studies in the related areas of experimental design and research
techniques.
Problems
13.1 A 5X5 Latin square was laid out to test the effects of 5 fertilizers on
the yield of potatoes. Perform a complete analysis of the data.
Column
"O i-v-vjr-r
Row
1
2
3
4
5
Totals
1
A 449
B 444
C 401
Z>299
292
1885
2..
B 463
C 375
Z>323
264
A 415
1840
3
C 393
Z?353
278
A 404
^425
1853
4
D371
E 241
A 441
B 410
C 392
1855
5 .
258
A 430
B 450
C 385
jD347
1870
Column
totals
1934
1843
1893
1762
1871
9303
Treatment totals
A: 2139
B: 2192
C: 1946
D: 1693
E: 1333
13.2 Shown below are the yields (cwt. per 1/40-acre plots) of sugar cane
in a Latin square experiment comparing fertilizers.
A 14
B 19
D23
C 21
23
E22
D21
A 15
#46
C 16
20
A 16
C20
24
C 18
23
B 18
>21
A 17
D25
C 18
23
A 18
B 19
A : No fertilizer
B : Complete inorganic fertilizer
C: 10 tons manure per acre
>: 20 tons manure per acre
: 30 tons manure per acre
What conclusions do you draw from this experiment?
13.3 Analyze the following data from a cacao experiment consisting of 3
separately randomized Latin squares. The 3 treatments were:
A : No fertilizer (check)
B: 1.5 Ibs. superphosphate per tree
C: 3 Ibs. superphosphate per tree
PROBLEMS
427
The field plans of the squares, together with plot yields in average
pods per tree, are as follows:
B
C
A
41
25
15
A
B
C
20
32
24
C
^
^
22
12
21
C
B
.4
27
28
3
^
C
B
4
17
9
B
A
C
22
4
17
A
C
B
11
15
17
B
A
C
24
14
33
C
B
A
22
20
15
Note: Do not consider a transformation because these are averages
(rounded) from the trees on 1/15-acre plots. The total numbers of
pods were large enough to approximate a continuous distribution.
13.4 Five levels of a fertilizer were tried in a 5X5 Latin square. This is
the analysis:
H> agrees of Mean
Freedom Square
Rows 4 25
Columns 4 20
Treatments 4 28
Error 12 15
The sums of the yields in the 5 plots of each level were:
Level 12345
Sum of yields 2 14 26 3O 28
Subdivide the 4 degrees of freedom for treatments into
d.f.
Linear regression 1
Second degree term 1
Remainder 2
Is any comparison significant?
428 CHAPTER 13 r OTHER DESIGNS
13.5 Crop wheat; Location R. W. Gt. Harpenden (175); Year 1935;
Type 6X6 Latin square; Comment yield in pounds of grain per
1/40-acre plot
Total
4
2
1
3
5
77.2
88.0
89.7
92.6
72.1
76.2
495.8
3
4
5
1
2
93.2
95.8
94.1
93.9
91.6
67.3
535.9
5
2
3
4
1
90.2
87.0
86.1
85.5
93.4
68.5
510.7
2
3
1
5
4
72.5
76.7
96.3
95.3
95.9
78.2
514.9
1
5
2
4
3
84.2
96.5
98.5
81.6
90.1
81.8
532.7
1
5
4
3
2
77.0
91.9
95.1
86.3
82.8
60.5
493.6
Total 494.3
535.9
559.8
535.2
525.9
432.5
3083.6
Treatment
Treatments Total
No (NH02SO 4 515 .5
1 (NH4) 2 SO 4 applied Oct. 26 at 0.4 cwt. of N/A 522.5
2 (NH4) 2 SO 4 applied Jan. 19 at 0.4 cwt. of N/A 480.9
3 (NHLOaSCX applied Mar. 18 at 0.4 cwt. of N/A 496.2
4 (NELOaSCU applied Apr. 27 at 0.4 cwt. of N/A 521 .9
5 (NHOaSO* applied May 24 at 0.4 cwt. of N/A 546.6
Analyze and interpret the above data.
13.6 We wish to conduct a field experiment to test the yielding ability of
6 varieties of soybeans and have available an area of land sufficient
for 36 plots. Indicate the proper subdivision of the total degrees of
freedom for the following experimental designs:
(a) completely randomized
(6) randomized complete block
(c) Latin square.
Indicate, by means of arrows, the proper ^P-tests for testing variety
differences in each design.
13.7 Given that the data shown below resulted from an experiment such
as described in Example 13.4, perform the analysis and give your
interpretations of the results.
PROBLEMS
CRUSHING STRENGTHS (CODED VALUES)*
429
Columns (Days)
Rows
(Batches)
1
2
3
4
5
1
257
230
279
287
202
2
245
283
245
280
260
3
182
252
280
246
250
4
203
204
227
193
259
5
231
271
266
334
338
* The treatments are assumed to have been Imposed exactly as shown In Example 13.4.
13.8 An experiment was conducted to assess the relative resistances to
abrasion of four grades of leather (A, B, C, >). A machine was used
in which the samples could be tested in any one of four positions.
Since different runs (replications) are known to yield variable results,
it was decided to make four runs. A Latin square design was utilized
and the following results obtained. Analyze and interpret the data.
Position
Run
1
2
3
4
1
2
3
4
118(5)
127 (D)
174(4)
130(C)
136(Z>)
141(3)
173(C)
170(4)
168(4)
129(C)
126CB)
125(Z>)
135(C)
151(4)
134(1?)
95(5)
13.9 Another experiment such as described in Problem 13.8 was con
ducted at a second laboratory. In this case, the data shown below
were obtained. Analyze and interpret. (NOTE: M represents a
missing observation.)
Run
4
2
1
3
2
3
1
4
-4(150)
2?(130)
(98)
Position
(145)
C(172)
Z?(132)
-4(171)
-4(170)
(115)
C(132)
C(133)
(127)
.4(170)
Z?(120)
13 10 The experiment described in Problem 13.8 was conducted once more,
this time at a third laboratory. Analyze and interpret the data which
follow. [HINT: Use Equation (13.8) and the iterative technique dis
cussed in Section 12.13.]
430
CHAPTER 13, OTHER
Run
Position
1
2
3
4
1
2
3
4
C(131)
>(139)
B(157)
,4(185)
.D(Af')
-4(196)
C(133)
(146)
.4(167)
(140)
Z>(140)
C(M")
5(136)
C(148)
,4(184)
D(150)
13.11 On checking the original data sheets, it was discovered that the
technician took two independent abrasion readings on the samples
tested in the experiment described in Problem 13.8. The second set
of readings is reproduced below. Pooling these data with those given
in Problem 13.8, analyze and interpret the complete results.
Run
Position
1
2
3
4
1
2
3
4
120(5)
125(Z>)
175C4)
132(C)
130(Z>)
142(5)
180(C)
17004)
165 (-4)
120(C)
120(5)
130(1?)
140(C)
14004)
140(1))
102(5)
13.12 An experiment was performed to compare the effects of three
catalysts on the yield of a chemical process. Three runs were started,
one using catalyst A, another using B, and the third C. After 3 days,
a sample was drawn from each run and an analysis performed. A
similar operation (i.e., taking samples and performing the analyses)
was performed after 5 days. The whole experiment was repeated four
times. Analyze and interpret the resulting data.
CODED YIELDS OF AN UNSPECIFIED CHEMICAL PROCESS
Catalyst
A
B
C
Replicate
3 days
5 days
3 days
5 days
3 days
5 days
1
68
82
90
96
82
88
2
83
79
68
80
71
78
3
66
75
70
91
68
78
4
66
76
84
92
74
80
13.13 A split-split plot design was used in an experiment concerned with the
yield of cotton. Four replications (or blocks) were involved. Each
main plot was subjected to one of two levels of irrigation, each sub
plot was subjected to one of three rates of planting, and each sub-
PROBLEMS
431
subplot was subjected to one of three levels of fertilizer application.
Analyze and interpret the following experimental yields.
CODED YIELDS FROM COTTON GROWING EXPERIMENT
Rate of
Block
Planting
(Density of
T'VrtiliT'^T
Irrigation
Plants)
JL t*.L LiiljoCJ.
Rate
1
2
3
4
Light
Thin
ISTone
9.0
8.2
8.5
8.2
Average
9.5
8.1
8.8
7.9
Heavy
10.6
9.4
8.8
8.6
Medium
None
9.0
9.7
11.1
7.8
Average
8.9
9.7
10.3
8.5
Heavy
9.3
10.4
9.1
8.6
Thick
None
8.1
7.4
8.2
8.5
Average
9.0
8.1
7.6
8.8
Heavy
9.6
7.5
9.4
8.4
Heavy
Thin
None
8.1
10.3
6.0
7.2
Average
8.6
10.8
10.4
11.6
Heavy
10.2
10.4
11.5
11.6
Medium
None
12.2
9.8
9.1
11.0
Average
11.0
9.5
11.7
13.2
Heavy
12.0
12.4
11.6
^3.0
Thick
None
7.9
13.4
12.0
11.7
Average
10.0
14.2
12.2
13.8
Heavy
12.5
14.0
13.8
13.4
13.14 The following data resulted from an unreplicated complete factorial.
Analyze and interpret. State all your assumptions.
CODED YIELDS OP A CHEMICAL PROCESS
Concentration of Solvent
JL dUJJCrfcLLUiC
Low
Medium
High
100
44
46
42
200
51
55
55
300
50
50
48
13.15
In a manufacturing company, the micrometers used in checking
quality are themselves checked by use of gauge blocks. However,
there are 5 departments and each has its own micrometers and
432
CHAPTER 13, OTHER DESIGNS
gauge blocks. Because of a suspicion that there is too much variation
among micrometers and/or gauge blocks, the quality control engi
neer ran a test utilizing a random sample of instruments. Analyze
and interpret the following data.
Gauge
Block
Micrometer
1
2
3
4
5
A
0.0110
0.0115
0.0130
0.0151
0.0121
B
0.0135
0.0127
0.0132
0.0155
0.0128
C
0.0127
0.0124
0.0132
0.0152
0.0130
13.16 An experiment was run to investigate the effect of temperature, type
of powder, amount of powder, and packing pressure on the function
time of an explosive actuator. An unreplicated complete factorial
yielded the following data. Analyze and interpret.
FUNCTION TIME (IN MILLISECONDS)
Type of Powder
A
B
C
Amount of Powder (Mg.)
5 10 15
5 10 15
5 10 15
Temperature
(F.)
Packing
Pressure
(psi)
-50
10,000
15,000
20,000
7.4 7.0 6.8
7.5 7.2 6.7
7.4 7.4 6.0
5.4 5.0 4.8
5.5 5.2 4.7
5.4 5.4 4.0
7.2 6.9 6.6
7.2 6.6 6.5
7.2 6.7 6.2
75
10,000
15,000
20,000
6.6 6.6 5.8
6.8 6.6 6.6
6.8 6.2 5.9
4.6 4.6 3.8
4.8 4.6 4.6
4.8 4.2 3.9
6.8 7.2 4.9
6.9 7.0 5.0
7.0 7.1 5.0
200
10,000
15,000
20,000
5.1 5.1 5.1
5.1 4.8 4.9
5.2 4.7 5.0
3.1 3.1 3.1
3.1 2.8 2.9
3.2 2.7 3.0
6.0 4.9 4.8
6.4 4.8 4.1
5.9 4.9 2.0
13.17 A complete but unreplicated factorial was used to investigate the
effects of type of metal (a qualitative factor), amount of primary
initiator (a quantitative factor), and packing pressure (a quanti
tative factor) on the firing time of explosive switches. Analyze and
interpret the following data:
PROBLEMS
433
FIRING TIMES (IN MILLISECONDS)
Metal
Primary
Initiator
(Mg.)
Packing Pressure (psi)
12,000
20,000
28,000
2 s al
5
12.3
10.6
15.2
10
10.4
9.5
15.0
15
8.8
9.1
14.5
teflon
5
12,4
11.7
15.0
10
11.0
11.0
14,6
15
11.0
9.8
14.6
13.18 A certain type of capacitor was to be tested to assess its perform
ance as a function of a number of specified factors. The four factors
considered were:
(a) ^potted (+) or not potted ( )
(6)== wedged (+) or not wedged ()
(c) impregnated (+) or not impregnated ( )
(d) = high temperature (+) or low temperature ( ).
The performance characteristic measured was the high voltage
breakdown when the capacitors were subjected to a voltage rise of
250v/sec. Some hypothetical data which could have resulted from
such an experiment are:
Capacitor
Level of Factor
High Voltage
Breakdown (kv)
abed
1
__
10.7
2
+ +
11.4
3
+ +
12.2
4
-j- 4_ _
13.0
5
+ +
10.6
6
-}- .4-
12.1
7
4- +
12.0
8
+ + + +
13.2
Analyze and interpret the one-half replicate of a 2* factorial described
above.
13.19 An experiment was to be performed to assess the effects of the
following factors on the surge voltage of a specific model of thermal
battery: temperature, humidity, amount of electrolyte, amount of
heat paper, and type of electrolyte. These five factors were denoted
as a, b y c, d, and e, respectively. Since this was only a preliminary
experiment (in the development phase) and since all three-, four-,
and five-factor interactions could be assumed to be negligible, a
434
CHAPTER 13, OTHER DESIGNS
one-half replicate of the 2 5 factorial was performed. Analyze and
interpret the following data.
Treatment Combination
Surge Voltage (volts)
(i)
14.0
ae
14.6
be
11.7
ab
16.3
ce
11.2
ac
16.6
be
15.6
abce
10.2
de
13.9
ad
13.8
bd
15.1
abde
13.2
cd
14.6
acde
14.3
bcde
12.6
abed
15.4
References and Further Reading
1. Anderson, H. E. Random permutations of selected powers of 2 and 3.
Sandia Corporation Monograph SCR-4%8, Sandia Corp., Albuquerque, N.
Mex., Aug., 1961.
2. Anderson, R. L. Complete factorials, fractional factorials, and confounding.
Experimental Designs in Industry. (Editor: V. Chew). John Wiley and
Sous, Inc., New York, 1958.
3 - ^ and Bancroft, T. A. Statistical Theory in Research. McGraw-Hill
Book Company, Inc., New York, 1952.
4. Anscombe, F. J. Quick analysis methods for random balance experimenta
tion. Technometrics, 1 (No. 2):195-209, May, 1959.
5. Bicking, C. A, Experiences and needs for design in ordnance experimenta
tion. Experimental Designs in Industry. (Editor: V. Chew). John Wiley and
Sons, Inc., New York, 1958.
6. Box, G. E, P. Multi-factor designs of first order. Biometrika, 39:49, 1952.
7. - . The exploration and exploitation of response surfaces; some general
considerations and examples. Biometrics, 10:16, 1954.
8. - , and Coutie, G. A. Application of digital computers in the explora
tion of functional relationships. Proceedings of the Institution of Electrical
Engineers, Vol. 103, Part B, SuppL No. 1, 1956.
, and Hunter, J. S. A confidence region for the solution of a set of
simultaneous equations with an application to experimental design.
metrika, 41:190, 1954.
10. - , and - . Multifactor designs. Report prepared under Office of
11.
12.
Ordnance Contract No. DA-36-034-ORD-1177 (RD), 1954.
and
Ann. Math. Stat., 27, 1956.
-. Multifactor designs for exploring response surfaces.
and
-. Experimental designs for exploring response surfaces.
Experimental Designs in Industry. (Editor: V. Chew). John Wiley and Sons,
Inc., New York, 1958.
REFERENCES AND FURTHER READING 435
13. , and Wilson, K. B. On the experimental attainment of optimum
conditions. Jour. Roy. Stat. Soc., Series B, 13:1, 1951.
14. , and Youle, P. V. The exploration and exploitation of response sur
faces; an example of the link between the fitted surface and the basic mecha
nism of the system. Biometrics, 11:287, 1955.
15. Brownlee, K. A. Statistical Theory and Methodology in Science and Engi
neering. John Wiley and Sons, Inc., New York, I960.
16. Budne, T. A. Random balance: Part I The missing statistical link in fact
finding techniques, Part II The techniques of analysis, Part III Case
histories. Industrial Quality Control, 15 (No's. 10-1 1-12) :5~10, 11-16, 16-
19, April, May, and June 1959.
37. . The application of random balance designs, Technometrics, 1
(No. 2):139-55, May, 1959.
18. Carroll, M. B., and Dykstra, O., Jr. Application of fractional factorials in a
food research laboratory. Experimental Designs in Industry. (Editor: V.
Chew). John Wiley and Sons, Inc., New York, 1958.
19. Chapin, F. S. Experimental Designs in Sociological Research. Harper and
Brothers Publishers, New York, 1947.
20. Chew, V. (editor) Experimental Designs in Industry* John Wiley and Sons,
Inc., New York, 1958.
21. . Basic experimental designs. Experimental Designs in Industry.
(Editor: V. Chew). John Wiley and Sons, Inc., New York, 1958.
22. Cochran, W. G., and Cox, G. M. Experimental Designs. Second Ed. John
Wiley and Sons, Inc., New York, 1957.
23. Connor, W. S. Experiences with incomplete block designs. Experimental
Designs in Industry. (Editor: V. Chew). John Wiley and Sons, Inc., New
York, 1958.
24. 9 and Young, S. Fractional Factorial Designs for Experiments With
Factors at Two and Three Levels. National Bureau of Standards Applied
Mathematics Series 58, U.S. Govt. Print. Of!., Washington, IXC., Sept. 1,
1961.
25. , and Zelen, M. Fractional Factorial Experiment Designs for Factors
at Three Levels. National Bureau of Standards Applied Mathematics Series
54, U.S. Govt. Print. Off., Washington, D.C., May 1, 1959.
26. Cox, D. R. Planning of Experiments. John Wiley and Sons, Inc., New York,
1958.
27. Daniel, C. Fractional replication in industrial research. Proceedings of the
Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 5
(Editor: J. Neyman). University of California Press, Berkeley, Calif., 1956.
28. Davies, O. L. (editor) The Design and Analysis of Industrial Experiments.
Second Ed. Oliver and Boyd, Edinburgh, 1956.
29. Davies, O. L., and Hay, W. A. The construction and uses of fractional
factorial designs in industrial research. Biometrics, 6:23349, 1950.
30. DeBaun, R. M., and Schneider, A. M. Experiences with response surface
designs. Experimental Designs in Industry. (Editor: V, Chew). John Wiley
and Sons, Inc., New York, 1958.
31. Federer, W. T. Experimental Design. Macmillan Co., New York, 1955.
32. Finney, D. J. The fractional replication of factorial experiments. Ann.
Eugenics, 12:291-301, 1945.
33. Fisher, R. A. Statistical Methods for Research Workers. Tenth Ed. Oliver and
Boyd, Edinburgh, 1946.
34. . The Design of Experiments. Fourth Ed. Oliver and Boyd, Edinburgh,
1947.
35 f an< i Yates, F. Statistical Tables for Biological, Agricultural and Me&i-
cal Research. Third Ed. Oliver and Boyd, Edinburgh, 1948.
36. Horton, W. H. Experiences with fractional factorials. Experimental Designs
436 CHAPTER 13, OTHER DESIGNS
in Industry. (Editor: V. Chew). John Wiley and Sons, Inc., New York,
1958.
37. Hunter, J. S. Determination of optimum operating conditions by experi
mental methods, Part II-1-2-3, models and methods. Industrial Quality
Control, 15 (No's. 6-7-8): 16-24, 7-15, and 6-14, Dec., 1958, Jan., and
Feb., 1959.
38. Kempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
39. National Bureau of Standards. Fractional Factorial Experiment Designs for
Factors at Two Levels. National Bureau of Standards Applied Mathematics
Series 48, U. S. Govt. Print. Off., Washington, D.C., April 15, 1957.
40. Plackett, R. L., and Burman, J. P. The design of optimum multi-factor experi
ments. Biometrika, 33:305-25, 1946.
41. Quenouille, M. H. The Design and Analysis of Experiment. Charles Griffin
and Co., Ltd., London, 1953.
42. Rao, C. R. General methods of analysis for incomplete block designs. Jour.
Amer. Stat. Assn., 42:541, 1947.
43. Satterthwaite, F. E. Random balance experimentation. Technometrics,
I (No. 2):lll-37; May, 1959.
44. Snedecor, G. W. Statistical Methods. Fifth Ed. The Iowa State University
Press, Ames, 1956.
45. Youden, W. J., Kempthorne, O., Tukey, J. W., Box, G. E. P., and Hunter,
J. S. Discussion of the papers of Messrs. Satterthwaite and Budne (including
authors 7 responses to discussion). Technometrics, 1 (No. 2):157 93, May,
1959.
CH APTE R 14
ANALYSIS OF COVARIANCE
IN PRECEDING CHAPTERS great emphasis has been placed on two very
important techniques, namely, regression analysis and analysis of vari
ance. Further, in certain Sections (11.11, 12.11, and 13.6) these two
techniques were combined to handle particular problems associated
with the exploration of response curves or surfaces. In the present
chapter we shall investigate another blending of these two fundamental
tools. This new technique, known as analysis of covariance, is one
which has proved very useful in many areas of research.
14.1 USES OF COVARIANCE ANALYSIS
Before discussing the actual methods of covariance analysis, let us
give one or two examples of situations in which the technique may be
profitably employed. These examples should, of course, indicate to the
reader the nature of the combination of the ideas of regression and
analysis of variance. For the first example, consider a case where the
researcher is interested in the effects of various rations on the weights
of hogs. If a randomized complete block design is utilized and the final
weights, F, of the animals after a specified number of days of feeding
are analyzed, the differences among the effects of the various rations
may or may not be significant. In either case, however, the good re
searcher will think more about the conduct of the experiment before
drawing any conclusions from the analysis of variance implied in the
preceding sentence. He might say to himself, "If the experimental ani
mals varied greatly with respect to their initial weights at the time the
experiment was started, how do we know that differences among final
weights reflect ration effects rather than just varying initial weights?
Calling the initial weights X, he might adjust the F-values according
to the associated X-values and then analyze and interpret the experi
mental data. The method by which this is carried out is known as Co-
variance analysis.
Another example is the following: When dealing with an experiment
to compare several methods of teaching statistics in which the criterion
is to be the final score, Y, obtained by the students, all of whom take
the same examination, final judgment concerning the various methods
of teaching should not be rendered until the I.Q. ratings, X, of the in
dividual students have been examined and the necessary corrections
(adjustments) made. Many other examples could be given, and the
reader is asked to formulate some in his own field of interest as an aid
in better appreciating the techniques to be presented.
4373
438 CHAPTER 14, ANALYSIS OF COVAR1ANCE
14.2 ASSUMPTIONS UNDERLYING ANALYSES OF
COVARIANCE
As would be expected, the assumptions one makes "when performing
a covariance analysis are similar to those required for linear regression
and analysis of variance. Thus we find the usual assumptions of inde
pendence, normality, homogeneous variances, fixed X's, etc. To be
more specific, we give the mathematical models associated 'with some
of the more common designs when a covariance analysis is contem
plated.
Completely randomized design
YH - f Hr Tt + pXv + <,-; i = 1, - - , k (14.1)
j = 1, - - - , n
Randomized complete block design
+ *; i = 1, ,r (14.2)
Latin square design
Y ijk = + Pi + yj + r k + 0Xvm + e/ ( *o; i = 1, - - - , m (14.3)
j = 1, - - - , m
k = 1, , m
Two-factor factorial in a randomized complete block design
Y-iik = + Pi + <*j + Vk + (&v}jk +P Xij k + eij k ', i = 1 ? - , r (14. 4)
j = 1, - , a
k = 1, - , c.
In practice it is more customary to express these equations in terms
of deviations of the X variable from its mean. When this is done, the
equations appear as
YV = M + T* + 0(Xt,- - X) + , y ; i = 1, - - - , k (14.5)
j = i, . . . , n
- M + Pi + TJ + /3(Jr<y - X} + , y ; i = 1, - - - , r (14.6)
F,-yJb = M + Pi + Vj + T k
+ /3(-y<yc*) - 3) + </); i = 1, , m (14,7)
y = i, - - - , m
k = 1, , ra
and
14.3 COMPLETELY RANDOMIZED DESIGN 439
y^jk = M + pi + dj H- y k + (coOy*
+ /3(Xv* - X) + </*; i = 1, - - - , r (14.8)
y = i, , a
z> = 1 ... r
** * ? ?
respectively, where /j,=*g-\-p'X, and 3 s is the arithmetic mean of all the
X's. The reason for writing the equations in this last form is that, by
so doing, we simplify the algebra of the mathematical solution. Conse
quently., it is in this form that you will find the model presented in
most texts. The usual assumptions are made concerning the various
terms in each of the preceding equations, and the reader is advised to
read again the appropriate sections in Chapters 8, 11, 12, and 13 to
refresh his memory on such matters.
Although not mentioned above, there is another item which is com
monly considered as a necessary assumption for a valid analysis of
covariance, namely, that the concomitant variable, X, should not
be affected by the treatments. That is, the treatments which have been
applied to the experimental units so that we may observe and judge
their effects on the Y variable should not influence the observed values
of -XT. However, this is too restrictive an assumption. Even though the
treatments do affect the JXT-values, a covariance analysis may be
profitably employed if proper care is exercised in the interpretation of
the experimental results. It is clear, then, that the inferences which
may be made are different in the two cases, depending upon whether
or not the X variable is affected by the treatments. The researcher is
therefore cautioned to be extremely careful when dealing with the inter
pretation of covariance analyses. Let us now consider some special
cases so that the reader will not only gain practice in the interpretation
of data amenable to an analysis of covariance but also become familiar
with the details of the computational procedure.
14.3 COMPLETELY RANDOMIZED DESICN
As our first example of a covariance analysis we shall make use of a
completely randomized design. Before giving a numerical example, let
us examine the problem in general form. Assuming that we have t
treatments, or groups, and that there are n^ observations on each of X
and Y in the ith group, it is customary to proceed as follows: Calculate
the following sums of squares and products and then complete Table
14.1 as indicated. (NOTE: There is a great similarity between Table
14.1 and Table 8.22.)
x 2 = corrected total sum of squares for X
/ * "^
\ ;=i y-i
y-y-^ ^= A ^ ' (14 * 9)
= 2^ 2^ x a
1=1 y==i
Pi
faO
p
i
o
a
o
a 1
g
o
ft
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OJ
<u
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PQ
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X- v
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4)
T
4J
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3
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3
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rt
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^fi
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r-H
b
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iT.P
1
1
4
*
H
Source of
Variation
Among treatments
Among experimen
tal units treated
alike (within
treatments)
Among treatments
+
Within treatments
( = total).
Difference for testir
* The symbols S x
other tables which
14.3 COMPLETELY RANDOMIZED DESIGN 441
xy = corrected total sum of products for X and Y
t ni \ / t m
2; 2:**) (2: 2:
i ,-i / \ .-i y-i
x r i ,-i / \ .-i y-i / (14. 10)
= 2^
2^ ni
y 2 = corrected total sum of squares for F
F ~. _ ( 14 - u >
t=-l
. = treatment sum of squares for
(ni \2 / * ni
T^ jc \ I T^ V x
Z-*t ^ 13 J \ Z-*i 2-*t ^ *}
-2. ^ ' - - 1 '7 1
i i
/ = treatment sum of products for X and Y
yy
2 t ni 2
(14.13)
T yy = treatment sum of squares for Y
\ 2
(14.14,
= experimental error sum of squares for X
2 T
-L xx
= experimental error sum of products for X and F
E yy = experimental error sum of squares for F
442 CHAPTER 14, ANALYSIS OF COVAR1ANCE
The proper F-ratio for testing the hypothesis that there are no dif
ferences among the true effects of the t treatments on the Y variable
after adjusting for the effect of the X variable is
F
A*
z;,.,- *-
i=l
with degrees of freedom v\ = t 1 and
It is customary, in addition to performing the jP-test just indicated, to
present a table of adjusted treatment means as an aid in the interpre
tation of the experimental results. The adjusted means may be found
using the formula
adj. Yt = Y; - b(Xt - X); i = 1, . - - , t, (14, 19)
where 6 is the regression coefficient calculated from the experimental
error sums of squares and products, that is, 'b = E axu /E aiX . The estimated
variance of an adjusted treatment mean is given by
V (adj. F,) = s^ E \- + ( ^~ T)2 1 (14 . 20)
Lni E xx J
and the standard error of an adjusted treatment mean by
f\
s y
(adj. F,) s y -- 1 -- ~ -- (14.21)
n* E xas
The estimated variance of the difference between two adjusted treat
ment means is, of course, given by
+ (Xi ~ ^T (14. 22)
V (adj. F, - adj. F y ) - 4 \- +
L^i n y
It should be clear that the regression coefficient, /?, in Equation (14.5)
has been assumed to be nonzero. If such were not the case, the intro
duction of the concomitant variable, X, into the calculations would be
an unnecessary complication. Sometimes the researcher will wish to
check on this assumption. That is, he will consider the hypothesis
H:& = Q, rather than the assumption =^0. When this is done, he will
be interested in testing the validity of H. The proper F-ratio is
S E
(14 . 23)
74.3 COMPLETELY RANDOMIZED DESIGN
443
which has degrees of freedom v- L = l. and
TABLE 14.2-Gains In Weight (F) and Initial Weights (X} of
Pigs in a Feeding Trial
Treatment
1
2
3
4
X
F
X
F
X
F
X
F
30
165
24
180
34
156
41
201
27
170
31
169
32
189
32
173
20
130
20
171
35
138
30
200
21
156
26
161
35
190
35
193
33
167
20
180
30
160
28
142
29
151
25
170
29
172
36
189
Total
160
939
146
1031
195
1005
202
1098
TABLE 14.3-Analysis of Covariance for Data in Table 14.2
Source of
Variation
Degrees
of
Freedom
Sum of Squares and Products
Deviations About Regression
2>
lL,xy
Z^ 2
^v >
Degrees
of
Freedom
Mean
Square
^ 2>
Among treat
ments
3
20
365.46
361.50
451.21
496.83
2163.13
5937.83
Among ani
mals treat
ed alike. . .
Total. .
5255.01
19
276.58
23
726.96
948.04
8100.96
6864 . 61
1609.60
22
3
536.53
Difference for testing among adjusted treatment
means
Example 14.1
Given the data of Table 14.2, the following calculations were made
and the results reported in Table 14.3.
]L> 2 = S xx = TX* + E xx = (30) 2 + . - - + (36) 2 - ^^ = 726.96
-*!,= (30) (165)
24
4- (36) (189) -
(703) (4073)
24
948.04
444
CHAPTER 14, ANALYSIS OF COVARIANCE
T yy + E vy = (165)*
h (189) 2 -
(4073) 2
24
8100.96
(160) 2 + (146) 2 + (195) 2 + (202) 2 (703) 2
365.46
* xy ===
T
-* vy
6 24
(160) (939) + (146) (1031) + (195) (1005) + (202) (1098)
__ (703) (4073)
""" 24
(939) 2 + (103 1) 2
= 451.21
(1005) 2
(1098) 2 (4073) 2
24
2163.13
= S** - T xx = 361.50
= S*v T xv = 496.83
^ *^yv * yy == 59o/.0v5.
Carrying out the F-test outlined in Equation (14.18), we obtain
F = 536. 53/276. 58 = 1.94 with degrees of freedom ?i = 3 and j> 2 = 19. This
is not significant at the 5 per cent level, and thus we are unable to reject
the hypothesis of no differences among the true effects of the 4 treat
ments on the gain in weight of pigs after adjusting for the varying
initial weights of the experimental animals. Incidentally, in this case
the same decision would have been reached had no adjustment been
made for the concomitant variable. However, in many instances the
conclusions may change considerably depending on whether or not the
covariance technique is used, and thus the researcher should always
see if it is applicable to the problem at hand. The adjusted treatment
means are presented in Table 14.4.
TABLE 14.4-Calculation of Adjusted Treatment Means From
Data of Table 14.2
(jr = 29.29, 7=169.71, 6 = 496.83/361.50=1.374)
Treatment
1
2
3
4
Xi
26.67
24.33
32.50
33.67
XiX
2.62
4.96
3.21
4.38
b(Xi T)
3.60
6.82
4.41
6.02
Yi
156.50
171.83
167.50
183 00
adj. ~Yi
160.10
178.65
163 09
176 98
Standard error of adj.
T f
7.17
8.06
7.35
7.80
14.4 RANDOMIZED COMPLETE BLOCK DESIGN
When our data conform to a randomized complete block design, the
appropriate mathematical model is as given in Equation (14.6). The
analysis to be performed is given in Table 14.5, the quantities R xx ,
T**, E x *, R VV) Tyy, and E yy being obtained as in any randomized com-
o
7
T-l
^
J
^ i
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eatment
+ error.
8
f^^^f^ **
k*
H
S
446 CHAPTER 14, ANALYSIS OF COVAR1ANCE
plete block design. The sums of products R xyj T xy , and E xy are found
using the following equations:
= corrected total sum of products for X and Y
(14. 24)
F<, F*
= replicate (block) sum of products for X and Y
F,A
^-^/ * *J I
(14.25)
t rt
,= treatment sum of products for X and Y
3 1 \ - , - ** /
(14.26)
(14.27)
The proper F-ratio for testing the hypothesis of no differences among
the true effects of the t treatments on the F variable after adjusting for
the effect of the X variable is
r rt
= experimental error sum of products for X and
(S T+E -
( }
As in the case of a completely randomized design, we will wish to
calculate the adjusted treatment means and, possibly, to test the hy
pothesis that of Equation (14.6) is 0. The calculation of the adjusted
treatment means is easily carried out using
adj. Y,, = F. y - b(X., -2); j = 1, - . - , t, (14.29)
where & is the regression coefficient computed from the experimental
error sums of squares and products, that is, b = E^/E^. The estimated
variance of an adjusted treatment mean is
V (adj. F.,-) = 4 f-1 + V-*- ^1 (14. 30)
L r E xx J
and the standard error of an adjusted treatment mean is
J adi. ?., = VF(adj
j. F. y ) = SE \/~ + (X \ ^ (14.31)
14.4 RANDOMIZED COMPLETE BLOCK DESIGN
447
The estimated variance of the difference between two adjusted treat
ment means is _
V (adj. F. y - adj. TV) = 4 - +
To test the hypothesis that 13 equals 0, we calculate
F =
(14.32)
(14.33)
which has degrees of freedom v= 1 and v 2 = (r 1) (t 1) - 1.
TABLE 14.6- Yields for 3 Varieties of a Certain Crop in a Randomized
Complete Block Design With 4 Blocks
(^ == yield of a plot in a preliminary year under uniformity
trial conditions; Y yield on the same plot in the
experimental year when the 3 varieties were used)
Varieties
"Rlorlr
Block
A
B
C
Totals
1
X
54
51
57
162
Y
64
65
72
201
2
X
62
64
60
186
Y
68
69
70
207
3
X
51
47
46
144
Y
54
60
57
171
4
X
53
50
41
144
Y
62
66
61
189
Variety. . . .
X
220
212
204
636
Totals . .
Y
248
260
260
768
Reproduced from Table 7 in John Wishart, Field Trials II: The Analysis of Covariance,
Tech. Comm. No. 15, Commonwealth Bureau of Plant Breeding and Genetics, School of
Agriculture, Cambridge, England, May, 1950. With permission of author and publishers.
Example 14.2
Consider the data of Table 14.6. These data have been examined in
considerable detail by Wishart (17); we shall, however, consider them
from a more limited point of view, which will be sufficient for our pur
poses. The required calculations are
396
12
(162)2 4- (186) 2 + (144) 2 4- (144) 2 (636) 2
12
448
CHAPTER 14, ANALYSIS OF COVARIANCE
(220) 2 + (212) 2 + (204) 2 (636) 2
12
32
514 396 32 = 86
+ ---- h (61)2 __
324
(201)2 + (207)2 + (171) 2 + (189) 2 (768) *
-
(248) 2 + (260) 2 + (260) 2 (768) 2
252
__
- 24
- 324 - 252 24 = 48
- (54) (64) -i ---- + (41) (61) -
286
(162) (201) + (186) (207) + (144) (171) + (144) (189) (636) (768)
(220) (248) + (2 12) (260) + (204) (260) (636) (768)
4 ~~ 12
= 286 264 (24) = 46,
and these are summarized in Table 14.7.
12
24
264
TABLE 14.7-Analysis of Covariance for Data of Table 14.6
Source of
Variation
Degrees
of
Free
dom
Sum of Squares
and Products
Deviations About
Regression
2> 2
][>:y
Zy
IZy* ^
-(2>30VZ>
Degrees
of Free
dom
Mean
Square
Replicates (blocks) .
Treatments (varie
ties)
3
2
6
396
32
86
264
-24
46
252
24
48
Experimental error.
23.4
5
4.68
Treatments + error .
8
118
22
72
67.9
44.5
7
2
22.25
Difference for testing among adjusted variety means . .
Before testing the hypothesis of no differences among the true effects
of adjusted varieties, let us test the hypothesis that the true regression
coefficient, 0, is 0. After all, it is more reasonable to examine this point
first, for unless we can reject such a hypothesis, that is, unless we can
safely conclude that /3 5^0, the decision to perform a regular analysis of
covariance is questionable. Accordingly, we compute F [(46) 2 /48]/4.68
= 9.42 with degrees of freedom *>i = l and v*=* 5. Since F = 9. 42 >F f 95(1,5)
= 6.61, we may reasonably assume that /J is not and thus be justified
in performing a covariance analysis.
We shall now examine the variety differences. First, we note that an
ordinary analysis of variance would give rise to F = (24/2)/(48/6)
= 12/8 = 1.5 with degrees of freedom j>i = 2 and j> 2 = 6, and this would
not permit us to reject the hypothesis of no differences among the true
14.5 LATIN SQUARE DESIGN 449
effects of the 3 varieties. Let us next observe what effect, if any, the per
formance of a co variance analysis will have on our inferences. To test
the hypothesis of no differences among the true effects of the 3 varieties
after adjusting for the effect of the natural fertility differences from plot
to plot, as measured by the uniformity trial, we compute F = 22. 25/4. 68
= 4.75 with degrees of freedom v\ = 2 and v 2 = 5. Since F=4.75 lies
between F. 90 (2,5) = 3.78 and F. 95(2,5) =5.79, the variety differences would
not ordinarily be called statistically significant. However, the reader is
cautioned again that the choice of the significance level is quite arbi
trary and the above results, therefore, may well indicate differences that
are of real importance. Apart from this, we note that the adjustment of
the yields for the unequal fertilities of the experimental plots has caused
the resulting F-value to approach more closely what is conventionally
thought of as a critical value. This should suggest to the researcher that
quite likely the fertility differences among the plots are tending to
obscure the true differences among varieties. If the experiment were
performed again with more replication, significant results might be
obtained.
14.5 LATIN SQUARE DESIGN
The performance of an analysis of covariance on data resulting
from a Latin square design introduces no new concepts. Thus, we shall
proceed immediately to outline the computational technique and
indicate the appropriate test procedures. The only calculations besides
those specified for an ordinary analysis of variance in a Latin square
are the sums of products. These are found as follows:
= corrected total sum of products for X and Y
_"* ***
2: 2:
m m
= row sum of products for X and F
m (14.35)
column sum of products for X and F
m / m \ / rn
z:( i:x-c*))( z:
/ 1 \ z-l / \ i-1
(14.36)
45O CHAPTER 14, ANALYSIS OF COVARIANCE
m m
TW = treatment sum of products for X and Y
m / j m m x / m rn \
Z (r*) G,r fc ) ( Z Z *<,<*, ) ( Z Z Ftfob, ) (14 . 37)
k=t \ tl jr'=l / \ i=l j=l /
m m*
E*y = 5>y ~ ^-v ~" C^ r^, (14.38)
where
xT^ = sum of all X observations associated with treatment k (14.39)
yT k == sum of all Y observations associated with treatment k. (14.40)
The results of these calculations are presented in Table 14.8. The test
of H : /3 = is given by
F = _ E **" /Exx _ = E ^ /Exx (14 41)
S E /[(m - l)(m - 2) - 1] ^ U ^
with degrees of freedom j>i=l and v^ (m~ l)(m 2) 1. To test
among adjusted treatment means we compute
= = +
^/[( -!)(- 2) - 1] 4 '
with degrees of freedom ^i = ?7^ 1 and ^ 2 = (m 1) (m 2) 1 .
The adjusted treatment means may be found using
adj. F..0b) - 7.. (Jfc ) - SC^.c*) - ^) (14.43)
where 6 is the regression coefficient associated with experimental error,
that is, b=E xv /E xx . The estimated variance of an adjusted treatment
mean is
V (adj. 7.. w ) - 4 |^ + v " w ' \ , (14.44)
and the estimated variance of the difference between two adjusted
treatment means is
f'Cadj. "F..C*) adj. 7.. (fc /)) = 4 ^ " (&/) L (14.45)
o
cr
CO
es
1
cr
to
<i>
tf
c
o
X
(U
Q
.*. A
Sf
a
cu
o
00
j
R
o
W
+
w
w
&|
p^
"s
4) *5
S'C
o ,rt
CO >
ft
e
I 1 I
I
03
W)
fi
S
W
452 CHAPTER 14, ANALYSIS OF COVARIANCE
14.6 TWO-FACTOR FACTORIAL IN A RANDOMIZED
COMPLETE BLOCK DESIGN
The performance of a covariance analysis when the treatments are
of a factorial nature follows the pattern established in the preceding
sections. The only refinement that occurs is one which is naturally
anticipated once we know we are dealing with a factorial setup: We
are now able to test among adjusted means for each of the factors and
for all the interactions. The appropriate calculations, in addition to
those outlined earlier are given below:
= corrected total sum of products for X and Y
r a b
= 23 23 23 XjjkYijk
a=l y=l fc=l
(14.46)
r a b \ / r a b
(r a b \ / r a b
z z z **) ( z z z Y ij
i=-l y=l =1 / \ i=l y1 A;*l
rab
= replicate sum of products for X and Y
b / a b
T / a b \ / a b
z( sz;*0(5:i:r*
i=*l \ y=l A=l / \ /! A:=l
ab
(14.47)
(r a & N. / r a b
Z Z Z x* ) ( Z Z Z F
4^.1 y=i <fe =1 / \ a-i yi AI
rab
= sum of products for X and Y for the a X b table
(r a, b \ / r a ' b
Z Z Z x*) ( Z Z Z
x=l y=i &*=! / \ i,-,! y*i fc=l
a, / r b \ / r b
Z ( Z Z x*) ( Z Z F
y1 \ i=l A 1 / \ t 1 A=-X
(r a b \. / r a b
Z Z Z xv) ( Z Z Z r<
^=1 yi A=I / \ i=i y=-i jfc=-i
(14.48)
(14.49)
a, / r b \ / r b
r5 (14.50)
raft
14.6 TWO-FACTOR FACTORIAL 453
J> / r a v / r a,
z; ( z z; *) ( z: i:
fc=l \ il Jl / \ i=l JW1
ra
(14.51)
(r g 5 \ / r a, b
z: z: z; * ) ( z; z: z r
1=1 y=i jb^i / \ twi y=i &=i
rob
B xy . (14.52)
These calculations are summarized in Table 14.9. Following are the
appropriate ^-ratios for testing the hypotheses: (1) no differences
among the true effects of the levels of factor a on the variable Y after
adjusting for the concomitant variable X, (2) no differences among the
true effects of the levels of factor & on the variable Y after adjusting
for the concomitant variable X, and (3) no interaction between factor
a and factor 6 as they affect the variable Y after adjusting for the effect
of the concomitant variable X, respectively,
- 1)
' /., , CTQ\
' (14 ' 53)
.
and
= (^H^- = M ,
The test of H:@ = Q is performed by calculating
F = 2^ (14.56)
with degrees of freedom n=l and j> 2 = (r l)(ab -1) 1.
The appropriate standard errors for the various mean effects are
found from the following estimated variances :
A-effect V (adj. T. y .) - ^~ + " < 14 ' 57)
B-effect V (adj. T..*) = ^ f + ( ^"* _" ^H (14. 58)
Lra^ J^x* J
f (adj. 7^) = *Z f- + (X ^~ X) 1 ' < 14 ' 59)
L ^ -^sx -I
8
ft
I
a
o
"B
-1
O
+->
O
1
I
I
O
-o
w
1?
w
sf
Co*
I
I I
-Cl
co" 1
T-H
I
I
J
O)
CD
O
PQ
1
PH
"&
cr^
CO
w
w
w
I
4
?*
II fc?
<*+
CCj
T
i
8
i I
I
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4J,
1 +
3 a
" a
a
,
s
-| s
^4+erro
Diff
fference
"S
a*
o
a
14.6 TWO-FACTOR FACTORIAL
455
Example 14.3
As an example of a covariance analysis in a randomized complete
block design where the treatments are of a factorial nature, consider
the data of Table 14.10, These data were originally examined by
Wishart (17), and the interested reader is referred to his study for a more
detailed discussion than we shall give here.
In discussing the data of Table 14.10, we shall consider the 5 pens as
5 replicates, and thus we have a 3X2 factorial in a randomized complete
block design. Following the calculational procedure outlined in Table
14.9, we arrive at the results presented in Table 14.11.
To test H:{3 = 0, we calculate
F =
(39.367) V442.93
0.2534
13.81
with degrees of freedom PI = 1 and v 2 =19, and this is significant at
a. = .01. The various treatment effects may also be tested for signifi-
TABLE 14.10-Initial Weights and Gains in Weight of Young Pigs
in a Comparative Feeding Trial
(X = initial weight in pounds; Y gain in weight in pounds}
Pen
Feeding Treatments
Totals
A
B
C
Male
Female
Male
Female
Male
Female
I
X
Y
38
9.52
48
9.94
39
8.51
48
10.00
48
9.11
48
9.75
269
56.83
II
X
Y
35
8.21
32
9.48
38
9.95
32
9.24
37
8.50
28
8.66
202
54.04
III
X
Y
41
9.32
35
9.32
46
8.43
41
9.34
42
8.90
33
7.63
238
52.94
IV
X
Y
48
10.56
46
10.90
40
8.86
46
9.68
42
9.51
50
10.37
272
59.88
V
X
Y
43
10.42
32
8.82
40
9.20
37
9.67
40
8.76
30
8.57
222
55.44
Totals
X
Y
205
48.03
193
48.46
203
44.95
204
47.93
209
44.78
189
44.98
1203
279.13
Reproduced from Table 11 in John Wishart, Field Trials II: The Analysis of Covariance,
Tech. Comm. No. 15, Commonwealth. Bureau of Plant Breeding and Genetics, School of
Agriculture, Cambridge, England, May, 1950. With permission of author and publishers.
456 CHAPTER 14, ANALYSIS OF COVARIANCE
TABLE 14. 11- Analysis of Covariance for the Data of Table 14.10
Source of
Variation
Degrees
of
Free
dom
Sum of Squares and Products
Deviations About Regression
2> I>3> Z? 2
(2>y) 2
Y%*
Degrees
of Free
dom
Mean
Square
- v
Replicates (pens) . .
Treatments
Food
4
2
1
2
20
605.87 39.905 4.8518
5,40 -0.147 2.2686
32.03 -3.730 0.4344
22.47 3.112 0.4761
442.93 39.367 8.3144
Sex
Food X sex
Experimental error
4.8155
19
0.2534
Food ~ f~ error
22
448.33 39.220 10.5830
7.1520
21
Difference for testing among
adjusted food means
2.3365
2
1 . 16825
Sex-j-error
21
474.96 35.637 8.7488
6.0749
20
Difference for testing among
adjusted sex means
1.2594
1
1.2594
(Food X sex)
-f- (error) .
22
465.40 42.479 8.7905
4.9133
21
Difference for testing among adjusted foodXsex effects . . .
0.0978
2
0.0489
cance, the appropriate variance ratios being
1.16825
Food: F =
Sex:
Food X Sex: F
0.2534
1.2594
0.2534 :
0.0489
0.2534 :
4.61
4.97
0.19
where the degrees of freedom are as given in Table 14.11. These F-ratios
(and the corresponding inferences) should be compared with those
resulting from an analysis of variance on the gains in weight taking no
account of the varying initial weights. Such comparisons will aid the
reader in understanding the principles of covariance analyses and, in
our example, will help to explain the effect of initial weights on weight
gains subject to the chosen experimental conditions. A table of adjusted
treatment means, together with the appropriate standard errors,
should also be presented to make the analysis complete.
COVARIANCE WHEN THE X VARIABLE IS
AFFECTED BY THE TREATMENTS
When the treatments being employed in the experiment are such
that they have an appreciable effect on the concomitant variate, X,
as well as on the Y variate, the researcher should proceed with caution.
Computationally, each step is carried through as before, but the final
14.7
14.8 MULTIPLE COVARIANCE 457
inferences must take account of the effect which the treatments have
had on the concomitant variate. For example, if the concomitant
variate in a feeding experiment had been "amount of feed consumed"
rather than the "initial weight/' it is quite possible that the different
treatments (feeds) "would have a significant effect on the food con
sumption. Thus any covariance analysis of gain in weight should take
cognizance of the weight-producing effects of the different feeds due to
increased (or decreased) consumption apart from any nutritional dif
ferences among the feeds. Other examples of covariance analyses in
volving similar problems are available in the literature and should be
studied critically by those who desire a fuller understanding of covari
ance techniques. The reader is especially referred to Cochran and
Cox (5) and Bartlett (2) for a discussion of this type of problem.
14.8 MULTIPLE COVARIANCE
As might be anticipated, procedures are available for performing
covariance analyses when data are collected on two or more concomit
ant variables. These procedures will follow the same pattern that has
been used in the preceding sections of this chapter, the only change
being that the sum of squares due to regression is calculated in accord
ance witli the principles outlined in Section 8.15. Rather than specify
an analytical approach for the general case, let us be content with a
numerical example to illustrate the ideas involved.
Example 14.4
Crampton and Hopkins (8) studied the effects of initial weight and
food consumption on the gaining ability of pigs when given different
feeds. The data are presented in Table 14.12.
To carry out a multiple covariance analysis, the first step is to find the
various sums of squares and products for treatments and for error and,
hence, for "treatments+error." These are determined to be
^ 2 x 2 = 28,404.9
Sr,*, = 90,792.3
>*,*, 119,197.2
^x, 2187.8
,, = 264<5.2
T*i*i- 509.2
*i*i= 368.4
Tyy = 5741.7
Eyy = 10,405.5
& I!BI = 877.6
T XlV = 1172.2
JSc lV = 1001.8
Syy - 16,147.2
T Xlk y = 11,596.5
E*rt = 24,508.7
2173.8 S Xjt y = 36,105.2 S^x* 4834.0
where
Y = final weight
X initial weight
Xz feed eaten.
The next step is to calculate the partial regression coefficients as
sociated with the multiple regression equation so that we may obtain
the sum of squares "due to regression" and thus, by subtraction from
the corrected total sum of squares, the sum of squares of the deviations
about regression. The required partial regression coefficients may be
5"
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14.8 MULTIPLE COVARlANCE
459
found using the methods of Chapter 8. Since we are dealing with a
case involving only two independent variables, it is simpler to solve the
following equations:
Error
36SAbuE + 2646.2&2J? = 1001.8
2646.2&1J? + 90,792.3^2^7 = 24,508,7
Treatment + error
87
4834.0&UJP+JB) 4- 119,197.2& 2 <T+*>
2173.8
36,105.2
giving
.9868,
= .2412,
1.0410 and
.2607.
Thus, the sums of squares due to regression (and, hence, the sum of
squares of the deviations about regression) for "error" and for "treat-
ment + error," respectively, are determined to be:
Error
S.S. due to regression = (.9868) (1001.8) + (.2412) (24,508.7)
= 6900.07
S.S. of deviations about regression = 10,405.5 6900.07
= 3505.43
Treatment + error
S.S. due to regression = (1.0410) (2173.8) + (.2 607) (3 6, 105. 2)
= 11,675.45
5\.S. of deviations about regression = 16,147.2 11,675.45
= 4471.75.
These results are then presented in Table 14.13. Notice that this time
we have ' lost' 7 two degrees of freedom rather than one degree of freedom
as in a simple covariance. If there had been k independent (concomitant)
variables, we would have "lost" k degrees of freedom. The reader will
note that this is in agreement with the procedures outlined in Chapter 8.
The F-test is then performed as before, and we obtain
F = 241, 58/103. 10 = 2. 34 with degrees of freedom vi = 4 and i> 2 = 34,
which is not significant at a. = .05. This should be compared with the
TABLE 14. 13- Abbreviated Analysis of Covariance for Data of Table 14.12
Source of
Variation
Degrees
of Free
dom
I>
S.S. Due to
Regression
S.S. Devia
tions About
Regression
Degrees
of Free
dom
Mean
Square
Treatments (T) . .
Error CE)
4
36
5741.7
10,405.5
6900.07
3505.43
34
103 . 10
T-f-72
40
16,147.2
11,675.45
4471.75
966.32
38
4
241.58
Difference for testing among adjusted treatment means . .
460
CHAPTER 14, ANALYSIS OF COVARlANCC
results obtainable from an analysis of variance of the final weights, and
the appropriate conclusions drawn. Adjusted treatment means and their
standard errors may be calculated by combining methods indicated in
Section 8.25 and earlier sections of this chapter.
Problems
14,1 An experiment using a randomized complete block design gave the
following corrected sums of squares and products:
Degrees
of Free
Source of Variation
dom
2> 2
Z>:v
Z? 2
b
Replicates
5
200
600
4000
Treatments
5
100
200
2500
2
Experimental error. . .
25
300
1200
7500
4
(a) Based on the experimental error sum of squares and products,
is the regression of Y on X significant at a: = .05?
(6) Are the differences among the treatment means for Y adjusted for
variation attributed to X significant at OL = .05?
(c) What conclusions do you draw from the above data about the
effects of treatments? Make any additional computations that
you consider necessary.
14.2 Given the following data:
Source of Variation
Degrees of
Freedom
X> 2
Z>;y
i:? 2
Replicates
4
100
140
400
Xreatments
10
100
100
900
Experimental error
40
400
900
2500
(a) What conclusions may be drawn about the effect of treatments
on F?
(6) Test the regression coefficient based on experimental error for
significance at the 5 per cent level.
14.3 Ten lines of soybeans were compared in randomized complete blocks
with 4 replications. The differences in yield, Y, were not significant,
but it was observed that the incidence of an infestation, X, differed
among the varieties. Following is the table of sums of squares and
products:
Source
Degrees
of
of Free
yjlx*
y^xv
> ^-y 2
Variation
dom
Lines
9
4684
532
112
Error
27
3317
65O
216
PROBLEMS
461
Test the hypothesis that the yields adjusted for infestation do not
differ in the sampled populations. What fraction of X)2/ 2 f r lines is
unexplained by the regression?
j.4.4 For an experiment involving 9 soil sterilization treatments, the effect
on the number of seedling alfalfa plants (X) and on the green weight
of plants at 3 weeks (F) is summarized by the following sums of
squares and products:
Source of
Degrees
of Free
Variation
dom
!> 2
T,*y
Z? 2
Replicates
5
4
16
96
Treatments
8
16
32
80
Error
40
20
40
160
Complete the analysis, making appropriate tests to indicate the
reason for your conclusions. If the mean of the X 9 s is 15 and the
mean of the Y's is 25, give the regression equation for error.
14.5 A study of eastern Iowa farms included one group of tenants who
were not related to their landlords, and another group of tenants,
each of whom was related to his landlord. It was assumed that soil
improvements would be more generally undertaken when landlord
and tenant were related. Hence, value of crops should be greater in
those situations. An analysis of variance was undertaken to examine
this hypothesis. Since size of farm could confuse the comparison, the
size of farm was introduced as a covariate. The following table was
prepared :
COVARIANCE ANALYSIS or VALUE or CROPS ON SIZE OP FARM TOR
EASTERN IOWA FARMS WITH LANDLORD AND TENANT RELATED
AND LANDLORD AND TENANT NOT RELATED
Source of Variation
Degrees
of Free
dom
I>*
JLxy
I>*
Total
59
125000
33000
36600
Sub-areas (replicates)
4
20000
14010
13600
Bet. grps of tenants
1
61000
13260
4200
Interaction ,
4
4000
13270
6100
"Within subclasses
50
40000
19000
12700
14.6
Value of crops has been coded for this analysis.
(a) Is the acceptance of the hypothesis, H (no difference between
groups of tenants) changed by the introduction of farm size as the
covariate?
(6) Is the error regression significant?
A sample of farms was taken in the eastern livestock area of Iowa for
462
CHAPTER 14, ANALYSIS OF COVARIANCE
the purpose of studying certain types of farm lease arrangements.
For this problem we are taking a portion of the data to study the dif
ference in "gross value of crops" produced on two groups of cash-
rented farms: (1) farms for which landlord and tenant are related,
and (2) farms for which landlord and tenant are not related. The
variates measured are value of crops produced (F) and size of farm
(X). The data presented are given for 3 hypothetical blocks. In
practice, these blocks might be strata, e.g., different counties, differ
ent soil areas, type-of-farming areas, or groups of farms enumerated
by different enumerators, for instance, 3 agricultural economics stu
dents for the 3 blocks of our example. The data are presented in the
table following. Perform an analysis of covariance on these data.
FARM DATA FROM EASTERN LIVESTOCK AREA, IOWA, FOR COVARIANCE
ANALYSIS OF VALUE OF CROPS AND SIZE OF FARM GROUPS To BE
COMPARED: FARMS WITH LANDLORD AND TENANT RELATED
AND LANDLORD AND TENANT NOT RELATED*
Block I
Block II
Block III
Farm
Y
X
Farm
Y
X
Farm
Y
X
No.
Related
No.
Related
No.
Related
22
6399
160
27
2490
90
17
4489
120
13
8456
320
24
5349
154
25
10026
245
20
8453
200
11
5518
160
1
5659
160
8
4891
160
34
10417
234
26
5475
160
21
3491
120
38
4278
120
4
11382
320
Not Related
Not
Related
Not
Related
31
6944
160
13
4936
160
20
5731
160
30
6971
160
1
7376
200
15
6787
173
11
4053
120
19
6216
160
7
5814
134
6
8767
280
32
10313
240
5
9607
239
16
6765
160
28
5124
120
25
9817
320
* Source: Agricultural Economics Dept., Iowa State College, 1951.
14.7 (a) What are the assumptions behind a covariance analysis?
(6) In the process of analyzing data by a covariance analysis, what
tests of significance are made?
(c) Explain the interpretation or inferences and the course of action
indicated when each of the above tests is significant; when each
is nonsignificant.
14.8 The following is an experiment involving randomized complete blocks
with 4 replications. Eleven lines of soybeans were planted. The data
are as follows:
X i = maturity, measured in days later than the Hawkeye variety
X z = lodging, measured on a scale from to 5
Y = infection by stem, canker measured as a percentage of stalks
infected.
PROBLEMS
463
Replicate 1
Replicate 2
Replicate 3
Replicate 4
Line
JSTi X^ Y
Xi X* Y
JSTi X 2 Y
-Yi X 2 F
Lincoln
9 3.0 19 3
10 20 29.2
12 3 10
925 6.4
A7-6102
10 3.0 10.1
10 20 34.7
9 20 14
930 5.6
A7-6323 ....
10 25 13 1
9 15 59 3
12 2 5 11
10 25 8.1
A7-6520
8 2.0 15 6
5 20 49
8 20 17 4
6 20 11 7
A7-6905
12 2.5 4.3
11 10 48.2
13 3 63
10 2 5 67
C-739 ... .
4 2.0 25 2
2 15 36 5
2 20 23 4
1 20 12 9
C-776
3 1.5 67.6
4 10 79 3
6 2 13 6
2 15 39 4
H-6150 .
7 2.0 35 1
8 2 4O
7 20 24 7
720 48
L6-8477
8 20 14
8 15 30 2
10 1 5 72
720 89
L7-1287
925 3.3
9 20 35 8
13 3 11
930 20
BEIV Sp
10 3 5 31
10 3 96
11 3 10
10 3 5 01
The principal objective is to learn whether maturity or lodging is
more closely related to infection. Determine this from the error
multiple regression. Test the hypothesis of no differences among
adjusted mean infection for the varieties.
14.9 Discuss the use of covarianee analysis. What factors must be con
sidered in interpreting the results of the analysis?
14.10 The data for this problem consist of 54 pairs of observations on the
calories consumed (Y) on one particular day by a respondent, and her
age (X) . The respondents were adult Iowa women over the age of 30
who were interviewed to obtain information on nutrition and health.
About 1000 women were so enumerated for this survey, and our
group of 54 is a subgroup from the total, which was taken so as to
make numbers in the subclasses equal.
Among the items observed for each respondent in addition to
caloric intake and age was place of residence (zone) and income class.
These are listed as
Zone 1 open country
Zone 2 rural place
Zone 3 urban
Income Group
1
2
3
4
5
6
0-$ 999
1000- 1499
1500- 1999
2000- 2999
3000- 3999
over 4000
Education, height, weight, national origin, marital status, family
composition, and many other factors were recorded for each re
spondent.
The nutritionists studying these data are interested in determining
how food intake and health are related to these other observed fac
tors. A few relevant hypotheses could be advanced. Preliminary
analysis consisted of preparing tables of means for several classifica
tions of the total sample and graphical analysis (plotting on scatter-
grams of a subsample of 60 stratified by age). A number of nutritive
factors exhibited an apparent negative regression on age. Age thus
seemed a useful covariate. Other factors, such as education, height,
and weight, seemed to indicate no relation to nutritive intake.
464
CHAPTER 14, ANALYSIS OF CO VARIANCE
With this background we shall use these data to undertake an
analysis of covariance for the purpose of testing hypotheses about
zone and income group effects after taking account of the regression
on age. The data table gives the 54 pairs of observations with the
sums, sums of squares, and sums of products. Both zone and income
group may be considered as fixed effects,
(a) Prepare the analysis of covariance table.
(6) Find the error regression of calories on age.
(c) Test the hypotheses that zone and income effects are = (sepa
rately, of course)*
(d) It would also be of interest to test for interaction of zone and in
come group. What do you conclude on this point?
(e) The regression of calories on age may not be homogeneous over
the zones. Indicate by a schematic analysis of variance of regres
sion how you would examine these regressions.
Zone
1
Zone
2
Zone
3
Income Group
Y
X
F
X
F
X
1
1911
46
1318
80
1127
74
1560
66
1541
67
1509
71
2639
38
1350
73
1756
60
2
1034
50
1559
58
1054
83
2096
33
1260
74
2238
47
1356
44
1772
44
1599
71
3
2130
35
2027
32
1479
56
1878
45
1414
51
1837
40
1152
59
1526
34
1437
66
4
1297
68
1938
33
2136
31
2093
43
1551
40
1765
56
2035
59
1450
39
1056
70
5
2189
33
1183
54
1156
47
2078
36
1967
36
2660
43
1905
38
1452
53
1474
50
6
1156
57
2599
35
1015
63
1809
52
2355
64
2555
34
1997
44
1932
79
1436
54
166416926
CT 156053200
-CT
= 10363726
4573454
4773496
200042
CT
157356
146016
*= 11340
REFERENCES AND FURTHER READING 465
References and Further Reading
1. Anderson, R. L,., and Bancroft, T. A. Statistical Theory in Research. McGraw-
Hill Book Company, Inc., New York, 1952.
2. Bartlett, M. S. A note on the analysis of covariance. Jour. Agr. Sci., 26:448,
1936.
3. Cochran, W. G. Analysis of covariance. Mimeo Series No. 6, Institute of
Statistics, University of North. Carolina, Chapel Hill, 1949.
4 u Analysis of covariance: its nature and uses. Biometrics, 13 (No.
3):261-81, Sept., 1957.
5. , and Cox, G. M. Experimental Designs. Second Ed. John Wiley and
Sons, Inc., New York, 1957.
6. Coons, Irma. The analysis of covariance as a missing plot technique. Bio
metrics, 13 (No. 3) :387-405, Sept., 1957.
7. Cox, D. R. Planning of Experiments. John Wiley and Sons, Inc., New York,
1958.
8. Crampton, E. W., and Hopkins, J. W. The use of the method of partial
regression in the analysis of comparative feeding trial data. Part II. Jour.
Nutrition, 8:329, 1934.
9. Federer, W. T. Experimental Design. Macmillan Co., New York, 1955.
10. Variance and covariance analyses for unbalanced classifications.
Biometrics, 13 (No. 3) :333-62, Sept., 1957.
11. Finney, D. J. Stratification, balance, and covariance. Biometrics, 13 (No.
3):373-86, Sept., 1957.
12. Kempthorne, O. The Design and Analysis of Experiments. John Wiley and
Sons, Inc., New York, 1952.
13. Quenouille, M. H. The Design and Analysis of Experiment. Charles Griffin
and Co., Ltd., London, 1953.
14. Smith, H. Fairfield. Interpretation of adjusted treatment means and
regressions in analysis of covariance. Biometrics, 13 (No. 3) :282 308, Sept.,
1957.
15. Truitt, J. T., and Smith, H. Fairfield. Adjustment by covariance and conse
quent tests of significance in split-plot experiments. Biometrics, 12 (No. 1):
23-39, Mar., 1956.
16. Wilkinson, G. N. The analysis of covariance with incomplete data. Bio
metrics, 13 (No. 3):363-72, Sept., 1957.
17. Wishart, J. Field trials II: The analysis of covariance. Tech. Comm. No. 15,
Commonwealth Bureau of Plant Breeding and Genetics, Cambridge, Eng
land, May, 1950.
18. Zelen, M. The analysis of covariance for incomplete block designs. Bio
metrics, 13 (No. 3):309-32, Sept., 1957,
CH APTE R 15
DISTRIBUTION-FREE METHODS
IN PRECEDING CHAPTERS the emphasis has been on those statistical
techniques which assume the sampled populations to be of known form.
However, because the analyst is not always certain of the validity of
such assumptions and/or because not all statistical techniques are
robust (i.e., insensitive to departures from such assumptions), much
work has been done in recent years to devise procedures which are free
of these restrictions. These new techniques, referred to as distribution-
free methods -, 1 will be the subject of this chapter.
Although most distribution-free methods have been developed only
recently, the literature in this area is already quite extensive. Further,
it continues to grow every day. Consequently, it will be impossible to
do more than mention a few of the more popular and useful methods
in this book. Those persons who wish to delve deeper into this area of
statistics are encouraged to consult the references listed at the end of
this chapter.
15.1 DISTRIBUTION-FREE METHODS INCLUDED IN
PREVIOUS CHAPTERS
Four widely used distribution-free methods have already been intro
duced in earlier chapters. These are: (1) Tchebycheff's inequality dis
cussed in Section 5.3, (2) the distribution-free tolerance limits referred
to in Section 6.13, (3) the chi-square goodness of fit test described in
Section 7.15, and (4) the measures of rank correlation described in
Section 9.11. Because these methods were discussed in the afore
mentioned sections, it would be superfluous to repeat their descriptions
at this time. It is recommended, however, that the indicated sections
be reread in the present context. Let us now proceed to the study of
some additional distribution-free methods that have been found useful
in a variety of situations.
15.2 THE SIGN TEST
In many experimental situations, the investigator wishes to compare
the effects of two treatments. When the data occur in pairs, one mem
ber of the pair being associated with treatment A and the other with
treatment B, one test of wide applicability is the sign test. Using in
equality signs to denote the relationship between the members of a
1 Many authors refer to distribution-free methods as nonparametric methods
and, although the expressions are not strictly equivalent, they have been, and
probably will continue to be, used interchangeably.
[466]
15.2 THE SIGN TEST 467
pair, whether the comparison be qualitative or quantitative. 2 the sign
test proceeds as follows:
(1) Examine each of the pairs (X iy Y.
(2) If Xi> Y i? assign a plus sign; if X* < Y if assign a minus sign; if
Xi = Yt, discard the pair.
(3) Denote the number of pairs remaining, that is, the number of
pairs resulting in either a plus or minus sign, by n.
(4) Denote by r the number of times the less frequent sign occurs.
(5) To test the hypothesis of no difference between the effects of
the two treatments, compare r with the critical values tabu
lated in Appendix 13.
(6) If the observed value of r is less than or equal to the tabulated
value for the chosen significance level, the hypothesis is re
jected; otherwise, it is not rejected.
Before giving numerical illustrations, it is appropriate that attention
be called to different hypotheses that can be tested in the manner
indicated above. Some of these are:
(1) Each difference X* F;, has a probability distribution (which
need not be the same for all differences) with median equal to
0, that is, H:P{Xi> Y,} = 0.5 for all i.
(2) If the underlying distributions are assumed to be symmetric,
the sign test may be used to test the hypothesis fl r : Mjr , = My ./.
(3) If it can be assumed that the underlying distributions' differ
only in their means, then a test of Hip, x ,=p Y . is equivalent
to testing the hypothesis that the probability distributions of
each pair are the same.
(4) Questions such as:
(a) Is A better than B by P per cent?
and
(b) Is A better than B by U units?
may also be studied by applying the sign test to the differences
D = A - (l+P/100) B and D = A (B+ C7), respectively.
Example 15.1
Consider once again the data given in Table 7.6 and discussed in
Example 7.21. We note that = 15 and r = 4. Assuming <* = 0.05, it is
seen that the hypothesis of equal hardness indications by the two' steel
balls cannot be rejected since the critical value of r tabulated in Appen
dix 13 was 3. You will note that this is the opposite decision to that
reached in Example 7,21. The reason for this is that, when normality
can be assumed, the sign test is less efficient (that is, less sensitive) than
"Student's" J-test.
2 If measurements are recorded, then X< > Y t will signify that, in the ith pair,
treatment A resulted in a higher reading than treatment B. If no measurements
are available, then Xi > Yi will signify that, in the fcth pair, treatment A resulted
in something larger than Cor better than or preferred over) treatment B.
468 CHAPTER 15, DISTRIBUTION-FREE METHODS
Example 15.2
In a marketing study, two brands of lemonade were compared. Each
of 50 judges tasted two samples, one of brand A and one of brand B,
with the following results: 35 preferred brand A, 10 preferred brand J3,
and 5 could not tell the difference. Thus, n = 45 and r = 10. Assuming
CK=0.01, we reject the hypothesis of equal preference (since r = 10 <13
= critical value) and conclude that brand A is preferred.
15.3 THE SIGNED RANK TEST
The sign test described in Section 15.2 was simple to apply. However,
when measurement data have been obtained, it is not the most efficient
distribution-free test available. A better test, sometimes referred to as
the Wilcoxon signed rank test and at other times more simply as the
signed rank test, is one which takes account of the magnitude of the
observed differences. It proceeds as follows:
(1) Rank the differences without regard to sign, that is, rank the
absolute values of the differences. (The smallest difference is
given rank 1 and ties are assigned average ranks.)
(2) Assign to each rank the sign of the observed difference.
(3) Obtain the sum of the negative ranks and the sum of the
positive ranks.
(4) Denote by T the absolute value of the smaller of the two
sums of ranks found in the previous step.
(5) To test the hypothesis of no difference between the effects of
the two treatments, compare T with the critical values tabu
lated in Appendix 14.
(6) If the observed value of T is less than or equal to the tabulated
value for the chosen significance level, the hypothesis is re
jected; otherwise, it is not rejected.
Before giving numerical illustrations, we should note that the signed
rank test is also applicable in the following situations :
(1) To test the hypothesis that the median of a population is
equal to some specified value, say M .
(2) To test the hypothesis that the median of a population of dif
ferences is equal to some specified value, say M .
It should be clear, of course, that in Case 1 the basic variable is
\X MO\, while in Case 2 it is \(XY)M Q \. Apart from this
obvious transformation, the procedure is exactly as specified above.
Example 15.3
Consider again the data of Table 7.6. These are reproduced in Table
15.1 for your convenience. Applying the procedure for the signed rank
test, it is seen that 7 7 = 18.5. Assuming a: = 0.05 and consulting Appendix
14, it may be verified that T=18.5 <Tc = 25, and therefore, the hy
pothesis of equal treatment effects is rejected. The reader should com
pare this result with those reached in Examples 7.21 and 15.1.
15.3 THE SIGNED RANK TEST
TABLE 15.1-Data Obtained in a Brinell Hardness Test
469
Sample
Number
Differences
(>)
Rank of |Z>|
Signed Rank
Positive
Negative
1
22
2
4
12
11
15
28
5
S
4
1
10
2
25
7
13
2.5
4.5
11
10
12
15
6
8
4.5
1
9
2.5
14
7
13
2.5
4.5
11
10
12
15
2 ....
3
4
5
6
7
8
6
9
8
4.5
10
11
1
9
- 2.5
12
13
14
14
7
15
Total
101.5
18.5
Example 15.4
Given the data of Table 15.2, test the hypothesis that the population
median equals 12. It is easily seen that the calculations, also shown in
Table 15.2 for convenience, lead to T=6 which, for n = 8 and oj=0.05,
tells us we are unable to reject the stated hypothesis.
TABLE 15.2-Hypothetical Data To Illustrate the Procedure
of the Signed Rank Test
"O rt -nTi- s\f
Signed
Rank
Ob servations
tx)
X Mo
jtvanK. 01
\X-M*\
Positive
Negative
12 55
55
3
3
14 62
2 62
8
8
12 93
93
4
4
12 46
0.46
2
2
11 95
05
1
1
14 55
2 55
7
7
13 11
1 11
6
6
10 90
1.10
5
5
Total
30
6
470 CHAPTER 15, DISTRIBUTION-FREE METHODS
15.4 THE RUN TEST
Among other things, the theory of runs may be used to test the
following two hypotheses:
(1) The observations have been drawn at random from a single
population.
(2) Two random samples come from populations having the same
distribution.
Because the mathematics of the theory of runs is quite involved, we
shall do no more than sketch the approach. Those persons who need to
know the details are advised to consult the references at the end of
the chapter.
Case 1
(a) List the observations in the order in which they were obtained,
that is, in their order of occurrence.
(b) Determine the sample median.
(c) Denote observations below the median by minus signs and
observations above the median by plus signs.
(d) Denote the number of minus signs by HI and the number of
plus signs by n^
(e) Count the number of runs 3 and denote this number by r.
(f) If r is less than or equal to the critical value tabulated in
Appendix 15 (Table 1) or greater than or equal to the critical
value tabulated in Appendix 15 (Table 2), the hypothesis is
rejected at the 5 per cent significance level.
Case 2
(a) List the r&i+n 2 observations from the two samples in order of
magnitude, that is, arrange them in one sequence according to
their values.
(b) Denoting observations from one population by x's and obser
vations from the other population by j/'s, count the number
of runs.
(c) Denote the observed number of runs by r.
(d) If r is less than or equal to the critical value tabulated in
Appendix 15 (Table 1), the hypothesis is rejected at the 5 per
cent significance level,
Example 15.5
Suppose a manufacturing process is turning out washers, and the
characteristic of interest is the outside diameter. In the first 40 washers
tested, there were 16 runs above and below the sample median. Noting
that ni = n 2 = 20, we refer to Appendix 15 and find that r I = 14<16
<28==r u -. Thus, at the 5 per cent significance level we are unable to re
ject the hypothesis that the 40 observations constitute a random sample
from a single population.
3 In terms of our symbols, a run is a sequence of signs of the same kind bounded
by signs of the other kind.
15.5 THE KOLMOGOROV-SMIRNOV TEST
471
Example 15.6
Consider the data of Table 15.3. Listing according to ranks, we have:
B, AAAAAA, B, A, B, AA, BBB, A, BBB } A, B } A. That is we have
r = 12 runs. In addition, 7^ = 12 and 72^ = 10. Reference to Appendix 15
(Table 1) tells us that, since r=12>r c = 7 ? we are unable to reject the
hypothesis that the two random samples came from populations having
the same distribution. In other words, using the run test and operating
at the 5 per cent significance level, we are unable to reject the hypothesis
that the two lines are producing equivalent product.
TABLE 15.3-Outside Diameters of Washers Produced
by Two Different Production Lines (Figures in
Parentheses Are the Ranks)
Line A
Line B
1.63 (6)
1.65 (8)
1 . 68 (9)
1.69 (10)
1 . 59 (4)
1.72 (13)
1.64 (7)
1.91 (21)
1.70 (11)
1.74 (14)
1.58 (3)
1.75 (15)
1.62 (5)
1.55 (1)
1.71 (12)
1.86 (17)
1.57 (2)
1.87 (18)
1.84 (16)
1.88 (19)
1.90 (20)
1.96 (22)
15.5 THE KOLMOGOROV-SMIRNOV TEST OF
GOODNESS OF FIT
An alternative to the chi~square goodness of fit test described in
Section 7.15 is provided by the Kolmogorov-Smirnov test to be de
scribed here. Since the Kolmogorov-Smirnov test is more powerful
than the ehi-square test, its use is to be encouraged. It proceeds as
f ollows :
(1) Let F(x) be the completely specified theoretical cumulative
distribution function under the null hypothesis.
(2) Let S n (x) be the sample c.d.f . based on n observations. For any
observed x, S n (x) =k/n where k is the number of observations
less than or equal to x.
(3) Determine the maximtim deviation, D, defined by
D = max [ F(x) S n (x) \ .
(4) If, for the chosen significance level, the observed value of D is
greater than or equal to the critical value tabulated in
Appendix 16, the hypothesis will be rejected.
472
CHAPTER 15, DISTRIBUTION-FREE METHODS
Example 15.7
To illustrate the Kolmogorov-Smirnov test of goodness of fit, we shall
apply It to the data of Table 7.11. These data are reproduced here as
Table 15.4. To test the hypothesis that the data constitute a random
sample from a Poisson population with a mean of 10.44, calculations
are carried out as shown in Table 15.4. The values of F(x) were found
by consulting Appendix 2 and using A = 10. 5. (TTOTE: If a more precise
evaluation is needed, F(x) should be determined using A = 10. 44. The
approximate value, 10.5, was used since A = 10.44 would require interpo
lation in Appendix 2.) Since > = max \F(x) S n (x) \ =0.013 < 1.63
/ VS75 0.027, the hypothesis may not be rejected at the 1 per cent
significance level. The reader should compare this result with that
obtained in Example 7.27.
TABLE 15. 4- Application of the Kolmogorov-Smirnov Goodness of Fit
Test to the Number of Busy Senders in a Telephone Exchange
Number
Busy
Observed
Frequency
Observed
Curn illative
Frequency
Relative
Cumulative
Frequency
S n (x)
Expected
Relative
Cumulative
Frequency
FW
\F(x*)-S n (^\
1
5
5
0.001
o
0.001
2
14
19
O.005
0.002
0.003
3
24
43
0.011
0.007
0.004
4
57
100
0.027
0.021
0.006
5
111
211
0.056
0.050
0.006
6
197
408
0.109
0.102
007
7
278
686
183
179
004
8 ...
378
1064
0.283
279
004
9
418
1482
0.395
397
002
10
461
1943
0.518
521
003
11. . . .
433
2376
0.633
639
006
12
413
2789
0.743
0.742
001
13
358
3147
0.838
0.825
013
14
219
3366
0.897
0.888
0.009
15
145
3511
0.935
932
003
16
109
3620
0.964
0.960
004
17
57
3677
979
978
001
18. .,
43
3720
991
988
003
19
16
3736
995
994
001
20
7
3743
0.997
0.997
o
21
8
3751
0.999
0.999
22
3
3754
1.000
0.999
O.OO1
Data Source: Thornton C. Fry, JProbability and Its Engineering Uses. D. Van No strand
Company, Inc., New York, 1928, p. 295.
PROBLEMS 473
15-6 MEDIAN TESTS
The procedures to be described in this section are of value when test
ing the following hypotheses;
(1) That k random samples were drawn from identically dis
tributed populations, fc>2.
(2) That, in a one-factor experiment, the k levels of the factor
have the same effect.
(3) That, in a two-factor experiment, (a) the a levels of factor a
have the same effect, (b) the b levels of factor 6 have the same
effect, and (c) there is no true interaction between factors
a and 6.
For our purposes, it is deemed sufficient to concentrate on Case 1.
Those persons wishing to investigate Cases 2 and 3 are referred to
Brown and Mood (4, 5) and Mood (22).
If k random samples consisting of n\, - , n k observations, respec
tively, are available, determine the numbers of observations in each
sample that are above and below the median of the combined samples.
These data may then be analyzed as a 2Xfc contingency table in the
manner specified in Section 7.17.
Example 15.8
Consider the two samples in Table 15.3. Examination of these data
leads to the 2X2 contingency table shown in Table 15.5. Using Equa
tion (7.27), we obtain
X 2 - 22(| (4) (3) - (8) (7) | - 11)Y(12)(10)(11)(11) - 2.07.
Since x 2 = 2.07 <X 2 95(1) =3.84, we are unable to reject the hypothesis
that the two random samples were drawn from identically distributed
populations.
TABLE 15.5-Contingency Table Formed From the Data
of Table 15.3
Line A
Line B
Total
Above median
4
7
11
Below median . ...
8
3
11
Total
12
10
22
Problems
15.1 Apply the method described in Section 15.2 to the following problems:
(a) 7.29
(6) 7.32
(c) 7.33
Discuss any differences between the method used here and that used
in Chapter 7.
474 CHAPTER 1 5, ^DISTRIBUTION-FREE METHODS
15.2 Apply the method described in Section 15.3 to the following problems:
(a) 7.29
(6) 7.32
(c) 7.33
Discuss your results with reference to those obtained in Problem 15.1
and in Chapter 7.
15.3 Apply the method described in Section 15.3 to the following problems:
(a) 7.1
(6) 7.2(d)
(c) 7.3
State any changes in the assumptions and the wording of the hy
potheses that you make. Discuss the implications of these changes.
Compare the results of using distribution-free tests with those ob
tained using parametric tests in Chapter 7.
15.4 Apply the method described in Section 15.4 to the data given in the
following problems:
(a) 6.1 (i) 7.30
(6) 6.5 (/) 7.31
(c) 6.20 (fc) 7.33
(d) 6.22 (Z) 7.35
(e) 6.23 (m) 7.36
Of) 7.1 (n) 7.37
(0) 7.2 (o) 7.39
(A) 7.3 (p) 7.40
In each case, state the hypothesis being tested and specify any as
sumptions you make.
15.5 Apply the method described in Section 15.5 to check on the assump
tion of normality in the following problems:
(a) 6.5
(6) 7.1
15.6 Apply the method described in Section 15.6 to the data given in the
following problems:
(a) 7.30 (gr) 7.40
(6) 7.31 (/O 11.9
(c) 7.35 (0 11.10
(d) 7.36 (j) 11.11
(<0 7.37 (Jfe) 11.12
CO 7.39
Compare the conclusions reached here with those reached in the earlier
chapters. Discuss any discrepancies.
References and Further Reading
1. Birnbaum., Z. W. Numerical tabulation of the distribution of Kolmogorov's
statistic for finite sample values. Jour. Amer. Stat. Assn., 47:42541, 1952.
2. . Distribution-free tests of fit for continuous distribution functions.
Ann. Math. Stat., 24:1-8, 1953.
REFERENCES AND FURTHER READING 475
3. Blum, J. R., and Fattu, N. A. Nonparametric methods. Rev. Educ. Res.,
24:467-87, 1954.
4. Brown, G. W., and Mood, A. M. Homogeneity of several samples. Amer.
Stat., 2:22, 1948.
5. , and . On median tests for linear hypotheses. Proceedings of
the Second Berkeley Symposium on Mathematical Statistics and Probability,
pp. 15966, University of California Press, Berkeley, Calif., 1951.
6. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineer
ing. John Wiley and Sons, Inc., New York, 1960.
7. Brunk, H. D. An Introduction to Mathematical Statistics. Ginn and Co.,
New York, 1960.
8. Cochran, W. G. The x 2 test of goodness of fit. Ann. Math. Stat., 23:315-45,
1952.
9. . Some methods for strengthening the common x 2 tests. Biometrics,
10:417-51, 1954.
10. Dixon, W. J. Power under normality of several non-parametric tests. Ann.
Math. Stat., 25:610-14, 1954.
11. , and Massey, F. J., Jr. Introduction to Statistical Analysis. Second
Ed. McGraw-Hill Book Company, Inc., New York, 1957.
12. , and Mood, A. M. The statistical sign test. Jour. Amer. Stat. Assn.,
41:557-66, 1946.
13. Festinger, L. The significance of differences between means without refer
ence to the frequency distribution function. Psychometrika, 11:97, 1946.
14. Fraser, D. A. S. Nonparametric Methods in Statistics. John Wiley and Sons,
Inc., New York, 1957.
15. Kendall, M. G. Rank Correlation Methods. Charles Griffin and Company
Limited, London, 1948.
16. Kruskal, W. H. Historical notes on the Wilcoxon unpaired two-sample test.
Jour. Amer. Stat. Assn., 52:356-60, 1957.
17. , and Wallis, W. A. Use of ranks in one-criterion variance analysis.
Jour. Amer. Stat. Assn., 47:583, 1952,
18. Mann, H. B., and Whitney, ID. H. On a test of whether one of two random
variables is stochastically larger than the other. Ann. Math. Stat., 18:50,
1947.
19. Massey, F. J., Jr. The Kolmogorov-Smirnov test for goodness of fit. Jour.
Amer. Stat. Assn., 46:68-78, 1951.
20. . The distribution of the maximum deviation between two sample
cumulative step functions. Ann. Math. Stat,, 22:12528, 1951.
21. Mood, A. M. The distribution theory of runs. Ann. Math. Stat., 11:36792,
1940.
22. . Introduction to the Theory of Statistics. McGraw-Hill Book Company,
Inc., New York, 1950.
23. Moses, L. E. Non-parametric statistics for psychological research. Psych.
Bui., 49:122-43, 1952.
24. Olmstead, P. S., and Tukey, J. W. A corner test for association. Ann. Math.
Stat., 18:495-513, 1947.
25. Savage, I. H, Bibliography of nonparametric statistics and related topics.
Jour. Amer. Stat. Assn., 48:844-906, 1953.
26. SchefTe*, H. Statistical inference in the non-parametric case. Ann. Math.
Stat., 14:305-32, 1943.
27. Siegel, S. Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill
Book Company, Inc., New York, 1956.
28. Smirnov, N. V. Table for estimating the goodness of fit of empirical distri
butions. Ann. Math. Stat., 19:279-81, 1948.
29. Smith, K. Distribution-free statistical methods and the concept of power
efficiency. Research Methods in the Behavioral Sciences (edited by L. Fes
tinger and D. Katz). Dryden Press, New York, 1953.
476 CHAPTER 15, DISTRIBUTION-FREE METHODS
30. Swed, Frieda S., and Eisenhart, C. Tables for testing randomness of group
ing in a sequence of alternatives. Ann. Math. Stat., 14:83-86, 1943.
31. Wilcoxon, F. Individual comparisons by ranking methods. Biometrics,
1:80-83, 1945.
32. . Probability tables for individual comparisons by ranking methods.
Biometrics, 3:119-22, 1947.
33. . Some Rapid Approximate Statistical Procedures. American Cyanamid
Co,, Stanford, Conn., 1949.
34. Wilks, S. S. Order statistics. Bui. Amer. Math. Soc., 54:6-50, 1948.
CHAPTER 16
STATISTICAL QUALITY CONTROL
SINCE THE EARLY 1940's the use of statistical methods in industry has
been on the upswing. This increased use of statistics has been par
ticularly noticeable in two areas: (1) research and development and (2)
reliability and quality control. Because the major part of this book
has been concerned with those statistical methods that have proved
most valuable in research and development, this chapter will be
devoted to a presentation of two special techniques especially useful
for controlling quality and reliability. (NOTE: This is not to imply
that these methods are of no value to those persons located in research
and development organizations; it is only a statement of relative
importance.)
The two techniques not discussed in the preceding chapters and
which are especially useful in controlling and improving quality and
reliability are control charts and acceptance sampling plans. Because
these two techniques are discussed at great length in books devoted
entirely to the subject of statistical quality control, only a brief out
line of each will be given. However, the material to be presented will
be sufficient to acquaint the reader with the basic concepts. Those
persons in need of more detailed explanations are referred to the publi
cations listed at the end of the chapter.
1 6.1 CONTRO L CHARTS
Although control charts may prove useful in many situations they
are most commonly employed in the analysis and control of production
processes. For this reason, discussion of these charts will be in terms
perhaps more familiar to the engineer than to the research worker.
It has long been recognized that some variation is inevitable in any
repetitive process. For example, Grant (16) states:
Measured quality of manufactured product is always subject to a certain
amount of variation as a result of chance. Some stable "system of chance
causes' 3 is inherent in any particular scheme of production and inspection.
Variation within this stable pattern is inevitable. The reasons for variation
outside this stable pattern may be discovered and corrected. . 1
Taking Grant at his word, what we seek, then, are tests for detecting
unnatural patterns in the plotted data.
The control chart, as conceived and developed by Shewhart (21), is
"a simple pictorial device for detecting unnatural patterns of variation
in data resulting from repetitive processes. That is, the control chart
1 E. L. Grant, Statistical Quality Control, Second Edition, McGraw-Hill Book
Company, Inc., KTew York, 1952, p. 3.
C477J
478 CHAPTER 16, STATISTICAL QUALITY CONTROL
provides criteria for detecting lack of statistical control. (NOTE:
When a process is operating under a constant system of chance causes,
it is said to be in statistical control.) Rather than go into details con
cerning the theory underlying control charts, we shall be content with:
(1) indicating the proper procedures for constructing the charts, (2)
stating criteria to be used for indicating unnatural patterns of vari
ation, and (3) giving numerical examples of the four most commonly
used charts.
Basically, all control charts appear as in Figure 16.1. The sample
points are, of course, plotted in a sequential manner, that is, as ob-
UPPER CONTROL LIMIT
o
1
/^-^ / \ I \
CENTER
o:
LJ
g
ID
O
I
CO
o:
O
\
LINE
-LOWER CONTROL LIMIT
1 2 3 A 5 6 7 8 9 1O 11 12
SAMPLE NUMBER
FIG- 16.1 -Illustration of the general appearance of a control chart.
tained. The plotted points are joined together solely as an aid to the
visual interpretation.
ISTow, what tests should be employed to detect unnatural patterns in
the plotted data? Depending on the degree of effectiveness desired,
there are many criteria that might be used. However, since our sole
purpose is to indicate the basic nature of the control chart technique,
only the most common tests will be mentioned. As mentioned earlier,
those persons wishing to investigate more sophisticated tests should
consult the references at the end of the chapter.
The most common tests for unnatural patterns are tests for insta
bility, that is, tests for determining if the cause system is changing. As
commonly employed, they refer to the A, B, and C zones shown in
Figure 16.2 With reference to these zones, the observed pattern of
variation is said to be unnatural, or the process is said to be "out of
16.1 CONTROL CHARTS 479
control/' if any one or more of the following events occurs: 2
(1) A single point falls outside of the control limit; i.e., beyond
zone A.
(2) Two out of three successive points fall in zone A or beyond.
(3) Four out of five successive points fall in zone B or beyond.
(4) Eight points in succession fall in zone C or beyond.
It should be noted that the above tests apply to both halves of a control
chart but they are applied separately to each half, not to the two halves
in combination.
CONTROL LIMIT
ZONE A
ZONE B
ZONE C
^ - CENTER LINE
FIG. 16.2 Diagram defining the A, B, and C zones used in control
chart analyses. (Each of the zones. A, B, and C, constitutes
one-third of the area between the center line
and the control limit.)
Before presenting numerical illustrations of control chart applica
tions, it is necessary that the four most commonly used charts be intro
duced and that formulas be given for the calculation of the center lines
and control limits. The charts most often encountered are: (1) the ~X
chart, (2) the R chart, (3) the p chart, and (4) the c chart. The first two
of these charts deal with measurement data while the last two deal
with attribute (enumeration) data. The pertinent assumptions and the
formulas for the control limits are specified in Table 16.1.
Example 16.1
Consider the data of Table 16.2. It may be verified that If T^SVfc
= 213,20/20 = 10.66 and R=* J2R/k = 31. 8/20 = 1.59. Then, using the
2 The tests described here may be used when the two control limits are reason
ably symmetrically located with respect to the center line. If the limits are de
cidedly asymmetric, the tests should be modified as described in (33).
48O
CHAPTER 16, STATISTICAL QUALITY CONTROL
TABLE 16.1 Assumptions and Formulas for the Most Commonly Used
Control Charts
Chart
Assumed
Distribution
Center
Line
Upper Control
Limit (UCL)
Lower Control
Limit (LCL)
~x
Normal
*X
X+A^R
XAolt
R
Normal
R
D*R
D*R
Binomial
P
p+3\/p(lp)/n
p 3Vp(l p)/n
c
Poisson
~c
Z-hS-x/f
"c 3\/zF
The constants A 2 , Z> 3y and D 4 are given in Appendix 8, while the quantities X, R, $,
and c are calculated from the sample data as shown in the numerical examples which
follow.
formulas given in Table 16.1, we see that, for the 3T chart, UCL = 10. 66
+ (0.58) (1.59) = 11,58 and LCL = 10.66- (0.58) (1.59) =9.74. Similarly,
for the R chart, UCL = (2.11)(1.59) =3.35 and LCL - (0)(1.59) =0. The
resulting charts are shown in Figure 16.3. The tests for unnatural
patterns have been applied to the X chart and the potential trouble
TABLE 16.2-Coded Values of the Crushing Strengths of Concrete Blocks
Sample
Number
Individual
Values
Mean
(3)
Range
X, X 2 X, X 4 X 5
1
11.1
9.6
9.7
10.1
12.4
10.1
11.0
11.2
10.6
8.3
10.6
10.8
10.7
11.3
11.4
10.1
10.7
11.9
10.8
12.4
9.4
10.8
10.0
8.4
10.0
10.2
11.5
10.0
10.4
10.2
9.9
10.2
10.7
11.4
11.2
10.1
12.8
11.9
12.1
11.1
11.2
10.1
10.0
10.2
10.7
10.2
11.8
10.9
10.5
9.8
10.7
10.5
10.8
10.4
11.4
9.7
11.2
11.6
11.8
10.8
10.4
10.8
9.8
9.4
10.1
11.2
11.0
11.2
10.5
9.5
10.2
8.4
8.6
10.6
10.1
9.8
11.2
12.4
9.4
11.0
10.1
11.0
10.4
11.0
11.3
10.1
11.3
11.0
10.9
9.8
11.4
9.9
11.4
11.1
11.6
10.5
11.3
11.4
11.6
11.9
10.44
10.46
9.98
9.82
10.90
10.36
11.32
10.86
10.58
9.52
10.56
9,96
10.44
10.96
11.14
10.04
11.44
11.84
11.14
11.44
1.8
1.4
0.7
2.6
2.4
1.1
0.8
1.2
0.5
1.9
1.5
2.4
2.8
1.0
1.5
0.8
2.1
1.0
2.7
1.6
2
3
4
5
6
7
8
9. .
10 .
11,
12
13
14
15
16.,..
17
18
19
20
Average
10.66
1.59
16.1 CONTROL CHARTS
481
10 -
12345
1O
SAMPLE NUMBER
20
2345
10
SAMPLE NUMBER
15
2O
FIG. 16.3-Control charts for the data of Table 16.2.
spots tagged in the customary manner. It is suggested that the reader
consider the application of these tests to the R chart.
Example 16.2
Consider the data of Table 16.3. Here we are dealing with enumera
tion data, namely, the number of defective fuses in samples of size 50
taken at random times during the production process. It is easily verified
that p= ^Cp/fc^ 1 - 68 / 40 ^ - 042 - Usin the formulas specified in Table
16.1, we obtain
0.127
UCL = 0.042 + 3VC0.042) (0.958) /SO =
and
LCL = 0.042 - 3V(0.042)(0.958)/50 = - 0.043.
(NOTE: Since the formula leads to a negative value for the lower control
limit and because the fraction defective is a nonnegative quantity, the
lower control limit is arbitrarily set at 0. This, of course, makes the
control limits asymmetric with respect to the center line, The tests for
TABLE 16.3-Number of Defective Fuses in
Random Samples of Size 50
Sample Number
Number of
Defectives
Fraction
Defective (p)
1
2
0.04
2
1
0.02
3. .
2
0.04
4
0.00
5
2
0.04
6
3
0.06
7
4
0.08
8
2
0.04
9. .
0.00
10
3
0.06
11
0.00
12
1
0.02
13
2
0.04
14
2
0.04
15
3
0.06
16
5
0.10
17
1
0.02
18
2
0.04
19
3
0.06
20
1
0.02
21
1
02
22. . ..
1
0.02
23
4
0.08
24
2
0.04
25
2
04
26
4
0.08
27
1
0.02
28
3
0.06
29
3
0.06
30
2
0.04
31
3
0.06
32
6
0.12
33
2
0.04
34
3
0.06
35
2
0.04
36
3
0.06
37
1
0.02
38
0.00
39
2
0.04
40
0.00
Average
0.042
16.1 CONTROL CHARTS 483
.= 0.127
. p=tO.O-42
T7T7 \ /\ A' V \ / " \/ v * v \ X
O.O2
" .... .. r - ^ . 1 , . , , . . . . . , j . . . _j j. ... m. . _ f i m ._ .. _j I
12345 10 15 20 25 3O 35 AO
SAMPLE NUMBER
FIG. 16.4 Control chart for the data of Table 16.3,
unnatural patterns specified earlier have, therefore, not been applied.
The application of the modified tests is left as an exercise for the reader.
The results of the preliminary analysis are presented in Figure 16.4.)
Example 16.3
Consider now a situation in which the characteristic of interest is the
number of defects per unit. In such a case, a Poisson distribution would
undoubtedly be assumed and a c chart would be appropriate. Given the
data in Table 16.4, it is easily verified that c= T^c/fc = 144/24 = 6.
Using the formulas specified in Table 16.1, we obtain UCL==6 + 3VB
= 13.35 and LCL = 6 3v^= 1.35. (NOTE: As in Example 16.2, the
negative lower control limit will arbitrarily be changed to since c is,
by definition, a nonnegative quantity.) The results of the analysis are
plotted in Figure 16.5.
12345 1O 15 2O 25
SAMPLE NUMBER
FIG. 16.5 Control chart for the data of Table 16.4.
484
CHAPTER 16, STATISTICAL QUALITY CONTROL
TABLE 16.4r-Number of Defects Observed in a Welded Seam (Each Count
Taken on a Single Seam; the Welder produced 8 Seams per Hour)
Sample
Number
Date
Time of
Sample
Number of
Defects (c)
1
July 18
8:00 A.M.
2
2
9:05 A.M.
4
3
10:10 A.M.
7
4
11 :00 A.M.
3
5
12 :30 P.M.
1
6
1 :35 P.M.
4
7
2:20 P.M.
8
8
3:30 P.M.
9
9
July 19
8:10 A.M.
5
10
9:00 A.M.
3
11
10:05 A.M.
7
12
11:15 A.M.
11
13
12:25 P.M.
6
14
1 :30 P.M.
4
15
2 :30 P.M.
9
16
3:40 P.M.
9
17
July 20
8:00 A.M.
6
18
8:55 A.M.
4
19
10:00 A.M.
3
20
11:10 A.M.
9
21 ...
12:25 P.M.
7
22
1 :30 P.M.
4
23
2:20 P.M.
7
24
3:30 P.M.
12
Total
144
Source: E. L. Grant, Statistical Quality Control, Second Edition, McGraw-Hill Book
Company, Inc., New York, 1952, p. 33. By permission of the author and publishers.
Examination of Figure 16.5 will reveal that some modifications have
been made in the usual method of presentation, namely, the various
half-days have been separated (i.e., not connected) to emphasize the
breaks between work periods. Now, it will be noted that two conclusions
are obvious: (1) no points plot above the upper control limit, and (2)
essentially the same pattern appears in each half-day. Study of the
recurring pattern suggests the existence of a fatigue factor. [NOTE: For
further discussion of this example, consult Grant (16, pp. 3235),]
Before leaving the topic of control charts, some additional remarks
are necessary. These will, however, be very brief and in no particular
order.
1. The primary reason for using control charts is to provide a signal
that some action is desirable.
16.2 ACCEPTANCE SAMPLING PLANS 485
2. Before control limits may be calculated with, any assurance of their
being reliable, at least 20 subgroups (samples) should be available.
3. Before the control limits, calculated from past production records,
are used to monitor future production, the process should be in
control.
4. If a process is in control and a point plots outside the control limits,
the taking of action may be viewed as committing a Type I error.
5. The control chart is not a panacea for all production problems; it is
only another useful tool.
16.2 ACCEPTANCE SAMPLING PLANS
Acceptance sampling, or sampling inspection, is of two types : lot-by-
lot sampling and sampling of continuous production. In this brief ex
posure to the concepts and procedures of acceptance sampling, only the
first of these two types will be discussed. Persons desiring information
on continuous sampling plans are referred to the publications listed at
the end of the chapter. In addition to the distinction between lot-by-
lot and continuous sampling, it is customary to classify sampling plans
as either attributes or variables plans. Attributes plans refer to those
cases in which eacli item is classified simply as eitlier defective or non-
defective ; variables plans refer to those cases in which a measurement
is taken and recorded (numerically) on each item inspected. In this
section only attributes plans will be considered. There is one other way
in which acceptance sampling plans may be classified, namely, as
single, double, or multiple (including sequential} plans. These categories
refer, of course, to the number of samples selected and will become
clear as the exposition continues.
An attributes single sampling plan which operates on a lot-by-lot basis
is completely defined by three numbers: the lot size, AT; the sample
size, n; and the acceptance number, a. 3 Such a plan operates as follows:
(1) A single sample of n items is selected, by chance, from a lot of
N items.
(2) Each item in the sample is then classified as either defective
or nondefective.
(3) If the number of defective items in the sample does not exceed
a, the lot is accepted.
(4) If the number of defective items in the sample exceeds a, the
lot is rejected.
As with all statistical procedures, the risks associated with decisions
(inferences) resulting from sampling inspection must be assessed. The
customary manner of presenting these risks is by means of a graph of
the OC function of the sampling plan, that is, by plotting the prob
ability of accepting the lot as a function of the fraction defective of the
lot. The protection afforded by various sampling plans may then be
compared by examining their OC curves.
3 In many publications, the acceptance number is denoted by c. However, I
prefer a for two reasons: it stands for the word acceptance and it permits an
easier extension to double and multiple sampling.
486 CHAPTER 16, STATISTICAL QUALITY CONTROL
For the single sampling plan specified previously, the OC function is
= C(D, d}-C(N - D,n- d}/C(N, n) (16.
where D represents the number of defective items in the lot and d
represents the number of defective items in the sample. Equation (16.1)
may, of course, be evaluated for Z) = 0, 1, - - * , N and the results
plotted as a series of ordinates erected at the corresponding values of
p = D/N, namely: 0, 1/-ZV, 2/N, , 1. It is clear, however, that these
calculations may prove onerous unless a high-speed computer is avail
able. Fortunately, helpful tables of the hypergeometric function have
been provided by Lieberman and Owen (19). In addition, extensive
catalogs of OC curves for lot-by-lot, single sampling (by attributes)
plans have been prepared by Wiesen (34) and Clark and Koopmans (9) .
If none of these publications is readily available and access to a high
speed computer is impossible, the OC function may be approximated by
C(n, d)p d (l - p^~ d (16.2)
d=Q
or
Pace S* 2L, e-^\ d /d\ (16.3)
where X = up. (NOTE : The reader is referred to Chapter 5 for discussion
of the accuracy and relevancy of these approximations.)
Persons familiar with the presentation of OC curves for sampling
plans, or those individuals who consult the references given in the
preceding paragraph, will realize that it is not customary to plot OC
functions as a series of ordinates (as stated in the preceding paragraph)
but to show smooth OC curves. That is, the functions are plotted as
though p were a continuous parameter. For lot-by-lot plans, this
minor "tampering with the truth" is not serious and the resulting gain
in ease of presentation far outweighs the inaccuracy of the graph.
Consequently, OC curves usually appear as in Figure 16.6.
Rather than plot the entire OC function, the practitioner frequently
contents himself with calculating two or three points on the curve. The
three points most commonly determined are:
Pi = the value of p for which P acc = 0.95,
p% = the value of p for which P acc = 0.50, and
Pz = the value of p for which P acc = 0.10.
These three values of p (that is, pi, p 2 , and z> 3 ) are usually referred to as
the acceptable quality level (AQL), the indifference quality } and the
16.2 ACCEPTANCE SAMPLING PLANS
487
PRODUCER'S RISK
:ONSUMER I S RISK
AQL RQL
p=LOT FRACTION DEFECTIVE
FIG. 1 6.6 Illustration of the general appearance of an OC curve,
(To aid in understanding the concepts, the AQL, RQL, consumer's
risk, and producer's risk are shown in this figure.)
rejectable quality level (RQL), 4 respectively. The AQL and RQL points,
as well as the associated expressions, consumer's risk and producer's
risk have been shown on Figure 16.6. (NOTE : If, in the definitions of p a
and j>3, the values 0.95 and 0.10 are replaced by 1 a. and 0, respec
tively, the reader will immediately see the close connection between
the ideas of this section and those discussed in Chapter 7. Incidentally,
the same remark may be made here as there, namely, the values as
signed to a, and are arbitrary; the use of = 0.05 and = 0.10 is only
a matter of custom.)
One other common way of presenting the performance ability of
acceptance sampling plans is to calculate and plot the average outgoing
quality (AOQ) function. This function, restricted to cases in which the
testing is nondestructive, depends on the assumption that all rejected
lots are submitted to 100 per cent inspection, with all defective items
being removed and replaced by nondefective items. Under this as
sumption, the average outgoing quality is determined to be
AOQ
(16.4)
4 Historically, the rejectable quality level (RQL) has been known as the lot
tolerance per cent defective (LTPD). However, in recent years, the expressions
rejectable quality level and objectionable quality level (OQL) have been suggested,
and I believe that rejectable quality level will soon be universally adopted. For
this reason, it is used in this book.
488
CHAPTER 16, STATISTICAL QUALITY CONTROL
where P acc is defined by Equation (16.1). The graph of a typical AOQ
function is shown in Figure 16.7. The maximum value of the average
outgoing quality is known as the average outgoing quality limit (AOQL).
From a practical point of view, the AOQL is, perhaps, the most im
portant descriptive measure associated with any acceptance sampling
plan. [NOTE: AOQ curves are also included in the catalogs of Wiesen
(34) and Clark and Koopnaans (9) . ]
AOQ
AOQL
O
P*LOT FRACTION DEFECTIVE
FIG. 16.7 Illustration of the general appearance of an AOQ curve.
Let us now consider double sampling plans. An attributes double
sampling plan which operates on a lot-by-lot basis is completely defined
by six numbers: the lot size, N; the size of the first sample, ni; the
acceptance number associated with the first sample, a\\ the rejection
number associated with the first sample, rij the size of the second
sample, n^* and the acceptance number associated with the combined
samples, a 2 . Such a plan operates as follows:
(1) A sample of n items is selected, by chance, from a lot of N
items.
(2) Each item in the sample is then classified as either defective or
nondef ective .
(3) If the number of defective items in the sample does not exceed
ai, the lot is accepted.
(4) If the number of defective items in the sample equals or ex
ceeds ri, the lot is rejected.
(5) If the number of defective items in the sample exceeds ai but
is less than ri, a second sample (of n 2 items) is selected, by
chance, from the remainder of the lot.
(6) If the number of defective items in the two samples combined
does not exceed a 2 , the lot is accepted.
(7) If the number of defective items in the two samples combined
exceeds a*, the lot is rejected.
Rather than go into the same detail as was done for single sampling,
only the bare essentials will be presented. The OC function is
16.2 ACCEPTANCE SAMPLING PLANS 489
r-l
C(D, dd-C(N - D, m - dd/C(N, + 2^ C(D, d$
C(N D,n-
d 2 =0
C(N ni D + di, n* d$/C(N n ly n%) (16.5)
where di represents the number of defective items in the first sample
and d 2 represents the number of defective items in the second sample.
Subject to the usual restrictions, this may be approximated by
C(n 2? J 2 )^(l p)-2-^2 (16.6)
^2=0
or
a i r i~ 1 a z~ d i
Paoc^ 2 - Xl xtV^il+ 12 - Xl xtV^iI S r-x*xtV^2l (16.7)
di=0 d 1 =a 1 4-l ^2=0
where \i = n^p and \2= : n 2 p. As before, the average outgoing quality is
given by
AOQ = p-P acc (16.8)
where P occ is defined by Equation (16.5). In addition, since the total
sample size is now a variable, it is also possible to assess the relative
"costs" of sampling plans by comparing their expected sample sizes.
To summarize, it is customary to calculate the average sample number
(ASN). Proceeding on the assumption that every item in each sample
is inspected, the average sample number for a double sampling plan is
given by
ASN = m + n 2 3 C(D,dJ-C(N Z>, i d^/C(N,n^. (16.9)
<Zl*=ai+l
The graph of a typical ASN function for a double sampling plan is
shown in Figure 16.8, where it is compared with the constant sample
size of an equivalent (in terms of the OC function) single sampling
plan.
Before proceeding to multiple sampling plans, it seems desirable to
list the advantages and disadvantages of double sampling plans rela
tive to single sampling plans. These are:
Advantages
(1) They have the psychological advantage of giving doubtful
lots a second chance.
490
CHAPTER 16, STATISTICAL QUALITY CONTROL
(2) On the average, they have (for the two extremes of good and
bad quality) the advantage of requiring fewer inspections.
Disadvantages
(1) They are said to be more difficult to administer. (Personally,
I do not subscribe to this point of view.)
(2) The inspection load is variable.
(3) The maximum number of inspections can (for intermediate
quality) exceed that for comparable single sampling plans.
ASN
DOUBLE
SAMPLING
n
SINGLE
SAMPLING
p = LOT FRACTION DEFECTIVE
FIG. 1 6.8 Illustration of the general appearance of ASN functions
for single and double sampling plans that exhibit
essentially the same OC curve.
Since there is little, if anything, new in multiple sampling plans that
has not been discussed in connection with single and double sampling
plans, the following remarks will be very brief. If one extends the con
cepts and procedures of double sampling to k > 2 samples, it may easily
be seen that such plans are completely defined by the numbers: AT,
ni, - , n k , ai, - - - , a*, ri, - - , r k = a,k+l' Proceeding in a manner
analogous to that followed for double sampling plans, one may obtain
OC, AOQ, and ASN functions. [NOTES: (1) The calculations will,
however, be quite involved and lengthy. (2) If each n,-= 1 and if k > oo 9
the multiple plan is usually referred to as a sequential plan.] For those
persons desiring more details on multiple and sequential acceptance
sampling plans, the appropriate references at the end of the chapter
are recommended.
Example 16.4
Suppose that lots of size 100 are submitted for acceptance. A sample
of size 2 is drawn from a lot and the lot is accepted if both the sample
items are nondefective; if one or both are defective, the lot is rejected.
The OC function for this plan is
Pace = CCA 0)-C(100 - D, 2)/CC100, 2)
and this may be evaluated for Z> = 0, 1, - - , 100.
PROBLEMS 491
Example 16.5
With reference to Example 16.4, it is easily verified that the AOQ
function is
AOQ = p-P^c - [Z>/100][C(D,0)-C(100 - D, 2)/C(100 3 2)]
where, once again, this may be evaluated for Z> = 0, 1, * , 100.
Problems
16.1 Define and discuss each of the following terms: (a) quality, (b) con
trol, (c) quality control, (d) statistical quality control.
16.2 Why should both the ~5c chart and the R chart be used when dealing
with measurement (as opposed to attributes) data?
16.3 A control chart for ~X has the upper control limit (UCL) and the
lower control limit (LCL) equal to 24.23 and 23.99, respectively.
The calculations were based on samples of size four. If the engineering
specifications had been given as 24.10 0.20, what percentage of the
sample means would you expect to plot out of control? State any
assumptions you find it necessary to make.
16.4 Given the following data, construct and interpret the appropriate
control charts.
SAMPLE MEANS AND RANGES (w 10) FOR LENGTH
OF LIFE OF LIGHT Bulges
(CODED DATA) __
Sample No. Mean (5T) Range (R)
1
69.4
45
2
63.4
48
3
55.0
72
4
64.0
48
5
57.4
36
6
82.0
81
7
85.0
78
8
33.4
42
9
46.0
69
10
112.4
84
11
93.8
48
12
95.6
75
13
117.8
51
14
113.6
84
15
74.8
54
16
80.8
45
17
71.8
57
18
53.2
75
19
74.8
48
20
59.2
63
21
65.8
129
22
109.6
42
23
44.2
51
24
73.6
51
25
51.4
27
Total 1848.0 1503
492 CHAPTER 16, STATISTICAL QUALITY CONTROL
16.5 Given the following sample data, construct control charts for the
sample mean and the sample range, that is, X and R charts, and
interpret the results.
Sample
1
2
3
4
5
6
7
8
9
Individual values. .
+ 2
+3
+ 1
1
1
+ 2
1
+ 1
-3
+ 2
+ 1
3
+4
+ 1
1
+ 3
2
+2
2
+ 1
2
1
^
+ 1
3
1
1
+1
3
+ 3
+ 1
Sample
10
11
12
13
14
15
16
17
18
Individual values. .
+ 1
2
+ 2
+ 2
2
+ 3
2
3
1
2
+4
1
+2
+ 1
2
+ 2
+ 1
4
1
+2
+ 1
+ 1
2
1
2
1
1
+ 1
+3
5
1
+3
+ 2
Sample
19
20
21
22
23
24
25
Individual values . .
2
+2
1
+ 1
2
1
+ 2
1
+ 1
1
1
-1
1
+ 1
+ 2
+ 1
4
1
+ 2
1
+ 1
_ i
3
1
1
+ 1
+ 1
+ 1
2
+ 1
PROBLEMS
493
16.6 Construct and interpret the appropriate control charts for the fol
lowing data.
Group No.
(-) (V (*> (d) (e)
X
R
1
.831
.829
.836
.840
.826
.8324
.014
2
.834
.826
.831
.831
.831
.8306
.008
3
.836
.826
.831
.822
.816
.8262
.020
4
.833
.831
.835
.831
.833
.8326
.004
5
.830
.831
.831
.833
.820
.8290
.013
6
.829
.828
.828
.832
.841
.8316
.013
7
.835
.833
.829
.830
.841
.8336
.012
8
.818
.838
.835
.834
.830
.8310
.020
9
.841
.831
.831
.833
.832
.8336
.010
10
.832
.828
.836
.832
.825
.8306
.011
11
.831
.838
.844
.827
.826
.8332
.018
12
.831
.826
.828
.832
.827
.8288
.006
13
.838
.822
.835
.830
.830
.8310
.016
14
.815
.832
.831
.831
.838
.8294
.023
15
.831
.833
.831
.834
.832
.8322
.003
16
.830
.819
.819
.844
.832
.8288
.025
17
.826
.839
.842
.835
.830
.8344
.016
18
.813
.833
.819
.834
.836
.8270
.023
19
.832
.831
.825
.831
.850
.8338
.025
20
.831
.838
.833
.831
.833
.8332
.007
21
.823
.830
.832
.835
.835
.8310
.012
22
.835
.829
.834
.826
,828
.8304
.009
23
.833
.836
.831
.832
.832
.8328
.005
24
.826
.835
.842
.832
.831
.8332
.016
25
.833
.823
.816
.831
.838
.8282
.022
26
.829
.830
.830
.833
.831
.8306
.004
27
.850
.834
.827
.831
.835
.8354
.023
28
.835
.846
.829
.833
.822
.8330
.024
29
.831
.832
,834
.826
.833
.8312
.008
494 CHAPTER 16, STATISTICAL QUALITY CONTROL
16.7 Using the data below, calculate limits and plot the ~X and R charts.
Apply the standard tests for unnatural patterns and discuss the
results. (NOTE: The sample size is 7^ = 5.)
X R X R
1
1.444
.09
26
1.424
.05
2
1.427
.08
27
1.434
.05
3
1.464
.08
28
1.414
.09
4
1.455
.08
29
1.406
.07
5
1.462
.10
30
1.418
.14
6
1.448
.05
31
1.438
.09
7
1.454
.04
32
1.416
.07
8
1.446
.08
33
1.419
.06
9
1.437
.12
34
1.406
.08
10
1.471
.11
35
1.428
.06
11
1.438
.09
36
1.430
.06
12
1.438
.05
37
1.421
.07
13
1.415
.12
38
1.434
.07
14
1.428
.12
39
1.408
.05
15
1.425
.08
40
1.414
.08
16
1.440
.09
41
1.410
.03
17
1.430
,05
42
1.406
.06
18
1.457
.09
43
1 .405
.07
19
1.444
.09
44
1.419
.10
20
1.432
.05
45
1.410
.07
21
1.438
.05
46
1.420
.05
22
1.404
.10
47
1.414
.05
23
1.409
.05
48
1.426
.11
24
1.400
.07
49
1.386
.06
25
1.425
.09
50
1.387
.08
PROBLEMS
495
16.8 Following are the number of defective piston rings inspected in 23
daily samples of 100. Calculate the control limits for the type of
control chart that should be used with these data. Interpret the
results.
Date
Number Defective
1
9
2
5
3.
10
4
10
5 ....
13
8
10
9
13
10
2
11
1
12 .
3
15
2
16
2
Date
Number Defective
17... . . .
3
18
6
19
3
22
3
23
3
24
25
5
26
5
29
3
30
2
31 . . . .
4
496 CHAPTER 16, STATISTICAL QUALITY CONTROL
16.9 Given the following data, construct the appropriate control chart.
Interpret the results.
NTJMBEU or DEFECTIVES IN SAMPLES or SIZE 100
Sample No.
Number Defective (cf)
1
3
2 . ....
1
3
4
4
4
5.
4
6
6
7
5
8
5
9
2
10
4
11. .
3
12
4
13
4
14. .
3
15
5
16
8
17
2
18
3
19
5
20
4
21
3
22
4
23
6
24
4
25
3
26
5
27
4
28
7
29
6
30
5
31
5
32
6
33
4
34
9
35
6
36
4
37
3
38
1
39
3
40 ; . .
1
PROBLEMS
497
16.10
At a certain point in the assembly process, TV sets are subjected to
a critical inspection. The following data resulted from the inspection
of 25 randomly selected sets. Plot and interpret the appropriate con
trol chart.
Set No.
Number of Defects
per Set
Set No.
Number of Defects
per Set
1
12
14
7
2
11
15
4
3
13
16
9
4
17
17
15
5
7
18
12
6
10
19
11
7
9
20
9
8
8
21
4
9
13
22
4
10
15
23
10
11
18
24
12
12
9
25
4
13
14
16.11 Phonograph records were selected at random times from a production
line, Given the following data, construct and interpret the appropri
ate chart.
Record No,
Number of Defects
per Record
Record No.
Number of Defects
per Record
1. . . ....
1
16
20
2
1
17
1
3
3
18
6
4. . . . ....
7
19
12
5
8
20
4
6
1
21
5
7 .
2
22
1
8
6
23
8
9
1
24
7
10 .
1
25
9
11
10
26 ,
2
12
5
27
3
13
28
14
14
19
29
6
15
16
30
8
16.12 (a) Plot the OC curve for Example 16.4.
(6) Determine the AQL and RQL points.
498 CHAPTER 16, STATISTICAL QUALITY CONTROL
16.13 (a) Plot the AOQ curve for Example 16.5.
(6) Determine the AOQL.
16.14 Discuss the basic purpose of acceptance sampling and the general
methods by which this purpose is approached.
16.15 Plot, on the same graph, the following OC curves:
(a) 7^ = 500, n = 5Q, a =
(6) AT
(c) AT
16.16 Plot, on the same graph, the following OC curves:
(a) AT = 500, ^ = 25, a = l
(6) AT = 500, 7i = 50, a = l
(c) N = 500, n = 100, a = 1
16.17 For the double sampling plan specified by AT 500, ni = 50, n 2 = 100,
ai = 5, n = 14, and a 2 = 13, determine and plot the OC, AOQ, and ASN
functions. Find the value of the AOQL.
16.18 The following truncated sequential sampling plan is proposed:
(a) If the first item is defective, reject the lot;
(6) If the first item is nondefective and there is no more than one
defective up to and including the tenth item, accept the lot;
(c) If the first item is nondefective and there are two defectives
prior to or including the tenth item, reject the lot.
Determine the OC and ASN functions.
References and Further Reading
1. American Society for Testing Materials. ASTM Manual on Quality Control
of Materials. Special Technical Publication 15-C, Philadelphia, Pa., 1951.
2. American Standards Association, Inc., Guide for Quality Control. Zl. 1-1958,
New York, 1958.
3 m Control Chart Method of Analyzing Data. Zl. 2-1958. New York, 1958.
4. . Control Chart Method of Controlling Quality During Production.
Zl.3-1958, New York, 1958.
5. Bazovsky, I. Reliability Theory and Practice. Prentice-Hall, Inc., Englewood
Cliffs, N.J., 1961.
6. Bowker, A. H., and Lieberman, G. J. Engineering Statistics. Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959.
7. Burr, I. W. Engineering Statistics and Quality Control. McGraw-Hill Book
Company, Inc., New York, 1953.
8. Chorafas, D. N. Statistical Processes and Reliability Engineering. D. Van
Nostrand Company, Inc., New York, 1960.
9. Clark, C. R., and Koopmans, L. H. Graphs of the Hyper geometric O.C. and
A.O.Q. Functions for Lot Sizes 10 to %&5. Sandia Corporation Monograph
SCR-121, Sandia Corp., Albuquerque, N. Mex., Sept., 1959.
10. Crow, E. L., Davis, F. A., and Maxfield, M. W. Statistics Manual. Dover
Publications, Inc., New York, 1960.
11. Dodge, H. F. A sampling inspection plan for continuous production. Ann.
Math. Stat., 14:264, 1943.
12. , and Romig, H. G. Sampling Inspection Tables: Single and Double
Sampling. Second Ed. John Wiley and Sons, Inc., New York, 1959.
13. Dummer, G. W. A., and Griffin, N. Electronic Equipment Reliability. John
Wiley and Sons, Inc., New York, 1960.
14. Duncan, A. J. Quality Control and Industrial Statistics. Revised Ed. Richard
D. Irwin, Inc., Homewood, 111., 1959.
15. Feigenbaum, A. V. Total Quality Control: Engineering and Management.
McGraw-Hill Book Company, Inc., New York, 1961.
REFERENCES AND FURTHER READING 499
; c . S^?^ ^^ ^^ **' McGraw-Hill Book
'
f ' J ' r "- Mc ^ fee >'N- J., Ryerson, C. M., and Zwerling, S. (editors)
ew York 1Q ^ SeC nd Ed ' Instit ^ of Radio Engineers, Inc.,"
20 " il 7 d ' ? K ;.' an Lipow, M. Reliability: Management, Methods and Mathe-
matics Prentice-Hall, Inc., Englewood Cliffs, N.J , 1962
21. Shewhart W A. Economic Control of Quality of Manufactured Product.
JJ. Van Nostrand Company, Inc., New York, 1931
22 ' . Q Statistical Method from the Viewpoint of Quality Control. The Gradu-
ate fachool, The Department of Agriculture, Washington, D.C., 1939.
lor, n in^:, Net W Y?rk, Z 'T9 e Il'. ^"^ f Statistical ^ethode. John Wiley and
24. Statistical Research Group, Columbia University. Selected Techniques of
Stato*ttcalA.nal y a t a. (Edited by C. Eisenhart, M. W. Hastay, and W. A.
Walhs). McGraw-Hill Book Company, Inc., New York 1947
Tvr + Sam ^ in ^ Z . ns P ec ^. on - (Edited by H. A. Freeman, M. Friedman, F.
Mosteller, and W. A. Walhs.) McGraw-Hill Book Company, Inc., New York,
J. 4
. .
26. United States Government, Department of Defense. Sampling Procedures
and Tables for Inspection by Variables for Percent Defective MIL-STD-414
Washington, D.C., 1957. * '
27 ' ~~. - ~ Multi-Level Continuous Sampling Procedures and Tables for Inspec-
tzon by Attrzbutes. Inspection and Quality Control Handbook (Interim)
H-106, Washington, D.C., 1958.
28 - - 1 - Mathematical and Statistical Principles Underlying Military Stand
ard 414. Technical Report, Washington, D.C., 1958.
29 ' ~^, - ' Sam P lin 9 Procedures and Tables for Inspection by Attributes. MIL-
STD-105C, Washington, D.C., 1961.
30. - . Statistical Procedures for Determining Validity of Suppliers' Attributes
Inspection. Quality Control and Reliability Handbook (Interim) H-109
Washington, D.C., 1960.
31. Wald, A. Sequential Analysis. John Wiley and Sons, Inc., New York, 1947.
32. ~- - , and Wolfowitz, J. Sampling inspection plans for continuous produc
tion which insure a prescribed limit on the outgoing quality. Ann. Math
Stat., 16:30, 1945.
33. Western Electric Company, Inc., Statistical Quality Control Handbook.
Second Ed. Western Ellectric Company, Inc., New York, 1958.
34. Wiesen, J. M. Extension of Existing Tables of Attributes Sampling Plans.
Sandia Corporation Technical Memorandum SCTM42-58(12)., Sandia Corp.,
Albuquerque, 1ST. Mex., Feb., 1958.
CH APTE R 17
SOME OTHER TECHNIQUES AND
APPLICATIONS
IN THE PRECEDING CHAPTERS those techniques most frequently em
ployed by users of statistical methods have been presented as integral
parts of a complete discipline. In this chapter a few special techniques
will be discussed and one or two interesting applications of probability
and/or statistics illustrated. It is hoped that these items will prove
useful to many of the readers and they they, the items, will stimulate
some of you to search out new and exciting applications in your own
area of specialization.
17-1 SOME PSEUDO t STATISTICS
Many times, the researcher may feel that the time and effort in
volved in the calculation of s, the sample standard deviation, is too
great for the benefit derived therefrom. Thus, it is not surprising that
techniques have been devised which use R, the sample range, in place
of s. One of these techniques involves the use of a pseudo t statistic
defined by
T, - & - ti/R. (17.1)
If we are willing to assume random sampling from a normal popula
tion, tests of hypotheses concerning ^ or confidence interval estima
tion of ju may be performed by considering the sampling distribution of
T-L and utilizing the values recorded in Table 1 of Appendix 17.
Example 17.1
Consider again the data and the hypothesis of Example 7.8. Using
Equation (17.1), we obtain r 1 = (1267 1260) /8 =0.875. Since this cal
culated value of r x exceeds the critical value (for n = 4), namely,
r i(.975) = -717, the hypothesis that /x = 1260 is rejected. (NOTE: This
is in agreement with the conclusion reached in Example 7.8)
Example 17.2
Utilizing the same data as in the preceding example, a 99 per cent
confidence interval for p. may be obtained as follows:
L = X - T I( 995y R 1267 1.316(8) = 1256.47
U * T + r 1(995y R * 1267 + 1.316(8) = 1277.53.
An even simpler statistic which may be used as a substitute for
"Student's" t is
[5003
17.3 EVOLUTIONARY OPERATION 501
(17.2)
Critical values of this statistic are given in Table 3 of Appendix 17.
Because of the similarity (in use) of this statistic to the one discussed
in the preceding paragraph, no numerical examples will be presented.
When dealing with the difference between two sample means, a third
pseudo t statistic may be utilized, namely,
r d = C^x - TsVCRx + 2). (17.3)
Critical values of r d are tabulated, for samples of equal size, in Table 2
of Appendix 17.
Example 17.3
Consider the data and hypothesis of Example 7.19. Using Equation
(17.3), we obtain r d = (4 7)/(7+7) = 3/14= 0.214. Since the cal
culated value to r d lies between T d (. 96 ) = .24:6 and r d( ^^^ = .246, we
are unable to reject the hypothesis that Mi =M2. (NOTE: This agrees with
the conclusion reached in Example 7.19,)
17.2 A PSEUDO F STATISTIC
As might be expected, it is also possible to utilize the range in place
of the standard deviation to provide a quick substitute for the familiar
F statistic. The suggested statistic is
= the ratio of the two sample ranges, (17.4)
and critical values are tabulated, for certain selected sample sizes, in
Table 4 of Appendix 17. As with F, the critical values for the lower end
of the distribution of Ri/R* may be found by interchanging ni and n 2
(the two sample sizes) and calculating the reciprocals of the tabulated
values. Because of the simplicity of this test and its similarity to those
discussed in the preceding section, no numerical examples will be given.
17.3 EVOLUTIONARY OPERATION
In Section 13.6 the reader was exposed to, but not indoctrinated in,
the subject known as "response surface techniques." In the present
section, a related technique wUl be introduced. This technique, known
as evolutionary operation (or EVOP), is an application of the concepts
of response surface methodology to the problem of improving the per
formance of industrial processes. Consequently, this technique should
be of great interest to those persons concerned with production proc-
Q^CI
Because no detailed discussion of response surface methodology was
undertaken in Chapter 13, a full description of EVOP cannot be given
here. However, in capsule form, the important elements of the tech
nique are:
(1) It is a method of process operation which includes a built-in
procedure for improving productivity.
502 CHAPTER 17, OTHER TECHNIQUES AND APPLICATIONS
(2) It uses some relatively simple statistical concepts.
(3) It is run during normal routine production by plant personnel.
(4) The basic philosophy of EVOP is: A process should be run
not only to produce product but also to provide information on
how to improve the process and/or product.
(5) Through the planned introduction of minor variants into the
process, the customary "static" operating conditions are
made "dynamic" in nature.
(6) Utilizing the elementary principles of response surface meth
odology and making small changes (in a prescribed fashion)
in the "controlled" variables of the process, the effects of the
forced changes can be assessed.
(7) If a simple pattern of operating conditions is employed (e.g.,
a 2 2 factorial plus a center point) 1 and if the operation of the
process under each of these conditions is termed a "cycle,"
the running of several cycles will yield sufficient information to
permit a judgment to be made as to what is a better nominal
set of operating conditions.
(8) Constant repetition of this program will lead to continual
improvement of the process.
(9) Two important items in an EVOP program are :
(a) All data should be prominently displayed on an Infor
mation Board.
(b) An Evolutionary Operation Committee (composed of re
search, development, and production personnel) should
make periodic reviews of the EVOP program if the maxi
mum benefit is to be derived from this approach.
Inasmuch as the preceding remarks are only a skeletal description
of the EVOP technique, those persons desiring more details must con
sult other sources. In particular, the publications by Box (3) and Box
and Hunter (4) are especially recommended.
17.4 TOLERANCES
In Section 5.14 some remarks were made with regard to the distri
bution of a linear combination of random variables. At this time, it is
appropriate to consider a specific application of those remarks,, namely,,
to the subject of tolerances.
Before embarking on this discussion, a distinction must be made
between specification limits (that is, a nominal value plus and minus
certain engineering tolerances) and natural, or statistical, tolerances. In
general, specification limits are set by the designer as a statement of
his requirements with respect to a certain dimension. Thus, in many
cases, the specification limits have little connection with production
capabilities. On the other hand, statistical tolerance limits reflect the
1 This center point will usually be that nominal set of operating conditions
specified in the engineering drawings or production manual.
17.4 TOLERANCES 503
capabilities of the process producing the dimensions in question. (See
Sections 6.11, 6.12, and 6.13). Therefore, from the point of view of
managerial decision making, the subject of statistical tolerances is of
great importance, for it bears directly on the success or failure of the
company's product.
In what follows, we are interested only in general concepts and a
method of approach that may prove helpful in the design of complex
equipment. According 
