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METHODS OF 
STATISTICAL ANALYSIS 



METHODS OF 

STATISTICAL ANALYSIS 



BY 

C. H. GOULDEN 

Senior Agricultural Sciential, Dominion Rutt Research Laboratory 

Honorary Lecturer in StatUtiet, Unitenity of Manitoba 

Winnipeg, Manitoba 



NEW YORK 
JOHN WILEY & SONS, INC. 

LONDON: CHAPMAN & HALL, LIMITED 



IN THE REPRINTING OF THIS BOOK, THE RECOMMEN- 
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AND OTHER IMPORTANT WAR MATERIALS. THE 
CONTENT REMAINS COMPLETE AND UNABRIDGED. 



COPYRIGHT, 1939 

BT 

CYRIL H. GOULDEN 



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be reproduced in any form without 
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PRINTED IN U. B. A. 



PREFACE 

From several years' experience in teaching classes in statistics and 
giving advice at various times to experimentalists, I have come to the 
conclusion that there is a distinct need for more than one type of text- 
book. On the one hand there are many who are interested only in 
knowing something of the theory and principles. In this class we find 
students who are endeavoring to obtain a broad knowledge of all sub- 
jects related to science and art, practicing technicians such as doctors 
of medicine and technical advisers in agriculture, and adminislirators of 
research activities. It would be idle to set students of this type to 
work on laborious practical examples. It would probably discourage 
them at the start, and by absorbing time would reduce the possibility of 
teaching them some of the very attractive philosophical phases of the 
subject. In a maze of calculations the principles might be lost sight of 
completely, and the student would emerge with a technique for mechan- 
ical operations and no ability to solve actual problems. At the begin- 
ning it is not training in actual methods that is required, but the build- 
ing up of a sound knowledge of fundamental principles. 

On the other hand, we have an increasing number of students who, 
having had some elementary training in statistics and some experience 
in research work, come to the point finally of requiring a practical 
knowledge of methods of analysis and some facility in the devices of 
calculation. There is no denying the fact that two or three years spent 
in studying the principles and theory of statistics will not fit the student 
to solve practical problems. To do so is to ignore the many complica- 
tions that are involved and that training in facility is necessary in order 
that statistical computations may be attacked with determination and 
completed in a reasonable length of time. One of the objections very 
often raised to the use of statistical methods is the time necessary to do 
the routine work. Frequently this sort of thing can be Attributed to 
insufficient training in the actual methods that should be employed and 
a lack of organization of the work. 

The basis of this book, therefore, is the supplying of a textbook in 
statistics for students who have passed the elementary stage; who have 
studied a fair amount of theory and principles and now wish to equip 
themselves for actual statistical work in their own field of research 
activities. The experiment station agronomist, the cereal chemist, the 
plant breeder, and the economic entomologist are all examples of research 
workers who require a practical knowledge of statistical methods, and 
undoubtedly there are many others in the same class. It has been my 



VI PREFACE 

experience that to acquire this knowledge the student must work 
through a comprehensive series of actual examples, and these should 
not be miniature examples as they are likely to give him a wrong impres- 
sion of what will actually be required of him at a later date. Most of 
the various examples and exercises in this book are therefore of actual 
size, but every effort has been made to keep them within such limits as 
will enable the student to work through a representative set in one 
academic year. 

This is not to say that a course in statistical methods should ever be 
given without emphasis on principles, and this applies particularly to 
the principles of experimental design. When studying practical meth- 
ods, the opportunity is prime for the student to acquire a solid ground- 
ing in this important phase of the subject. The discussions in the 
greater part of the book, therefore, are worked out so that they have a 
direct bearing on the principles of the design of experiments. The first 
half, for example, while containing material that involves a repetition 
of elementary work that has already been covered, is nevertheless written 
so that, in reviewing, the student is brought into contact immediately 
with the structure of actual experiments. Also in this portion of the 
book are certain routine calculations which are designed mainly to give 
the student some facility in calculation before he comes to the heavier 
problems in the latter part. 

There are many to whom I owe thanks in the preparation of this 
book, but in the first place I must acknowledge a very great debt to 
Professor R. A. Fisher, who has been mainly responsible for the develop- 
ment of the methods that are set forth. Furthermore, he has been very 
generous of his own time in explaining how new problems may be solved 
and in clearing up doubts as to the exact application of previously estab- 
lished methods. I wish also to thank the staff of the Statistical Labora- 
tory at Ames, Iowa, for advice and suggestions, especially Dr. G. W. 
Snedecor, who in addition has given me permission to use, wholly or in 
part, any of the tables or material in his excellent new textbook, " Sta- 
tistical Methods." Thanks are due to many who have called attention 
to errors in the preprint edition, and to ways in which the explanations 
and examples could be improved. This applies particularly to my stu- 
dents, who have taken a special interest in suggesting improvements of 
this kind. They have also taken a particular interest in checking the 
calculations in order that the book should be as nearly perfect as possi- 
ble in this respect. In typing the manuscript I must acknowledge the 
untiring assistance of Misses E. J. Stewart and M. G. White. 

C. H. GOULDEN. 
February, 1939. 



CONTENTS 

CHAPTER PAGE 

' I. INTRODUCTION 1 

II. CALCULATION OF THE ARITHMETIC MEAN AND STANDARD DEVIATION- 
FREQUENCY TABLES ANI> THEIR PREPARATION . . 9 

III. THEORETICAL FREQUENCY DISTRIBUTIONS 20 

IV. TESTS OP SIGNIFICANCE WITH SMALL SAMPLES 33 

^W. THE DESIGN OF SIMPLE EXPERIMENTS 45 

X YE LINEAR REGRESSION 52 

WI. CORRELATION 65 

III. PARTIAL AND MULTIPLE REGRESSION AND CORRELATION 78 

IX. THE x 2 (CHI-SQUARE) TEST 88 

X. TESTS OF GOODNESS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES. 101 

XI. THE ANALYSIS OF VARIANCE 114 

XII. THE FIELD PLOT TEST 142 

XIII. THE ANALYSIS OF VARIANCE APPLIED TO LINEAR REGRESSION 

FORMULAE 210 

XIV. NON-LINEAR REGRESSION 219 

XV. THE ANALYSIS OF COVARIANCE 247 

XVI. MISCELLANEOUS APPLICATIONS 261 

INDEX 273 



vii 



METHODS OF STATISTICAL ANALYSIS 



CHAPTER I 
INTRODUCTION 

1. The Logic of Statistical Methods. Applying statistical methods 
to experimental work involves the use of certain logical ideas appropriate 
to experimental procedure. The problems of statistics are, therefore, not 
entirely mathematical problems; in fact they are very largely problems 
based on the technique and requirements of the research worker. This 
important point has not always been clearly understood and hence we 
find, in the history of the development of statistical methods^ various 
attempts to solve the problems of the experimentalist by the application 
of purely mathematical methods of reasoning and derivation. Thus we 
find prodigious attempts being made to apply the method of inverse prob- 
ability to the testing of the significance of results obtained in experi- 
ments. This theory has to do with the evaluation of the probability of 
the occurrence of certain specified events on the basis of what has hap- 
pened in some previous event. For example, if 8 balls are drawn from 
an urn containing black and white balls, and are found to consist of 3 
white and 5 black balls; to derive from this result an exact statement of 
the probability of obtaining a white ball in drawing another single ball 
is a problem in inverse probability. Everyone will agree that, on the 
basis of the ratio of white to black balls in the sample drawn, in drawing 
another ball one's expectation tends towards black, but very few will 
agree that this expectation can be put in the form of an exact statement 
of mathematical probability. On first thought, one might be inclined 
to think that this type of problem is the same as the statistical one of 
taking samples and reasoning from these samples to the populations 
from which they were drawn. We shall see, however, that there is a 
very essential difference between the two situations; that to regard 
these two situations as the same is merely to misunderstand the true 
nature of the methods of obtaining new information by experimental 
methods. To illustrate these points in further detail we shall follow 
through the procedure of operating a very simple experiment, in which 
the statistical method will arise as a natural consequence of the efforts 
of the investigator to get the most out of his experiment. 



2 INTRODUCTION 

2. A Simple Experiment in Identifying Varieties of Wheat. This 
hypothetical experiment is modelled after the famous tea-tasting 
experiment described by R. A. Fisher (1), but in some respects the pro- 
cedure is simplified. Fisher's hypothetical experiment will undoubtedly 
remain as a classic in statistical literature, and after following through 
the experiment described here the student will do well to make a similar 
study of the tea-tasting experiment as it discusses certain aspects of this 
type of problem that cannot be presented here. 

A wheat expert claims that, if he is presented with grain samples of 
two particular varieties which we shall designate as A and B, he can 
distinguish between them. He does not claim the ability to identify 
either one of the varieties, if they are presented to him separately, and 
further there is no special mention of an ability to differentiate between 
these samples at all times and under all conditions with perfect accuracy. 
The claim is for a certain power of differentiation, and we must proceed 
in the. planning of the experiment accordingly; that is, we must plan 
the experiment in such a way that any reasonable power of differentia- 
tion possessed by the operator will be demonstrated. With this 
knowledge we can proceed to set up the experiment. 

It will be obvious with a little study that, in order to plan the experi- 
ment correctly, it will be necessary to anticipate the possible results. 
Suppose that we presented the operator with only one pair of samples 
and he classified them correctly. Without any knowledge whatever of 
wheat varieties he could, by pure guesswork, name the varieties cor- 
rectly in 50 per cent of the cases. This follows from the fact that there 
are only 2 ways of classifying them, and if the operator has no power of 
differentiating them, these 2 ways are equally likely. Thus in about 
half of the cases he would place them correctly, and in the remainder of 
the cases incorrectly. Our conclusion must be that 1 pair of samples 
would not be sufficient to produce a clear-cut result, regardless of the 
efficiency or the inefficiency of the operator. What will be the effect of 
increasing the number of pairs of samples? Obviously, the operator 
would be much more unlikely to place several pairs of samples correctly 
than he would just 1 pair. Can this statement be put in more definite 
terms? Let us assume that 6 pairs are being used and see if we can 
calculate the probability of a correct result, or, in other words, the 
proportion of the cases in which the operator, without any power of 
differentiation of the samples, could be expected to reach a correct 
placing. If there are 6 pairs of samples, each pair may be placed either 
rightly or wrongly, so that there are just 7 different kinds of results. 
These are: 6 right, 5 right, 4 right, 3 right, 2 right, 1 right, and none 
right. The pairs may be thought of as being presented to the operator 



SIMPLE EXPERIMENT IN IDENTIFYING VARIETIES OF WHEAT 3 

one at a time, so there are 2 ways of placing the first pair (right or 
wrong), 2 ways of placing the second pair, and so forth for all the pairs. 
Each result for a pair may occur with either result for another pair, so 
that for 2 pairs we would have 2X2 possible combinations of placings. 
These are: both right; first pair right and second pair wrong; second 
pair right and first pair wrong; and both wrong. Continuing with this 
reasoning, it turns out that for 3 pairs the possible number of combina- 
tions of placings is 2 X 2 X 2; and, finally, for 6 pairs the total number 
is 2 6 = 64. If now the operator places all 6 pairs of samples correctly, 
we are in a position to place an evaluation on this result. There is only 
1 way of placing all pairs correctly, so that if the operator has no knowl- 
edge whatever of wheat varieties he would be expected to place them 
correctly in only 1 out of 64 trials. This would be a rather odd chance, 
and we would therefore be inclined, in the event of a successful placing, 
to attribute it to the ability of the operator in differentiating the 
varieties. Another way to regard this is to consider the consequences of 
adopting as a standard, in the examination of a large number of opera- 
tors, that all pairs must be placed correctly. Then in 1 out of 64 cases 
we could be expected to attribute to the operator a power of differentiat- 
ing the varieties that he did not actually possess. This would seem to be 
a fairly safe standard. In fact it would undoubtedly be argued from 
the standpoint of the operators being tested that the standard was 
much too high. In general practice, it is usual to adopt a ratio of 1/20 
as an arbitrary level for discriminating between real and chance effects. 
That is, an event is not regarded as significant unless it would only 
occur by chance variation in not more than 1 out of 20 trials. 

We now have to consider the interpretation that would be made if 
the operator were to obtain such a result as 5 pairs right and 1 pair 
wrong. In the above case there was only 1 way of placing 6 of the 
pairs correctly, but the situation is different now in that any one of the 
6 pairs may be the one that is incorrectly placed, making a total of 6 
ways, out of the grand total of 64, in which the samples may be placed 
5 right and 1 wrong. Then, in considering the experiment from the 
standpoint of the possibility of its indicating a power of differentiation 
on the part of the operator, we must also take into consideration the 
number of ways of placing 6 pairs correctly. That is, we must enumer- 
ate the number of ways in which the operator can place 5 pairs of samples 
correctly, or any other result more favorable to his claim. This makes 
a total of 1 + 6 = 7 out of 64 ways in which such a result or one more 
favorable to the operator could occur, and if the operator has no power 
of differentiation this result will be expected to occur in just that pro- 
portion of the cases. In approximate figures the ratio 7/64 is equal to 



INTRODUCTION 



1/9; and we note that this is larger than the ratio 1/20, which, as pointed 
out above, is accepted as a general level of significance. To accept 
the ratio of 1/9 as indicating a power of differentiation would be to 
take the risk of being wrong in 1 out of 9 similar trials, and this would 
probably be too great a risk for most investigators to accept. It might, 
however, be taken as a sufficient indication to justify further experi- 
mentation. 

It will be found convenient, in experiments of this type, to set up in 
the form of a table all the possible results with the corresponding 
number of ways in which each can occur. Another column of the table 
may be used to show the ratio that we have taken above to indicate the 
significance of each result. The figures for this experiment are given in 
Table 1. Why do we not give more values in the third column? 

TABLE 1 

POSSIBLE RESULTS, NUMBER OF COMBINATIONS, AND RATIO OF SIGNIFICANCE, 
FOB A SIMPLE EXPERIMENT IN DIFFERENTIATING Six PAIRS OF SAMPLES 



Possible Results 


No. of Combinations 


Ratio of Significance 


6 right wrong 


1 


1/64 


5 1 


6 


7/64 


4 2 


15 


22/64 


3 3 


20 




2 4 


15 




1 5 


6 




6 


1 




Total 


64 











The procedure in this simple experiment may now appear to be 
quite clear and apparently straightforward in every respect. The 
reader will then be surprised to learn that we have been guilty of a very 
serious omission. We have said that, if the operator has actually no 
power of differentiation, the 64 ways of arranging the pairs are all 
equally likely to occur. Suppose now that the samples are presented to 
the operator in pairs with variety A to his left hand and variety B to 
his right hand. On the off chance that there may be such a systematic 
arrangement of the pairs, the operator decides to guess this order and 
then adhere to it throughout the experiment. The result is that the 
most probable arrangements are 6 right, or 6 wrong, and our theory as 
to the probable frequency of the different possible results is completely 



DEFINING SOME STATISTICAL TERMS 5 

broken down. Another possibility that we have omitted to consider 

so far is that the 2 samples may show differences as to weight or quality 

which are actually quite independent of the variety characteristics. 

Here again the operator may, by guessing, obtain a result that is either 

all wrong or all right. We could go on and point out a number of factors 

that would tend to upset our calculations, and in the end the reader 

might despair as to the possibility of carrying through any experiment 

that would lead to valid conclusions. Why not take into consideration 

such factors as we have mentioned and work out the theoretical fre- 

quencies of the different combinations accordingly? A little thought 

will show that this is quite impossible. The vagaries of the minds of 

operators, for example, in taking advantage of certain orderly arrange- 

ments of the pairs, would be quite beyond the possibility of definite 

enumeration. The situation is not hopeless, however, as there is 

always at hand an extremely powerful method of overcoming this 

difficulty. The method is to arrange all factors that may enter into 

the results, completely at random. Thus, in presenting the pairs to the 

operator, a random arrangement would be followed that would be 

determined beforehand by throwing coins, drawing cards, or from a 

book of random numbers. It could then be stated with absolute con- 

fidence that, on the hypothesis that the operator has no knowledge of 

differentiating the samples, all possible arrangements would be equally 

likely to occur. It would be possible, for example, to use different 

colors of trays as containers for the samples. In each pair 1 tray might 

be red and 1 blue, and, if the varieties are assigned to the trays at 

random, it will still be true that all possible arrangements are equally 

likely. Of course a word of caution is needed here. Different colored 

trays, or any other disturbing influence on the ability of the operator to 

differentiate the samples, are not recommended, as they tend to reduce 

the efficiency of the test ; but at the same time if such factors are properly 

randomized they do not affect the validity of the test of significance. 

3. Defining Some Statistical Terms. In describing our simple 
experiment, statistical terms were avoided as much as possible. Such 
terms are, however, a kind of shorthand and will be found very convenient 
as we proceed to the consideration of more intricate problems. The 
6 pairs of samples of grain constitute in themselves ^SW^ i n the true 
statistical sense. We were Aot particularly interested in what the 
operator did with the 6 pairs except in so far as it indicated his ability 
to differentiate the varieties in general. In other words, we were trying 
to obtain an estimate of what would happen if he were presented with a 
very large group of such pairs. This l^ 



number of pairs might be said to constitutetEepopulcrfton that we are 



6 INTRODUCTION 

sampling. The general problem, of statistics, therefore, is the estimation 
of values for populations by means of determinations made on samples 
drawn at random from these populations. Assuming that the final 
result of our experiment was 5 pairs of samples placed correctly, the best 
estimate we would have for what our operator might do with a very large 
sample is that he would place f of the pairs correctly. This value is the 
mean number of successful placings that the operator would make in a 
population of similar pairs. A value such as this, calculated from a 
sample, is said to be a statistic. The population value of which the 
statistic is an estimate is referred to as a parameter. Statistics are sub- 
p5^"Tb^va^ ^U g e {. (jfffg^t results with different 

samples. The populations sampled are regarded for convenience as 
being infinite; and therefore for any one variabk, such as the number of 
successful placings, there is only 1 value of the parameter. 

In all experiments there is a hypothesis to be tested. It will have 
been noted that in the description oTflieljimple experiment we repeatedly 
used the words ''if the operator has no power of differentiation. " This 
points to the fact that the hypothesis we were testing was just that. In 
statistical parlance our hypothesis is now, owing to the pertinent sug- 
gestion of Professor Fisher (1), referred to as the null hypothesis. This 
null hypothesis was the basis for the calculation of the number of ways 
out of the total that certain results would be obtained, it being assumed, 
owing to randomization of the experiment, that all the possible ways 
were equally likely, v 

4. Summary of Principles. We have now worked through an actual 
experiment, which, although it was extremely simple, has introduced us 
to the main principles of the statistical method and has allowed us to 
obtain an easy introduction to many of the common statistical terms. 
It will be convenient after this discussion to return to some of the gener- 
alizations of Section 1. 

It will have been noted that the logic employed in tests of significance 
is clearly that of the experimentalist. This is true whether or not the 
experimenter has any knowledge of mathematics. Always, if he is 
critical of his results, he asks himself whether or not they could have 
arisen as a chance variation, and on this basis arrives at some conclusion 
as to their significance. The statistical method, therefore, does not 
introduce anything new in this sense, but merely supplies him with the 
technique for planning his experiment so that it is justifiable to ask such 
a question, and then furnishes him with a method of measuring the 
confidence to be placed in the findings. 

The results from one sample are not used to obtain a statement as 
to the probability of obtaining a given result in drawing another sample, 



THE FUNCTIONS OF STATISTICAL ANALYSIS 7 

but they are used to obtain an estimate of the population from which the 
sample was drawn. 

A test of significance is, essentially, the use of the data provided by 
the sample to test any hypothesis that may be set up. In such tests we 
do not always realize that a hypothesis is involved, but nevertheless this 
is true. When we ask the question, "Is my result due to some real 
effect or to a chance variation?" we can answer this question only by 
setting up the hypothesis that there is no effect; and determining whether 
or not the results agree or disagree with the hypothesis. 

The mathematical derivations involved in statistical tests arise 
from attempts to state the proportion of cases, according to a given 
hypothesis, in which the results obtained will occur. Thus, in the 
experiment described above, the hypothesis was that the operator had 
no power of differentiating the varieties; and on this basis we inquired as 
to the proportion of cases in which a result of 6 right would occur. The 
order in which the samples were presented having been randomized, it 
was possible to state that all placings were equally likely; and hence we 
were able to derive by strictly mathematical methods the proportion of 
cases in which a given placing would occur. 

5. The Functions of Statistical Analysis. The chief functions of 
statistical analysis as applied to experimental procedure may now be 
enumerated as follows: 

(a) To provide a sound basis for the formulation of experimental 
designs. 

(6) To provide methods for making tests of significance and 
trustworthy estimations of the magnitude of the effects indicated 
by the results. 

(c) To provide adequate methods for the reduction of data. 

The discussion of the previous sections will have given a reasonably 
clear picture of the manner in which the principles of statistics are made 
use of in designing experiments. Since this is the most recent develop- 
ment in this field, it is natural that -it is with respect to experimental 
design that the beginner is most likely to err. Frequently an elementary 
knowledge of statistics, consisting merely of an outline of the facts of 
variability and the various methods of measuring this variability, is 
taken as a sufficient knowledge for applying statistical methods to 
experimental work. The results of this practice are often disastrous. 
It is the reason why the consulting statistician is frequently presented 
with a set of data collected from an experiment which has been very 
badly designed. At the best, in such an experiment, there will be a loss 



8 INTRODUCTION 

of precision and information; but in addition there may be a decided 
bias in the results and as a consequence the whole or at least a part of the 
data may have to be discarded. It is not exaggeration, therefore, to 
state that to the experimentalist a study of statistical methods is futile 
unless he endeavors to apply these methods not only to the analysis of 
data but also to the structure of proposed experiments. 

The necessity for tests of significance has already been dealt with, 
but very little emphasis in the above discussion was placed on methods of 
estimation. It was pointed out, however, in the hypothetical example, 
that, if the operator's result was 5 right placings out of a possible 6, this 
would have to be taken as the best estimate available of the proportion of 
correct placings the operator could be expected to make if presented with 
a large series of samples. Obviously the experiment was so small that 
this may not be very close to the proportion that the operator would 
actually accomplish, and hence in this respect the experiment was not 
sufficiently extensive. The methods of statistics are concerned very 
vitally, therefore, with methods of estimation; and here again we cannot 
avoid noting the importance of experimental design, in that by careful 
design we can very largely determine beforehand the accuracy with which 
a particular estimate can be made. 

The necessity for the reduction of data is perfectly obvious, but it 
may not be clear as to the various methods employed in statistics for 
bringing this about. It is impossible to list these here, but we can 
classify them into three general groups: viz., tables, graphs, and statistics. 
The tables are usually prepared first, and from these we draw graphs to 
illustrate the main features of the data, and calculate statistics. The 
statistics are single expressions such as the mean or average which 
express the general characteristics of the samples studied. 

REFERENCE 

1. R. A. FISHER. The Design of Experiments. Oliver and Boyd, London and 
Edinburgh, 1937. 



CHAPTER II 

THE ARITHMETIC MEAN AND STANDARD DEVIATION- 
FREQUENCY TABLES AND THEIR PREPARATION 

1. The Arithmetic Mean. This is our first example of a. statistic. It 
is called a statistic because we regard it in statistical practice as a value 
calculated from a sample, and an estimate of the mean of the population 
from which the sample was drawn. Values for the means of samples will 
be expected to vary from sample to sample, and are Jheref ore not essen- 
tially different from individual variates in that rqspect. It is for this 
reason that it is not consistent terminology to speak of the mean or any 
other statistic calculated from a sample as a constant. The only con- 
stant values in statistical theory and practice are the values representing 
the infinite populations from which the samples are drawn. These, as 
we shall see later, are usually referred to in modern statistical litera- 
ture as parameters. 

It is often said of the arithmetic mean that it is the best single value 
that can be applied to the sample as a whole. Thus we find that the 
agronomist refers to the average yield of a variety, and not to the indi- 
vidual yields of a series of plots. Many other instances of this kind could 
be cited; in fact, it is an everyday usage and needs no further explana- 
tion. 

For a sample of N variates where z, represents any one variate, the 
mean is given by: 

Xl + X2 + Xz + + Xi + " + X n 



which for the sake of abbreviation is written : 

* - IT (1) 

If the values for three variates are 6, 8, and 1, the mean is obviously: 

6 + 8 + 1 _ 15 
3 ~ 3 



10 THE ARITHMETIC MEAN AND STANDARD DEVIATION 

Using the short formula means simply that the summation of the three 
quantities is understood, and, instead of writing out all the values and 
connecting them with plus signs, we merely write 15/3 = 5. According 

N 

to strict mathematical usage, S(z) should be written S(aQ, to show that 

N values are summated, but the simpler form may be used when the 
number of summations is obvious. 

f One of the most interesting properties of the mean is that the sum of 
the deviations of all the individual variates from the mean is zero. 
Again representing an individual variate by #,-, an individual deviation 
from the mean will be (a:,- x). Then summing all these we get: 



2 (a -*) = (a?i - $) + (x 2 -*) + ... 

= (Xl + X2 + . . . + X w ) - NX 



And since 



N 
It is clear that 

S(3 - ) = 
Using the summation sign to shorten the algebra we would have 

2(s - $) = 2(x) - 2(4) = 2(x) - NX 
And since 



It is again clear that 

2(s - x) = 

2. The Standard Deviation. In using the mean of a sample to 
represent the sample as a whole, it must occur to us that the reliability 
of this method will depend on the degree of variation among the indi- 
vidual variates that make up the sample. If there is no variation the 
mean would represent the whole set perfectly; but as the variation 
becomes greater the single value of the mean is less and less descriptive 
of the entire group, and it becomes more and more necessary in order to 
iescribe the sample completely that we have some measure of variability. 
rhe average deviation from the mean might suggest itself, but we have 
jeen that the sum of the deviations from the mean is zero, and from this 
t follows that the mean deviation is also zero. For this reason the sta- 
istic that has been adopted as a measure of variability is the root mean 
iquare deviation, commonly known as the standard deviation. The 



THE STANDARD DEVIATION 11 

formula for the standard deviation, which is usually represented by the 
Greek letter sigma (<r), is: 



<r = 



(2) 



The direct method of calculating the standard deviation is to find all the 
deviations from the mean, square them, summate, divide by N, and 
then extract the square root. For example, if we have the three figures 
6, 8, and 1, for which the mean is 5, the standard deviation would be: 



I 2 + 3 2 + 4 2 26 
3 = Vy 

When there are more variates in the sample, and especially when the 
deviations contain decimal figures, a much shorter method can be used. 
The main part of the work is to find the sum of squares of the deviations, 
and it can be shown very easily that: 

l^WJ /n\ 

\pj 



Applying this to our miniature example we have: 

- ' 2(a? - z) 2 = (6 2 + 8 2 + I 2 ) - 15 2 /3 = 26 

This formula is especially useful for machine calculation and is now used 
almost exclusively in statistical laboratories. 

We now have to consider a point which is very important in the prac- 
tical application of statistical methods, and one over which there is often 
a great deal of confusion. It was pointed out above that the mean of a 
sample is taken as the best possible estimate of the mean of the parent 
population. This practice of estimating values for parent populations is 
the main object of calculating values for samples. With a little thought 
this point should be quite clear. We determine the reaction of a crop to 
a given fertilizer on a sample of plots which may not be more than 6 to 10 
in number. It cannot be stated, even by the wildest stretch of the 
imagination, that we are primarily interested in the reaction to the 
fertilizer on those 6 to 10 plots. What we are attempting to find out is 
the general reaction to the fertilizer under fanning practice, and bence 
we must picture a very large population of plots for the mean reaction of 
which we are trying to obtain an estimate. If we let this population, for 
purposes of clarity of thinking, be regarded as infinite, it follows that the 



12 THE ARITHMETIC MEAN AND STANDARD DEVIATION 

mean and thes^ are fixed values and 

henSTw^call them parameters. If the mean of the parent population 
isjlenoted bv m> then x, jthe mean of thfl s^mpl^ is nn Pst.ima.te ^f jhe 
parameter m._ Similarly if <r is the standard deviation of the parent 
population, the value which we calculate from the sample must also be 
the best possible estimate of <r. Actually this estimate is not the root 
mean square deviation that we have defined above. This arises from 
the fact that, if m is the mean of the parent population, the test estimate 

ofcris: 

^ _,_,-- m) 2 



N 

but since we do not know m we use x instead, and it can be shown by a 
simple algebraic derivation that the best estimate of a is given by: 



(4) 



wherein we put this expression equal to sin that it is not <r but the best 
possible estimate of a. We keep to this symbolism throughout in order 
to distinguish the standard deviation calculated from a sample from the 
true value which is a parameter of the parent population. The divisor 
(N 1) is known as the number of degrees of freedom available for 
estimating the standard deviation. We shall learn more of this term in 
later chapters. 

3. Standard Deviation of a Sample Mean. If we take a series of 
samples and determine a mean for each one, it is obvious that the means 
for these samples will vary from sample to sample, and that the degree 
of variation among these means will be related to the degree of variation 
among the individual variates. If one particular sample is taken, the 
exact relation is given by the equation: 

(5) 



*~ VN 

where * is the standard deviation of the mean of the sample, s is the 
standard deviation for the sample as a whole, and N is the number in the 
sample. The standard deviation of a mean is therefore inversely 
proportional to the square root of the number in the sample. 

4. The Frequency Table. This is a table which shows, for the 
sample of variates studied, the frequencies with which they fall into 
certain clearly defined classes. If the sample is very small the frequency 



SELECTION OF CLASS VALUES 13 

table may not be necessary, and even if prepared may not mean very 
much; but for moderately large samples it is usually desirable to begin 
the reduction of the data with a table of this kind. The frequency table 
provides the values for easy graphical representation," and from it such 
Statistics as the fog&tt Md ^tttmdltttf' deviation 'may "TRT calculated with 
much greater ease than from the, original set of individual values. 

6. Selection of Class Values. Frequency tables may deal with 
either continuous or discontinuous variables. A continuous variable is 
one in which a single variate may take any value within the range of 
variation. Thus the yield of a plot of wheat may take any value within 
the range from the lowest-yielding plot to the highest. A discontinuous 
variable can take only certain specified values. For example, in tossing 
5 coins we can have 5, 4, 3, 2, 1, or heads, and no other values can occur. 

A frequency table for the number of heads in tossing 5 coins 100 times 
might be as follows: 

Class Values Frequency 

5 heads 3 

4 heads 16 

3 heads 28 

2 heads 31 

1 heads 17 

heads 5 

Total = 100 

The class values to be selected for such a table are obvious, and this is 
usually true for discontinuous variables. In some examples, however, 
it may be necessary to form the class values such that the class interval 
is greater than unity. In tossing coins 20 at a time, we might use the 
classes 0-2 heads, 3-5 heads, and so forth. 

If the variable is continuous, the classes for which the frequencies are 
to be determined must be chosen arbitrarily, the choice depending on the 
accuracy required in the computation of statistics from the table, the 
range of variation which is, of course, the difference between the lowest 
and the highest value of the sample the number in the sample or total 
frequency, and the facility with which these classes can be handled in 
computation. In the first place, the greater the number of classes the 
greater the accuracy of the calculations made from the table. But there 
must be a limit to the number of classes we can handle conveniently, 
and these two opposing factors must be balanced up. A good general 
rule is to make the class interval not more than one-quarter of the stand- 
ard deviation. Of course we do not as a rule know what the standard 
deviation is before the table is made up, but it is possible to make a 



14 THE ARITHMETIC MEAN AND STANDARD DEVIATION 



rough estimate of its value from the range of variation. Tippett (3) 
has published detailed tables on the relation between the range of varia- 
tion and the standard deviation, and these have been summarized in a 
short table prepared by Snedecor (2). The following values are taken 
from Snedecor's table after rounding off the figures to two significant 
digits. V-- 

TABLE 2 

VALUES OF THE RATIO, RANGE DIVIDED BY THE STANDARD DEVIATION (SD) t 
FOR SAMPLE SIZES FROM 20 TO 1000 



Number in Sample 


Range/Z> 


Number in Sample 


Range/57) 


20 


3.7 


200 


5.5 


30 


4.1 


300 


5.8 


50 


4.5 


400 


5.9 


75 


4.8 


500 


6.1 


100 


5.0 


700 


6.3 


150 


5.3 


1000 


6.5 



Now suppose that we have a sample of 500 variates and the range of 
variation is 0.25 to 2.63. The difference is 2.38, and if we were to 
divide this by the standard deviation our table tells us that we would get 
a quotient of approximately 6.1. In order to make the standard devia- 
tion about one-quarter of the class interval, it is clear that its magnitude 
will have to be about 2.38/6.1 X 4 = 0.098. It is more convenient to 
have an odd number for a class interval than an even one, since it means 
that the midpoint of the interval does not require one more decimal 
place than we have in the values that define the class range. In the end 
we should probably decide in this case on an interval of 0.11. In making 
up the classes it is usual to begin with the lower boundary of the first 
class slightly below the lowest value, so that our classes and midpoints 
would finally be set up somewhat as follows: 



Class Range 
0.19toO 29 
0.30 to 0.40 
0.41 to 0.51 
0.52 to 0.62 
etc. 



Class Value, or Midpoint 
of Class Range 
0.24 
0.35 
0.46 
0.57 
etc. 



By following the above rules we ensure a sufficient degree of accuracy 
in any statistics that are calculated from the frequency table; but, if 
the frequency table is required mainly for the preparation of a graph as 



SORTING OUT VARIATES AND FORMATION OF TABLES 



15 



described below, this method may give classes that are too small, in that 
some of the classes may contain only very small frequencies or perhaps 
none at all. It is desirable in such cases to make the class interval from 
one-half to one-third of the standard deviation. 

In statistical literature one may come across references to Sheppard's 
corrections for grouping. These are designed to remove bias from 
certain statistics that are calculated from grouped data instead of from 
the individual values. Thus, in calculating S(x x) 2 /N 1, it has 
been shown that the bias is positive and equal approximately to 1/12 of 
the class interval. In the tests for abnormality described in Chapter 
III, and in certain other specific calculations, it is necessary to make the 
adjustments, but in general practice they are usually ignored and in 
many tests of significance it is more correct to omit them altogether. 

The student should note carefully at this point that Sheppard's cor- 
rections are for the purpose of removing a definite bias and in no sense do 
they make allowance for inaccuracies introduced by using groups that 
are too large. 

6. Sorting out the Variates and Formation of the Frequency Table. 
Sorting is greatly facilitated by writing the value of each variate on cards 
of a convenient size for handling. The class ranges are first written out 
on cards and arranged in order on a table. The sorting can then be done 
rapidly, and after it is finished it is very easy to run through the piles 
and obtain a complete check on the work. It is very important to have 
perfect accuracy at this point. In a series of studies a misplaced card 
may give a great deal of trouble at a later stage in the work. The fre- 
quency table is finally made up by entering the frequencies opposite the 
corresponding class values. 

Table 3 is a sample of a frequency table. It represents data on the 
carotene content of the whole wheat of 139 varieties. The class values 
are in parts per million of carotene in the whole wheat. In this instance 
a great deal of accuracy in the calculations was not desired, and it will 
be noted that the class values are larger than they would be if the rules 
for the formation of these values as outlined above had been followed. 
Check this point by reference to Table 2. 

TABLE 3 

FREQUENCY TABLE FOR PARTS PER MILLION OF CAROTENE IN THE 
WHOLE WHEAT OF 139 VARIETIES OF WHEAT 





fO 85 


96 


1 07 


1.18 


1.29 


1.40 


1.51 


1.62 


1.73 


1.84 


1.95 


2.06 


2.17 


Class Values. . . 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 




[0.95 


1.06 


1 17 


1.28 


1.39 


1.50 


1.61 


1.72 


1.83 


1.94 


2.05 


2.16 


2.27 


Frequency ... . 


2 


6 


14 


21 


24 


37 


13 


10 


4 


3 


2 


2 


1 



16 THE ARITHMETIC MEAN AND STANDARD DEVIATION 

7. Graphical Representation of a Frequency Table. Graphs of two 
types are in general use. The best type of graph and the one most 
commonly used is the histogram. It is a diagrammatic representation of 
a frequency table in which the class values are represented on the hori- 
zontal axis, and the frequencies by vertical columns erected in their 
appropriate positions on the horizontal axis. The histogram is most 
useful when a curve for some theoretical distribution is being fitted. The 
nature of any disagreement between the theoretical distribution and the 
actual frequencies can be located readily when the theoretical curve is 



35 



30 



25 



90 101 112 123 134 145 156 167 178 189 2OO 211 222 
CAROTENE - PARTS PER MILLION 

FIG. 1. Histogram for the data of Table 3. 



superimposed on the histogram. As an example the histogram for the 
data of Table 3 is shown in Fig. 1. 

The other type of graph is usually known as a frequency polygon. A 
straight line is erected for each frequency at the midpoint of the corre- 
sponding class value, and the ends of these connected in sequence by 
straight lines. It does not give as accurate a picture for the sample as 
the histogram, but tends in its shape towards the smooth curve of the 
population from which the sample was drawn. 

8. Calculation of the Mean and Standard Deviation from a Frequency 
Table. After the frequency table has been formed, we add two more 
columns as indicated in the small example given below: 



COEFFICIENT OF VARIABILITY 



17 



Class Value 




Frequency 


Frequency 


or Midpoint of 


Frequency 


Multiplied by 


Multiplied by Square 


Class Range 




Class Value 


of Class Value 


(*) 


/ 


/X(*) 


/XOr 2 ) 


1 


2 


2 


2 


2 


4 


8 


16 


3 


7 


21 


63 


4 


6 


24 


96 


5 


1 


5 


25 


Totals 


20 =* N 


60 = S(x) 


202 - S(x 2 ) 











On summating the last three columns we get N, (#), and S(x 2 ), which 
are the values necessary for the calculation of the mean and the standard 
deviation. The mean is given by: 



N 



and the standard deviation by : 



m = 



N- I 



(6) 



It will be noted that the numerator of the standard deviation is 
2(x x) 2 , and that to obtain it we have made use of the identity given 
in formula (3). 

The class values are very frequently numbers containing two to four 
digits, in which case a great deal of labor can be saved by replacing them 
by the series of natural numbers 1, 2, 3, 4, etc. By this method we 
obtain a mean and a standard deviation that we shall designate by x' 
and s f , respectively. These can be converted into the true values by 
means of the following identities: 

s = s'i (8) 

where i is the class interval and X i is the first true class value. 

9. Coefficient of Variability. This is the term applied to the stand- 
ard deviation when it is expressed in percentage of the mean of the 
sample. It is a statistic of very limited usage owing to the difficulty of 
determining its reliability by statistical methods. The formula is 
obviously: 



C (coefficient of variability) = a (\ 



(9) 



18 THE ARITHMETIC MEAN AND STANDARD DEVIATION 



10. Exercises. 

1. Substitute the natural numbers 1, 2, 3, 13 for the class values of Table 3, 
and calculate the mean and the standard deviation. Convert the calculated values to 
actual values using formulas (7) and (8). 



5.597 



1.406 



s' = 2.196 8 = 0.2416 



2. Table 4 gives the yields in grams of 400 square-yard plots of barley. Make 
up a frequency table and histogram for these yields, using a class interval of 11, and 
make the first class 14 to 24. 

3. The areas in arbitrary units of 500 bull sperms are given in Table 5. 1 Prepare 
the frequency table and histogram, using 16 classes, making the first class 123 to 125. 

4. For either one of Exercises 2 and 3 above, calculate the mean and the standard 
deviation from the frequency table, using actual class values. Then replace the 
actual class values by 1, 2, 3, 4, , and recalculate the mean and the standard 
deviation. 

Ex. 2 ' = 13.055 x = 151.60 s f = 2.880 s = 31.68 



Ex. 3 z' = 7.852 



144.56 s' 



2.576 



7.728 



6. For the data in Tables 4 and 5, determine the class values that should be used 
to give a high degree of accuracy in the calculations. 
6. Prove the identity: 

S(* - x) 2 - 2(* 2 ) - I2(z)] 2 /tf 



TABLE 4 
YIELDS IN GRAMS OF 400 SQUARE- YARD PLOTS OF BARLEY 



185 


162 


136 


157 


141 


130 


129 


17ff 


171 


190 


157 


147 


176 


126 


175 


134 


169 


189 


180 


128 


169 


205 


129 


117 


144 


125 


16 f 


170 


153 


186 


164 


123 


165 


203 


156 


182 


164 


176 


176 


150 


1216 


154 


184 


203 


166 


155 


21$ 


190 


164 


204 


194 


148 


162 


146 


174 


185 


171 


181 


T98 


147 


165 


157 


180 


165 


127 


186 


133 


170 


134 


177 


109 


169 


128 


152 


165 


139 


146 


144 


178 


188 


133 


128 


161 


160 


167 


156 


125 


162 


128 


103 


116 


87, 


123 


143 


130 


119 


141 


174 


157 


m 


195 


180 


158 


139 


139 


168 


145 


166 


118 


171 


143,. 


132 


126 


171 


176 


115 


165 


147 


186 


157 


187 


174 


172 


19f 


155 


169 


139 


144 


130 


146 


159 


164 


160 


122 


175 


156 


119 


135 


116 


134 


157 


182 


209 


136 


153 


160 


142 


179 


125 


149 


171 


186 


196 


175 


18 


214 


169 


166 


164 


195 


189 


108 


118 


149 


178 


171 


151 


192 


127 


148 


158 


174 


191 


134 


188 


248 


164 


206 


185 


192 


147 


178 


189 


141 


173 


187 


167, 


128 


139 


152 


167 


131 


203 


2*1 


214 


177 


161 


194 


141 


161 


124 


130 


112 


122 


192 


155 


196 


179 


166 


156 


131 


179 


201 


122 


207 


189 


164 


131 


211 


172 


170 


140 


156 


199 


181 


181 


160 


184 


154 


200 


187 


169 


155 


107 


143 


145 


m- 


4-73 


162 


123 


189 


194 


146 


,22 


160 


107 


70 


84 


112 


162 


124 


156 


138 


101 


138 


141 


'143 


135 


163 


183 


99 


118 


150 


151 


83 


136 


171 


191 


155 


164 


98, 


136 


115 


168 


130 


111 


136 


129 


122 


120 


179 


172 


192 


m 


151 


142 


193 


174 


146 


180 


140 


137 


138 


194 


109 


120 


124 


126 


126 


147 


115 


148 


195 


154 


149 


139 


163 


118 


126- 


127 


139" 


174 


167,, 


175 


179 


172 


174 


167 


142 


169 


122 


163 


144 


147 


123 


160 


137 


161 


122* 


101 


158 


103 


119 


164 


112 


57 


94 


106 


132 


122 


164 


142 


155 


147 


115 


143 


68 


184 


183 


167 


160 


138 


191 


133 


160 


156 


122 


111 


153 


148 


103 


131 


180 


142 


191 


in 


146 


18* 


111 


110 


154 


176 


163 


175 


175 


146 


148 


167 


106 


123 


121 


154 


148 


91 


93 


74 


113 


79 


131 


119 


96 


80 


97 


98 


106 


107 


69 


86 


94 


129 












A,, 






-f. 


"\ 


?' 


Tfr 


'"S 




*t 






x 







\ 

1 Data by courtesy of A. Savage, Department of Animal Pathology, t[niversity 
of Manitoba. 



REFERENCES 



19 



TABLE 5 
AREAS IN ARBITRARY UNITS OF 500 BULL SPERMS 



140 


138 


140 


140 


140 


139 


138 


138 


138 


138 


133 


132 


140 


140 


138 


139 


139 


145 


145 


145 


147 


147 


147 


149 


149 


155 


160 


159 


159 


160 


139 


143 


142 


142 


141 


141 


145 


145 


144 


146 


140 


148 


147 


147 


149 


148 


148 


148 


149 


149 


153 


153 


153 


153 


155 


141 


149 


149 


149 


149 


149 


149 


149 


148 


147 


147 


148 


159 


161 


161 


158 


157 


157 


141 


141 


143 


143 


143 


143 


142 


141 


141 


141 


139 


138 


159 


161 


155 


137 


136 


144 


144 


145 


144 


144 


146 


145 


145 


144 


146 


138 


149 


149 


148 


148 


148 


162 


162 


153 


153 


144 


144 


144 


146 


145 


145 


145 


146 


141 


143 


134 


124 


124 


134 


132 


136 


137 


125 


123 


134 


146 


146 


139 


138 


138 


138 


140 


140 


146 


139 


139 


139 


152 


150 


150 


150 


152 


151 


149 


149 


149 


149 


149 


154 


154 


153 


155 


149 


149 


149 


161 


160 


159 


135 


154 


154 


154 


155 


154 


154 


142 


141 


141 


141 


141 


142 


142 


142 


142 


141 


136 


136 


135 


137 


136 


135 


135 


137 


137 


137 


146 


146 


146 


145 


140 


140 


140 


138 


138 


140 


155 


154 


153 


153 


153 


153 


153 


153 


153 


155 


143 


142 


142 


142 


147 


147 


150 


152 


152 


150 


134 


131 


130 


129 


131 


130 


129 


129 


134 


134 


140 


139 


139 


139 


127 


137 


134 


132 


133 


133 


148 


148 


147 


147 


147 


147 


147 


149 


148 


147 


149 


149 


149 


149 


149 


146 


146 


146 


145 


146 


137 


136 


136 


137 


137 


136 


134 


136 


135 


129 


139 


139 


152 


152 


152 


152 


151 


150 


152 


152 


136 


136 


137 


145 


144 


146 


146 


145 


145 


145 


153 


153 


155 


135 


158 


158 


157 


157 


157 


158 


150 


150 


150 


150 


151 


151 


150 


150 


152 


151 


133 


133 


134 


129 


130 


141 


143 


142 


141 


141 


134 


132 


127 


137 


128 


125 


136 


141 


143 


143 


147 


147 


169 


165 


162 


162 


149 


144 


144 


144 


146 


145 


145 


144 


146 


145 


144 


146 


146 


146 


135 


137 


137 


127 


134 


132 


135 


135 


127 


126 


151 


148 


148 


147 


147 


149 


149 


149 


150 


151 


145 


144 


144 


146 


144 


143 


143 


143 


143 


142 


157 


156 


137 


137 


137 


137 


136 


137 


135 


133 


150 


150 


152 


152 


152 


152 


152 


152 


152 


151 


141 


141 


143 


142 


142 


142 


138 


140 


140 


140 


143 


143 


144 


144 


144 


144 


146 


146 


140 


139 


144 


146 


145 


145 


145 


138 


140 


139 


138 


153 


146 


146 


146 


146 


146 


145 


145 


145 


146 


145 


134 


135 


157 


156 


156 


157 


157 


157 


157 


156 


151 


151 


151 


150 


150 


150 


150 


150 


150 


152 


142 


141 


142 


141 


141 


142 


142 


142 


141 


143 


135 


133 


133 


150 


151 


149 


139 


139 


139 


138 


138 


140 


140 


153 


153 


148 


147 


147 


156 


158 


158 


152 


141 


141 


142 


141 


143 


139 


139 


139 



REFERENCES 

1. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London and Edinburgh, 1936. 

2. G. W. SNEDBCOR. Statistical Methods. Collegiate Press, Inc., Ames, Iowa, 1937. 

3. L. H. C. TIPPBTT. Biometrika, 17: 386, 1926. 



CHAPTER III 
THEORETICAL FREQUENCY DISTRIBUTIONS 

1. Characteristics of Frequency Distributions of Biological Variates. 

A frequency table may be used to f urfcish an estimate of the frequency 
distribution of the population from which the sample has been taken. 
For example, we could take any one of the frequency tables of Chapter 
II and draw a smooth curve through the upper ends of the columns of the 
histogram. We would draw a smooth curve because the parent popula- 
tion is assumed to be infinite and each point on the base line could be 
represented by a frequency, or, to be more specific, the height of the 
perpendicular line from any point on the base line to the curve would 
represent the proportion of the total frequency of the population having 
the value represented by the point. This method, however, would not 
be very satisfactory, as the position of the curve would be, to a consider- 
able extent, a matter of individual judgment. Also, the sample studied 
might indicate, owing to errors of sampling, certain irregularities and 
lack of symmetry which might be entirely absent in the population. 
Furthermore, to be consistent in our logic, it follows that we are not so 
much interested in drawing a curve that fits the sample as we are in 
setting up a theoretical curve as a hypothesis and then determining 
whether or not the data of the sample agree with the theoretical fre- 
quencies. In setting up our theoretical curve, it is of course natural 
that we set up one that is likely to agree fairly well with the data of the 
sample, and this is only saying in other words that we should set up a 
reasonable hypothesis. We could set up a whole series of theoretical 
curves, the majority of which would have no resemblance whatever to 
the histogram of the sample; but obviously this would be a mere waste 
of time. To deduce a theoretical distribution into which our sample is 
likely to fit, it is necessary to study the characteristics of the frequency 
tables for biological variates as a whole and work out a logical theory for 
setting up the theoretical values. If we examine the histograms of 
Chapter II for three different kinds of biological variates, we find that 
they have certain characteristics in common. Close to the mean, the 
variates occur with much greater frequency than they do at some dis- 
tance from the mean; but the reduction in the frequencies from the mean 
to the extreme tails of the distribution is not uniform, with the result that 

20 



THE BINOMIAL DISTRIBUTION 21 

if a smooth curve i& drawn through the tops of the columns of the histo- 
grams it is seen to resemble an isosceles triangle but with a rounded top 
and very much flattened base. A curve of this type is found to resemble 
very closely a definite type of mathematical curve; but to understand 
more easily the reasoning behind the derivation of this curve it is neces- 
sary for us to look into the characteristics of another theoretical dis- 
tribution that is appropriate for discontinuous variables. 

' 2. The Binomial Distribution. In Chapter I we derived a theoretical 
distribution for the experiment on identifying varieties of wheat. This 
will be found in Table 1. Each theoretical frequency was derived by the 
direct application of elementary theorems of probability, and if, instead 
of dealing with specific numbers of pairs of samples, we had dealt with 
the problem as a general one for any number of pairs of samples we 
would have derived the binomial distribution. Thus the theoretical 
frequencies of Table 1 can be written out at once from the terms of the 
expansion of the expression ( + ^) 6 . These are: 

JL A i 15 L 

64 64 64 64 64 64 64 

wherein we note that the theoretical frequencies are stated as propor- 
tions of the total number and express directly the probabilities of par- 
ticular combinations. In general for similar problems where there are 
alternative possibilities such as right or wrong placings of pairs of 
samples, heads or tails in the tossing of a coin, an ace or any other num- 
ber in the throwing of a die, etc., the theoretical distribution can be 
written down directly by expanding the binomial (p + g) n , where n is 
the number of events in any 1 trial, p is the probability of the occurrence 
of the event in 1 way, q is the probability of the occurrence of the 
event in the alternative way, and p + q = 1. If p = we obtain a 
symmetrical distribution, but if p is not equal to q the distribution is 
asymmetrical or skewed. 

There are many applications of the binomial distribution in statistical 
analysis, and one application of particular interest will be dealt with in 
Chapter X. For the present it is sufficient to note that the form of the 
distribution is somewhat similar to the actual distributions of Chapter 
II, which we have concluded are fairly typical for biological variables in 
general. However, the binomial distribution is not suitable as a theoret- 
ical distribution for continuous variables, as in itself it is essentially 
discontinuous; so that if we make any use of it for continuous variables 
it must be as a stepping stone to some more general type of distribution. 
The biological variables we have studied indicated from the samples for 
which histograms were made that the parent populations were essen- 



22 



THEORETICAL FREQUENCY DISTRIBUTIONS 



tially symmetrical. The comparable situation for the binomial dis- 
tribution would occur when p = q. Starting from this point, therefore, 
let us suppose that n is infinitely large; and, in graphing the histogram 
for the theoretical distribution, the columns which will also be infinite 
in number are represented by vertical lines only. The result will be a 
smooth curve, and by carrying through this procedure algebraically and 
making certain approximations we can arrive at an equation for a 
smooth curve. This is the expression for what is commonly known as 
the normal frequency distribution. 

3. The Normal Distribution. Most variables dealt with in biological 
statistics show in their actual distributions only minor deviations from 
the theoretical normal distribution defined by: 



y = 



N 



where <r is the standard deviation of the population, N is the total num- 
ber of variates, e is the base of the Napierian system of logarithms, and 
y is the frequency at any given point x, where x is measured from the 




FIG. 2. Sketch of a normal curve, the base line measured in units equal to the 

standard deviation (<r). 

mean of the population. ^The curve expresses, therefore, the relation 
between y and #, with y as the dependent variable. Figure 2 is a sketch 
of a normal curve. It illustrates the measurement of x from the mean 
of the population which is located at the point where the dotted line has 



THE NORMAL DISTRIBUTION 23 

been erected. For the value of x taken, y is the perpendicular distance 
from that point to the curve. 
Equation (1) may be written: 



2) 
V ZTT 

and putting z for y (<r/N) we have: 



and since x/a varies in actual practice only from to 6, the values of z 
have been tabulated for all the values of x/0 from to 6 proceeding by 
intervals of 0.01. Any given value of z can then be transformed to y by 
multiplying by N/v for the particular population with which we are 
dealing. In other words, for a given population for which N and <r are 
known, we can proceed with a set of tables to plot the theoretical smooth 
curve. 

A smooth curve plotted by the above method is an estimate of the 
form of the infinite population from which the sample has been drawn; 
but what we often require is the theoretical frequency distribution corre- 
sponding to the actual frequency distribution of the sample. That is, 
we require the theoretical normal frequencies for the arbitrarily chosen 
class values of the actual distribution. For this purpose, if N is taken 
as 1, equation (1) becomes: 

y = 



which can be integrated from x = minus infinity to x = any assigned 
value. This gives the area under that portion of the curve, and we will 
represent it as -^(1 + a). The integration is started at x = minus 
infinity, because the normal curve never actually touches the base line 
although, at a* = 6, y is an exceedingly small value. The reason for 
expressing the area as ^(1 + ) or ^ + ^a will be seen from an exam- 
ination of Fig. 3. For any assigned value of x the area within the 
limits of x is represented by a. Therefore, from x = minus infinity to 
x = any assigned value, if the total area of the curve is 1, the area is 



The tabulated values of z and ^-(1 + a) for values of x/cr from to 6 
are given in Sheppard's " Tables of Area and Ordinate in terms of 
Abscissa." These are commonly referred to as Sheppard's tables of the 



24 



THEORETICAL FREQUENCY DISTRIBUTIONS 



probability integral. The detailed application of these tables to a prac- 
tical example is described below under Section 4. 
4. Methods of Calculation. v 




FIG. 3. Sketch of a normal curve showing ordinates erected at x/a =4-1, and 
x/tr 1. The unshaded area = a, and the shaded area = (1 ). 



Example 1. The calculations necessary to fit a normal curve to an actual 
frequency distribution and to determine the normal frequencies corresponding to 
the actual frequencies are given in Table 6. The data are for the transparencies of 
400 red blood cells taken from a patient suffering from primary anemia (4). The 
transparency is taken as the ratio of the total light passing through the cell to the 
area of the cell. For this distribution ' = 7.06 and a = 2.45. 

The calculations can best be described by considering each column of the table. 
The columns have been numbered at the head of the table for convenient reference. 

Column (1): The class ranges are as described in Chapter II. Note that 
unit class intervals have been used. This is necessary in obtaining y, but makes 
no difference to the remainder of the calculations. After setting up the class 
ranges, the actual frequencies may be entered as in column (10), but it is of no 
consequence when these are entered as they are not used in the calculations. 

Column (2): In order to understand clearly the meaning of the class limits, 
refer to any histogram as in Chapter II, Fig. 1, or Exercises 2 and 3. The limits 
correspond with the lines bordering the columns of the histogram. The mean of 
the sample is placed according to the class range in which it falls. In this case 
the mean is 7.06 and must be placed opposite the class range 6.6-7.5. The limits 
are then entered by passing in both directions from the mean. The class in 
which the mean falls will have two limits, but for each of the others we take only 
the one farthest from the mean. 



METHODS OF CALCULATION 



25 



TABLE 6 

CALCULATION OF ORDINATES FOE FITTING A NORMAL CURVE, AND 
THEORETICAL NORMAL FREQUENCIES 



(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


Class 
Range 


Class 

Limits 


d 


d/a 


9 


V"* 

(=> 


id-ho) 


N 
-(! + > 


Theoretical 
Normal 
Frequencies 


Actual 
Frequencies 






9.56 


3.90 


0.0002 


0.03 


1.0000 


400.00 










8.56 


3.49 


0.0009 


0.15 


9998 


399.92 


0.08 








7.56 


3.08 


0.0035 


0.57 


0.9990 


399.60 


0.32 








6.56 


2.68 


0.0110 


1.80 


0.9963 


398.52 


1.08 




6- 1.5 


1.5 


5.56 


2.27 


0.0303 


4.95 


0.9884 


395.36 


3.16 


4 


l.G- 2.5 


2.5 


4.56 


1.86 


0.0707 


11.54 


9686 


387.44 


7.92 


11 


26-35 


3.5 


3,56 


1 45 


0.1394 


22.76 


0.9205 


370 60 


16.84 


17 


3 6- 4,5 


4 5 


2 56 


1.04 


2323 


37.92 


8508 


340 32 


30.28 


29 


4.6- 5.5 


5.5 


1 56 


0.64 


0.3251 


53.08 


0.7389 


295.56 


44.76 


43 


5.6- 6.5 


6.5 


0.56 


0.23 


0.3885 


63.43 


0.5910 


236.40 


59.16 


56 


6 6- 7.5 


7.06 


0.00 


0.00 


0.3989 


65.12 


0.5000 


200.00 


64.96 


58 


7.6- 8.5 


7.5 


0.44 


0.18 


3925 


64.08 


0.5714 


228.56 


60.40 


63 


8.6- 9.5 


8.5 


1.44 


0.59 


3352 


54.72 


0.7224 


288.96 


47.56 


61 


9.6-10.5 


9.5 


2.44 


1 00 


0.2420 


39.51 


8413 


336.52 


31.16 


25 


10 6-11.5 


10.5 


3.44 


1.40 


0.1497 


24.44 


9192 


367 68 


18.24 


20 


11.6-12.5 


11.5 


4.44 


1.81 


0.0775 


12.65 


9648 


385.92 


8.80 


9 


12.6-13.5 


12 5 


5 44 


2 22 


0339 


5.53 


0.9868 


394 72 


3.56 


4 


13 6-14.5 


13 5 


6.44 


2.63 


0126 


2 06 


9957 


398.28 


1.24 








7.44 


3.04 


0039 


0.64 


9988 


299.52 


0.36 








8.44 


3.44 


0011 


18 


0.9997 


399.88 


08 








9 44 


3.85 


0.0002 


0.03 


0.9999 


399.96 


0.04 








10.44 


4.26 


0000 


00 


1 . 0000 


400.00 




















Total 


400 


400 



Column (3): The deviation of the class limit from the mean. Note that this 
corresponds to x in the discussion above. 

Column (4) : Figures in previous column divided by the standard deviation. 
The latter is calculated using tmit class intervals, and from the formula 



Column 
Column 
Column 
Column 
Column 



(5): 
(6): 
(7): 
(8): 
(9): 



N 



Values of z from Sheppard's "Tables." 
Corresponding z values multiplied by N/<r. 
Values of j|(l -f a) from Sheppard's "Tables." 
Corresponding |(1 + a) values multiplied by N. 
Differences between consecutive values in column (8). 



Begin 



at 400 at each end and go towards the center. At the center the two differences 
are added. Note that the theoretical frequencies are not kept in line with the 
values in column (8), but are lined up with the corresponding actual frequencies 
in column (10). 

Column (10) : The actual frequencies. 



26 



THEORETICAL FREQUENCY DISTRIBUTIONS 



8. Probability Calculations from the Normal Curve. We have 

observed from the previous exercises and examples that most biological 
variables tend to follow the normal distribution and that methods are 
available for making, for any particular sample, an estimate of the form 
of the normal distribution from which the sample was drawn. Since 
the normal distribution can be expressed by a mathematical equation, 
the area of any section of the curve cut off by an ordinate can be deter- 
mined readily by integration of the equation, and for all practical 
problems this work has been performed and tabulated in Sheppard's 




1/2(1-'-)- 0228 



FIG. 4. Sketch of a normal curve showing the proportions of the total area below 
and above the ordinate erected at d/o- = -f- 2. 

"Tables." It remains to show how these facts form the basis for tests of 
significance in statistical problems. 

If a variable is normally distributed and the mean and standard 
deviation of the population are known, we can draw the curve and erect 
an ordinate at any point. Suppose that such an ordinate is erected at a 
point which is at a distance, on the positive side of the mean, exactly 
equal to twice the standard deviation. Thus d/v = 2, and from 
Sheppard's "Tables" we find that (! + a) = 0.9772. Taking the total 
area of the curve as 1, the area to the left of the ordinate is 0.9772, and 
that to the right of the ordinate is (1 0.9772) = 0.0228. Assuming 
a population of 1000 variates, it is obvious that 22.8 of these variates 
would be greater than the mean by an amount equal to 2 or more times 
the standard deviation. Hence if one variate is selected at random from 



TE8T8 OF DEPARTURE FROM NORMALITY 27 

the 1000, it is obvious that the probability that this variate will exceed 
the mean to the extent of 2 or more times the standard deviation is 
22.8/1000. Reference to Fig. 4 will make this point clear. 

Looking at the same problem from another angle, we inquire as to 
the probability, in selecting a variate at random, that this variate shall 
fall outside the limits of plus or minus twice the standard deviation. 
We erect two ordinates, one at d/a * 2, and one at d/<r = + 2; and 
our problem is to find the area in both tails of the curve. Obviously 
this will be [1 - |(1 + a)] X 2 - (1 - 0.9772) X 2 = 0.0456. The 
probability that a single variate selected at random will deviate by an 
amount equal to or greater than db2 is 45.6/1000, or approximately 
1/22. 

Probability results are sometimes expressed in terms of odds. If the 
probability is 1/22, the odds are 1 out of 22, or, as usually stated, 1 to 21. 

For the case above, where the deviations in both directions are con- 
sidered, note that the probability is given directly by [1 (! + )] X 
2 = 1 a. The odds are given by a/ (1 a) : 1. 

Some examples follow that should make the whole procedure per- 
fectly clear. 

Example 2. The mean (ra) of a population is 26.4, and the standard deviation (?) 
is 2.0. Find the probability that a single variate selected at random will be 29.4 or 
greater. 

The deviation (d) 29.4 - 26.4 - + 3.0. Hence d/<r - f - 1.5. For d/<r - 1.5, 
J(l -f a) = 0.9332. The probability (P) <= (1 - 0.9332) - 0.0668. 

Example 3. For the above population, find the probability that a single variate 
selected at random will deviate from the mean to the extent of 3.5 or more. 

d - 3.5 - - - 1.75 

<r 2 

For d/<r - 1.75, (1 + a) 0.9599. a - (0.4599 X 2) =-.0.9198 
Hence P - (1 - a) - (1 - 0.9198) - 0.0802. 
Example 4. Determine the value of d/a corresponding to P = 0.05. 
p _ (i _ a ) = 0.05 
a (1 - 0.05) - 0.95 
j(l + ) - (0.5 + 0.4750) - 0.9750 

From Sheppard's "Tables," <*/<r'= 1.96. 

6. Tests of Departure from Normality.* The x 2 test of Chapter 
IX, Example 19, on the goodness of fit of actual to theoretical normal 

* Students studying statistics for the first time are advised to pass over the 
remainder of this chapter and come back to it at a later date. 



28 THEORETICAL FREQUENCY DISTRIBUTIONS 

frequencies is a general test of the normality of a distribution, and, by 
noting those classes that make the greatest contribution to x 2 , we can 
come to some decision as to the type of departure from normality. The 
test described here is one that involves the calculation of two statistics 
that are direct measures of the type and degree of abnormality, Fisher 

a). 

Types of Abnormality. Frequency distributions that depart signifi- 
cantly from the normal may be divided roughly into three classes: 

(a) Skew Distributions. The degree of skewness of a given distribu- 
tion is indicated approximately by the measure 

Mean Mode 
Skewness = - 
0* 

where the mode is the position on the base line, or x ordinate, of a per- 
pendicular line drawn to the maximum point of the curve. This 
measure is obviously zero for the normal distribution, as the curve is 
symmetrical and the mean and the mode coincide. When the mode is 
greater than the mean we have negative skewness, and when less than 
the mean, positive skewness. 

(b) Platykurtic, or flat topped. The shoulders of the curve are filled 
out and the tails depleted. 

(c) Leptokurtic, or peaked. At the center the curve is higher and 
more pointed than the normal, and the tails are extended. 

In certain distributions we may have skewness as well as kurtosis as 
indicated by (6) and (c). 

Test for Abnormality. The type of abnormality of a distribution can 
be determined directly by calculating two statistics known as g\ and 02. 
These are calculated from the k statistics fci, fez, 3, and fc 4 , that are in 
turn derived from the sums of the powers up to 4 of the deviations from 
the mean. 

One of the most convenient methods for the calculation of the k 
statistics is to obtain first a series of values a\ a^ defined as follows : 






From ai a*, we calculate a series of statistics Joiown as the momenta 
(v i 04), which in this form are unconnected for grouping in the fre- 
quency table. 



TESTS OF DEPARTURE FROM NORMALITY 29 



The k statistics are then given by: 

N \ 

N 2 



|f3 



= A r2 

4 " 



r 

- 2) [ 



(AT - 1) (TV - 2) # - 3 

Two of the k statistics 2 and 4 require correction for the interval of 
grouping of the frequency distribution. For a unit interval the cor- 
rected values are given by : 

&2 = k 2 ^ , and &4 = & 4 

Corrections for other intervals will, of course, not be necessary; as it is 
always possible to use a unit interval for the purpose of calculating the fc 
statistics. 

The measures of curve type g\ and #2 are given as follows, with their 
standard errors: 



^ - 2) (N + 1) (# + 3) 
9z = 7^> 



= J 
>f 



- 3) (JV - 2) (JV + 3) (JV + 5) 

For normal distributions both gi and g% are zero. The former is a 
measure of symmetry and has the same sign as (mean mode). Figure 5 
illustrates positive and negative skewness as indicated by positive and 
negative values of gi. A positive value of #2 indicates a peaked curve, 
and a negative value a flat-topped curve. These two types are also 
illustrated in Fig. 5 (see page 31). 



30 



THEORETICAL FREQUENCY DISTRIBUTIONS 



Example 6. We shall take as an example to which to apply the test for normality 
the frequency distribution given in Table 7, which also contains the necessary cal- 
culations. We get: 

0i = + 0.184 SEgi =0.227 



02 - + 0.0188 



0.451 



The signs of g\ and g% indicate that the curve departs slightly from normality in 
having a slight positive skewness and in being slightly peaked, but the values of 
g\ and g% are very much less than twice their standard errors so we conclude that 
there is no evidence of a significant departure from normality. 

When the number of classes is fairly large it is desirable to calculate the k statistics 
using an assumed mean. We measure x in terms of the deviations from the assumed 
mean and proceed exactly as in Table 7. Table 8 is an example of the calculation 
of the k statistics by this method, using the same data as in Table 7. 



TABLE 7 
CALCULATION OF THE k STATISTICS 



X 


Frequency 


fx 


/* 2 


/* 8 


fx* 


1 


1 


1 


1 


1 


1 


2 


6 


12 


24 


48 


96 


3 


13 


39 


117 


351 


1,053 


4 


25 


100 


400 


1,600 


6,400 


5 


30 


150 


750 


3,750 


18,750 


6 


22 


132 


792 


4,752 


28,512 


7 


9 


63 


441 


3,087 


21,609 


8 


5 


40 


320 


2,560 


20,480 


9 


2 


18 


162 


1,458 


13,122 



3007 



17,607 



0.697 
(2.4265)* 

0.111 
(2.4265) 2 



+0.184 



= + 0.0188 



SEgi = 0.227 



0.451 



110,023 



ai...a 4 4.911,504 


26.6106 
-24.1229 


155.814 
-392.094 
236.959 


973.655 
-3061.124 
3851.545 
-1745.739 


V1...V4 4.9115 
ki...k* 4.9115 


2.4877 
2.5098 


0.679 
0.697 


18.337 
0.103 


Corrections 
ki'...k* f 4.9115 


-0.0833 
2.4265 


0.697 


0.008 
0.111 



TESTS OF DEPARTURE FROM NORMALITY 



31 





MO ME 
POSITIVE SKEWNESS 



ME MO 
NEGATIVE SKEWCSS 





LEPTOKURTIC PLATYKURTIC 

FIG. 5. Illustrating types of abnormality in frequency distributions. 
MO = mode, and ME = mean. 

TABLE 8 
CALCULATION OF k STATISTICS USING AN ASSUMED MEAN 



Deviations (d) 
from Assumed 


X 


/ 


Mean 


/<* 


fd* 


fd* 


fd* 


1 


1 


-4 


- 4 


16 


- 64 


256 


2 


6 


-3 


-18 


54 


-162 


486 


3 


13 


-2 


-26 


52 


-104 


208 


4 


25 


-1 


-25 


25 


- 25 


25 


5 


30 












6 


22 


1 


22 


22 


22 


22 


7 


9 


2 


18 


36 


72 


144 


8 


5 


3 


15 


45 


135 


405 


9 


2 


4 


8 


32 


128 


512 


2(d)...Z(*) 


(AT- 


113) 


-10 


282 


2 


2,058 


fll . . . #4 






-0.088,496 


2.4956 


0.017,699 


18.212 










-0.0078 


0.662,552 


0.006 












-0.001,386 


0.117 














-0.000 


t?2 #4 








2.4878 


0.679 


18.335 


fo &4 




etc. 




2.5098 


0.697 


0.103 



32 THEORETICAL FREQUENCY DISTRIBUTIONS 

7. Exercises. 

1. Calculate the ordinates (y) and the theoretical normal frequencies for the 
frequency distribution of either Chapter II, Exercise 2, or Chapter II, Exercise 3. 
Totalling the theoretical frequencies will provide a check on the calculations. 

2. Make two graphs for Exercise 1. 

(a) Histogram of actual frequencies and smooth normal curve. 
(6) Histogram of theoretical frequencies and smooth normal curve. 

3. Examine equation (1) in Section 3 above, and show how the value of <r affects 
the shape of the curve. 

4. If the mean of a population is 21.65 and <r is 3.21, determine the probability 
that a variate taken at random will be greater than 28.55 or less than 14.75. 

P - 0.03. 

6. If, for the population described in Exercise 4, the standard deviation of the 
mean of a sample of 400 variates is <r/\/400, find the probability that the mean of 
any one sample of 400 taken at random will fall outside the limits 21.33 to 21.97. 

P = 0.045. 

6. Determine d/a values corresponding to the P values of 0.001, 0.01, 0.02, 0.10, 
0.20, and 0.50. 

7. Test the following distributions for departure from normality. 



(a) x.. 


1 2 


3 


4 


5 


6 7 


8 


9 10 11 


12 


13 


14 


15 16 


/... 


1 57 


185 


217 


177 


126 87 


54 


30 20 14 


11 


13 


5 


1 2 


(6) x.. 


1 2 


3 


4 


5 


6 7 


8 


9 10 11 


12 


13 


14 


15 16 


/... 


2 3 


4 


5 


7 


11 17 


30 


50 34 21 


10 


7 


5 


2 2 


(c) x.. 


1 2 


3 


4 


5 


6 7 


8 


9 10 11 


12 


13 


if 


15 16 


/... 


1 7 


13 


19 


23 


26 27 


28 


26 24 22 


17 


14 


9 


4 1 


(a) 01-1, 


.360, 02 


-2.143. (6)0i = 


- 0.327, 


02 = 


0.939. (c)0i 


= 0.107,02 = 


-0.766. 



REFERENCES 

1. R. A. FISHEK. Statistical Methods for Research Workers. Sixth Edition 

Oliver and Boyd, London and Edinburgh, 1936. Reading: Chapter III, 
Sections 11, 12, 13, 14, 18, and Appendix A. 

2. RAYMOND PEARL. Medical Biometry and Statistics. W. B. Saunders and Com- 

pany, Philadelphia and London, 1923. Reading: pages 235 to 245. 

3. KARL PEARSON. Tables for Statisticians and Biometricians. Part I. 

4. A. SAVAGE, C. H. GOULDEN, and J. M. ISA. Can. J. Research, 12:803-811, 1935 

5. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter II, Sections 2.63, 2.7. 

6. G. UDNY YULE. An Introduction to the Theory of Statistics. Charles Griffin 

and Company, Ltd., London, 1924. Reading: Chapter XV. 



CHAPTER IV 
TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

1. The Estimation of the Standard Deviation. In Chapter II, Sec- 
tion 2, it was pointed out that the best estimate of the standard deviation 
of a population from which a sample has been drawn is A/S(x m) 2 /N, 
where m is the mean of the population and N is the number in the sample. 
Since we never know the value of w, we use x instead ; but the substitu- 
tion of x in the above formula will not give us the best possible estimate 
of a; actually it will give us an estimate that is too small. In other 
words, if we take a large number of samples and calculate a standard 
deviation for each one, the average value of our standard deviations will 
be low, and this will be true regardless of how many samples we take. 
As a matter of fact, if we take a large enough number of samples, we can 
predict with accuracy the extent of the negative bias in the average of 
the standard deviations. To the beginner these facts often appear 
somewhat mysterious, particularly the fact that the bias, in our estimate, 
can be removed, as pointed out in Chapter II, by using the formula 
v S(x x) 2 /N 1. It may seem peculiar that the bias can be 
removed in so simple a manner. Now, it is easy enough to work out 
this proposition algebraically, but this does not settle the question 
necessarily for the beginner, as it is quite possible to work through a 
derivation and follow all the steps without really understanding the 
situation. Consequently, we shall not use the algebraic method here, 
but will try instead to point out why a bias should exist and why it is 
reasonable that it should be removed by dividing the sum of squares of 
the deviations from the sample mean by 1 less than the number in the 
sample. 

In the first place, we have noted already that the sum of the devi- 
ations from the mean of a sample or of a population is zero (Chapter II, 
Section 1). We shall now note that the sum of the squares of the devia- 
tions from the mean is a minimum. If the mean of the population is m 
and we take a large number of samples of size N and in each case we 
determine 2(x m) 2 , it follows that the sum of all these will be the 
same as if we had merely gone through the whole population without 
considering any portion of the variates as a sample. Then, on dividing 
this total sum of squares by the total number and extracting the square 

33 



34 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

root, we would have the value of <r for the whole population. It obvi- 
ously does not matter whether we divide the population into samples 
and determine cr for each one and then average, or merely take the whole 
population as one sample. However, this procedure is possible only in 
theory, as m is actually unknown. For each sample, therefore, suppose 
that we calculate V 2(x x) 2 /N and then average. Now, since the 
sum of the squares of the deviations from the mean is a minimum, the 
use of x will give a minimum value for the sample; but, since the values 
of x vary from sample to sample, it is perfectly clear that S(x x) 2 for 
any one sample will be as large as S(x m) 2 for the same sample only 
if x happens to be equal to m. No matter how slightly x varies from w, 
the sum of the squares of the deviations from the mean of the sample will 
be smaller than the sum of the squares of the deviations from the popu- 
lation mean, and hence the value of the standard deviation is under- 
estimated by the formula which has N as a, divisor. Now let us con- 
sider the extent of the bias and how it may be removed. There are N 
values in a sample, and in theory each of the N variates contributes 
equally to the estimate of the standard deviation; but in calculating 
S(# x) 2 we use one value, x, which is determined by the sample, and 
hence the effective weight of the sample is equal to N 1 instead of N. 
All the values of one sample may be large, and if we could calculate 
S(x m) 2 these values would contribute more to the total sum of 
squares than a set of values in another sample which are closer to m. 
Actually, since we take the deviations from the mean of the sample, the 
first sample would not necessarily contribute any more than the second 
sample. This brings out the idea that the mean used is fixed by the 
sample and to the extent of reducing the effective weight of the sample 
by 1. Thus we have the term introduced by R. A. Fisher, "degrees of 
freedom." When a sample of N variates are used for purposes of estima- 
tion, its weight is only that of the number of degrees of freedom. For 
every statistic calculated from the sample and utilized in forming the 
estimate, there is a loss of one degree of freedom. Thus, in the present 
example of estimating the standard deviation, the statistic calculated 
from the sample is x, and there is a corresponding loss of one degree of 
freedom. This principle will be found to hold throughout all statistical 
procedure. 

2^Terminology and Symbols for Populations and Samples 
Introducing the Term Variance. As pointed out above, we speak of 
population parameters which are true and undeviating values, and 
statistics which are estimates, from the samples, of the population para- 
meters. The statistics we have discussed so far are the mean x and the 
standard deviation s; and the corresponding parameters are m and <r. 



TERMINOLOGY AND SYMBOLS FOR POPULATIONS AND SAMPLES 35 

Very frequently in statistical procedure the square of the standard devia- 
tion, usually referred to as the variance, is the more convenient of the 
two statistics. Most tests of significance can be made by means of the 
variance, in which case the extraction of the square root in order to 
obtain the standard deviation is an unnecessary operation. In general, 
all discussions of methods of estimation refer equally to the standard 
deviation and the variance, and consequently in Example 6 below we 
confine our attention to the variance. 

Before proceeding with Example 6 it may be of assistance to sum- 
marize the symbols and terms that have been used up to this point, and 
any others that have not been used but are relative to those already dis- 
cussed. This summary is as follows: 

PARAMETEKS STATISTICS 

Mean ra Mean x 

Standard deviation <r Standard deviation s 

Standard deviation of a mean. <r m Standard deviation or stand- 
Variance a* or V ard error of aliiean s^ 

Variance of a mean <r or V m Variance or mean square .... s 2 or v 

Variance of a mean s| or t 

Number in sample N or n' 

Degrees of freedom n 

Special notice should be taken of the term standard error, which is coming into 
general use in place of the standard deviation of a sample mean. 

Example 6. The Use of Degrees of Freedom in Estimating the Variance. 
In Table 9 we have a set of random numbers taken from Tippett's tables (6), arranged 
in 10 groups of 20 numbers each. The variation in these numbers may be assumed 
to be made up of two portions: (1) within the groups, and (2) between the groups. 
But if the numbers have been selected at random these two sources of variation will 
be equally balanced. They would be unbalanced if, for example, some groups had 
all small numbers and the other groups a.11 large numbers. The random selection 
of the numbers ensures that this shall not be the case. In terms of variance, the 
above statement with respect to variation is simply that the variances for within 
groups, between groups, and the total variance will all be equal within the limits of 
random sampling. Now, if for a particular set of numbers, as in this set, the variance 
for between groups is adjusted until it is almost exactly equal to the total variance, 
it follows that the variance within groups must also be almost exactly equal to the 
total. We can determine, therefore, the variance within each group, and if our 
method is correct these should give an average value very close to that for the whole 
sample. 

The calculation of the variances within groups has been performed in Table 10 
by two methods. There are 20 numbers in each group, so that in each group we 
have 19 degrees of freedom for the estimation of the variance. In column (7) of 
Table 10 the sums of squares are divided by the degrees of freedom, but in column (8) 
they are divided by 20, the number in the sample. At the foot of the table the total 
variance is again calculated by two methods. In the first case we divide by 199 and 



36 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

TABLE 9 





A 


B 


C 


D 


E 


F 


G 


H 


/ 


J 


1 


29 


45 


14 


25 


11 


47 


28 


35 


18 


25 


2 


45 


39 


49 


32 


32 


36 


24 


27 


16 


39 


3 


16 


29 


29 


18 


46 


42 


10 


18 


34 


24 


4 


37 


11 


31 


28 


44 


36 


27 


44 


18 


30 


5 


50 


12 


19 


20 


28 


38 


11 


25 


30 


24 


6 


10 


37 


20 


44 


40 


21 


42 


33 


29 


36 


7 


26 


44 


49 


41 


27 


41 


22 


49 


35 


31 


8 


19 


15 


15 


10 


28 


26 


30 


11 


35 


10 


9 


24 


50 


11 


43 


27 


17 


17 


17 


42 


13 


10 


10 


14 


22 


19 


11 


50 


33 


39 


50 


43 


11 


10 


22 


23 


10 


48 


30 


44 


26 


21 


27 


12 


48 


20 


41 


13 


21 


39 


32 


29 


11 


20 


13 


22 


46 


40 


31 


44 


21 


23 


16 


45 


39 


14 


13 


14 


12 


45 


16 


46 


25 


47 


18 


30 


15 


28 


21 


39 


39 


36 


22 


27 


10 


31 


18 


16 


32 


15 


43 


23 


42 


34 


16 


20 


26 


11 


17 


10 


37 


31 


11 


12 


50 


20 


12 


34 


46 


18 


11 


26 


34 


22 


48 


13 


47 


42 


22 


43 


19 


30 


29 


49 


35 


30 


46 


38 


50 


24 


44 


20 


37 


22 


37 


49 


30 


47 


12 


34 


42 


24 


Totals.. . 


507 


548 


608 


558 


621 


702 


528 


584 


581 


577 



in the second case by 200. We have, therefore, four determinations of the variance 
as shown below. Note that the last line is calculated independently and does not 
come from totalling the values above except for columns (2) and (3). 

By the first method we obtain for the average variance within groups a value 
that is 99.94% of the total. By the second method the average variance is only 
95.43% of the total, and therefore underestimates the true value by 4.57%. Where N 
is the number of variates in a sample, it follows therefore that the correct estimate 
of the variance is given by 2(x x) 2 /N 1. 

3. The Distribution of the Estimates of the Standard Deviation. If 

a large population is being sampled and each sample contains 100 
variates, we will get a series of varying values for the standard deviation 
calculated from these samples. But, if, instead of taking samples of 
100 variates, we take samples of 10, it is to be expected that in the second 
case we will get values for the standard deviation fluctuating more widely 
than in the first case. This is the same as saying that the distribution 
of the standard deviation is dependent on the number of degrees of 
freedom in the sample. In this respect it is very much the same as a 
mean. In order to obtain from one sample a value for the mean that 



DISTRIBUTION OF ESTIMATES OF STANDARD DEVIATION 37 



TABLE 10 
CALCULATION OF VARIANCE VALUES OF FIGURES IN TABLE 9 

BY GROUPS OF 20 AND FOR WHOLE GROUP 



(1) 


(2) 
T x 


(3 2 


(4) 


(5) ' 


(6) 


(7) 2 


(8) 


A 


507 


16,159 


25.35 


12,852.45 


3,306 55 


174.0289 


165.3275 


B 


548 


18,110 


27.40 


15,015.20 


3,049.80 


162.8842 


154.7400 


C 


608 


21,602 


30.40 


18,483.20 


3,118 80 


164.1474 


155.9400 


D 


558 


18,620 


27.90 


15,668.20 


3,051.80 


160.6210 


152.5900 


E 


621 


22,189 


31.05 


19,282.05 


2,906.95 


152.9974 


145.3475 


F 


702 


27,208 


35.10 


24,640.20 


2,567.80 


135.1474 


128.3900 


G 


528 


16,132 


26.40 


13,939.20 


2,192.80 


115.4105 


109.6400 


H 


584 


20,306 


29.20 


17,052.80 


3,253.20 


171.2210 


162.6600 


I 


581 


19,043 


29.05 


16,878.05 


2,164.95 


113.9447 


108.2475 


J 


577 


19,045 


28.85 


16,646.45 


2,398.55 


126.2395 


119.9275 












Av. 


= 147.6642 


140.2810 






T 


S( 2 ) 


T 2 /AT 


Z( -) 2 


S(:r-) 2 /199 


Sfo ~5) 2 /200 






















5814 


198,414 


169,012.98 


29,401 02 


147.7438 


147 0051 



Average within Groups . 
Total 



Method (1) 

Using Degrees 

of Freedom 

147.66 

147.74 



Method (2) 

Using Number 

in Sample 

140.28 

147.00 



is quite close to the mean of the parent population, we must take a large 
sample. Small samples will give us unbiassed estimates, but they will 
be more variable estimates. 

Now in Chapter II we observed that, if a population is normally dis- 
tributed and we know its standard deviation and mean, we can make a 
direct calculation of the probability of drawing from that population a 
sample with a mean of a given magnitude. This is, in a sense, a test of 
the significance of the mean of a particular sample, since if the prob- 
ability is very small we should conclude that the sample was not drawn 
from the population in question, but from some other population. 
However, the standard deviation of the population cannot be deter- 
mined, and the only value we have is the estimate s which has been cal- 
culated from the sample and varies from sample to, sample. This 
brings us therefore to the general question of making tests of significance 
from the data of samples of any size. 



38 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

4. Tests of Significance. The method of Chapter II for making 
probability determinations arose from our knowledge that the ratio of a 
mean of a sample to the standard deviation of the population from which 
the sample is drawn is normally distributed. This follows, of course 
because, if the mean is normally distributed and the standard deviation 
is constant for the population, the ratio of the two will also be normally 
distributed. Suppose, however, that we take the ratio of the mean of a 
sample to the estimate of the standard deviation s. Since s is more vari- 
able for small samples than for large ones, the ratio will obviously have 
a distribution that is dependent on the size of the sample, and, in order to 
determine the probability of the occurrence of any particular value of 
this ratio, we must know its distribution. This was worked out by 
"Student" (4) in 1908, and for the first time practical statisticians had 
placed in their hands a tool which could be applied in tests of significance 
for samples of all sizes. "Student" gave first a set of tables for the 
distribution of x/s, which he designated by the letter Z. Later he 
prepared a table based on the distribution of t, which is x/s^. Fisher, 
in "Statistical Methods for Research Workers," gives a compact table of 
t for degrees of freedom varying from 1 to 30, and the probability levels 
P = 0.01, 0.02, 0.05, 0.10, and 0.90. These are the most convenient for 
general use, and are reproduced in part in Table 94. 

Example 7. Two varieties of wheat are compared in 4 pairs of plots, there being 
1 plot of each variety in each pair. Referring to the two varieties as A and B, we 
determine the difference in yield A-B for the 4 pairs of plots, and the results are as 
follows in bushels per acre: 

Pair 1234 

A-B 2446 

The differences are all positive and are therefore in favor of the variety A, but we 
wish to make a test so as to be able to state whether or not the data are in agreement 
with any hypothesis that we may set up. The obvious hypothesis here is that the 
varieties are not different in yielding quality, and consequently our theoretical dis- 
tribution is built up on that basis. If the varieties are not different, the data will 
be expected to give a value of t that is not improbable. If they are different, we will 
expect the data to give a value of t which will occur by random sampling in only a 
small proportion of the cases. Let us proceed to the calculation of t. 

We note first thai the mean difference is 4, and that the sum of the squares of the 
deviations of the individual values from the mean is 8. We then have 2 * 8/3, 
the numerator being the number of degrees of freedom available for estimating the 
standard deviation s. Then s = Vs/3, and 8% Vs/3 X 4, which simplifies to 
\/2/3. Finally t = 4 X Vi/2 = 4.87. Now if we examine Table 94 it is observed 
that the 5% value of t for 3 degrees of freedom is 3.18, and the 1% value of * is 5.84. 
Thus the value of t given by the data would occur according to the hypothesis in 



FIDUCIAL LIMITS 39 

less than 5% and somewhat more than 1% of the cases. Our conclusion is that the 
difference observed is due to a real varietal effect, and is not a chance occurrence. 

It may be argued that in an example such as tfce above we are not actually 
testing the significance of the mean difference, because we are basing it on the distri- 
bution of 2, wherein an exceptional value of t may be due to extreme deviations in 
either the mean difference or the standard error. This point is actually only of 
academic interest, because in either case the two samples are proved to be different 
regardless of which factor brings about the exceptional value of t. When we consider 
the actual problem of testing the difference in yield of two varieties, it is obvious 
that a real difference in the variation of the yields from plot to plot is so unlikely a 
factor that in general we can disregard this viewpoint, and assume that the significant 
value of t is at least mainly due to a significant difference in the mean yields. 

5. Fiducial Limits. Stress has already been laid on the principle of 
estimation; and we come now to a method of setting up limiting values 
according to given probability levels, such that it can be said with a 
reasonable degree of certainty that the true value which is being esti- 
mated lies between these limits. In the example above, the difference 
between the yields of the two varieties was found to be significant; but 
no attempt was made to set up two limiting values, one on each side of 
the mean difference of 4 bushels, and to state that according to a given 
probability level, the true mean difference was between these limits. 
If we can perform such an operation it will obviously be of great prac- 
tical value, because in the end we are not really concerned with being 
able to say only that one variety is a higher yielder than the other. 
Unless we can make a reliable estimate of this difference our experiment 
is not contributing information of value in actual practice. 

It was emphasized in Chapter I that a test of significance involves 
setting up a hypothesis and determining the agreement between the 
hypothesis and the data of the experiment, and furthermore that any 
hypothesis whatever can be set up. In the example above, the hypoth- 
esis was that the mean difference in yield between the varieties was zero, 
and what we actually did was to find the value of t from the expression 
( m)/Sj, where m t the mean of the parent population according to 
the hypothesis, was taken to be zero. We can, however, take m equal 
to any value that we please, and we might choose for example to take m 
equal to 2. Then t = (4 - 2) X \/3/2 = 2.46, and this value is less 
than the 5% point. The inference from this test is that there is no 
definite evidence that the true difference is greater or less than 2. We 
begin to see therefore that, though our difference is significant, we cannot 
specify very closely the range within which the true value lies. Suppose 
now that we can locate a lower limit such that, if we substituted it for m 
in the t test, the value of t obtained would be exactly equal to its 5% 
point, and we determine in addition a similar upper limit. The observed 



40 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

difference could then be said to differ significantly from either of the 
limiting values, and we could say with a reasonable degree of certainty 
that the true value lies between these limits. The procedure is simple, 
as all we have to do is to set up the equation for t with m as an unknown 
and t equal to its value at the 5% point. Thus: 

3.18 - (4 - m) X 

Solving for m we get an answer of 1.40, and our limits are 0.60 to 3.40. 
It is now clear that, although our experiment gave a significant result, it 
did not enable us to estimate very accurately the true difference in yield 
between the two varieties. These limiting values have been very aptly 
termed by R. A. Fisher the fiducial limits, and in the present example we 
would describe them as the fiducial limits at the 5% point. 

6. General Methods for Testing the Significance of Differences. 
One of the most common problems in statistics is the testing of the sig- 
nificance of a difference between two means. The reasoning behind 
such tests involves picturing an infinite population of differences for 
which the mean is zero. We have two samples for which the means are 
different; and we wish to know in what proportion of the cases on the 
average, in the procedure of taking pairs of samples, we will get a differ- 
ence as large as or larger than the one observed. Tests of this kind fall 
into two classes: 

(a) Samples are distinct and the variates are not paired in any way. 
If there are two blocks of land and we take the yields of a group of plots 
from each block, and we wish to test the significance of the difference 
between the means for the blocks, we have a problem that falls into 
this class. The number of variates in the two samples may be either 
the same or different. Let the samples be designated as 1 and 2; then: 

xi = mean of sample 1. 
X2 = mean of sample 2. 
1 2 = mean of difference to be tested. 

n\ = degrees of freedom for sample 1 which contains, 
therefore, n\ + 1 variates. 

nz = degrees of freedom for sample 2 which contains, 
therefore, n% + 1 variates. 

The calculations are carried out as follows: 

S(xi xi) 2 = sum of squares for sample 1. 
S(XL> x%) 2 = sum of squares for sample 2. 



TESTING THE SIGNIFICANCE OF DIFFERENCES 41 



n\ 



+n 2 + 2 



(3) 



We enter the table of t under n = ni + n2- If the samples contain an 
equal number of variates, we have: 

(m + 1) = (n + 1) = N 



and = x/ 1 2(N _ t) ^ (4) 



(5) 

The table of t is entered under n = 2(N 1). 

Example 8. Let i 196.42 and x 2 = 198.82; then (xi - x 2 ) = 2.40. The 
samples are taken independently, and consequently there is no reason for assuming 
that xi and 22 are correlated. In sample 1 we have taken 9 variates, and in sample 2 
we have 7 variates. Hence HI = 8 and n% = 6. We calculate first S(#i i) 2 
and 2(^2 a) 2 . We will assume that this is done, and we get: 

-i) 2 = 26.94 
-2 2 ) 2 = 18.73 



Then: 



and 



Entering the table of / under n = 14 we find that a J value of 2.62 corresponds almost 
exactly with a P value of 0.02. Between the means of the tw.o samples a difference 
of 2.40 would occur by chance in only 2 cases out of 100. 

(6) Variates are paired; that is, each value of x\ is associated in some 
logical way with a corresponding value of #2- Thus, if two varieties of a 
field crop are being tested in pairs of plots, each pair containing one plot 
of both varieties, we would have a problem of this kind. There will, 




42 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

of course, be the same number of variates in the two samples so that, 
if there are N pairs, there will be N 1 degrees of freedom available 
for the comparison. This follows logically from the fact that we are 
now dealing with individual differences and there is one difference for 
each pair of variates. 
The calculations are : 



(Ti - r 2 ) 2 . 

s = \/ Z(zi X2) 2 T- / N 1 (6) 

N 



t = - , same as formula (3) 



If the student should be confused to find later that s 2 as computed 
above is not the same as when obtained by the analysis of variance, it 
may be just as well to adopt the following method, which is identical 
with that of the analysis of variance. The value of t obtained by the 
two methods is, of course, the same. 



(8) 



t = same as formula (3). 

Example 9. In this example assume that the variates are paired, as in a feeding 
experiment where a series of animals are paired up according to initial weight. 
One animal in each pair is given ration 1 and the other one ration 2. There are 
10 pairs of animals, and the difference between the mean gains per 100 pounds of 
feed at the end of the feeding period is 1.42 pounds. We shall assume that 



- xzY - ' r = 15 08 

L 2 ^ J 

Then 

s = ^ / ~ = 1 30 

and 

1 42 



2 /10 _ 
0\ 2 



130V- 244 



EXERCISES 43 

Entering the t table under n = 9, we find that the P value is between 0.05 and 0.02, 
but closer to the former. We can take P = 0.05 as approximately correct, so that 
the difference between the two means could only occur by chance in about 1 out of 
20 trials. 



7. Exercises. 

1. The figures below are for protein tests of the same variety of wheat grown in 
two districts. In district 1 the average for 5 samples is 12.74, and in district 2, the 
average for 7 samples is 13.03. If these are the only figures available, test the 
significance of the difference between the average proteins for the two districts. 



District 1 12.6 13.4 

District 2 13.1 13.4 



Protein Results 
11.9 12.8 13.0 
12.8 13.5 13.3 12.7 12.4 

t = 1.04 P = 0.3, approximately. 



2. Mitchell (2) conducted a paired feeding experiment with pigs on the relative 
value of limestone and bonemeal for bone development. The results are given in 
Table 11 below. 

TABLE 11 

ASH CONTENT IN PERCENTAGE OF SCAPULAS OF PAIKS OF PIGS 
FED ON LIMESTONE AND BONEMEAL 



Pair 


Limestone 


Bonemeal 


1 


49.2 


51.5 


2 


53.3 


54.9 


3 


50.6 


52.2 


4 


52.0 


53.3 


5 


46.8 


51.6 


6 


50.5 


54.1 


7 


52.1 


54.2 


8 


53.0 


53.3 


Mean. . . . 


50.94 


53 14 



Determine the significance of the difference between the means in two ways: (1) by 
assuming that the values are paired, and (2) by assuming that the values are not 
paired. On the basis of your results, discuss the effect of pairing. 



(1) Paired: t - 4.42, P 

(2) Unpaired: t - 2.48, P 



less than 0.01. 
> approximately 0.02. 



3. In a wheat variety test conducted over a wide area, the mean difference 
between two varieties was found to be 4.5 bushels to the acre. The standard error 



44 TESTS OF SIGNIFICANCE WITH SMALL SAMPLES 

of this difference s^ was 1.5 bushels per acre, and was determined from 100 pairs of 
plots. Set up the fiducial limits at the 5% probability level for the mean difference 
in yield between the two varieties. 

Note that t can be taken as 1.96, then fiducial limits are 1.56 to 7.44. 

REFERENCES 

1. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London and Edinburgh, 1936. Reading: Chapter V, Sections 23, 24, 24.1. 

2. H. H. MITCHELL. Proc. Am. Soc. An. Prod., 63-72, 1930. 

3. G. W. SNEDECOR. Statistical Methods. Collegiate Press Inc., Ames, Iowa, 1937. 

Reading: Chapters II, III, and IV. 

4. STUDENT. Biometrika, 6:1, 1908. 

5. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London. Reading: Chapter V, Sections 5.1, 5.2, 5.3. 

6. L. H. C. TIPPETT. Random Sampling Numbers. Cambridge University Press 

London, 1927. 



CHAPTER V 
THE DESIGN OF SIMPLE EXPERIMENTS 

1. What is Experimental Design? In Chapter I some ideas relative 
to experimental design were presented, but in view of what we have 
now learned of the t test it should be worth while at this point to repeat 
some of these ideas, and at the same time introduce any new concepts 
that have arisen out of later discussions. An experiment can be said to 
have a definite design if it has been carefully planned in advance, and if 
due attention has been paid to possible results and their interpretation. 
The latter point is probably the most frequently neglected. A great 
deal of time may be spent on the various details of procedure, and full 
preparations made for carrying the experiment through to completion. 
This may be assumed to be sufficient to ensure a successful experiment, 
but a long list of such experiments that contribute neither positive nor 
negative information is good evidence that careful planning of the pro- 
cedure is in itself incomplete. Only by thinking in terms of the various 
types of results that an experiment can yield is it possible to obviate 
some very costly mistakes. If these possibilities are thoroughly worked 
out it is self-evident that a complete failure is impossible. 

2. Planning to Remove Bias. Qne of the commonest mistakes in 
experimental design is the failure to guard against biased results. Such 
experiments may give good results but their great weakness is that they 
are not beyond criticism; and regardless of the truth and importance of 
the results obtained the investigator may never feel quite happy about 
presenting them with conviction. Let us examine hypothetical plans of 
experiments that are subject to a bias of some sort. 

Suppose that we are to conduct an experiment on the value of feeding 
milk to school children. There are two neighboring schools, and milk is 
given to the children in one of the schools and not to those in the other. 
At the end of the experiment the children are compared on the basis 
of height, weight, etc., by means of the t test. The children from the 
school in which milk was given are found to be significantly heavier than 
those from the other school. The error in design is so obvious here that 
it is scarcely necessary to point it out. The experiment has shown that 
the children of the two schools are significantly different in weight, but 
this might easily have been the case if no milk had been given or even if 

45 



46 THE DESIGN OF SIMPLE EXPERIMENTS 

the order of giving the milk had been reversed. In fact the experiment 
is not at all what it seemed to be at first. It consists actually of just 
two variates which are the two schools, and no determination of the 
error of such an experiment is possible. 

Now let us endeavor to improve the plan, and we will confine the 
giving of milk to pairs of boys or girls, one getting the milk and the other 
not. The pairs are selected at random, and in each pair the milk is 
given to the younger and not to the elder child. The reader will object 
that we are again introducing a bias in that the difference observed might 
easily be due to age and not to the effect of milk in the diet. This is 
perfectly true, so in order to overcome this defect we decide to give it to 
the younger child in one case and the elder child in the second case, 
alternating in this way throughout the entire group. Now the experi- 
ment seems to be perfect, and in truth it is much improved, but with a 
little thought it should be clear that we have succeeded in removing only 
the gross defects those that are obvious to us at the outset and which 
anyone can remove with a little thought and a general knowledge of the 
problem being investigated. The chief trouble with our design is not 
that we have knowingly allowed some factor to bias the experiment, but 
that we have not planned it in such a way that it is impossible for bias to 
enter in. A definite method is available for this purpose, which has 
already been referred to in Chapter I. It involves merely assigning at 
random which member of each pair of children is to receive milk. This is 
a simple device and one which is absolutely trustworthy in the matter 
of removing bias. 

Numerous examples may be cited of experiments that are designed 
so that bias may enter in. One of the most common is the field plot test 
in which the varieties or treatments are arranged systematically in the 
blocks or replications. It is not possible to discuss this particular prob- 
lem and deal with it fully until we have made a study of the methods of 
the analysis of variance, but we can consider the simple type of experi- 
ment in which only two varieties or treatments are being tested and they 
are arranged in pairs of plots. Here we are dealing with a series of 
differences, and we set up a hypothesis as, for example, that the mean 
difference is normally distributed about zero. On the basis of this 
hypothesis we can determine the proportion of the trials in which a dif- 
ference as great as or greater than the one observed will occur. The 
validity of our test depends on its being designed so that if the hypoth- 
esis is true the distribution of the results from a large number of trials 
will be normal and will have a mean of zero. What would anyone after 
a little thought say of an experiment designed so that, if the varieties 
being tested are actually equal in yield, the result turns out according 



DESIGNS THAT BROADEN THE SCOPE OF THE EXPERIMENT 47 

to a large series of tests, either definitely positive or definitely negative? 
Yet this is just the kind of result that may be expected if the principle 
of randomization is not used in setting up the experiment. This applies 
particularly to the position of the varieties or treatments in the pairs. 

3. Designs that Broaden the Scope of the Experiment This is 
another subject than cannot be treated fully at this stage, but a few of 
the general principles may be pointed out. Suppose that the all-inclu- 
sive subject of the experiment is the effect of milk in the diet of young 
animals. Most of us would reject this as a proper subject for experi- 
mental investigation at once, because we can see that it is one for which 
there is no possibility of obtaining a resolt that will be of practical value. 
In one group of animals the milk may be beneficial and in another group 
it may be of no value or even harmful, so that unless the experiment is 
repeated with all possible kinds of animals and the results with each 
kind studied separately we cannot expect to gather any valuable infor- 
mation. The decision with regard to an experiment of this type is likely 
to be that we should select one kind of animal in which we are particu- 
larly interested, and then confine the tests to a limited age group. In 
the first case the subject of the investigation called for an experiment of 
such enormous scope that the entire proposition was absurd. Now we 
have limited the scope of the experiment, but we have not gone as far 
as we might. Let us suppose that the investigator decides on pigs as 
the kind of animal to be tested, then he decides to use pigs of one age 
within the limits of one week, and finally that they shall be from the 
same litter. He has now gone to the other extreme and has set up an 
experiment such that, no matter how significant the results, they will 
not be of any value except within a very narrow range. It cannot be 
assumed that the results will apply to other age groups, to other breeds, 
or perhaps even to other litters, as it may easily be that the litter selected 
is peculiar in some respect with regard to the reaction of the individuals 
of the litter to milk in the diet. No amount of mathematical knowledge 
will help the investigator over the difficulty encountered here, of setting 
up an experiment that will not have too great a scope but will at the 
same time give results that can be interpreted on a fairly wide basis. 
Only his own experience and general knowledge of the problem that is 
to be investigated will give the clue to the correct form for the experi- 
ment to take. In this instance there may be one breed of pigs that is 
predominant in the area in which the investigator is interested, and con- 
sequently it is quite justifiable to confine his experiment to this breed. 
Again, there will be a definite range in age at which farmers will be con- 
cerned with feeding milk, and only this range need be represented. It 
will not be wise, however, to use only pigs from one litter; in fact it 



48 THE DESIGN OF SIMPLE EXPERIMENTS 

would seem to be desirable to have as many litters as possible represented 
in order that the experimental material will be representative of pigs as 
a whole in the area in which they are being raised. An obviously de- 
sirable plan will be to take pairs of pigs of nearly equal weight and con- 
dition from a number of litters, assign the alternative diets at random 
to the members of each pair, and then feed the pigs individually so that 
in<Jptdual records may be kept of food eaten and gains made. 

4. Replication and the Control of Error. The value of replication 
in experimental design is easily understood. In the first place, replica- 
tion increases the accuracy and scope of the experiment; in the second 
place, it enables us to determine the magnitude of the uncontrolled varia- 
tion that is usually referred to as the error; and in the third place it 
allows for designs that give us an effective control over error. The in- 
crease in accuracy due to replication is expressible in terms of a mathe- 
matical equation. In Chapter II, Section 3, we noted that the standard 
deviation of a mean is reduced in proportion to the square root of the 
number in the sample. In ordinary experiments any one treatment is 
represented by a sample which is made up of one unit in each replication. 
Therefore in general the accuracy of an experiment, as expressed by the 
standard error of a mean of any one treatment, is increased in proportion 
to the square root of the number of replications. This statement should 
not be interpreted to mean that results of twice the value are obtained 
by multiplying the replications by 4. This depends on what we mean 
by the value of the results. In terms of work done or energy expended 
on an experiment to bring about a given reduction in the standard error 
this is true, but it may be that the expenditure of additional energy in 
order to increase the accuracy of the experiment is unnecessary, in which 
case the value of the results is not enhanced. More will be said on this 
subject later; but for the present we should note that replication is the 
primary tool at our disposal for increasing the accuracy of the experi- 
mental results. 

Another phase of the increased accuracy due to increased replication 
arises from the distribution of t for different-degrees of freedom. From 
Table 94 we note that, for 1 degree of freedom, t at the 5% point is 
12.706 while for 60 degrees of freedom the corresponding value of t is 
2.00. In the first case a much larger difference would be necessary to 
represent a significant effect than in the second case. In a paired ex- 
periment the number of degrees of freedom available for estimating the 
error of the experiment is equal to 1 less than the number of pairs. Sup- 
pose then that we have one experiment with 3 pairs and another one 
with 10 pairs. For the first experiment we would require for significance 
a difference that is 4.30 times the error, and for the second experiment a 



REPLICATION AND THE CONTROL OF ERROR 49 

difference that is 2.26 times the error, these being the values of t at the 
5% point for 2 and 9 degrees of freedom respectively. It is important 
for the beginner to note carefully that this increase in accuracy due to 
increased replication is entirely distinct from that discussed above which 
results from dividing the standard error of the experiment by the square 
root of the number of replications in order to determine the standard 
error of a mean. Both factors act together and in the same direction 
but they arise from different sources. 

The manner in which replication increases the scope of the experi- 
ment will be evident from the discussion of Section 3. In the example 
discussed there it was decided purposely to make the replications some- 
what different, in order that the results might be of general application. 
The importance of this is sometimes overlooked, and we will find field 
plot investigators looking for an exceptionally uniform patch of soil on 
which to carry out an experiment and putting all the replications on this 
same patch. No criticism is offered of this procedure provided that the 
investigator is not under the impression that by doing so he is necessarily 
improving the experiment. Within each replication it is desirable to 
have as much uniformity as possible, but between the replications it 
does not improve matters to have a great deal of uniformity; and from 
the standpoint of increasing the scope of the experiment it may even be 
harmful. To put these ideas into concrete form let us assume that two 
soil treatments are being compared in paired plots. On the field that 
is available for the experiment there are several types of soils, and we 
shall assume for the purpose of argument that all the soil types are 
present that occur in the area for which the results of the experiment are 
to apply. The investigator has three choices. The pairs of plots can 
be placed all on one soil type, an equal number of pairs on each type, or 
at random over the field. Placing the pairs all on one soil type and 
close together in the field has in its favor compactness and economy of 
space; but the results obtained on the one type of soil may not apply 
to the other types, and consequently to get full information on the 
problem a separate test must be planned for each condition. This 
may be beyond the scope of the facilities of the investigator, so he turns 
his attention to the other possibilities. Placing an equal number of 
pairs on each soil type has decided advantages. For example, if there 
are at least four pairs in each location it is possible to regard each set 
as an individual but very rough experiment, capable of yielding an 
approximate measure of the particular reaction of the two treatments 
on the soil type represented. The average yields of the two treatments 
over the whole field will, however, be representative for the whole area 
in which the results are to be put to practical use only if in that area 



50 THE DESIGN OF SIMPLE EXPERIMENTS 

there are about an equal number of acres belonging to each type. This 
statement, of course, implies that the treatments will give different 
results under the various substratum conditions, but experience tells us 
that this is very likely to be the case. We turn now to the third method, 
that of randomizing the pairs of plots over the whole field. The process 
of randomization will ensure that the various soil conditions represented 
in the field will have an equal chance of being used in the experiment. 
As nearly as possible, therefore, we are obtaining a random sample of 
the infinite population for which we are endeavoring to obtain an un- 
biassed estimate of the difference between the two treatments. The 
only possible criticism of this method is that some of the soil types will 
not be represented, and hence certain information will be lost. The 
answer is that with a given type of experiment we cannot perform two 
functions at once. Without enlarging it considerably we cannot design 
an experiment that will give us a general average result for the whole 
area under consideration, and at the same time give us detailed informa- 
tion on the reactions of the treatments under varying conditions. In- 
formation regarding the whole area is not lost, but gained, by placing 
the pairs at random and perhaps missing some of the types. On the 
assumption that the field is representative of the larger area being sam- 
pled it gives us a more correct measure than if we assumed without 
proper information that each of the types is equally represented. 

This somewhat theoretical discussion does not bear precisely on the 
practical problem with which the investigator is faced, because it is im- 
possible to obtain a field that is really representative of a large area. 
However, it serves to bring out some very important points that may 
be put into practice in tests of this kind. Any investigator who gives 
the problem serious thought will take note of the limitations of one test 
carried out under very uniform conditions, and at the same time will 
realize the importance of replication in widening the scope of field plot 
experiments. 

The second important function of replication is to enable us to obtain 
a measure of the experimental error. This follows directly from the 
principles of the t test. If there is only one plot of treatment A and 
one of B there can be only one difference, and the number of degrees of 
freedom available for estimating the standard error is zero. In non- 
statistical terms there is only one value, the difference between the two 
plots, and this difference is the only measure we have of both soil varia- 
tion and the effect of the treatments. We cannot compare a difference 
with itself; therefore, we say that there are no degrees of freedom avail- 
able for estimating the error of the difference. This defect in an experi- 
ment is obviously overcome as soon as we introduce replication. Even 



REFERENCE 51 

if we have only two plots of A and two of B we have at least one degree 
of freedom available for estimating the error, and by means of the i 
test an unbiassed comparison of the treatments can be made. 

The third function of replication has to do with the control of error. 
Another hypothetical example will make this clear. Again we can sup- 
pose that two soil treatments are being compared in paired plots. The 
measure of error is determined from the variation in the differences 
within the pairs. Suppose now that the plots are all distributed at 
random over the field, and the pairs are made up simply by taking the 
two plots of A and B that happen to fall together in another random 
selection. This can have only one effect, and that is to increase the 
variability of the differences, and consequently the accuracy of the test 
is reduced. A question that may be asked here is whether or not the 
method that increases the variability of the differences will also increase 
the average difference between the two treatments. Yes, the average 
difference will also be increased but it must be remembered that this is 
due in actual practice to two components. A part is due to the real 
difference between the treatments and a part to the variability of the soil. 
The latter component will be increased in the same proportion as the 
error, but the former will not, and consequently the precision of the 
experiment becomes correspondingly less as the error component 
increases. 

The benefits to be obtained from the arrangement of treatments in 
replications wherein each replication contains one of each of the treat- 
ments is fairly well known to experimentalists, especially in agronomic 
research. Variety trials are therefore arranged in compact blocks so 
that the plots within the blocks are as nearly alike as possible. There 
are, of course, many applications of the same principle in other types of 
experimentation; but this subject will be discussed more fully under the 
heading of the analysis of variance. 

REFERENCE 

1. R. A. FISHER. The Design of Experiments. Oliver and Boyd, London and 
Edinburgh, 1937. Reading: Chapters I, II, III. 



CHAPTER VI 
LINEAR REGRESSION 

1. General Observations. In the previous discussions emphasis was 
placed on the variations that occur in any one variable, such as the yield 
of wheat plots, the weight of animals, or the height of students. Some- 
times the values of one variable are classified in two or more ways, in 
which case we may be interested in the joint variation of the pairs or 
groups of values so formed. For example, in Chapter V a problem was 
discussed in which pairs of plots of two varieties were arranged in differ- 
ent ways over a field. The interest there was largely in the differences 
between the members of pairs, but it was also pointed out that if the 
plots were close together they would tend to yield alike, or in other 
words they would vary together. The present chapter, however, deals 
with examples wherein there are paired variates but of two different 
kinds of variables, and in general one of the variables may be regarded 
as independent and the other as dependent. In a study of the effect of 
rainfall on yields of field crops, we would have a typical example of a 
dependent and an independent variable, in that the interest would lie 
in the degree to which rainfall, acting as an independent variable, would 
have an effect on yield, the dependent variable. It would be useless, 
of course, to think of this problem in any other terms, as we could not 
imagine the yield of field crops having any effect on rainfall. 

It is not difficult to see that, for any set of data for paired variates, 
it should be possible to obtain a measure of the physical relation between 
the two variables. Suppose that the data are arranged as in Fig. 6, 
which shows graphically the average yields of groups of plots of Marquis 
wheat for given percentages of infection with stem rust. It would not 
be difficult to draw a straight line so that it would represent the general 
trend of decreasing yield with increasing percentages of infection, and 
we could then read off the approximate decrease in yield for a given 
increase in infection. This, of course, would be a very crude method, 
as the fitting of the line would be purely a matter- of eye judgment and 
different individuals would place the line in slightly different places. 
Then to develop from the graph a general expression for the relation 
between the two variables, from which the line could be reconstructed 
at any time and which could be used for predicting the effect on yield 

52 



FITTING THE REGRESSION LINE 



53 



of given percentages of infection, it would be necessary to draw out the 
graph very accurately and make an average of a number of measure- 
ments. In order to arrive at a more precise method of fitting the line, 
recourse is had to the " method of least squares." This means that 
a line is fitted such that the sum of the squares of the deviations of the 
points in the graph from the straight line is a minimum. It gives us 
a statistic known as the regression coefficient, which expresses the in- 
crease or decrease in the dependent variable for one unit of increase 
in the independent variable. From the regression coefficient we can set 
up a regression equation, which can be used to make predictions; and 
also it defines the straight line known as the regression straight line. 



50 



^ 30 



20 



10 



n 




20 



60 



70 



80 



30 40 50 

PERCENTAGE RUST 

FIG. 6. Regression graph for yields of Marquis wheat on degree of 
infection with stem rust. 

The essential difference between the treatment of different kinds of 
variables that are thought to be related and pairs of variables that 
merely vary together will now be clear. In the first case our concern is 
to determine a function, in the present case a straight-line function, that 
will express the average relation between the two variables. In the 
latter case the function will obviously not be of very much value; we 
will probably be better satisfied with some expression giving the com- 
bined effect of the variables on each other or perhaps, if we cannot think 
in such terms, the degree to which both variables are acted upon by 
outside influences that cause them to vary together. Of this second 
condition we shall learn more in the next chapter. 

2. Fitting the Regression Line. Let the two variables be represented 
by x and y, where x is independent and y dependent. Then, if the 



54 LINEAR REGRESSION 

relation between x and y can be represented by a straight line, the equa* 
tion of the line will be of the form: 

Y = a + bx (1) 

where a and 6 are constants and F represents the values of y estimated 
from the equation. For any one value of z, say x t -, the corresponding 
value of y estimated will be F,-, and the error of estimation will be 
(|/* F). The value of y* would be represented on the graph as in 
Fig. 6 by one of the points, and the corresponding estimated value F,- 
would be a point on the straight line. To fit the line, it is required that 
the sum of the squares of the errors of estimation 2(y F) 2 shall be a 
minimum. It is best to begin with x and y measured from their means, 
so that our regression line is actually: 

(F - y) = a + b (x - x) (2) 

whence the error of estimation is given by 2[(F y) (y y)] 2 = 
!Z(y F) 2 , the same as before. Minimizing by the method of least 
squares f or S(?/ F) 2 , we obtain the equations: l 



Na + S(z - x)b = S(y - y) 
S(z - x)a + S(z - x) 2 b = 2(y - t/)(z - x) 

and solving we have: 

a = 

2(y - y)(s - f) ... 

6 = sr r _ r \ (3) 

Li\X X) 

In equation (3) we note the expression S(y y)(x x), which is 
usually referred to as the Bum of products. For two variables, it is the 
expression that corresponds to the sum of the squares of the deviations 
from the mean for one variable. We know that the variance for a single 
variable is given by: 



N -I 

and now we learn that the covariance for two variables is given by: 



<. 



1 For the method-of-least-equares technique see any good textbook on elementary 
calculus. If it is confusing to apply these methods to expressions containing the 
summation sign, 2), write out one or two sets of values and proceed with them con- 
secutively. The procedure for the entire set of values summated will then be clear. 



TESTS OF SIGNIFICANCE OF THE REGRESSION COEFFICIENT 55 

In (3) if the numerator and denominator are divided by N 1 the 
equation becomes: 

___ Covariance (xy) 

Variance (x) 
Going back now to (2) above: 

Y - y = frfcr - x) 

and: Y = g + b(x - x) .~ , f (6) 

or: Y = (y - bx) + bx '*** (7) 

the last being the form in which this expression is most frequently used. 
It is known as the linear regression equation, and b in the equation is the 
regression coefficient. 

3. Properties of the Regression Coefficient. In the equation 
Y = y + b (x x), b expresses the probable relation between x and y 
in terms of the values in which x and y are measured. The coefficient in 
this equation is usually represented as b vxj which means that it is the 
regression coefficient for the regression of y on x\ and thus in any sample 
of paired variates studied it represents a kind of average of the increase in 
y for a given increase in x. Thus if y is bushels per acre and x is tons of 
fertilizer applied, b yx is an estimate of the increase in yield to be expected 
from one ton of fertilizer. 

For every example where we study the regression of y on x, there is 
also the theoretical possibility of studying the regression of x on y] but as 
stated above the theory of linear regression is best confined to examples 
where we can think clearly in terms of the effect of one variable on the 
other, and consequently the investigator is concerned with only one 
aspect of the regression. 

The regression coefficient is a measure of the slope of the regression 
line, but only relative to the class values of the two variables and their 
range of variation. Suppose that, in a study of the effect of rainfall on 
yield, the rainfall varies from to 9 and the yields from 20 to 30, and 
the mean yield is 25 and the mean rainfall 5. In a graph such as Fig. 6 
the units could be of the same length for the two variables, and if the 
regression coefficient is 1 the regression line would go from one diagonal 
to the other and would have a slope of 1 ; that is, it would lie at an angle 
of 45 degrees. However, if rainfall varied from to 20 the slope would 
be less than 1, even for the case where yield is completely dependent on 
rainfall. 

4. Tests of Significance of the Regression Coefficient. The sam- 
pling error of the regression coefficient is related to the error of estimation 



56 LINEAR REGRESSION 

measured by 2(y F) 2 . Thus we have the standard error of estimate 
given by: 



and the standard error of the regression coefficient by: 



s b = s e /?(x - )* (9) 

The value of S(y F) 2 can best be calculated by equating it to 
- ) 2 - 6 2 S(x - x), or S(y - y) 2 - bV(y - y)(x - ), depend- 
ing on which form is the more convenient at the time. In these equali- 
ties it is understood that the regression coefficient is b vx . 

Then to make the test of significance t is given by: 



Sb S e 

and the table of t is entered under N 2 degrees of freedom. There are 
N 2 degrees of freedom because both y and 6 yx are statistics calculated 
from the sample. 

The test for the significance of the difference between two regression 
coefficients is based on their respective standard errors. For the two 
regression coefficients bi and &2, with standard errors calculated as in (9) 
above, the standard error of the difference would be: 

i-2 = V? + 4 (11) 

and 

t = (bl " &2) (12) 

Sl-2 

The two coefficients may be calculated from different numbers of paired 
values, so that there would be a total of (N\ 2) + (N% 2) degrees 
of freedom available for the comparison of the coefficients, where Ni 
and N% are the numbers of pairs respectively from which 61 and &2 are 
calculated. 

A special case arises when there are two sets of values of the depend- 
ent variable. If these are y\ and 1/2, there are two regression coefficients 
6 tfl and 6y 2X ; and it may be necessary to test the significance of the 
difference between them. The simplest and most direct method is to 
form a new variable from (y\ 3/2) and calculate b (vi ^. v ^ xt which may 
be tfcsted in the ordinary way. 

6. Methods of Calculation. It will be remembered from formula 
(3) that the numerator of the regression coefficient is the sum of products 



METHODS OF CALCULATION 57 

of the deviations from the means of the two variables, and is expressed 
algebraically as S(i/ y)(x ). The denominator of the coefficient 
is the already familiar sum of squares of the deviations from the mean, 
for the independent variable usually indicated by x. Our problem, 
then, is to learn the most convenient method of calculating the sum of 
products. The method follows from the identity: 

~ (13) 



where ^(xy) is the sum of the products of the original'values of x and y, 
taken of course by pairs, and T x and T y are the totals for all the original 
values of x and y, respectively. The latter are somewhat more conveni- 
ent symbols for the familiar S(x) and S(2/). Given a series of paired 
values, therefore, for which a regression coefficient is to be calculated, 
the first step is to determine T x and T y . Then each value of x is multi- 
plied by each value of y (or vice versa), and the sum of the products 
accumulated in the machine. This gives us %(xy), and if we subtract 
from this T x T y /N y the remainder is the required sum of products of the 
deviations. 2 (re x) 2 is, of course, calculated in the manner indicated 
in Chapter II. 

In many examples the labor of calculation can be reduced by coding 
the data. This involves either subtracting a uniform quantity from the 
values of each individual variate or dividing by a constant quantity, or in 
certain cases both devices are employed at the same time. Supposing 
that the actual values are as given below on the left; the values on the 
right are examples of how the coding may be carried out. 



UNCODED 


CODED 


x 


y 


'X 


y 




2402 


2785 


240 


278 


Dividing by 10 and rounding off last figure. 






40 


78 


Subtracting 200. 


198 


196 


8 


6 


Subtracting 190 from each value. 


195 


193 


5 


3 




256 


274 


56 


74 


Subtracting 200. It is quite permissible to 


229 


198 


29 


-2 


have negative values, but usually they compli- 










cate the calculations slightly and if a machine 










is available for calculation most workers avoid 










them. 



The regression coefficient having been calculated, the next step is to 
determine the regression equation, Y = (y bx) + bx. The portion 



58 LINEAR REGRESSION 

(y bx) is constant and is computed once and for all. Putting the 
result for this portion equal to a, we have the working equation: 

F = a + b v *x (14) 

from which all the Y values that are necessary can be obtained. 

It must be remembered that, if the regression equation is calculated 
from coded data, the resulting equation itself must be decoded before it 
can be used for prediction purposes. If the data have been coded by 
subtraction only, the only correction required is to the means of x and y 
and this correction must be made while the equation is in the form given 
in equation (7). If in the coding the x and y values are divided by a 
different constant value, then a correction must be made to the regression 
coefficient as well as to the means of x and y. For example, if x has been 
divided by A and y by B, then the regression coefficient calculated from 
the coded data must be multiplied by B/A. 

Example 10. Calculation of the Regression Coefficient and Regression Equa- 
tion from a Small Series of Paired Values. In a hypothetical example the values 
from 10 pairs of variates are as given below: 



9 8 7 7 6 5 3 3 1 1 T 7 * = 50 
9 9 8 6 6 5 4 3 1 1 !T y = 52 



Values for the totals are given at the end of each line and N = 10. To find the sum 
of products, and the sum of squares of x, we proceed as follows: 



(9 X 9) + (8 X 9) + (7 X 8) + + (1 X 1) - 335.0 

T X T V /N (50 X 52) /10 _ - 260.0 



Difference = S(y 


- y)(x - 5) 


- 75.0 


S(x 2 ) - 9 2 -f 8 2 - 


f 7 2 -f 7 2 -f 6 2 -h - - -f I 2 


- 324.0 


Tf/N - 50V10 




- 250.0 


Difference = S(x 


-x) 2 


74.0 



Then by, 75.0/74 - 1 014. 50/10 5.0. y - 52.0/10 - 5.2. 
Also a (5.2 - 1.014 X 5.0) 13. 

Finally the regression equation is F = 0.13 -p- 1.014*. 

In order to use this equation for predicting values of y from given values of x, 
it is only necessary to insert the required value for x and determine the resulting 
value of F. For example, if we take x equal to 2 the calculated value of F is 
0.13 -f 1.014X 2 - 2.158. 

Example 11. Calculation of the Regression Coefficient and Regression Equa- 
tion from a Large Series of Paired Values. When dealing with large numbers of 
variates, we found that it was convenient to make up a frequency table in order to 
summarize the data and reduce the labor of calculating the mean and the standard 
deviation. Similarly, in regression studies, if a large series of paired values is avail* 
able it is desirable to make up a ttJble which is a combination of the frequency dis- 
tributions of the two variables. From long usage such a table has become known as 



METHODS OF CALCULATION 



59 



a correlation table, and we shall see in the next chapter that it is likewise of value for 
calculating the correlation coefficient. 

To prepare a correlation table the best plan is to copy the paired values on cards 
of a size that can be handled conveniently. Thus, if we decided to make up a table 
for the yields of plots in adjacent rows of Table 4, Chapter II, we would make our 
cards as follows: 



First card 



x 185 
y 162 



Second card 




and proceed until all the pair? had been entered. After deciding on the class values 
in very much the same mannei as described in Chapter II, Section 5, we would dis- 
tribute the cards for one of the variables and then distribute each pile for the second 
variable. Table 12 is the final result of distributing all the cards for the yields of 
adjacent plots as taken from Table 4. The classes here are somewhat larger than 
they should be, in order to save space and to make the table more convenient to use 
as an example. The cards were first distributed for x, giving the frequency distribu- 
tion as shown in the last row of the table. The 4 cards falling in the first class were 
then distributed in the vertical column according to the values of y, and so on for 
each pile. When all the piles were distributed, the cards in each small pile were 
counted, and the frequencies entered in the table Notice also that the natural num- 
bers have been inserted in the table to replace the class values. This is the device 
introduced in Chapter II for reducing the labor of calculating the mean and standard 
deviation from frequency tables. It may be used here in the same way, in order to 
reduce the labor of calculating the regression coefficient. It will be noted that this is a 
form of coding, and consequently the regression coefficient and the regression equa- 
tion will require correction if they are calculated from a table of this kind. 

The next step is to prepare Table 13, in which the first four columns are entered 
directly from the correlation table. For the column headed " totals for y arrays " 
we proceed to obtain the totals for each array as follows, where the first array of y is 
the distribution in the y classes of the variates that fall in the first class for x. 

1st array (2 X 3) + (1 X 6) + (1 X 8) - 20 

2nd array (2 X 3) + (4 X 4) + (5 X 5) + (1 X 6) + (1 X 7) 60 

The total for this column is obviously T V) the grand total of y. In the same way we 
proceed to obtain the totals for the x arrays and T*, the grand total of x. There are 
two columns headed 2(xy), the object being to calculate S(xy) in two ways BO as to 
have a complete check on the calculations. The entries in these columns are obtained 
by multiplying the totals for the y arrays by the corresponding class values of x, and 
the totals for the x arrays by the corresponding class values of y. Summating at the 
foot of the columns we obtain S(#y). 

Finally from the correlation table we have to calculate S(x 2 ), and the method is 
the same as in Chapter II for any frequency distribution. Tabulating our calcula- 
tions we have: 

2(xy) 5448 

S(x 2 ) - 3952 

T x - 850 

T y - 1246 

N -200 



60 



LINEAR REGRESSION 



Then: 1,(y - g)(x - ) 5448 - (850 X 1246)/200 - 152.50 

And: 2(x - f) 2 - 3952 - 850 2 /200 =339.50 

The regression coefficient is given by b vx = 152.50/339.50 = 0.4492 

TABLE 12 

CORRELATION TABLE FOR THE YIELDS OF ADJACENT BARLEY PLOTS 

x 



A& 
Cl 


turned 
asses 


1 


2 


3 


4 


5 


6 


7 


Fre- 
quency 

y 


Actual 
Classes 


66 

88 


89 
111 


112 
134 


135 
157 


158 
180 


181 
203 


204 
226 


1 


20 
42 








1 








1 


2 


43 
65 






1 










1 


3 


66 

88 


2 


2 


2 










6 


4 


89 
111 




4 


4 


3 


2 


1 




14 


5 


112 
134 




5 


11 


4 


11 


3 




34 


6 


135 
157 


1 


1 


12 


15 


17 


5 


2 


53 


7 


158 
180 




1 


12 


16 


13 


11 


2 


55 


8 


181 
203 


1 




1 


9 


13 


4 


2 


30 


9 


204 
226 










3 


1 




4 


10 


227 
249 












2 




2 


Frequency x 


4 


13 


43 


48 


59 


27 


6 


200 



METHODS OF CALCULATION 



61 



TABLE 13 
CALCULATION OF THE REGRESSION COEFFICIENT 



y 


Frequency 
V 


X 


Frequency 

X 


Totals 
for y 
Arrays 


Zto) 


Totals 
for x 
Arrays 


Sto) 


i 

2 
3 
4 
5 

6 
7 

8 


1 
1 
6 
14 
34 
53 
55 
30 


I 
2 
3 
4 
5 
6 
7 


4 
13 
43 

48 
59 
27 
6 


20 
60. 
243 
307 
387 
187 
42 


20 
120- 
729 

1228 
1935 
1122 
294 


4 
3 
12 

48 
132 
228 
247 
143 


4 
6 
36 
192 
660 
1368 
1729 
1144 


9 
10 


4 
2 










21 
12 


189 
120 




















200 

N 




200 

N 


1246 

T 
* y 


5448 
S(*y) 


850 
T, 


5448 

2 to) 



In order to set up the regression equation, the means of x and y are required. These 
are x = 850/200 = 4 25, and y -= 1246/200 = 6.23, and the regression equation is 
written: 

Y = (6 23 - 4492 X 4.25) + 0.4492* 

= 4.3209 -0.4492z 

Since the regression equation has been calculated from coded values, the necessary 
corrections must be applied. To correct the means we apply formula (7), Chapter I/, 
obtaining: 

y (6.23 - 1) X 23 + 31 - 151.29 

x (4.25 - 1) X 23 + 77 = 151.75 

Since the class value is 23 for both variables, the regression coefficient does not 
require any correction, so the new equation is: 

Y = (151 29-0 4492 X 151.75) - 0.4492s 
= 83 12 - 0.4492z 

In order to plot tjie regression straight line, we require only two points on the graph, 
preferably as far apart as possible. It is simpler to use the coded regression equation 
to find any values of Y required, and also the graphing may be done in the coded 
values and the actual values inserted when everything is completed. The end 
points of the line are 

Yi - 4.3209 - 0.4492 X 1 - 4.77 

F 2 - 4.3209 - 0.4492 X 7 - 7.46 



62 



LINEAR REGRESSION 



The graph is finally as in Fig. 7. If such a graph is required in the final presentation 
of the results, it would be necessary only to substitute the actual class values for the 
assumed values. The means of the y arrays are, of course, obtained by dividing the 
totals for the y arrays by the corresponding frequencies. These may be converted 




Means of 
y arrays 

5 00 
4.62 
5 65 
6.40 
6.56 
6.93 
7.00 



1234567 

YIELD 

FIG. 7, Regression graph for yields of adjacent plots showing regression 
line and means of y arrays. 

to actual values by means of the formula for correcting means as described in 
Chapter II, and used above for finding the true values of x and ?/. 
To test the significance of the regression coefficient we find 



S, = 



j 
t = 



b\"2(x 



0.4492V339.50 



41 7.42 - 0.4492 2 X 339.50 
198 



rt0 
= o 2o. 



1.3275 



s e 1.3275 

from which it is clear that the regression coefficient is highly significant. 



6. Exercises. 

1. Table 14 gives the results obtained in an experiment with 25 wheat varieties 
on the number of days from seeding to heading and the number of days from seeding 
to maturity. Calculate the regression equation for the regression of days to mature 
on days to head, and test the significance of the regression coefficient. Code the 
data before beginning your calculations by subtracting 50 from the days to head 
and 85 from the days to mature. Find the fiducial limits at the 5% point of the 
regression coefficient, and decide as to the practicability of using days to head to 
replace days to mature on the basis of the data provided by this sample. 

Regression coefficient = 105.23/125.68 - 0.8373. (Coded data.) 



REFERENCES 



63 



TABLE 14 
DATA ON DAYS TO HEAD AND DAYS TO MATURE OF 25 WHEAT VARIETIES 



Variety 


Days 
to 
Head 


Days 
to 
Mature 


Variety 


Days 

to 
Head 


Days 
to 
Mature 


1 


60.0 


94.4 


14 


58.2 


92.4 


2 


53 6 


89 


15 


58.0 


91 6 


3 


59.0 


94.0 


16 


59.4 


94 


4 


61.8 


95 4 


17 


55.4 


90 8 


5 


53.8 


88.2 


18 


61.6 


95.2 


6 


57.8 


93 4 


19 


63 


97.2 


7 


57 8 


93.6 


20 


60 2 


94.6 


8 


58 4 


92 


21 


61.6 


96 


9 


57.8 


92 8 


22 


57.6 


92 6 


10 


59.0 


93.4 


23 


60 8 


95.4 


11 


59 2 


93.8 


24 


61.2 


94.4 


12 


59 


92.8 


25 


58.2 


94 


13 


58.6 


94.2 









2. Table 15 contains data on the carotene content determined by two methods for 
139 wheat varieties. By one method carotene was determined on the whole wheat, 
and by the other method, on the flour. The figures for carotene in the wheat are 
lower than for carotene in the flour, which is of course the reverse of the actual 
condition. This was due to a different method of extraction used for the whole 
wheat which gave lower but relative results. 

Make out cards, one for each pair of values, and prepare a correlation table, 
letting the flour carotene represent the dependent variable y. In order to reduce 
the labor of calculation make the classes fairly large; for example, let the first class 
for x be 0.85 to 0.95, and the first class for y be 1.33 to 1.49. Also do not forget to 
replace the actual class values by the natural numbers, beginning at 1, before going 
ahead with the calculations. Determine the regression equation and prepare a 
graph similar to Fig. 7. b yx - 438.39/665.96 = 0.6583. (Coded data.) 



3. Prove: (a) 



- *) 



(b) Sfc, - 



REFERENCES 

1. M. EZEKIEL. Methods of Correlation Analysis. John Wiley & Sons, New York, 

1930. Reading: Chapter V. 

2. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter V, Sections 25, 26, 26.1. 

3. G. W. SNEDECOR. Statistical Methods. Collegiate Press, Inc., Ames, Iowa, 1937. 

Reading: Chapter VI. 

4. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate Ltd., 

London, 1931. Reading: Chapter VII, Section 7.22 and Appendix. 



64 



LINEAR REGRESSION 



TABLE 15 
CAROTENE CONTENT OF FLOUR AND WHOLE WHEAT FOR 139 VARIETIES 



Variety 
No. 


Carotene 
in Flour 


Carotene 
n Wheat 


Variety 
No. 


Carotene 
in Flour 


Carotene 
n Wheat 


Variety 
No. 


Carotene 
in Flour 


Carotene 
n Wheat 


1 


2.39 


1.18 


48 


1.71 


.16 


95 


1.97 


1.33 


2 


3.11 


2.13 


49 


1.93 


.14 


96 


1.83 


1.14 


3 


2.15 


1.41 


50 


1.81 


.30 


97 


2.00 


1.51 


4 


1.96 


1.42 


51 


1.89 


.32 


98 


1.96 


1.28 


5 


2.02 


1.50 


52 


1.65 


.32 


99 


2.00 


1.33 


6 


1.76 


1.25 


53 


1.93 


.28 


100 


2.02 


1.32 


7 


2.10 


1.65 


54 


2.12 


.48 


101 


1.78 


1.17 


8 


2.12 


1.24 


55 


2.25 


.50 


102 


1.83 


1.10 


9 


2.28 


1.48 


56 


1.92 


1.42 


103 


1.93 


1.22 


10 


1.86 


1.35 


57 


2.25 


1.66 


104 


2.14 


1.44 


11 


2.60 


1.58 


58 


2.25 


1.63 


105 


2.15 


1.54 


12 


2.11 


1.45 


59 


1 65 


1.18 


106 


2.13 


1.46 


13 


2.30 


1.74 


60 


1.63 


1.14 


107 


1.97 


1.40 


14 


1.80 


1.42 


61 


1.70 


1.22 


108 


1.83 


1.11 


15 


2.00 


1.45 


62 


1.61 


1.20 


109 


2.10 


1.40 


16 


2.05 


1.87 


63 


1.83 


1.33 


110 


1.84 


1.19 


17 


2.09 


2.00 


64 


1.60 


1.13 


111 


1.98 


1.39 


18 


2.33 


1.65 


65 


1.37 


.92 


112 


2.31 


1.60 


19 


2.29 


1.64 


66 


1.96 


1 20 


113 


2 29 


1.53 


20 


2.30 


1.62 


67 


1.88 


1.2G 


114 


2 15 


1.45 


21 


1.97 


1.55 


68 


1.92 


1.34 


115 


1.96 


1.44 


22 


2.36 


1.68 


69 


1.89 


1 04 


116 


1.98 


1.40 


23 


1 73 


1.32 


70 


1.99 


1.26 


117 


1.89 


1.30 


24 


1 72 


1.47 


71 


1.82 


.98 


118 


2.08 


1.33 


25 


1.70 


1.53 


72 


2.12 


1.31 


119 


2.00 


1.42 


26 


1.63 


1.50 


73 


2.16 


1.16 


120 


2.06 


1.44 


27 


1.93 


1.48 


74 


2.14 


1.04 


121 


1.96 


1.36 


28 


1.50 


1.25 


75 


1.63 


.88 


122 


2.07 


1.38 


29 


1.77 


1.33 


76 


2.76 


1.91 


123 


2.24 


1.51 


30 


1.60 


1.40 


77 


2.07 


1.20 


124 


2.15 


1.38 


31 


2.31 


1.49 


78 


1.67 


1.07 


125 


1.83 


1.18 


32 


2.17 


1.42 


79 


2.78 


1.80 


126 


1.84 


1.20 


33 


2.10 


1.35 


80 


3.40 


2.02 


127 


2.03 


1.45 


34 


2.90 


1.58 


81 


3.67 


2.10 


128 


1.87 


1.05 


35 


2.17 


1.50 


82 


2.41 


1.61 


129 


2.24 


1.44 


36 


2.15 


1.40 


83 


2.23 


1.38 


130 


2.14 


1.06 


37 


2.01 


1.40 


84 


3.07 


1.93 


131 


2.13 


1.10 


38 


2.35 


1.67 


85 


2.22 


1.44 


132 


2.03 


.98 


39 


2.34 


1.62 


86 


2.55 


1.58 


133 


2.25 


1.31 


40 


2.00 


1.47 


87 


2.12 


1.39 


134 


2.33 


1.08 


41 


2.18 


1.55 


88 


1.94 


1.27 


135 


2.01 


1.14 


42 


2.47 


1.73 


89 


1.95 


1.41 


136 


1.89 


1.41 


43 


2.25 


1.62 


90 


1.59 


1.08 


137 


3.00 


2.20 


44 


1.77 


1.39 


91 


2.00 


1.30 


138 


2.16 


1.73 


45 


1.68 


1.34 


92 


1.77 


1.22 


139 


2.29 


J.61 


46 


2.46 


1.29 


93 


1.98 


1.26 








47 


1.86 


1.28 


94 


1.97 


1.30 









CHAPTER VII 
CORRELATION 

1. Covariation. This is a term that is very expressive with respect 
to the fundamental situation regarding two variables, from which the 
methods of correlation arise. In the previous chapter it was pointed 
out that, when two variables are so related that one may logically be 
considered as being dependent on the other one, the method of regression 
is completely applicable to a study of this relation; but when the two 
variables cannot be considered in the light of dependence and inde- 
pendence, the method of regression does not appear to be satisfactory. 
Suppose that a study is to be made of the relation between the heights of 
brothers and sisters. It would not be logical to consider the height of 
one member of the pair as being dependent on the height of the other 
one, yet we may be fairly certain that there is a relation of some sort 
and we may wish to estimate what this relation is. The question that is 
asked with respect to two such variables seems to be this. "To what 
extent do the heights of brother and sister vary together "? Thus we 
have the term covariation, and the conventional statistic for the measure- 
ment of covariation is the correlation coefficient. 

2. Definition of Correlation. In Table 16 there are three sets of 
figures that may be taken as measurements on two variables that we 
shall designate as x and y. On examining these three sets of values it 
will be noted that the relation between x and y is different in each case. 
In set 2 we have high values of x associated with high values of y> and 
in set 3 we have high values of x associated with low values of y. In 
both cases there is an obvi ous relation but one is the reverse of the other. 
In set 1, on the other hand, there is no apparent relation between the 
two variables. These sets may be regarded as samples from infinite 
parent populations of paired variates. In the population from which 
set 2 is drawn, whenever a pair of variates is selected, we expect to find, 
if the pair contains a high value of x, that there will be a high value of y 
associated with it. In the population represented by the sample in 
set 3 it is to be expected that high values of x will be found associated 
with low values of y. These two opposite situations are referred to as 
positive and negative correlation. Set 1 represents still another situa- 
tion. High values of x do not appear to be associated with either high 

65 



66 



CORRELATION 



TABLE 16 

THREE SAMPLES OF PAIRED VARIATEB ILLUSTRATING THE 
PHENOMENON OF CORRELATION 



Set! 


x. . . . 


7 


7 


1 


6 


5 


3 


8 


9 


3 


1 


Total = 50 




y 


5 


9 


6 


1 


3 


1 


9 


4 


6 


8 


Total = 52 


Set 2 


X. ... 


9 


8 


7 


7 


6 


5 


3 


3 


1 


1 


Total = 50 




y 


9 


9 


8 


6 


6 


5 


4 


3 


1 


1 


Total = 52 


Set3 


x. ... 


1 


1 


3 


3 


5 


6 


7 


7 


8 


9 


Total - 50 




y 


9 


9 


8 


6 


6 


5 


4 


3 


1 


1 


Total = 52 



or low values of y. In other words, we shall expect that in the parent 
population the two variables vary independently. A graphical picture 
of the results with these three samples is given in Fig. 8, For each 
sample we have prepared what is usually known as a dot diagram. The 
values of y are represented as ordinates and the values of x as abscissae, 
so that each pair can be represented by a dot on the diagram. The final 



SET I 



SET 2 



9 
8 
7 
6 

Y 5 
4 
3 
2 
I 



123456789 
X 



I 23456789 
X 



SET 3 



9 
8 

7 
6 

Y 5 
4 
3 
2 
I 



I 23456769 
X 



FIG. 8. Dot diagrams for the sets of values given in Table 16. 

result is a figure which represents in a general way, by the scatter of the 
dots, the relation between the two variables. For set 1 the dots are 
scattered more or less uniformly over the whole surface. For sets 2 
and 3 there is a definite relation between the variables, as shown by the 
tendency for the dots to arrange themselves in a straight line along the 
diagonals of the square. We are reminded here of the regression graphs 
of the previous chapter. The difference is that we are not now studying 
the effect of one variable on the other, but rather the degree to which 



THE MEASUREMENT OF CORRELATION 67 

the variables vary together owing presumably to influences that are 
common to both. If such measurements represented heights of brothers 
and sisters, it is apparent that this common influence might be the simi- 
larity of their genes. 

This rough illustration is sufficient to give a general idea of the nature 
of correlation, but it is not adequate to give a complete picture of cor- 
relation as it occurs in nature. The student who is specially interested 
in this subject should make a thorough study of the references given at 
the end of this chapter. Each writer on this subject presents the situa- 
tion in a somewhat different manner, and after a study of several view- 
points the student will begin to grasp the fundamental points very 
clearly. We are concerned here mainly with the viewpoint that cor- 
relation is a measure of the degree to which two variables vary together, 
as we believe this to be the most useful viewpoint from the standpoint 
of the research worker. Since we have become acquainted with the 
variance and the standard deviation as measures of variability, it is of 
interest now to inquire how the combined variation of two variables can 
be measured, and how much of the variability of one variable is tied up 
with the variability of some other variable. In the first place, however, 
we must consider a few points that are fundamental to the methods of 
measurement that will be employed. 

The dot diagrams given in Fig. 8 will result from combining the fre- 
quency distributions of two variables. Since they represent samples 
only, they give merely an estimate of the combined frequency distribu- 
tions of the two variables in the parent populations. The single or 
univariate distributions are represented by a curve, but the combined or 
bivariate distributions must be represented by a surface. On extending 
the diagrams of Fig. 8 to much larger samples it is evident that the dots 
will begin to form into swarms of some definite shape, depending on the 
degree of correlation between the variables. If the correlation is high 
the swarm will evidently be of the greatest density along the diagonal 
of the figure; if there is no correlation the swarm is likely to be almost 
circular in shape. The theoretical bivariate frequency distribution will 
obviously be represented by a volume, in contradistinction to that of 
the univariate distribution which is represented by an area. These 
points give us some clue as to how we may obtain a measure of corre- 
lation. 

3. The Measurement of Correlation. Figure 9 illustrates the shape 
of the swarm in a correlation surface for three different degrees of cor- 
relation. The circular swarm at (a) represents zero correlation. In (c) 
the swarm falls entirely on the diagonal and must represent perfect 
correlation. In (b) we have a condition between the other two extremes. 



68 



CORRELATION 



Now each surface is divided into quadrants by lines erected at the posi- 
tions of the means, and in each quadrant are plus and minus signs that 
represent the signs of the products of the x and y deviations from their 
means. Thus in the upper right-hand quadrant (1) the deviations of 
x and y are both positive so that the product of the deviations is positive. 
Therefore we have positive products in quadrants (1) and (3) and nega- 






FIG. 9. Correlation surfaces showing the variation in the shape of the swarm 

with increasing correlation. 

tive products in quadrants (2) and (4). Now if we obtain the sum of 
the products it is obvious that in (a) the plus and minus products will 
cancel each other and the sum will be zero. In (c) all the products will 
be positive so that their sum will be a maximum. In (6) the condition 
is intermediate between (a) and (c). The plus products are greater than 
the negative products; hence we have a positive but not a perfect 
correlation. 

Let us consider now the sets of figures in Table 16. If we calculate 
the sum of the products S(a? x)(y y) for each set we should find an 
agreement with the theory outlined above. To carry out these calcula- 
tions we shall make use of the identity: 



-x)(y-y} = 



(1) 



where T x is the total of the x values, T y the total of the y values, and N is 
the number of pairs. Our calculations then come out as follows : 



Set 1. 
Set 2. 
Set3. 



262 
335 
186 



T X T V /N 
260 

260 
260 



- y) 



2 

75 

-74 



The result is in perfect agreement with the theory that the sum of 
products is a measure of correlation. 



THE MEASUREMENT OF CORRELATION 69 

The sum of products is an absolute measure of correlation but will 
not serve as a relative measure, since it is dependent on several factors 
that have nothing to do with the correlation between the two variables 
with which we are concerned. It depends on the number of pairs of 
measurements or variates, on the units in which the two sets of variates 
are measured, and on the variability of both of the variables. The first 
objection can be overcome by dividing by the number of pairs of vari- 
ates, and we now find that we have 2 (a* x)(y y)/N, which was 
defined in the previous chapter as the covariance cv of x and y. The 
covariance, however, is still not a relative measure of correlation, as it is 
affected by the units of measurement and the variability of x and y. To 
overcome this difficulty it is clear that the covariance must be divided by 
some factor which measures the variability of x and y and is expressible 
in terms of the units in which these variables are measured. The first 
factor which suggests itself is the product of the two standard deviations, 
and this actually gives the formula for the correlation coefficient, usually 
designated by the symbol r. Thus we have: 

- y)/N 



Another formula can be given using the variances of x and y in place of 
their standard deviations. This must of course be: 

Z(x - x)(y - y)/N 

xy / W 



where v x is the variance of x and v y is the variance of y. Formula (3) 
shows also the algebraic relationship between the regression coefficient 
b yx and the correlation coefficient. Since: 



it follows that: 



and is obviously the regression coefficient bxy where x is taken as the 
v v 

dependent variable instead of y. Of course in all regression problems 
there are two regression coefficients, although, in the type of problem we 
have referred to in the chapter on regression, one of these will be of 
theoretical interest only. The correlation coefficient is finally: 

(4) 




70 CORRELATION 

In other words, it is merely the geometric mean of the two regression 
coefficients. 

A brief inspection of the formula of the correlation coefficient will 
show that it has a maximum value of + 1 and a minimum value of 1 
under conditions that we would ordinarily take to represent perfect 
correlation. (1) Let yi = kx it where y* and #,- represent any pair of 
values of y and x, and k is a constant. We have therefore a constant 
positive relationship between x and y. 

Then 

(yi if) = (kxi **) = k(*i *) 
and 

(yi 
Hence 

S(x 
Also 

Therefore 
And finally 



<r<r y fctr, 0% 

(2) Let y+ ~ kx>. Here we have a constant negative relationship 
between x and y. Then 

(2/ - #) = - (tei - kx) = - k(xi - x) 
and 

(y<- #(*<-*) ^-^C^-x) 2 
Hence 

S(x - x)(y - y) 
Also 

S(2/ - J/) 2 
Therefore 

ir y 5= ka 
Finally 

f)(y - y)/N - 



These two conditions that we have postulated are those for which we 
should expect a satisfactory coefficient to give us a maximum value of 
+1 and a minimum value of 1. Between these two extremes we 



THE MEASUREMENT OF CORRELATION 71 

should expect the coefficient to give us values varying between +1 and 
1, and this is what it actually does. Our proof as given above indicates 
this also, but it is not a rigid proof in that particular respect. 

Having satisfied ourselves that when we have perfect positive corre- 
lation the coefficient will be +1, and when we have perfect negative 
correlation the coefficient will be 1, it remains to decide how the 
coefficient will measure correlations that fall within this range. As a 
matter of fact it is easy to state this proposition, but quite difficult to 
explain it in a simple and satisfactory manner. Perhaps the best inter- 
pretation arises from considerations that actually are more closely 
related to the theory of linear regression than that of correlation. For 
example, if we take y to be the independent variable, then we can work 
out the relation between the correlation coefficient and the two vari- 
ances, the total for t/, and the variance of the errors of estimation. As 
pointed out in the previous chapter, the sum of squares of the errors of 
estimation is S(y F) 2 , where Y represents points on the regression 
straight line corresponding to each value of y. The variance of the 
errors of estimation is therefore given by: 



Now the variance of y is related to the above variance in the manner 
indicated by the following equations: 

J f-f (6) 

(1 - r*)2(y - g) 2 
"' = AT^2 <7) 

From which it follows that the ratio of the two variances is: 



On tne same basis, if we examine the relation between v v and the variance 
due to the regression function, the latter being given by: 



= lfcS(z - f) s /l or i* vy t y/ (9) 

we find that: 

, _ j) ( 10 ) 



72 CORRELATION 

Finally, the ratio Vb/v e is given approximately by: 

v* r 2 



v e (1 - 



(N - 2) (11) 



The variance v e is frequently taken as representing that portion of the 
variation in y which is independent of x\ hence we note that from this 
standpoint equation (8) is the most important. If v e is expressed in per- 
centage of v v , then it is clear from (8) that thip percentage is almost pro- 
portional to (1 r 2 ). This is another way 01 expressing the commonly 
known fact that differences between high correlation coefficients are 
much more significant than similar differences between small correla- 
tion coefficients. As a measuring stick for general use it is therefore 
much more convenient to think in terms of r 2 than in terms of r. For 
example, if we have a correlation coefficient of 0.5, the ratio v e /v v = 0.75, 
and the ratio does not fall to 0.5 until r reaches 0.75. 

Considerable space might be devoted to further viewpoints on the 
interpretation of the correlation coefficient, and the student who is 
especially interested in this phase of statistics should refer to the discus- 
sions in the references cited at the end of this chapter. Special notice 
should be taken of the discussions by R. A. Fisher (1) of the distribution 
of the correlation coefficient; by G. W. Snedecor (4) of the relation 
between "common elements" and the correlation coefficient; and by 
A. E. Treloar (6) of many phases of the entire subject of correlation. 
** 4. Testing the Significance of the Correlation Coefficient. R. A. 
Fisher (1) has shown that for small samples the distribution of r is not 
sufficiently close to normality to justify the use of a standard error or a 
probable error to test its significance. A more accurate method has 
been developed by Fisher, based on the distribution of t. For a correla- 
tion coefficient: 

r\/n 

t = v . (12) 



where n = the number of degrees of freedom available for estimating the 
correlation coefficient. The degrees of freedom can always be taken 
equal to N 2, because there is a loss of one degree of freedom for each 
statistic calculated from the sample in order to obtain r. These are y 
and b vx (the regression coefficient). Although b vx may not actually have 
been calculated, it is involved in the formula of the correlation coefficient 
through the sum of products 2(x x)(y y). This point will be 
clear from a consideration of equation (8) which shows that the ratio 
vjv v is a function of the correlation coefficient. Now v e measures the 



CALCULATION OF THE CORRELATION COEFFICIENT 73 

discrepancies between individual values of y and the corresponding 
values of Y estimated from the regression equation. It follows from 
this that the correlation coefficient can measure only that portion of the 
relation between x and y which is represented by the regression equation. 

Since the use of t provides a correct method of testing the significance 
of a correlation coefficient regardless of the size of the sample, in general 
practice one uses this method for samples of all sizes. For large samples 
one might calculate a standard error of r, but even this procedure would 
be subject to criticism if the value of the correlation coefficient was 
high. 

For testing the significance of the difference between two correlation 
coefficients t is not suitable, and Fisher (1) has developed an accurate 
method which involves transforming the values of r as follows : 

z' = i{log.(l + r) -loged -r)} (13) 

The values of z r can be shown to be normally distributed even for small 
samples and with a standard deviation given by: 

(14) 



To test the significance of the difference between two correlation coeffi- 
cients ri and T2, we proceed as follows: 

I + ri) log c d ~ n) } 



z i Z 2 = difference 



/ 1 
'.-i ~ \ N _ 



- 3 



(15) 



where Ni and N% are the numbers in the two samples from which ri and 
TZ respectively have been calculated. Finally : 



The table of t is entered under N\ + N^ 6 degrees of freedom. 

6. Calculation of the Correlation Coefficient. From the previous 
chapter, the methods for calculating the sum of products 
S(x )(y y}, either directly from paired values or from a correla- 
tion table, will have been noted. It is sufficient therefore to note that 



74 CORRELATION 

the formulae given in (2) and (3) may be written as follows in convenient 
form for calculation. 



r - xy 

r xv > ....... "" ':"" ,= (JO) 



Formula (17) is the most direct, but (16) and (18) are perhaps better 
suited to machine calculation. In (18) there are no divisions in either 
the numerator or the denominator; and after all the preliminary calcu- 
lations of the values of 2(xy), T x , T v , S(z 2 ), and S(i/ 2 ) have been per- 
formed, each of the three factors in the formula may be obtained without 
removing any figures from the machine and recording them elsewhere. 

The methods of calculating 2(xy), T x , T V1 2(z 2 ), and S(r/ 2 ) will of 
course be the same as described in Chapters II and VI. They may be 
calculated either from the correlation table or directly from the paired 
values. For N = 50 or less it is probably best to proceed directly, as 
setting up the correlation surface is not likely to save any time. When 
the numbers are fairly large it is nearly always best to have a correlation 
table, as we shall learn later of a test to determine the agreement between 
the actual data and the straight line fitted by the regression equation, 
and to carry out this test the correlation table must be set up. 

Example 12. Direct Calculation of the Correlation Coefficient from Paired 
Values. For the sets of paired values given in Table 16 the calculations of 2(xy) 
were performed and the results given in Section 3 of this chapter. Let us assume 
that we wish to calculate the correlation coefficients using formula (17). 

Set 1. 2( X y) - T S T V /N 262 - 260 - 2.0 

S(x 2 ) - Tl/N 324 - 250 74.0 
- Tl/N - 360 - 270.4 - 79.6 
2.0 



-+ 0.026 



V74.0X79.6 

Set 2. Z( X y) - T X T V /N - 335 - 260 76.0 

- TS/tf 

Same as Set 1 

= +0.997 




CALCULATION OF THE CORRELATION COEFFICIENT 75 

Set 3. *L(xy) - T X T V /N - 186 - 260 - - 74.0 

I Same as Set 1 

r zy ~ 74 ' = - - 0.964 
V74.0X79.6 

To calculate r^ for Set 2 using formula (18) we would write directly: 

10 X 335 - 50 X 52 

V(10 X 324 - 50 2 )(10 X 350 - 52 2 ) 

and performing one operation with the machine for each factor we obtain: 

750 



A/740 X 796 
By one more operation we find the denominator and have: 



Example 13. Calculation of the Correlation Coefficient from a Correlation Table* 
Suppose that we wish to calculate the correlation coefficient for Table 12, Chapter VI. 
The first step is to prepare Table 13, which we have already used in Example 11 to 
calculate the regression coefficient. From this table we have: 



- 5448 T s - 850 

- 3952 T y = 1246 
8180 

And making use of formula (18) above we calculate: 

200 X 5448 - 850 X 1246 

T * V V(200 X 3952 - 850 2 )(200 X 8180 - 1246 2 ) 

30,500 _ 30,500 _ ^ 



X/67,900 X 83,484 260.58 X 288.94 

Example 14. Tests of Significance. Although the correlation coefficients cal- 
culated in Example 12 were for only 10 pairs of values, the t test will give a reliable 
measure of their significance. The t values are determined as follows: 



Setl. r xv ** + 0.026 t y == - 0.07 

Vl - 0.026 2 

0.977V8 

Set 2. rxy -+ 0.977 t , v - 129.7 

Vl-0.977 2 

0.964\/8 

Set 3. r* - - 0.964 t - , v - 102.5 

VI- 0.964 2 



76 CORRELATION 

Turning to Table 94 we note that for n = 8 and P 0.05 the value of t required 
is 2*3,06. The coefficient 0.026 is therefore quite insignificant, but the other two are 
highly significant. 

Example 16. The Significance of Differences between Correlation Coefficients. 
In a study of the relation between the carotene content of wheat flour and the crumb 
color of the bread, Goulden, et al. (2), obtained the following results with 139 
varieties. 

Carotene in whole wheat with crumb color, ri = 0.4951 

Carotene in flour with crumb color, f2 0.5791 

The most accurate method for this test is to make use of Fisher's z' transformation. 
For the z f test we write: 

z \ . | {log. (1 + 0.4951) - log* (1 - 0.4951)} 
- % (log, 1,4951 - log, 0.5049) 



" * loge vFo " * loge 2 * 9612 " logl 2 * 9612 x 1 ' 

= 0. 47147 X 1.1513 - 0.5428 
4 - i (tog. 1.5701 - log. 0.4209) 

| log, 3. 7517 0.57423 X 1.1513 - 0.6611 



it - z\ - 0.6611 - 0.5428 = 0.1183 

0.1213 



Since the difference is less than its standard deviation it is not significant. 

Note that in writing out the formula for z' we pay no attention to the sign of r 
as it is the numerical difference between the coefficients that we are testing. 



6. Exercises. 

1. The figures in Table 17 are the physics and English marks 1 for home economics 
students in the University of Manitoba. Determine the correlation coefficient for 
the relation between the marks in the two subjects. Use the direct method, and 
test the significance of the coefficient. r + 0.705. 

2. For the same 50 students the correlation coefficient for the marks in art and 
clothing is +0.7300, and for art and physics it is +0.6491. Is this a significant 
difference? 

3. Determine the correlation coefficient for days to head and days to mature of 
25 wheat varieties, using the data from Table 14. Find the fiducial limits at the 
5% point for this coefficient. r + 0. 946. 

1 By courtesy of the Registrar, University of Manitoba. 



REFERENCES 



77 



TABLE 17 

MARKS IN PHYSICS AND ENGLISH OF 50 STUDENTS IN HOME ECONOMICS 
OF THE UNIVERSITY OF MANITOBA 



Studen 


Physics 


English 


Student 


Physics 


English 


Student 


Physics 


English 


1 


20 


21 


18 


26 


29 


35 


23 


26 


2 


25 


26 


19 


24 


27 


36 


26 


27 


3 


24 


27 


20 


19 


26 


37 


22 


21 


4 


22 


24 


21 


25 


25 


38 


26 


25 


5 


27 


27 


22 


18 


20 


39 


23 


21 


6 


26 


28 


23 


20 


24 


40 


29 


26 


7 


26 


24 


24 


23 


24 


41 


23 


19 


8 


32 


26 


25 


22 


20 


42 


20 


19 


9 


22 


24 


26 


23 


26 


43 


23 


30 


10 


27 


26 


27 


31 


27 


44 


33 


32 


11 


22 


23 


28 


24 


25 


45 


21 


19 


12 


22 


25 


29 


28 


30 


46 


28 


30 


13 


22 


24 


30 


25 


28 


47 


24 


21 


14 


29 


30 


31 


26 


32 


48 


26 


28 


15 


26 


30 


32 


24 


25 


49 


24 


22 


16 


24 


28 


33 


27 


30 


50 


30 


25 


17 


25 


29 


34 


28 


30 









REFERENCES 

1. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter VI, Sections 30, 31, 33, 34, 35, 36. 

2. C. H. GOULDEN, W. F. GEDDES, and A. G. 0. WHITESIDE. Cereal Chem., 

11:557-566, 1934. 

3. RAYMOND PEARL. Medical Biometry and Statistics. W. B. Saunders Co., 

Philadelphia, 1923. Reading: Chapter XIV, first section. 

4. G. W. Snedecor. Statistical Methods. Collegiate Press, Inc., Ames, Iowa, 

1937. Reading: Chapter VII. 

5. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter VII, Sections 8.1, 8.2, 8.21, 8.22. 

6. A. E. TRELOAR. An Outline of Biometric Analysis. Burgess Publishing Co., 

Minneapolis, 1936. Reading: Part I, Chapters X, XI, XII, XIII. 

7. G. UDNY YULE. The Theory of Statistics. Charles Griffin and Company, Ltd., 

London, 1924. Reading: Chapter IX. 



CHAPTER VIII 

PARTIAL AND MULTIPLE REGRESSION AND 
CORRELATION 

1. The Necessity for Dealing with More Than One Independent 
Variable. In many regression problems the investigator is concerned 
purely with the effect of one variable on another, and this holds true 
regardless of other complicating factors. Suppose that a new rapid 
method has been developed for determining the protein content of grain 
samples and this method is to be compared with an older and thoroughly 
tested method which is known to give very accurate results. The two 
methods are used on a large series of samples and for the entire series 
the linear regression equation is determined for the regression of protein 
by the old method on protein by the new method. Regardless of how 
these two variables are related, from the practical standpoint of studying 
the efficiency of the new method as a substitute for the old method, it is 
clear that the investigator is concerned purely with the closeness of the 
relationship between the two variables. The new method may not ac- 
tually measure protein content but some other factor that is so closely 
associated with protein content that if we know one we know the other. 
Hence, although the relation between the two variables may be indirect, 
it is the total relation with which we are concerned, as we require merely 
a measure of the accuracy with which we can predict one variable from 
individual measurements of the other variable. In examples of a some- 
what different nature it may be quite misleading to study only the total 
relation between two variables. Suppose that we find a correlation of 
+0.60 between the yield of wheat and temperature. Can we conclude 
from this result that, if all other conditions remain constant, there will 
be an increase in yield with increases in temperature? The answer is 
no, because temperature may be associated with some other factor in- 
fluencing yield and this second factor may be the one that is actually 
causing the variations in yield. Suppose that the second factor is rain- 
fall, which is probably the most important of the meteorological factors 
influencing the yield. If rainfall is itself associated with temperature, 
it is clear that there must also be a correlation between yield and tem- 
perature. The latter correlation, however, does not provide us with 
any information of a fundamental nature with respect to the actual 

78 



DERIVATION OF PARTIAL REGRESSION METHODS 79 

changes in yield brought about by changes in temperature. What we 
require here is a measure of the association between yield and tempera- 
ture when the rainfall remains constant. To the extent that the rela- 
tions between the three variables in a problem of this kind can be ex- 
pressed by linear functions, the measure that we require can be obtained 
by the method of partial regression or partial correlation. Thus the 
partial correlation of yield and temperature will measure the degree of 
covariation for these two variables with a constant rainfall. The partial 
regression coefficient for yield and temperature will give the actual in- 
crease in yield for one unit of increase in temperature when the rainfall 
is constant. If the correlation coefficients for the three variables are 
as follows: 

T v t (yield and temperature) = + 0.60 

r yr (yield and rainfall) = + 0.82 

r tr (temperature and rainfall) = + 0.78 

the partial correlation coefficient for yield and temperature with rainfall 
constant may be represented by r y t. rj in which the variable placed after 
the period is the one that is held constant. Applying the partial corre- 
lation method as illustrated below we find r yt . r = + 0.09. Therefore 
the actual effect of temperature when rainfall is constant is practically 
nil. 

It is just as well to emphasize by means of this example that the 
method of partial regression and partial correlation as we are considering 
it here has to do only with the linear relation between the variables. If 
the effect of temperature on yield is not the same for a constant low 
rainfall as it is for a constant high rainfall, then the linear measures are 
inadequate to express the actual relation. 

2. Derivation of Partial Regression and Partial Correlation Methods. 
The method of simple correlations is derived from the regression equa- 
tion : 

y y = t> vx (x - ) 

where b yx is the regression coefficient. Similarly, when there are three 
variables y, x, z, the regression equation is: 

V - y = b vz (x - x) + b vt (z - z) 

In order to simplify the writing of these equations we use x\ for the 
dependent variable and x%, x$ x n for the independent variables. 
Also 612 represents the regression coefficient for xi on #2, and to abbrevi- 



80 PARTIAL AND MULTIPLE REGRESSION AND CORRELATION 

ate further we write xi for (x\ xi) and x 2 for (x2 #2). Hence the 
general regression equation for n variables is: 



+ 613X3 + 6i 4 X 4 + ' ' ' + 6l w X (1) 

The error in estimating x\ from this regression equation will be: 

(Xl 612X2 613X3 6l n X n ) 

and it is required to find values of the regression coefficients such that 
the sum of the squares of these errors is a minimum. That is, we must 
find values of the regression coefficients such that 



2(Xl - 6l2X2 6l3X3 ' ' - 6l n X n ) 2 

is a minimum. For 4 variables this leads by mathematical treatment 
to the following "Normal Equations" 

2(xix 2 ) = 6i 2 2(z 2 ) 2 + 6i 3 2(x 2 x 3 ) + 6i 4 2(z 2 x 4 ) 

2(xix 3 ) = 6i 2 2(x 2 x 3 ) + 6i 3 2(x 3 ) 2 + 6i4S(x 3 x 4 ) (2) 

2(xix 4 ) = 6i 2 2(x 2 x 4 ) + 6i 3 S(x 3 x 4 ) + 6 14 2(x 4 ) 2 

For a set of n variables there are (n 1) simultaneous equations for 
which the sums of squares and sums of products are known, and by 
solving these we arrive at the values for the regression coefficients. 
Any partial correlation can then be determined as follows : 



^12-3.. n = V 612.3- --n X6 21 . 3 ... w (3) 

For three variables xi, X2, X3, the normal equations are as follows: 

2(xjx 2 ) = 6i 2 2(z 2 ) 2 + 6i 3 2(x 2 x 3 ) 
2(X!Z 3 ) - 6i 2 2(x 2 x 3 ) + 6 13 2(a; 3 ) 2 
from which it can be proved that 

7*12 "" 7*13 '7*23 
7*12.3 = 



Similarly 

ri3. 2 = / *" == (4) 

V (1 r 2 2 )(l - 

and 

T23 T\2'T] 



- r? 8 ) 



DERIVATION OF PARTIAL REGRESSION METHODS 81 

This is the most rapid method of obtaining the partials for only 
three variables. For four or more variables it is best to make use of the 
fact that the normal equations can be written as follows, taking as an 
example the equations for five variables: 

7*12 = ft 12 + ft 137*23 + 0147*24 + 0157*25 

7*13 = 012723 + 013 + 0147*34 + 0157*35 

(5) 
7*14 = 0127-24 + ^137-34 + 014 + 0157*45 

7*15 = 0127*25 + 0137*35 + 0147-45 + 015 

The correlation coefficients are the known values, and the beta (0) Values 
the unknown. The latter can be used as illustrated below to compute 
the partial correlation coefficients. 

Tabular methods of solving these equations for the beta values have 
been devised which reduce the labor to a minimum. The beta values 
are defined by: 



6l2.3 - n = ( I 012-3 - - 
\ff2/ 



(6) 

\ff2/ 

and 

&21-3 n = I ) 021-3 - . n (7) 

\*1/ 

Hence, on referring to equation (3) above, we find that : 

012-3 - . n'021-3 . . . w = ( ) &12-3 - . n X ( ~ ) &21-3 n 

\*1/ \**J 

= bl2-3 - - n* i>21.3 - n = 1*12-3 - - - n (8) 

And hence: 

v 012-3 n' 021*3 - n ~ 7*12.3 n (9) 

In order to obtain all the beta values, it is necessary to rewrite the 
normal equations in different ways and solve. For example, in order to 
obtain 02i, the equations for five variables must be written. 

7*21 = 021 + 0237*13 + 0247*14 + 0257*15 

7-23 = 0217*13 + 023 + 0247*34 + 0257*35 

7*24 =* 0217*14 + 0237"34 + 024 + 0257*45 

T25 = 0217*15 + 0237*35 + 0247*45 + 025 



82 PARTIAL AND MULTIPLE REGRESSION AND CORRELATION 

Correlation coefficients are often referred to as coefficients of the 
pth order, where p is the number of variables held constant. Thus the 
simple coefficient r\% is of zero order, arid the partial coefficient ri2-345 is 
of the third order. 

3. Example 16. Calculation of Partial Regression and Partial Correlation 
Coefficients. The simple correlation coefficients in Table 18 were obtained in a study 
(2) of the effect of the physical characteristics of wheat on the yield and quality of 
flour, 

TABLE 18 

SIMPLE CORRELATION COEFFICIENTS FOR THE 
RELATIONS BETWEEN Six VARIABLES 



6 


0.6412 -0.3190 -0.4462 


-0.3511 -0.3092 


5 


-0.3123 0.2861 0.1467 


0.1882 


4 


-0.3947 0.0429 -0.0655 




3 


-O.5612 0.3114 




2 


-0.4589 




where 1 = yield of straight grade flour. 


4 = per cent immaturity. 


2 * per cent bran frost. 


5 = per cent green kernels. 


3 per cent heavy frost. 


6 weight per bushel. 



In order to use the above method to determine the effects on yield of flour of any 
one of the forms of damage or of weight per bushel, it is necessary to determine the 
partial correlation coefficients: 

f 12 -3466, ^ 13- 2456, ^14-2356, ^15-2346, H6.2345 

For which we will require 

012 *021, 018* 031, 014-041, 015-061, 010-061 

We solve for these by the method illustrated in Table 19. It is a tabular method of 
solving the simultaneous equations and is best understood from a study of the table. 

Note that the calculations of Table 19 give 0i 2 , 0i 8 , 0n, 0is, and 0i, and that in 
order to obtain the other beta values the simple correlation coefficients must be 
rearranged and the calculations repeated. The rearrangement in the order 6; 5, 4, 3, 
1, 2, will give 2 i, 023, 024, 025, 020. The next logical rearrangement is 6, 5, 4, 1, 2, 3, 
giving 032, 031, 034, 035, 030- We continue rearranging the simple correlation coefficients 
until all the beta values have been calculated. Then they are put together in a table 
and we select those necessary in order to give the required partials. 

The following instructions will be found useful in carrying through the tabular 
method of solving the equations. 

(1) Rule a sheet of paper as in Tabb 19. 

(2) Enter all the correlation coefficients as indicated in lines 1, 3, 7, 12, and 18. 



DERIVATION OF PARTIAL REGRESSION METHODS 83 

(3) Sum the correlation coefficients to obtain values given in column S. 
Note that the first sum, line 1, is rei + ^2 4- rea + r4 + r& + fee* the sum in line 
3 is rgi + f62 + fas + r&4 + *"65 + 7*58, the sum in line 7 is ru -f r& + r& + f"44 + 
+ f46 + f4, etc. 

The S column provides a check for all the preceding work. The values 1.0662 
and 1.1789 must check with the sum of the values in lines 5 and 6 respectively. 
There are similar checks in the S column of lines 10 and 11, 16, and 17, and 23 
and 24. All these checks are approximate, and therefore the values obtained in 
the check column will not agree with those calculated from the body of the table 
to the last decimal figure. 

(4) The last value calculated in line 24 is ftn with its sign changed. It is 
written below in line 1 of the reverse with the correct sign, and also in column 2 
line 1 of the reverse. The remaining values in column 1 come from lines 17, 11, 
6, and 2, of the same column but with their signs reversed. 

In column 2 the values are: 



Pi2 X (17 -2) 
fci X (11 -2) 
012 X (6-2) 
012 X (2. 2) 

In line 2 (reverse) add from right to left and obtain /Sis, then the remaining 
values in column 3 are: 

ft* X (11 -3) 

0wX(6-3) 
Pit X (2*3) 

In line 3 (reverse) add from right to left and obtain 14, then the remaining 
values in column 4 are: 

0u X (6.4) 

014 X (2.4) 

In line 4 (reverse) add from right to left and obtain 0i&, then the remaining 
value in column 5 is: 

016 X (2.5) 

In line 5 (reverse) add from right to left and obtain 0i. 

After completing the calculations as in Table 19 the correlation coefficients are 
arranged in the order 6, 5, 4, 3, 1, 2, in a new table and the calculations carried out as 
before. For 6 variables there will be 6 tables to calculate, each table giving 5 of the 
total of 30 beta values. When the latter have all been calculated they can be tabu- 
lated, and all that remains is to work out the partials. It is convenient to make a 
table such as Table 20 for entering the beta values and the corresponding partial 
correlation coefficients. 



84 PARTIAL AND MULTIPLE REGRESSION AND CORRELATION 






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CALCULATION OF MULTIPLE CORRELATION COEFFICIENTS 85 



TABLE 20 
BETA VALUES AND PABTIAL CORRELATION COEFFICIENTS 



Subscript 


ft 


Subscript 


ft 


r Subscript 


r 


12 
13 


-0.2575 


21 
31 


0.5844 


12.3456 
13.2456 


-.388 


14 




41 




14.2356 




56 


- 


65 




56.1234 





4. Tests of Significance. The t test is applicable to partial correla- 
tions in the same way as to simple correlations but the degrees of free- 
dom are different. If p is the number of variables held constant, for 
partial correlation coefficients we have 



t 



(10) 



6. Multiple Correlation. In our example, if we consider not the 
separate but the total effect of weight per bushel and the different forms 
of damage on the yield of flour, the problem is one of multiple correlation. 
Since all these variables have some effect on flour yield the more infor- 
mation we have on them the more closely we can predict the flour yield 
of a particular sample of wheat. 

A simple correlation coefficient measures the relation between a de- 
pendent and one independent variable. A multiple correlation coeffi- 
cient measures the combined relation between a dependent variable and 
a series of independent variables. 

Equation (1): 

is in reality a multiple regression equation as it may be used to predict 
values of *x\ from the known values of #2, #3, 4 x n . 

6. Calculation of Multiple Correlation Coefficients. Two methods 
are in use for the calculation of the multiple correlation coefficient. 
These arise from the two equations (11) and (12) below: 

1 - ?.23 . . . . - (1 - f*2)d - f*8.2)(l - 1*4-28X1 - >*?5.234) . - - 

(1- 1*1.28.... -l) (ID 

R 2 012 'fl2 + 018 TIB + 014T14 + . . + 01* 'fin (12) 



86 PARTIAL AND MULTIPLE REGRESSION AND CORRELATION 

Method (11) can be used only when all the partial correlation coeffi- 
cients of the first, second, third, to the (n 2) order are known, and 
hence it is impossible when the partials have been obtained by solving 
the normal equations. It is very useful, however, when only three 
variables are being studied. For three variables we have: 



Method (12) is directly applicable when the partial correlation coeffi- 
cients have been obtained by solving the normal equations for the beta 
values. 

7. Testing the Significance of Multiple Correlations. It should be 
noted that, in equation (11) above, any one of the factors such as 
1 i^ia.a cannot be greater than unity, since the square of a correlation 
coefficient cannot be less than zero. Hence if we compare 

(1 - R?. 28 . . . ) and (1 - r? 2 ) 



giving 

-Bl-23 . . 1 > ff 2 "~ 

or 



Similarly for any other factor on the right of the equation; hence: 

72i-23 n > *12, r !3'2> M23 n~l 

The multiple correlation coefficient is greater therefore than any of 
the constituent coefficients; and its minimum value is zero and not 
1, as is the case with a simple or partial coefficient. For this reason 
a special table must be used for testing the significance of multiple cor- 
relations. 1 The calculation of t values, standard errors, or probable 
errors will give entirely erroneous results. Two tables that may be used 
are in the references given below. 

8. Exercises. 

1. Complete the calculation of the partial correlation coefficients begun in 
Example 16. The following values will assist in checking the work: 

TIJ.MW "" 0.3177 
r i 6 . 2846 0.3367 
fas- 1240 = 0.0393 
rw-iaai - -0.1373 

1 A test is described in Chapter XIII that is baaed on the analysis of variance. 



REFERENCES 87 

2. If N is 36, determine the minimum value of a fourth order correlation coefficient 
that is significant. Put r in terms of t and the number of degrees of freedom. 

The value obtained should be 0.3493. 

3. Calculate the multiple correlation coefficient #1.23456 for the same data as in 
Example 16, and determine its significance. R - 0.7936. 

4. Write the simultaneous equations for three variables in the same form as (5) 
above. Then prove: 



f12 ' 3 



REFERENCES 

1. R. A. FISHBR. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter VI, Sections 32, 33, Example 28. 

2. W. F. GEDDBS, G. J. MALLOCH, and R. K. LARMOUR. Can. J. Research, 6:119-155, 

1932. 

3. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter XI, Sections 11.1, 11.2, 11.5, 11.6, 11.7. 

4. H. A. WALLACE and G. W. SNEDECOE. Correlation and Machine Calculation. 

Iowa State College, Bull. 4, 1931. 

6. J. WISHART. Table of the Significant Values of the Multiple Correlation Coeffi- 
cient. /<wr. Roy. Met. Soc., No. 54, 1928. 



CHAPTER IX 
THE x 2 (CHI-SQUARE) TEST 

1. Data Classified in Two Ways. On reviewing the types of prob- 
lems that have been presented in the previous chapters, it will be recalled 
that they have dealt with data of two kinds. In the first place we 
studied an example in which an operator attempted to classify grain 
samples according to variety. The samples were placed either rightly 
or wrongly, and there was no intermediate condition. The power of 
the operator to differentiate the samples was therefore measured in 
terms of the number of samples placed correctly. With a little thought 
it will be clear that a great many problems must occur in which the data 
are of this type. Thus, in describing the health of a population, an 
obvious criterion will be the proportion of the population that are ill, or 
perhaps the percentage dying within the year. Again, a set of varieties 
of a cereal crop may be differentiated by the number of seeds that are 
viable, and so forth. In further examples the data were of a different 
type as in the case of yields of wheat plots, weights and heights of men, 
and degree of infection. We may be reminded, by these remarks, of 
the classification of variables as continuous and discontinuous, wherein 
the distinction between the two is fairly clear cut. Will data arising 
from discontinuous variables always fall into the first class mentioned 
above, and data from continuous variables into the second class? The 
answer is that they will not be so easily separated in this way, as we can 
easily imagine a situation in which data for a continuous variable may 
be treated by the two methods. We may take as an example a com- 
parison of the yields of two varieties of wheat. In the first place, if 
there are a sufficient number of plots we may compare the two varieties 
according to the number of plots that fall into an arbitrarily determined 
low-yielding class, or an arbitrarily determined high-yielding class; or 
better still we may compare the numbers of plots falling into both 
classes. In the second case we may simply compare the average yields 
of the two varieties on all the plots. Which method shall we use? This 
question is also very easily answered, as it will be clear that the first 
method applied to an example of this kind is cumbersome and unwieldy, 
and will be used only when the numbers are fairly large and the method 

88 



TESTS OF GOODNESS OF FIT 89 

of classifying the plots according to yield is only approximate. For 
example, in a comparison of two varieties as grown by farmers it may 
be impossible to obtain accurate yields, but it may be possible to classify 
the fields quite accurately into the groups low-yielding and high-yielding. 
Then, with a fairly large number of fields to work with, a good com- 
parison of the varieties may be made simply by determining the number 
in each group. For discontinuous variables, on the other hand, com- 
parisons will usually be found to be most conveniently made by the first 
method, and this is particularly true if the character with which we are 
concerned is definitely not measurable in a quantitative manner. Thus 
people may be classified only as dead or alive ; and although there may be 
a theoretical situation existing for a short period in which this classi- 
fication is uncertain, it is certainly of no practical significance in describ- 
ing what has happened to two populations as a result, say, of their 
having received two different treatments. 

In this chapter we are concerned mainly with methods of applying < 
tests of significance in examples where the data are in the form of fre- ' 
quencies as in the first class mentioned above. Snedecor (4) has very 
aptly used the term enumeration data to describe data of this type. 

2. Tests of Goodness of Fit. In many problems the test that is 
, required is a comparison of a set ofactual frequencies with a correspond- 
ing set of theoretical frequencies. Thus in experiments in genetics an 2 
population may be classified into two groups, as in a wheat experiment 
in which the F% population of 131 plants is classified as 106 that are 
resistant to rust and 25 that are susceptible. The predominance of 
resistant plants can be explained by the well-known theory of dominance 
of the genes for rust resistance coupled with the supposition that rust 
reaction is determined by only one pair of genes, one parent having con- 
tributed the gene for rust resistance and the other parent the gene for 
susceptibility. This is plainly an hypothesis which gives a general 
explanation of the results, and as such may be subject to testing in the 
same manner as the familiar null hypothesis of Chapter I. The pro- 
cedure of this test follows from the following considerations. 

In a population for which the hypothesis is true, if a large number of 
samples of 131 plants each are taken, these will be found to vary around 
a mean value for the frequencies of resistant and susceptible plants 
which will be directly calculable from the hypothesis. Thus in the 
present example it is easily demonstrated that the mean of such a popu- 
lation will be 98.25 resistant plants and 32.75 susceptible plants. In 
taking samples from this population, it is to be expected that owing to 
random variation some of these samples will exhibit quite wide varia- 
tions from the mean of the population, but a large proportion of them 



90 THE x 2 (CHI-SQUARE) TEST 

will, of course, be fairly close to the mean of the population. If we 
knew the theoretical distribution of such samples around the mean, we 
could calculate for samples the same size as ours the numbers of resistant 
and susceptible plants which would occur as the result of random varia- 
tions in only 5% of the trials. This would establish for us the 5% level 
of significance that is, if our actual sample fell outside of the range of 
this_5% level wejsmuIcLiay that the dat^ 

h^rgothesis, in fact it is fairly convincing evidencfTtlmt the hypQthesisJs 
not true. If our sample fell well jwithin the 5% level we would then 
say that there was good agreement between the data and the hypothe- 
sis, but the hypothesis would not necessarily be proved Now the dis- 
tribution of the samples can be calculated directly by methods similar 
to those used in Chapter I, and we shall see in Chapter X that if the 
sample is small it may be advantageous to proceed on this basis. How- 
ever, for general application a much easier method is available. This 
method involves the calculation for the data of the sample a statistic 
known as x 2 (chi square) which is distributed in a known manner depend- 
ing on the number of degrees of freedom available for its estimation. 
For the general case x 2 is given by: 






where a represents the actual frequencies and t the corresponding theoret- 
ical frequencies. Thus in the present example the actual frequencies 
are 106 and 25, and the corresponding theoretical frequencies are 98.25 
and 32.75. The two values of a t are therefore both equal to 7.75, 
and x 2 = 7.75 2 /98.25 + 7.75 2 /32.75 = 2.445. 1 The number of degrees 
of freedom available for the estimation^)? x 2 is 1. In this respect the 
problem is similar to the t test for the differences between paired values. 
Here we have two pairs of differences as represented by the two values of 
a t, and consequently there is only one degree of freedom. Another 
concept of the degrees of freedom arises from the fact that there are 
only two classes, resistant and susceptible. The total number in the 
sample being fixed, if the number in any one class is fixed the number in 
the other class must also be fixed. There is therefore only one class 

1 For simple ratios a direct formula suggested by F. R. Immer for calculating 
X 2 may be used. This formula is: 



xN 

where the theoretical ratio is x : 1, ai is the actual frequency corresponding to x 
and 02 is the actual frequency corresponding to 1. N is the total frequency. 



TESTS OF GOODNESS OF FIT 91 

which can be arbitrarily assigned a given frequency, and this means that 
there is only one degree of freedom. 

The next step in the test is to examine the tables that give the dis- 
tribution of x 2 and find the value at the 5% level for one degree of 
freedom. We enter Table 95 and find that the value of x 2 at the 
5% point is jyS^L., Our conclusion is that Jfa 



ca^^ Of course, we can if necessary go 

further and determine approximately in what proportion of cases such 
a result as ours would be obtained. The x 2 value of 2.445 falls between 
the two values of x 2 that correspond to the 10 and 20% levels of P. By 
interpolation our value is found to correspond to the 13% point, and 
consequently we can say that a sample showing a deviation from the 
theoretical as great as or greater than the one observed would be expected 
to occur in 13% of the trials. The observed deviation is therefore not 
very important and does not in any sense disprove the hypothesis. 

It should be noted at this point that the possible deviations from the 
theoretical may occur in both directions, and that in the test of signifi- 
cance both these possibilities have been taken into account. Since 
there is very often a good deal of confusion on this point, it may be just 
as well to emphasize here that it is absolutely necessary, in testing the 
hypothesis set up, to take into account possible deviations in both 
directions. Our hypothesis involves picturing a population deviating 
about a mean of 98.25 resistant to 32.75 susceptible plants. Accord- 
ing to the theory, deviations of 7.75 in either direction are equally likely, 
and in our sample the deviation happened to be positive for the resistant 
group and negative for the susceptible group. If we should determine 
the proportion of the trials in which a positive deviation as great as or 
greater than the one observed would occur, it is clear that this proportion 
would be exactly half of the proportion determined above, or about 6|%. 
But this would not be a test of agreement with the hypothesis, any 
more than it would be to determine the proportion of the trials, say, 
in which a deviation of +7.75 to + 8.00 would occur. The proportion 
would be very small, but it would in no way indicate disagreement with 
the hypothesis. Another way to consider this problem is to examine 
the possible consequences of accepting as a test of significance the 5% 
level, taking into account positive deviations only. On a large series 
of samples the investigator would expect to classify 5% of the samples 
as giving a significant disagreement with the hypothesis, even when the 
hypothesis is true. If positive deviations only are considered he would 
classify only 2f % of the samples in this way, and consequently would 
not be setting up the level of significance at the 5% but at the 2^% 
point. In certain cases, as we shall see later in the next chapter, it is 



92 



THE x 2 (CHI-SQUARE) TEST 



legitimate as a test of significance to take into consideration the devia- 
tions at one end of the distribution only; but these are special cases and 
not comparable to the example given above. 

Example 17. In a cross of two wheat varieties. Reward and Hope, the following 
results were obtained for the frequencies of resistant, semi-resistant, and susceptible 
plants in the Fa generation. 

Resistant 111 

Semi-Resistant 232 
Susceptible 1181 

The theoretical frequencies according to two hypotheses are as follows: 





Single Factor 


Two Complementary 




Difference 


Factors and an 






Inhibitor 


Resistant 


381 


119 


Semi-resistant 


762 


238 


Susceptible 


381 


1167 



If we wish to test the two hypotheses by comparing the actual with the expected 
frequencies in each case, the work may be set up and carried through as follows: 



Single Factor 
Hypothesis 



Complementary and Inhibiting 
Factor Hypothesis 



Actual 


Theoretical 


(a - 0V* 


Actual 


Theoretical 


(a - ff/t 


111 


381 


191.3 


111 


119 


0.5378 


232 


762 


368.6 


232 


238 


0.1513 


1181 


381 


1679.8 


1181 


1167 


0.1680 



X 2 =2239.7 n - 2 P - 0.0000 



x 2 - 0.857 n - 2 



P - 0.65 



We have two degrees of freedom in each case, and we find for the first case that such 
a large value of x 2 is not given in the table. The largest value under n = 2 is 9.21, 
which corresponds to a P of 0.01. We can conclude, therefore, that the probability 
of obtaining deviations, due to chance variation, as great as or greater than those 
observed is too remote .to be considered. In the second case, x 2 a 0.857 and this 
corresponds approximately to P * 0.65. The fit here is very good since deviations 
as great as or greater than those observed may be expected in at least 50% of the 
cases. The final conclusion is that the single factor hypothesis is quite inadequate 
to explain the type of segregation observed, but there is good evidence to support the 
second hypothesis baaed on a pair of complementary factors and an inhibiting factor. 



TESTS OF GOODNESS OF FIT 



93 



Example 18, In an assumed cross between parents of the constitution BBcc and 
bbCC, the F* population is classified as follows: 



BC 
1260 



Be 
625 



Cb_ 
610 



cb 
5 



Total 
2500 



According to a theoretical 9:3:3:1 ratio, the theoretical frequencies would be: 

BC Bc_ Cb_ cb_ Total 

1406 469 469 156 2500 



The actual results differ very widely from the expected as indicated by calculating 
x 2 . In this case we find x 2 = 255.60 and referring to Table 95 and entering at 
n * 3 we note that 11.34 is the highest value given. It is clear that the fit is very 
poor; so we proceed to analyze the data for the source of the disturbance, and develop 
a hypothesis more in accordance with the facts. In the first place the assumption is 
made when the 9 : 3 : 3 : 1 ratio is built up that the ratio of B to 6 is 3 : 1, and that of 
C to c is also 3:1. A discrepancy in either one of these ratios will result in a poor fit 
to the 9 : 3 : 3 : 1 for the whole set. Consequently we set up the two actual ratios 
and calculate x 2 for each. 



B jb_ 

1885 615 

(1885 - 3 X 615) 2 /3 X 2500 
0.2133 



1870 630 

(1870~3X630) 2 /3X2500 
0.0533 



Now x 2 values may be added together or separated into components. In this case we 
can add the two x 2 values, obtaining a new x 2 of 0.2666. Similarly we add the 
degrees of freedom, obtaining n = 2. On looking up the tables we find that the P 
value is between 0.95 and 0.50 but closer to the latter, hence the fit is good and the 
discrepancy of the actual from the theoretical 9:3:3:1 ratio is not due to the segre- 
gation of the individual pairs of factors, but to the behavior of the factor pairs in relat- 
ion to each other. In other words, there must be a tendency for the factors to be linked 
in inheritance. It is a common procedure in such cases to calculate the linkage 
intensity. An approved method (1) for examples of this type gives 9% of crossing 
over, and on that basis we can determine a new set of expected frequencies. These are 
set up below with the actual frequencies and another value of x 2 determined. 



Classes 


Actual 


Theoretical 


(o - 0V< 




Frequencies 


Frequencies 




BC 


1260 


1255 


0.0199 


Be 


625 


620 


0.0403 


cB 


610 


620 


0.1613 


cb 


5 


5 


0.0000 








X 8 0.2215 



94 



THE x 2 (CHI-SQUARE) TEST 



The theoretical frequencies in this table have been calculated on the basis of 9% 
crossing over, a value which was determined from the sample itself. Therefore, we 
lose one degree of freedom and must enter the table under n = 2. In this case we 
find P approximately 0.90. There is a very close agreement between the two sets 
of frequencies, but it would not be correct to consider this a very satisfactory fit. 
Such close agreement could only occur by chance on the basis of the hypothesis 
being tested in 10% of the cases. However, the agreement is not sufficiently close to 
prove that the original data were selected to give a good fit. If we had obtained a 
P of 0.95, it would have been worth while investigating the data to determine the 
reason for the very unusual agreement. 

Example 19. * The goodness of fit test may be useful in determining the agree- 
ment between actual and theoretical normal frequency distributions. In Chapter III, 
Example 1, we calculated the normal frequencies corresponding to the actual fre- 
quencies for the transparencies of 400 red blood cells. In Table 21, these two dis- 
tributions are repeated, and the third column gives the calculation of x 2 . 

TABLE 21 

ACTUAL AND NORMAL FREQUENCIES FOR TRANSPARENCIES 
OF 400 RED BLOOD CELLS, AND CALCULATION OF x 2 



Actual 


Theoretical 
Normal 


(a - <)*/< 


4 


4.64 


.0883 


11 


7.92 


1.1978 


17 


16.84 


.0015 


29 


30.28 


.0541 


43 


44.76 


.0692 


56 


59.16 


.1688 


58 


64.96 


.7457 


63 


60.40 


.1119 


61 


47.56 


3.7980 


25 


31.16 


1.2178 


20 


18.24 


.1698 


9 


8.80 


.0045 


4 


5.28 


.3103 


400 


400.00 


X 2 = 7.9377 



In connection with a test of this kind, two important points should be noted. 

(1) At the tails of the distribution the theoretical frequencies and corresponding 
actual frequencies are grouped. The object is to avoid very small theoretical values 
which, if present, to some extent invalidate the x 2 test. The general rule is to avoid 
having theoretical frequencies less than 5. This point is discussed in greater detail in 
the following chapter on tests of goodness of fit and independence with small samples. 

(2) The theoretical frequencies are determined from the total frequency and the mean 
and standard deviation of the sample, so we must deduct one degree of freedom for 
each. Thus three degrees of freedom are absorbed in fitting, and since there are 13 
classes we have 10 degrees of freedom for the estimation of x 2 . 



TESTS OF INDEPENDENCE AND ASSOCIATION 



95 



In the present example we enter the * 2 table therefore under n = 10, and note 
that a x 2 of 7.9377 corresponds approximately to a P value of 0.65. Consequently 
the fit may be considered a very good one. 

3. Tests of Independence and Association. From a cross of two 
wheat varieties 82 strains were developed and tested for their agronomic 
characters. One set of data for these strains is given in Table 22. On 

TABLE 22 

CLASSIFICATION OF 82 STRAINS OF WHEAT FOR 
YIELD AND CHARACTER OF AWNS 



Yield Classes weight in grains 




151-200 


201-250 


251-325 


Total 


Awned 


6 


7 


21 


34 


Awnless 


18 


21 


9 


48 


Total 


24 


28 


30 


82 



examining the frequencies in the 3X2 table, we note that there seems 
to be a tendency for the awned types to give higher yields than the 
awnless ones. To test the significance of such a result, we have to 
determine the probability of its occurrence if the two characters are 
entirely independent. For this particular problem we have to find the 
percentage of cases in which the above distribution, or one emphasizing 
still more the difference in yield of the two classes of varieties, would be 
obtained if there were no tendency whatever for awned varieties to 
yield higher or lower than awnless ones. Such a test could be applied 
by calculating x 2 if we could obtain the theoretical frequencies for each 
cell representing complete independence of the two characters. A 
reasonable basis for the calculation of these theoretical frequencies is to 
assume that, if the distributions are independent, they will be distributed 
within the table in the same proportion as they are in the totals. Thus 
in the cell in Table 22 containing 6 strains, we should have, on the basis 
of complete independence, x strains where x : 24 : : 34 : 82. Hence 
x = (24 X 34)/82. In the cell below, x = (24 X 48)/82. In the same 
manner all the theoretical frequencies can be calculated, and then we 
can proceed to the calculation of x 2 . This is the direct method of cal- 
culating x 2 ; but a shorter method for general use is given below under 
Section 5. 



THE X z (CHI-SQUARE) TEST 



4. Degrees of Freedom in X 2 Tables. In goodness of fit tests where 
the theoretical frequencies are determined according to some chosen 
hypothesis, the degrees of freedom can usually be equated to (N 1) 
where N is the number of cells in the table. In certain cases, however, 
as in Example 19 above, additional statistics calculated from the sample 
are utilized to determine the theoretical frequencies, and one degree 
of freedom must be subtracted for each of such statistics. 

In tests of independence or association, the subtotals of the classes 
into which the variates are distributed are used to determine the theoret- 
ical frequencies, and obviously these must be treated as statistics, so 
far as they themselves absorb degrees of freedom. Examining Table 22, 
we note that originally we have 5 degrees of freedom in the table, but 1 
of these is absorbed by the awning subtotals and 2 for the yield sub- 
totals. Therefore we have finally only 2 degrees of freedom left for the 
estimation of x 2 - Another method of determining the degrees of free- 
dom is to make an actual count of the number of cells that can be filled 
up arbitrarily. To do this we must assume that the subtotals are 
chosen first. Then, as in Table 22, any two cells such as those contain- 
ing 6 and 7 may be filled up arbitrarily but all the rest are fixed. The 
two cells that can be filled arbitrarily represent 2 degrees of freedom. 

In m X n fold tables the degrees of freedom can be equated to 
(m l)(n 1) for the general case with which we are dealing. Special 
cases will of course arise where this rule will not hold, but usually it is 
easy in such cases to arrive at the correct number by some such method 
as that described above. 

6. Methods of Calculation for Independence and Association Tests. 
(a) For (m X n) fold tables. The generalized x 2 table may. be repre- 
sented as follows: 

C 



1 


11 12 13 In 


TBI 


2 


21 22 


T Bt 


B 3 




TB* 


. 


etc. 





m 




T., 




T T TU. T*- 


T 



INDEPENDENCE AND ASSOCIATION TESTS 



97 



In order to determine x 2 we must calculate the theoretical frequency for 
each cell. For cell 11 we find t (T c \-T B i)/T, and for cell 12, 
i = (T C 2-T B i)/T, and so forth for all the cells. We then set up the 
theoretical frequencies with the corresponding actual frequencies and 
calculate x 2 = S[(a - t)*/t]. 

(6) For (8 X ri) fold tables. A table of this type may be represented 
as follows : 

1 2 3 n 



We can calculate x 2 for this table in exactly the same manner as for the 
(m X ri) fold table above, but a short-cut method giving x 2 directly 
without calculating the theoretical frequencies is given by Brandt and 
Snedecor, as follows: 



Each frequency in either of the rows is squared and divided by the cor- 
responding subtotal. These are summated and the correction term 
subtracted as shown in the formula. The remainder is multiplied by 
the quotient of the square of the total frequency by the product of the 
two subtotals on the right. This formula shows as each value of 
b 2 /T 8 is calculated the contribution of each pair to the value of x 2 - 

(c) For (2 X *) fold tobies. Representing the (2 X 2) fold table 
as follows: 



h bt 



Cl 



n 



X 2 is givefa by 



(3) 



We multiply diagonal frequencies and find the difference between the 
two products. The difference is squared and multiplied by the grand 
total, and the result is divided by the product of the subtotals. 



98 THE x 2 (CHI-SQUARE) TEST 

6. Coefficient of Contingency. It will have been noted that the 
methods employed in tests of independence and association are com- 
parable to the method of correlation, with this essential difference, that 
in the former the categories are either descriptive or numerical. If the 
categories are numerical and of equal magnitude, we can calculate a 
correlation coefficient for any of the tables to which we usually apply x 2 
with the reservation that if the categories are very broad we will get 
only an approximation to the true value of the correlation coefficient 
even if corrections are made for grouping. The necessity for the use 
of x 2 arises, therefore, from material which can be classified, at least 
for one character, only in descriptive categories, or in numerical cate- 
gories that are not of equal magnitude. For tables to which only x 2 
methods can be applied, some investigators feel that in addition to the 
X 2 test, which is essentially a test of significance, they should have some 
measure of association comparable to the correlation coefficient. A 
measure of this type is Pearson's coefficient of contingency (C) given by: 



where N is the total number of observations (not the number of classes). 
Since it is a function of x 2 . the significance of the coefficient of con- 
tingency must be the same as for x 2 - It is not necessary, therefore, to 
have a standard error of C in order to test its significance. 

7. Exercises. 

1. Test the goodness of fit of observation to theory for the following ratios: 



(1) 
(2) 
(3) 
(4) 



Observed Values 


Theoretical 


Ratio 


A 


a 


A 


a 




134 


36 


3 


1 




240 


120 


3 


1 




76 


56 


1 


1 




240 


13 


15 


1 




The 


x 2 values you 


should obtain are: 


(1) 


1.32 








(2) 


13.33 








(3) 


3.03 








(4) 


0.53 



2. In an F% family of 200 plants segregating for resistance to rust, if resistance is 
dominant and susceptibility recessive, find the ratio that gives a P value of exactly 
0.05 when fitted to a 3 : 1 ratio. 

There are two possibilities, the ratios being 138 : 62 or 162 : 38. 



EXERCISES 



99 



3. In a certain cross the types represented by BC, Be, bC, and be are expected to 
occur in a 9 : 3 : 3 : 1 ratio. The actual frequencies obtained were: 

BC Be bC be 

102 16 35 7 

Determine the goodness of fit, and if the fit is poor analyze the data further to dis- 
close the source of the discrepancy. 

X 2 = 9.86; P is less than 0.01. Hence the fit is poor. 
In further analysis, test the segregation for each factor separately. 

4. Test the goodness of fit of the actual to the theoretical normal frequencies for 
either of the distributions from Chapter II, Exercise 2, or Chapter II, Exercise 3. 
Watch the grouping of the classes at the tails of the distributions in order that the 
theoretical frequency in any one class is not less than 5. 

For Exercise 2, x 2 = approximately 10. 

For Exercise 3, x 2 as5 approximately 2.6. 

6. Table 23 gives the data obtained during an epidemic of cholera (3) on the 
effectiveness of inoculation as a means of preventing the disease. Test the hypothesis 
that in the inoculated group the number of persons attacked is not significantly less 
than in the not inoculated group, and the number not attacked is not significantly 
greater. Note carefully how this hypothesis is worded. 

TABLE 23 

FREQUENCIES OF ATTACKED AND NOT ATTACKED 
IN INOCULATED AND NOT INOCULATED GROUPS 

Not attacked Attacked 



Inoculated 
Not inoculated 



6. Calculate x 2 and locate the approximate P value for Table 22 given in Section 
3 above. x 2 = 15.87. 

7. The data in Table 24 were obtained in a cross between a rust-resistant and a 
susceptible variety of oats. The ^3 families were compared for reaction to rust in 
the seedling stage, and in the field under ordinary epidemic conditions. 

TABLE 24 

CLASSIFICATION OF SEEDLING AND FIELD REACTIONS 
OF 810 Fs FAMILIES OF OATS 

Seedling Reaction 
Resistant Segregating Susceptible 



Resistant 

Field Reaction Segregating 
Susceptible 



192 
113 


4 
34 



142 


51 


3 


13 


404 


2 


2 


17 


176 



100 THE x* (CHI-SQUARE) TEST 

Test the significance of the association in this table, and calculate the coefficient of 
contingency. 

X 2 - 1127.87. (This result will vary according to the accuracy with which 
the t values are calculated. To check approximately with the value given here 
calculate the t values to at least two decimal figures.) 



REFERENCES 

1. R. A. FISHER and BHAI BALMUKAND. Journ. Gen., 20: 79-92, 1928. 

2. R. A. FISHER. Statistical Methods for Research Workers, Oliver and Boyd, 

London, 1936. Reading: Chapter IV. 

3. M. GREENWOOD and G. U. YULE. Proc. Roy. Soc. Med., 8: 113, 1915. 

4. G. W. SNEDECOR. Statistical Methods. Collegiate Press, Inc., Ames, Iowa, 1937. 

Reading: Chapters 1, 9. 

5. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter IV. 



CHAPTER X 

TESTS OF GOODNESS OF FIT AND INDEPENDENCE WITH 

SMALL SAMPLES 

1. Inadequacy of the x 2 Criterion and the Correction for Continuity. 
The method of x 2 is based on the smooth curve of a continuous distribu- 
tion and, when the numbers are large, gives probability results that are 
very close to the true values. When the numbers are small, and espe- 
cially when only one degree of freedom is involved, the x 2 method is 
quite inaccurate. One reason for this will be clear from an examination 
of Fig. 10, representing the distribution obtained by expanding the 




RATIO 



FIG. 10. Frequency distribution of (| + f ) 8 and corresponding smooth curve. 
Shaded areas indicate the need for a correction to x 2 for smaD samples. 

binomial $ + -|) 8 . Given a theoretical ratio of 1 : 1, say, for the suc- 
cess or failure of an event, the binomial distribution as in Fig. 10 would 
give the theoretical frequency of the successes through the total range 
from to 8. If we wished to determine the probability of obtaining 
6 or more successes in 1 trial of 8 events, we would find the ratio of the 
dotted area of the figure to that of the whole. A x 2 test of the 6 : 2 
ratio, however, would be based on the smooth curve shown in Fig. 10, 
and the probability would be the ratio of the cross-hatched urea to the 
whole. The cross-hatched area is obviously less than the dotted area, 

101 



102 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 

by an amount equal approximately to one-half the area of the 6 : 2 ratio 
column. Consequently the x 2 test will give a probability result that is 
too low. 

In order to correct for the above-mentioned discrepancy in the x 2 
test, Yates (8) has suggested a correction which he proposes to call the 
correction for continuity. In the ordinary case x 2 is given by S(a t) 2 /t, 
where a represents the actual and t the theoretical frequencies. Yates's 
correction is applied by subtracting | from each value of (a t), but it 
must always be subtracted in the direction that reduces the numerical 
value of (a t). In Fig. 10 the application of the correction would 
result in extending the cross-hatched area to the line bordering the col- 
umns representing" the 5 : 3 and 6 : 2 ratios, and must obviously bring 
about an improvement in the estimate of probability. 

It should be noted in connection with tests of significance applied 
to ratios that the x 2 method is exactly equivalent to the use of the 
standard deviation to determine the significance of a deviation from 
the mean. Likewise the correction for continuity must be made when 
the numbers are small. As will be evident from Fig. 10, the correction 
is simply a matter of subtracting f from the deviation from the mean. 
To test the significance of a 6 : 2 ratio when the theoretical is 1 : 1 or 
4 : 4, we would take the deviation equal to (6 _ 4 ^) = 1.5. The 
standard deviation of a binomial distribution is Vpgn = Vf X ^ X 8 = 
1.4142, and we can test in the usual way, using tables of the probability 
integral. 

The x 2 test for ratios is also inaccurate when applied to samples from 
populations having a definitely skewed distribution. In the case of 
ratios of successes to failures where the theoretical ratio is not 1:1, this 
inadequacy of the x 2 test becomes obvious. Table 25 gives the true 
probabilities calculated from the binomial distribution of obtaining from 
16 to successes when each trial consists of 16 events. These are worked 
out for two cases: (1) when the theoretical ratio is 1 : 1, and (2) when the 
theoretical ratio is 3 : 1. The corresponding \P l values obtained by 
calculating x 2 with and without Yates's correction are given in the same 
table. For the symmetrical binomial distribution it will be noted that 
the |P values for x' 2 with Yates's correction agree very well with the 
correct values except at the extreme tails of the distribution where x' 2 
tends to overestimate the probability. For the asymmetrical distribu- 
tion the agreement is not good anywhere in the range. In both cases it 

1 |P is used here to indicate that the probability is calculated from the area of 
only one tail of the distribution. As the problem is stated in terms of "15 or more 
successes," etc., it is obvious that only one tail of the distribution must be considered. 



INADEQUACY OF THE * 2 CRITERION 



103 



will be observed that x 2 uncorrected gives a very decided underestimate 
of the probability through practically the whole range. 

TABLE 25 
PROBABILITY OF n SUCCESSES IN A SAMPLE OP 16 EVENTS 





Distribution (\ -f i) 16 


Distribution - (f -f |) 16 




Corrected 


Uncorrected 




Corrected 


Uncorrected 


Successes 


I P (Bin) 


1 P (x' 2 ) 


iP(x 2 ) 


i P (Bin) 


\ P (x' 2 ) 


*/>(x 2 ) 


16 


0.000,015 


0.000,088 


0.000,032 


0.010,023 


0.021,656 


0.010,461 


15 


0.000,259 


0.000,577 


0.000,233 


0.063,477 


0.074,457 


0.041,638 


14 


0.002,090 


0.002,980 


0.001,350 


0.197,112 


0.193,248 


0.124,109 


13 


0.010,635 


0.012,224 


0.006,210 


0.404,988 


0.386,406 


0.281,837 


12 


0.038,406 


0.040,059 


0.022,750 








11 


0.105,056 


0.105,650 


0.066,807 


0.369,812 


0.386,406 


0.281,837 


10 


0.227,248 


0.226,627 


0.158,655 


0.189,653 


0.193,248 


0.124,109 


9 


0.401,809 


0.401,294 


0.308,538 


0.079,556 


0.074,457 


0.041,638 


8 








0.027,129 


0.021,656 


0.010,461 


7 


0.401,809 


0.401,294 


0.308,538 


0.007,469 


0.004,687 


0.001,946 


6 


0.227,248 


0.226,627 


0.158,655 


0.001,644 


0.000,748 


0.000,266 


5 


0.105,056 


0.105,650 


0.066,807 


0.000,285 


0.000,087 


0.000,027 


4 


0.038,406 


0.040,059 


0.022,750 


0.000,038 


0.000,008 


0.000,002 


3 


0.010,635 


0.012.224 


0.006,210 


0.000,004 


0.000,001 


0.000,000 


2 


0.002,090 


0.002,980 


0.001,350 


0.000,000 


0.000,000 


0.000,000 


1 


0.000,259 


0.000,577 


0.000,233 


0.000,000 


0.000,000 


0.000,000 





0.000,015 


0.000,088 


0.000,032 


0.000,000 


0.000,000 


0.000,000 



In probability tests applied to 2 X 2 frequency tables, the same 
difficulties arise with regard to the application of x 2 as for testing the 
goodness of fit of simple ratios. Since only one degree of freedom is 
involved, the number of possible combinations of the frequencies of 
unlike probability is relatively small and the theoretical distribution is, 
therefore, definitely discontinuous. The error is not significant when 
the frequencies are large, but with small frequencies it is very decided. 
The skewness factor is not so important for 2 X 2 tables as for simple 
ratios, as the x 2 curve adopts itself within certain limits to the shape of 
the theoretical distribution. After correction for continuity the remain- 
ing discrepancy may be regarded as due to the comparison between a 
histogram and a smooth curve which gives an approximate fit. 

The method of making the correction for continuity is to determine 
the larger of the two products &iC2 and &2Ci, and for the larger subtract- 
ing 0.5 from the two factors, and for the smaller adding 0.5 to the two 



104 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 

factors. After making these corrections the usual formula may be 
applied. 

Table 26 has been prepared to show the relation between the values 
of |P calculated for the 2 X 2 table: 



12 
6 




8 



using (a) a direct method for determining the exact probability, (b) x 2 
without correction, and (c) x' 2 > or that obtained by using the correction 
for continuity. The direct method was devised by R. A. Fisher (1) and 
will be described below under "Methods of Calculation." The prob- 
ability value for the modal frequency has been omitted since it may be 
considered as belonging to either tail of the distribution. 

It will be noted that at the extreme tails of the distribution x 2 tends 
to overestimate the probability, but that in the range where significance 
may be in doubt the agreement is fairly good. On the other hand, as 
indicated by the %P values for x 2 > unless the correction for continuity is 
made there is a very decided underestimation of the probability through- 
out the whole range. 

For 2X3 frequency tables, the correction for continuity is not so 
important as for 2 X 2 tables. With 2 degrees of freedom the number 
of possible combinations is much greater than for 1 degree of freedom, 
and the agreement between the smooth curve and the histogram must be 
much better. With more than 2 degrees of freedom the correction for 
continuity would hardly be necessary in any case. It must be remem- 
bered, however, that the tendency, especially when the numbers are 
small, is to underestimate the probability; and it may be necessary in 
certain cases to check the probability by direct calculation, or if this is 
impractical, by an analytical study of the larger table made by breaking 
it up into parts or condensing it into a single 2X2 table. The direct 
calculation of probabilities, even in a 2 X 3 table, is slightly complicated; 
so that in most cases the best practice is to endeavor to make an applica- 
tion of x 2 such that we are reasonably sure of a fair approximation to the 
true probability. 



METHODS OF CALCULATION. EXAMPLE 20 



105 



TABLE 26 
PROBABILITIES FOE ALL THE COMBINATIONS OF A 2 X 2 TABLE 



Combination 


^ F Calculated by 


Direct 
Method 


x 2 


x' 2 


12 
6 8 


0.00192 


0.00082 


0.00325 


11 1 

7 7 


0.02828 


0.01087 


0.03084 


10 2 
8 6 


15585 


0.07460 


0.15475 


9 3 
9 5 


0.43707 


0.27756 


0.49346 


8 4 
10 4 








7 5 
11 3 


0.24577 


0.13251 


0.24557 


6 6 
12 2 


06124 


02459 


0.06188 


5 7 
13 1 


00741 


00241 


0.00835 


4 8 
14 


0.00032 


00012 


0.00059 



2. Methods of Calculation. Example 20. In a study of the blood groups of 
some North American Indians, Grant (2) obtained the results given in the following 
table: 





Blood Groups 




Band of Indians 

















A 


B 


AB 




Fond du lac 


18 


6 


5 





29 


Chipewyan 


13 





1 





14 




31 


6 


6 





43 



106 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 



It appears that pure Indians tend towards a very high percentage of individuals 
having the blood group O, but the group at Fond du lac had an obviously larger 
percentage of white blood as indicated by other characteristics. The essential prob- 
lem in this case is to test the significance of the distribution of the two bands into 
two main groups, O and not O. We form, therefore, a 2 X 2 table, as below: 

O notO 



Fond du lac 
Chipewyan 



18 
13 


11 
1 


29 
14 


31 


12 


43 



Either the x 2 test with the correction for continuity or the direct probability method 
would be applicable to this table. In order to indicate the methods of calculation we 
shall apply the test in both ways. 

(a) x 2 corrected for continuity. If a 2 X 2 table is represented as follows* 



the corrected value of x 2 is given by 




where T/2 always reduces the numerical value of (6iC2 < 
equivalent to the method described on page 103. 

Applying the corrected formula to our example, we have 
9 (13 X 11 - 18 - 



(1) 



This is of course 



31 X 12 X 14 X 29 



3.0499 



Using Yules table of "P for divergence from independence in the fourfold table" 
(9), we look up 

X 2 - 3.0 P - 0.08326 

X 2 - 3.1 P - 0.07829 



Difference - 0.00497 



and by direct interpolation P - 0.08077 and J P - 0.0404. 

In order to obtain P more accurately we can make use of the fact that the dis- 
tribution of x 2 is normal for one degree of freedom, and V x 2 ? t the value for 
Catering tables of the probability integral. Here V x 2 - V3.0499 - 1.7464, and in 
Sheppard's table of the probability integral we look up 



METHODS OF CALCULATION. EXAMPLE 20 



107 



I m 1.74 J(l + a) - 0.9590705 
t - 1.75 (1 -f- a) 0.9599408 

Difference - 0.0008703 

and interpolating directly for t = 1.7464 we have $(1 -fa)- 0.959,6275. Since 
we want | P we take $ P - 1 - f (1 -f a) - 0.04037. 

(6) Direct probability method for a 2 X 2 table. Representing a 2 X 2 table 
as above, R. A. Fisher (1) has shown that, for any particular combination of 61, 
&2, ci, C& the direct probability of its occurrence is given by 

t H ^ :W \ (2) 

Tl / V&i! W d! c*!/ 

The easiest method of performing the calculations is by means of a table of 
logarithms of factorials. The different combinations that can occur are as follows: 



17 12 
14 


18 11 
13 1 


19 10 
12 2 


20 9 
11 3 



and so forth 



all other combinations having the same probability and occurring with equal fre- 
quency with one of the above. In this case, therefore, we require the sum of the 
separate probabilities of the first two combinations. These are given by: 



"31 IX 12! X 29! X 14! 

43! 
"31 !X 12! X 29! X 14! 



X 



18! X 11! X 13!_ 
1 



L 43! 17! X 14! X 12!J 

When a series of such terms are to be calculated, labor is saved by first calculating 
the logarithm of the constant factor. The logarithms of the terms are then obtained 
by subtracting the logarithms of the factorials in the numerator of each term. 
In this example, log constant factor 31.701,1593 
The logs to be subtracted are 33.201,7770 and 34.171,8139, giving: 

log term 1 - 2.499,3823 Term 1 - 0.031,578 
log term 2 - 5.529,3454 Term 2 - 0.003,338 



The values of 



Total - $P - 0.0349 
obtained by the two methods are in fairly close agreement. 1 



1 The student may use this example in order to straighten out in his mind the 
reason why for certain tests it is correct to base the decision on the value of $P 
instead of P. Actually the hypothesis being tested here is that Indians having an 
admixture of white blood do not contain a greater percentage of individuals with the 
blood group O than Indians that are relatively pure. If the hypothesis is stated 
differently for example, that the two groups of Indians are random samples drawn 
from the same population with respect to the distribution of the blood group O 
then it would be necessary to use the full value of P in order to make the test. The 
test based on the value of \ P arises from the knowledge that the Fond du lac group 
had an obviously larger percentage of white blood than the Chipewyans. 



108 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 

Example 21. For a certain disease we will assume that it has been shown that 
recovery or death is a certainty and that without treatment about half of the patients 
recover. A new treatment tried out in 16 cases gives 12 recoveries and 4 deaths. 
Is this a significant demonstration of the efficacy of the treatment? 

This problem can be solved by the direct calculation of probabilities according 
to the binomial distribution, or since the theoretical distribution is symmetrical the 
X 2 test corrected for continuity will give a fairly close approximation. Both methods 
will be used in order to demonstrate methods of calculation. 

(a) x 2 corrected for continuity. For ratios the short formula for determining 
X 2 as in Chapter IX, Section 2, is modified as follows to correct for continuity. 



- 02* 



where the theoretical ratio is x : 1, a\ is the actual frequency corresponding to x, and 
02 is the actual frequency corresponding to 1. AT is the total frequency or (ai+ 02), 

and ~ always reduces the numerical value of (ai a$t). In the present example: 



From Yule's table of P we find \ P - 0.0401. The odds are about 25 : 1 against the 
occurrence of a 12 : 4 ratio due to chance alone. 

(6) Direct probability from the binomial. Let p represent the probability of 
recovery and q the probability of death. We know that p = q ^, and we require 
the first five terms of the expansion of (p + 9) n where n 16. The expansion of 
(P + 0) n is given by: 

(P + 5) n - P n + nCip"- l q + n C 2 p n -y + - + C w g n (4) 

where n C r n( * " 1)( " " 2) ' " (n ~ r + *> * ! 

1-2-3 -T r!(n-r)I 

In our example we have: 



u 



In each term we have the constant factor () 16 . We determine the logarithm of this 
factor in the ordinary way and proceed to determine the logarithms of the coefficients 
by means of a table of the logarithms of factorials. The work is as shown in Table 27, 
which is self-explanatory with the possible exception of the last column. The term 
values give the probabilities of obtaining in one trial the number of recoveries (or 
deaths) shown in the same line. In general, however, we do not ask that question. 
We inquire, for example, as to the probability of obtaining 12 or more recoveries in a 
sample of 16, and hence we must add the probabilities for 12, 13, 14, 15, and 16 
recoveries. These summations have been performed and are given in the last column 
under the heading |P. Again, since we have summated for one tail of the distribu- 
tion only, we represent the probability by ^P. 

The answer to our problem is given in the line representing 12 recoveries. The 
corresponding value of \P is 0.0384, and this compares reasonably well with 
|P - 0.0401, obtained by the %* method* 



METHODS OF CALCULATION. EXAMPLE 22 



109 



TABLE 27 
CALCULATION OF PROBABILITIES FROM THE BINOMIAL 



Recoveries 


Log 


Log 


Log 


Term 


p 




nC r 


P n Y 


Term 






16 




5.183,5200 


5.183,5200 


0.000,015 


0.000,015 


15 


1.204,1200 


u 


4.387,6400 


0.000,244 


0.000,259 


14 


2.079,1812 


u 


3.262,7012 


0.001,831 


0.002,090 


13 


2.748,1880 


if 


3.931,7080 


0.008,545 


0.010,635 


12 


3.260,0714 


n 


2.443,5914 


0.027,771 


0.038,406 



Example 22. In the example above, let us assume that without treatment the 
ratio of recoveries to deaths is 3 : 1 instead of 1 : 1, and in the group of 16 patients 
receiving treatment the actual ratio is 14 : 2. Test the significance of the treatment. 

This problem differs from the first, in that the theoretical distribution is skewed, 
and what has been said about the x 2 method being remembered, it may be taken for 
granted that x 2 will not give a good approximation to the true probability. We must 
solve this problem, therefore, by a direct calculation of the probability from the 
binomial distribution. 

Since the ratio of recoveries to deaths is 3 : 1, p = f and q =* ^, and we must 
calculate the first three terms of the expansion of (f + ) 16 . Using the formula given 
we have: 

6 16!/3\ 16 /1\ 16! 



3 1\ 16 /3V 

4 + V ~ W 



Noting for convenience in calculation that: 



6 " 

3 



* 



The factor (J) 16 is constant, and when several terms are to be calculated this trans- 
formation results in a saving of labor. 

The calculations are given in Table 28. In the JP column representing 14 recov- 
eries we have |P = 0.1971, or the odds are only about 5 : 1 that the treatment is 
beneficial. This is an indication of a beneficial effect but it cannot in any sense be 

TABLE 28 
CALCULATION OF PROBABILITIES FROM THE BINOMIAL (f + J) 16 



Recoveries 


Log 


Log 


Log 
Term 


Term 


p 


16 
15 
14 


1.204,1200 
2.079,1812 


2.000,9808 
3.523,8595 
3.046,7382 


5.000,9808 
2.727,9795 
1.125,9194 


0.010,023 
0.053,454 
0.133,635 


0.010,023 
0.063,477 
0.197,112 



110 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 



considered a proof. It would be sufficient evidence to warrant further investigation, 
but the practical aspect of such a problem must not be lost sight of, in that the actual 
gain in recoveries is very small and further investigation might best be directed 
along the line of trials with other treatments. 

3. Selection of Method for Tests of Significance. Some confusion 
may arise as to when to apply x 2 and when to apply the direct method 
of calculating probabilities. Also when applying x 2 the question arises 
whether or not the correction should be applied. In general these points 
can be made clear by the consideration of some hypothetical examples. 

Example 23. The following is a 2 X 4 fold table of frequencies 

A B C D 



I 
II 



28 
68 


46 
43 


83 
12 


126 
1 



The numbers are large, and the theoretical frequencies in each cell are large. 
The x 2 criterion may be applied to the whole table, and no correction is required. 

Example 24. If some of the numbers in a 2 X 4 fold table are small, as in the 
table below, the table must be rearranged. 



B 



1 
II 



26 
94 


84 
18 


2 
1 


1 
4 



Obviously the classification of the I and II frequencies into C and D is meaningless, 
and the rearrangement is either a matter of adding these frequencies to B or elimi- 
nating them altogether. Assuming that they can be eliminated we have a 2 X 2 table 




To this table it is perfectly legitimate to apply the x 2 test, and, although the numbers 
are fairly large, the correction for continuity will improve the results slightly. Obvi- 
ously it would be very laborious to make a direct calculation of the probability, BO we 
would not even consider the method in this case. 



EXERCISES 
Example 25. We have a 2 X 2 table in which the numbers are small: 



111 




For this case the direct method is the most accurate and is not difficult. 

Example 26. Given a theoretical ratio of 1 : 1 for the occurrence of A and B in a 
series of events, we obtain in 100 trials 60 A 's and 40 B's. What is the significance of 
this result? 

The numbers are large so that the direct calculation of the probability will be 
very cumbersome. Therefore, we use x 2 with the correction for continuity, or we 
calculate the ratio of the deviation (also corrected for continuity) to its standard 
deviation and get the probability from tables of the probability integral. The cor- 
rection for continuity is not important, but it is bound to give a slight improvement. 

Example 27. In a test of the goodness of fit of a ratio, we have a very skew 
distribution. For example, the theoretical ratio of successes to failures is 15 : 1, and 
the actual results are 5 failures out of 160 events. The direct method is the only one 
that will give an accurate probability result in this case, and we must calculate the 
last six terms of the expansion of (15/16 + 1/16) 180 . When the numbers are large, 
the calculations are somewhat laborious, but in most cases it is sufficient to determine 
whether the result is or is not significant; and it will only be necessary in working 
from one end of the distribution to calculate enough terms such that their sum (F) 
is 0.05. If the observed deviation is within that range it is not significant. If the 
deviations in both directions are to be considered, we work from both ends of the 
distribution until the sum of the terms at each end is equal to 0.025. 

Example 28. The theoretical ratio gives a skew distribution, but the numbers 
are small. Calculate the probability by the direct method as in Example 27. 



4. Exercises. 



) 8 and (f -f f ) 12 , and calculate the value of 



1. (a) Expand the binomials (f 
each term. 

(6) If there is an equal probability of the birth of male and female rabbits de- 
termine the probability in a litter of 8 of the occurrence of 2 females and 6 males. 
(c) Plot the histogram for the expansion of (f 4- i) 12 - A bag contains white 
and black balls in the ratio of 3 white to 1 black. Show that, if a sample of 12 balls 
is taken at random, the probability of obtaining 12 white balls is different from that 
of obtaining 6 or more black balls, although both cases represent an equal deviation 
from the expected 9 white to 3 black. 

(a) In order to check the work add all the terms and the sum should be 
very close to 1.000. 

(6) p == 0.1094. (Note that this is not a test of significance. It is merely 

a question of determining the probability of the occurrence of one particular ratio.) 

(c) Illustrates the problem of making tests of significance in skew distributions. 

2. Koltzoff (3) performed an experiment on the control of sex in rabbits. Sperms 
were placed in a physiological solution in a tube and an electrical current passed 



112 TESTS OF FIT AND INDEPENDENCE WITH SMALL SAMPLES 



through the tube. A female impregnated with sperms taken from the anode produced 
6 females and males, and another female impregnated with sperms from the cathode 
produced 1 female and 4 males. Test the significance of this result. 

Using the direct method P is 0.0152. 

3. From a study of the position of the polar bodies in the ova of the ferret, 
Mainland (4) gives the frequencies in the following table: 



Similar Different 



5 

1 


1 
6 



!OM apart 

More than 10;* apart .... 



Test the significance of the apparent association between similarity and position 
of the polar bodies. JP - 0.025 calculated by the direct method. 

4. Neatby (6) studied the association, in a random sample of lines from a wheat 
cross, of resistance to different physiologic forms of the stem rust organism. Two 
tables from his results are given below. Test the significance of the association in 
each case. 

Form 21 Form 21 

SR S SR S 



Form R. 
27 8R. 



28 


41 


Form R 


17 


15 


57 S 









46 



40 
16 



= 0.93 



R (resistant) # (semi-resistant) S (susceptible) 
x 2 - 13.50 



7 
3 


3 
9 



5. Twenty-two animals are suffering from the same disease, and the severity of 
the disease is about the same in each case. In order to test the therapeutic value of a 
serum it is administered to 10 of the animals and 12 remain uninoculated as a control. 
The results are as follows: 

Recovered Died 



Inoculated 

Not inoculated .... 



Determine the probability in such an experiment of obtaining this or a result more 
favorable to the treatment. By the direct method \P * 0.0456. 

6. An experiment is conducted similar to that in Exercise 5 but no uninoculated 
animals are available for a control. Previous results, however, indicate very strongly 
that the proportion of recoveries to deaths without treatment is 1 to 3. Again, the 
result is 7 recoveries to 3 deaths when 10 animals are treated. Test the significance 
of this result, and explain why it differs from that obtained in Exercise 5. 

\P m 0.0035. 

In the problem of Exercise 5 the theoretical ratio is itself estimated from the 
sample. 



REFERENCES 113 



REFERENCES 

1. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter IV, Sections 21.01, 21.03. 

2. J. C. B. GRANT. National Museum of Canada, Bull. 64, 1930. 

3. N. K. KOLTZOFF and V. N. SCHROEDER. Nature, 131, No. 3305, 1933. 

4. DONALD MAINLAND. Am. Jour. Anat., 47, No. 2, 1931. 

5. DONALD MAINLAND. The Treatment of Clinical and Laboratory Data. Oliver 

and Boyd, London, 1938. Reading: Chapter III. 

6. K. W. NEATBY. Sci. Agr., 12: 130-154, 1931. 

7. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter III, Section 3.8. 

8. F. YATES. Journ. Roy. Stat. Soc., Suppl. L, No. 2, 1934. 

9. G. U. YULE. The Theory of Statistics. Charles Griffin and Co., London, 1924. 



CHAPTER XI 
THE ANALYSIS OF VARIANCE 

1. The Heterogeneity and Analysis of Variation. If we consider 
the variation in such a character as stature in man, it is obvious that 
this variation in general is not homogeneous. Two races may differ 
decidedly in their average stature, and the individuals of each race will 
vary around a common mean. Also, with reference to the variation 
within each race, there are regional and genetic differences between cer- 
tain groups so that even within the race the variation is not strictly 
homogeneous. In actual fact we can conclude with a reasonable degree 
of certainty that variation cannot be strictly homogeneous unless it is 
purely random, i.e., caused by a multiplicity of minor factors that cannot 
be distinguished one from another. In experimental work the hetero- 
geneity of variation is usually predetermined by the plan of the experi- 
ment. One set of results is obtained, for example, under a given set of 
conditions and another under distinctly different conditions, the object 
being to compare the two groups of results. Here the heterogeneity of 
the variation is the factor that is being tested, and the degree of its ex- 
pression determines the significance of the findings of the experiment. 
It would seem to be a necessity, therefore, in studies of variation, to be 
able to differentiate the variation according to causes or groups of causes, 
especially in experimental work where such differentiation is an essential 
part of the analysis of the results. The analysis of variance supplies the 
mechanism for this procedure and in addition sets out the results in a 
form to which tests of significance can be applied. 

The points mentioned above may be made more obvious by the con- 
sideration of a theoretical exainple. Suppose that, for two races of men 
that we shall designate as A and 5, the mean stature of race A is 66 
inches and that of race B is 68 inches. Histograms are prepared for 
the frequency distributions of stature for the two races, and one histo- 
gram is superimposed on the other. The two distributions will undoubt- 
edly overlap, but are very likely to show two distinct peaks at the means 
of the two populations. The variation over all the individuals com- 
prising the two races could then be fairly definitely described as hetero- 
geneous. We might now endeavor to picture what the situation might 
be if we were dealing with several races instead of only two. There 
might be a number of peaks, perhaps as many peaks as there are races; 

114 



THE HETEROGENEITY AND ANALYSIS OF VARIATION 115 

but it is more likely that some of the groups will so nearly coincide as 
to be indistinguishable. Now that we have in mind several races, how- 
ever, it is probably easier to think in terms of the total variability of 
all the individuals concerned being divided up into two portions. One 
portion is that which occurs within all the races. To get a mental pic- 
ture of this, we might suppose the frequency distributions for all the 
races superimposed on one another in such a way that the means of the 
different races would coincide. The resulting distribution would be a 
sort of average of all the separate distributions. The second portion 
of the variability would be that resulting from the differences between 
the means, and if we had a sufficient number of these means we could 
make up another frequency distribution for them. For each type of 
distribution a standard deviation or a variance could be calculated, and 
it becomes clear at once that a comparison of two such statistics would 
be valuable in coming to a conclusion as to the degree of heterogeneity. 
To make this point still more obvious, let us imagine a series of samples 
being taken from a homogeneous population. As we have already 
learned, these samples will have different means, but these differences 
will result merely from random sampling. They will be large or small 
according to the magnitude of the variation in the population from which 
they are drawn. This is a very important generalization and one which 
is fundamental to an understanding of the analysis of variance. If the 
original population has a very small variation, the means of the samples 
drawn from it will also have a small variation. If the population has 
a large variation, it is to be expected that this will be reflected in the vari- 
ations of the means of the samples. In fact, without going into the in- 
tricacies of an algebraic proof it seems reasonable to assume that, on 
the average, the variance of the means of the samples will be equal to 
that in the original population, provided of course that we multiply this 
variance by the number in the samples. Thus, if the variance of the 
population is v, the variance of the sample means is expected to be v/n, 
where n is the number of individual determinations entering into each 
mean. 

The next step in the development of these ideas is to consider what 
the situation would be if, in taking a series of samples, we did not know 
that they were being taken from a homogeneous population. The 
variance of the population is unknown; hence it must be estimated 
from the values in the samples. The most logical estimate is that aris- 
ing from the variations within each sample, from its own mean. Sup- 
pose that this estimate is vi and the estimate of the variance of the 
sample means is v*/n. Multiplying the latter by n we have t>2, which 
we shall expect to be very dose to v\ if the population is homogeneous, 



116 THE ANALYSIS OF VARIANCE 

but which may be very much larger than vi if the population is hetero- 
geneous and this heterogeneity has corresponded with the method of 
taking the samples. This suggests to us that there may be a technique 
here for making a test of significance. The null hypothesis is that all 
the samples have been drawn from the same population, and therefore 
that V2 does not differ significantly from vi. For example, if we take 
the ratio V2/vi, a test of significance could be made if, for a given example, 
we could determine the proportion of the trials in which a value as large 
as or larger than v%/v\ would be obtained owing entirely to random sam- 
pling fluctuations. We are indebted to Dr. R. A. Fisher for many of the 
recent developments in statistical methods, but especially for the solu- 
tion of this particular problem. If there are only two samples it will be 
noted that we have already discussed a solution, in that we may apply 
the t test to the significance of the difference between the means. How- 
ever, if there are more than two samples the t test does not apply, and we 
must use the technique of the analysis of variance as developed by R. A. 
Fisher (3). The details of this technique are best learned by the con- 
sideration of actual data. 

2. Division of "Sums of Squares" 1 and Degrees of Freedom. As 
pointed out in previous chapters the variance is a measure of variation, 
and it consists of a sum of squares of deviations from the mean divided 
by the corresponding degrees of freedom. In a set of observations, if 
the total sum of squares of the deviations from the mean can be divided 
up according to some scheme suggested by the data, and the degrees of 
freedom can be divided correspondingly, it is clear that a variance can 
be calculated for each group as well as for the total. It is through the 
comparison of such variance values that we obtain a true picture of the 
variation in the entire set of observations. 

With respect to the division of sums of squares, the best way to ob- 
serve this and to follow the method is to deal with actual data. The 
figures given below are yields in bushels per acre of 6 plots of wheat. 
Three of these plots are of variety A and three of variety J5. 

A 27.6 32.4 23.4 
n 19.2 18.6 16.5 

The total sum of squares is made up of the sum of the deviations of the 6 
plots from the general mean. A logical division of this total is to sepa- 
rate it into one part due to variation within the varieties, and another 

1 "Sums of squares' 1 written thus is an abbreviation for (sums of squares of devi- 
ations from the mean), but in general throughout this book the quotation marks are 
omitted. 



"SUMS OF SQUARES" AND DEGREES OF FREEDOM 117 

part due to variation between the varieties. Let the general mean be #, 
which in this case is 137.7/6 = 22.95. And the mean of A is = 27.8, 
and the mean of B is x* = 18.1. Then subtracting 22.95 from each 
value, squaring and summating, we have: 

2(z - x) 2 = 185.715 
i 

6 

where S indicates that 6 deviations are summated. Now, to obtain 

the sum of squares for within the varieties, we must repeat the above 
operation for each variety and add the two sums of squares together. 
Thus for A we subtract 27.8 from each of the A values and square and 
summate . This gives : 

S(o: - x a ) 2 = 40.560 
i 

and for B we have 

l(x - x b ) 2 = 4.020 

2 3 

Then SSO - x*) 2 = 40.560 + 4.020 = 44.580, where the double 
i i 

summation indicates the process of adding together the two sums of 
squares, and represents the mean of one group. 

The next step is to calculate the sum of squares for between the 
varieties. This is given by 

3 X [(27.8 - 22.95) 2 + (18.1 - 22.95) 2 ] = 141.135 

Note that we obtain the deviations of the means of A and B from the 
general mean and then square and summate, but we multiply the whole 
sum by 3 because each value such as 27.8 represents the mean of 3 single 

plots. 

2 
The formula for this sum of squares will be 3 S( ) 2 - 

Now if we add the sums of squares 

Within Between Total 

44.580 -f 141.135 - 185.715 

S S(* - *<) 2 + 3 Z(*< - *) 2 - SO* - *) 2 
11 i i 

we note that the within and between sums are exactly equal to the total. 
That the sums of squares can always be divided in this way is very 



118 



THE ANALYSIS OF VARIANCE 



easily proved for the general case. A set of observations classified in 
one direction may be represented as follows 



1 

2 

Groups 3 



Xn 12 13 ln 

21 22 23 X 2n 

Xn 82 33 3n 



Xkl XkZ XkZ Jkn 



where there are k groups and n single observations in each group. 
For any one observation, say xn, we can write 

(xn - ) = (xn - xi) + (ft - x) 
where ft is the mean of group 1. Then 

(xn - x) 2 = (xn - xi) 2 + (xi - )* + 2(x u - xiXft - x) 
And summating for all the values in group 1 we have 

n 

t n(n^t ^*|2 I . O/'5* ^1^5'i'y ^*i^ 
7l\Xl X) ^T 4\Xl Xjt\X X\) 



2(x - 



2(x - xi) 2 
i 



The last term is zero because the sum of the deviations from the mean 
must be zero and each deviation is multiplied by a constant factor. 
The second last term is written n(xi ) 2 because the factor (ft x) 2 
is constant and we merely summate it n times. Finally we have 



n(ft - 



Now we repeat this for each 'group, and summating over all the k 
groups we have 

n* In k 



1 1 1 

which is exactly equivalent to the equation given above with the actual 
sums of squares. 

The division of degrees of freedom corresponding to the sums of 
squares follows easily. In the example for two varieties we have a 
total of 5 degrees of freedom, for within varieties we have 2 in each 
group making a total of 4, and for between varieties we have only 1. 
Thus 

Total Within Between 

5 .44-1 



TESTS OF SIGNIFICANCE 



119 



In the general case as outlined above the degrees of freedom correspond- 
ing to the sums of squares of equation (1) are 

Total Within Between 

(nk - 1) k(n - 1) + (k - 1) 

3. Setting up the Analysis of Variance. For the practical example 
with two varieties we can now set up an analysis of variance as follows: 



Source of 
Sum of Squares 


Sum of Squares 


Degrees of 
Freedom 


Mean Square 
or Variance 


Within varieties 


44.580 


4 


11.14 


Between varieties 


141 135 


1 


141 1 










Total 


185.715 


5 













As would be expected from the difference between the means of A and 
By the variance for between varieties is very high as compared to that 
for within varieties. Reference to Chapter IV on tests of significance 
with small samples will recall that the variance for within varieties is 
the variance which is converted into a standard error in order to test 
the significance of the difference between the means. This variance 
can be termed, therefore, the error variance and can be used as a measure 
of the significance of the variance for between varieties. 

4. Tests of Significance. In the typical analysis of variance we 
have an error variance with which we wish to compare one or more other 
variances. Strictly speaking, all these variances are estimates of the 
true value, and this is, of course, the reason why to obtain them we must 
divide the sums of squares by the degrees of freedom. In order to under- 
stand the test of significance it is necessary to consider in the first place 
the condition that would obtain on the average if the variance we are 
testing is subject to exactly the same source of variability as the error 
variance. Let the sum of squares for error be represented by Si and 
the sum of squares for the variance to be tested by $2. The correspond- 
ing degrees of freedom are n\ and n?, and the estimates of variance are : 

= Si = S2 

HI n% 

and let F = V2/vi. 

Suppose that t>2 represents the variance for between the varieties 
A and B as in the actual example above. If there is no real difference 
between A and B, the differences between the means that occur will be 



120 



THE ANALYSIS OF VARIANCE 



due to soil heterogeneity which is the sole cause contributing to the 
error variance. On the average, therefore, vi = t>2, or F *= 1. But if 
the experiment, still assuming that A is not different from $, is repeated 
a number of times, F will be subject to random fluctuations and will be 
distributed in some regular manner. Thus in any one experiment if 
F =B 2.6 we could judge the significance of this value if we could deter- 
mine the exact percentage of cases in which an F of 2.6 would occur as 
the result of random sampling fluctuations. The problem is therefore 
one of determining the distribution of F and tabulating the results in such 
a way that they can be used to determine probabilities. R. A. Fisher (3) 
has worked out the distribution of F and in tests of significance re- 
places it by z % log F. The distribution of z depends entirely on 
the degrees of freedom n\ and n^ from which the variances are estimated. 
Its use therefore does not involve any assumptions regarding the popu- 
lation and is equally applicable for large and small samples. Tables 
have been prepared giving the values of z at the 5% and the 1 % points 
for different values of n\ and n^ In comparing v\ and 02, if we find that 
z is equal to the value given at the 5% point, this means that the observed 
F value would occur owing to random sampling fluctuations in only 5% 
of the cases. 

Snedecor (11) has calculated tables of F for the 5% and 1% points, 
and this enables us to make a test of significance directly without looking 
up logarithms. Table 96 is a copy of Snedecor's table of F. 

5. Multiple Classification of Variates. In the simple example we 
have considered, the variates were classified according to variety only. 
They may, however, be classified in several ways, and it is only rarely 
that they are not classified in two or three ways. We shall consider 
two-fold classifications first. The general case may be represented as 
follows: 

Classes 
1 2 3 - n 



1 


xn 


xn 


Xlf-Xln 


2 


xn 


#22 


#23' ' 'X2n 


Groups 3 


xn 


xn 


#33" * 'Xzn 


k 


Xkl 





XkZ' ' 'Xkn 



in which, the variates are in k groups and n classes. The essential dif- 
ference 1>etween this arrangement and that illustrated under Section 2 



MULTIPLE CLASSIFICATION OF VARIATES 121 

above is that the variates in any one class have something in common 
in that they can be logically placed together and recognized as a definite 
unit. In field experiments the groups may be varieties and the classes 
blocks or replicates. In a chemical experiment the groups may repre- 
sent formulae and the classes different temperature or moisture condi- 
tions under which the formulae are tried. In medical or nutritional 
work the groups may be different foods and the classes different quanti- 
ties or times of feeding. 

The equations representing sums of squares and degrees of freedom 
for the twofold classification are as follows: 

Within Groups Between Between 

Total and Classes Groups Classes 

nt Jt n 

)2 + n ( *< -*? + k 2(. - *) 2 (2) 



where x q is the mean of a group and x c is the mean of a class. Note that 
in this case the sum of squares for within groups and classes is rather 
complex and in corresponding form to equation (1) should be written 
with a triple summation. The form used, however, is more convenient 
and expresses the idea successfully. It is customary in analyses of 
experiments to refer to the within sum of squares as that due to error 
as it gives rise to the variance with which the other estimates of variance 
can be compared. 

In order to picture a threefold classification, we can assume that in 
the previous example there are m classes and n subclasses. Graphically 
the arrangement will be : 



m 



1 2---n 1 2---n 1 2---n 1 2- 



The analysis of data of this type introduces a new factor in the sums of 
squares, in that we must consider the interactions of the three classes 
with one another. This is best studied, however, from actual examples, 
and the same applies to still more complex types of classifications. 



122 THE ANALYSIS OF VARIANCE 

6. Selecting a Valid Error. Significance is a relative and not an 
absolute term. Differences are found to be significant or insignificant 
in relation to the variability arising from a source which is arbitrarily 
selected according to the interpretation that is to be put on the result. 
To make these points clear let us assume that an experiment is being 
conducted involving chemical determinations. Two kinds of material 
are being tested; the method is to draw samples from each kind of 
material, and in the laboratory each sample is being tested in duplicate, 
obviously here we have two sources of error. The first arises from 
sampling the material, and the second from differences between the 
results for duplicate determinations arising purely from errors in the 
laboratory technique. These two sources of error are independent and 
therefore may be of the same magnitude or widely different. If 20 
samples are taken from each kind of material the analysis of variance 
will be of the following form : 

DF Variance 
Materials (A and B) 1 m 

Between A samples 19 ] QC a 

>oo 



Between B samples 19 j b 

Between duplicates 40 d 

Total 79 

For the purpose of this discussion it can be assumed that the variances 
a and b are of the same magnitude and can be considered together, say 
as variance s. Now we wish to test the significance of the difference 
between the two kinds of materials, and we will suppose that d is very 
small in comparison to s. It is not difficult to see that the variance m 
is contributed to by the variability in the samples, or in other words 
that on the average if there is no difference between the two materials 
the variance m will be equal to the variance s. Since d is very small 
it is clear that to use it to test m is quite erroneous, as even when there 
is no difference between the materials the ratio of m to d will be quite 
large. What will the situation be, however, if d is much larger than s? 
With a little thought it will be plain that this would be a very unlikely 
situation as s is in itself contributed to by the factors that result in the 
variance d. Putting it another way, if there is no variation whatever 
due to sampling, s will on the average be equal to d. The question 
therefore has no point, and we must consider the only other possibility, 
and that is that d and s are of about equal magnitude. The inference, 
then, is that s results largely from the differences between the duplicates, 
and that the sampling error is in itself insignificant. The obvious 
course here is to use d in order to test w, and at the same time we take 



SELECTING A VALID ERROR 123 

advantage of the greater precision due to d being represented by a larger 
number of degrees of freedom than s. 

Another hypothetical experiment may be considered in which the 
situation is slightly different. Again two materials are being compared, 
but it can be assumed that the material is sufficiently homogeneous that 
the sampling error is negligible. There is a possibility of error in the 
laboratory technique and also there is a possibility of personal error in 
that no two operators can be expected to get exactly the same results. 
In making out the plan of the experiment it is decided that six different 
operators shall be used, all of whom perform exactly the same test on 
the same two materials. Also each operator makes his determination 
in triplicate in order that a measure may be obtained of the error in the 
technique. The analysis of variance for the results will be as follows: 

DF Variance 

Materials 1 m 

Operators 5 o 

Error due to operators 5 e 

Error of determination 24 d 



Total 35 

The variance e now requires some consideration in order to note its rela- 
tion to the significance of the results. If we set up the mean results for 
each operator in a table it will be of the following form : 

Operators 
123456 

A 
Materials 

B 



where 01, for example, represents the mean of three determinations made 
by operator 1 on material A. 

Now the variance e results from differences between such values as 
(ai bi) and (02 62). There being 6 of these values, there are 
5 degrees of freedom available for estimating the variance. If each 
operator gets the same result for the difference between A and Bj the 
variance e will be zero ; but if the operators get widely varying differ- 
ences the variance e will be very high. Suppose now that the experiment 
is presumed to be a sample of a large population of operators making 
similar determinations on these same two materials, then the variance 
m, which represents the difference between the two materials, will be 



d\ &2 flg 04 

61 &2 &a &4 


65 b& 



124 THE ANALYSIS OF VARIANCE 

contributed to by the factors that produce e\ and hence, if there is no 
difference between the materials, m will be equal to e. In sampling 
such a population, therefore, and testing the significance of the results, 
it will be necessary to use e as an error variance to test the significance 
of m. This fact may be more obvious if we consider the disastrous 
results of not using the variance e as a measure of error. The variance d 
may be quite low owing to extreme care in the standardization of the 
technique as applied to any one individual operator, and we shall assume 
that it is much lower than e. Using d as an error we find that, although 
m is very little greater than e, it is very significant if compared with d. 
The results are used therefore to prove that, for example, A gives a 
much larger result than 5, and on this basis the two materials are util- 
ized in some industry for manufacturing purposes. The manufacturers, 
however, in utilizing the material may have to employ a large number of 
operators; and hence the error that was neglected in the laboratory 
creeps in and it turns out in actual practice that the two materials give 
the same result, and the so-called carefully controlled experiment of the 
laboratory is discredited. This mistake would have been avoided if the 
investigator had carefully considered the exact nature of the population 
that was being sampled and made his test of significance accordingly. 
Of course it might happen that only one operator was used in the experi- 
ment, in which case the reader will recall the discussion of Chapter V on 
the scope of experiments and will realize that this would be another 
example of an experiment so planned that it did not have sufficient 
scope to answer the questions that it was supposed to answer. 

A point that may now be raised is this. If the error resulting from 
the determinations made by individual operators is not to be used to 
test the significance of the difference between the materials, what benefit 
is to be derived from making the determinations in triplicate and includ- 
ing the variance d in the analysis? The answer to this is that if there 
is an appreciable error in the determinations, the variance e will be con- 
tributed to by this source of variation, and hence, if there is no variation 
due to the operators, on the average e will be equal to d. The variance 
d, therefore, enables us to apply a test of significance to e; and, further- 
more, if d is appreciable, it reduces the precision of the experiment by 
making its contribution to e. In the latter case, improvement in the 
technique of the determination may result in a considerable improve- 
ment in the precision of the experiment. 

A variance such as e in the hypothetical example given above is 
usually referred to as an interaction variance. It gets this name because 
if it represents a fairly large effect it may be taken as an indication 
of an interaction between the two factors that are concerned. In con- 
sidering operators and materials, for example, we may conclude if e is 



SELECTING A VALID ERROR 



125 



very large that the materials respond quite differently in the hands of 
different operators. As a matter of fact, if we are willing to use more 
than one word to describe such an effect, it might be more appropri- 
ate to speak of an interaction as a differential response. Let us assume 
that, in general, material B gives a higher result in the determinations 
that are being made than material A. This may appear more rea- 
sonable if we assume that A and B are not different in quality but 
in quantity, in which case it is customary to refer to A and B as repre- 
senting two different levels of one of the interacting factors. The 
more appropriate symbolism then would be to represent A and B by 
such symbols as X \ and X%, the same letter indicating that there 
are no qualitative differences between the two, and the subscripts 
indicating that this factor is at two different levels. Now if Xz gives a 
higher value in the determinations than Xi 9 this is plainly a case of 
response to quantity, and if there were several levels of X instead of only 
two the result would recall the phenomena observed in the study of 
regression. It is now easy to visualize what is meant by a differential 
response. Some of the operators may be able to obtain the maximum 
response whereas others may obtain a much smaller response. In 
certain instances it may easily turn out that with some operators the 
response will be positive and with other operators it will be negative. 
This type of effect would be likely to result in a very large interaction 
variance. 

The meaning of interactions will be discussed in further detail in the 
consideration of actual examples. For the present it will suffice for the 
student to have a clear conception of the idea of differential responses, 
and to realize that frequently an interaction variance is in reality a true 
error variance and therefore must be used to test the significance of the 
results of the experiment. 

Example 29. Simple Classification of Variates. Table 29 given the yields of 
four plots each of three varieties of wheat. We shall use the analysis of variance to 
determine the significance of the differences between the varieties. 

TABLE 29 
YIELDS OF 4 PLOTS EACH OF 3 VARIETIES 



Plot Yields 


Totals 


A 
B 
C 


29.2 
32.7 
18.7 


36.4 
39.3 
23.1 


22.4 
28.6 
21.3 


27.6 
29.3 
19.6 


115 6 
129.9 
82.7 


Total 


328.2 



126 THE ANALYSIS OF VARIANCE 

The first step is to decide on the form of the analysis and to allocate the degrees of 
freedom to each component according to the scheme decided upon. In this case we 
are concerned merely with comparing the variety variance with a variance for error, 
and the most logical error variance is one arising from within the varieties. The 
form of the analysis is therefore 

Sum of Squares DF 

Between varieties 2 

Within varieties (error) 9 

Total 11 

The second step is to calculate the sums of squares. The best plan is first to obtain 
the total sum of squares. A formula has been given above, but this is not the best 
formula for actual calculation. It is much better to make use of the identity 

nk nk 7 7 J 

2(* - f) 2 = 2(* 2 ) - -~ (3) 

i i nk 

nk 

where T x is the total of all the values of x or 2 ($) 

i 

Therefore we merely square and summate the actual values and subtract from 
this sum the square of the grand total divided by the number of variates. The 
figures are 

Total sum of squares 9452.50 - 8976.27 = 476.23 

The calculation of the sum of squares for between varieties is carried out with the 
assistance of a similar identity 

% 2(*<) m2 

i n nk 

where jT represents the total for a variety. The formula consists therefore of squar- 
ing and summating the totals, dividing by the number of variates entering into each 
total, and then subtracting the same term as for the total sum of squares. The 
figures are 

Between varieties = 9269.16 - 8976.27 = 292.89 

To determine the sum of squares for within varieties we can perform a separate 
calculation for each variety: 

Within 4 - 3441.12 - 115.6 2 /4 100.28 
" B 4290.23 - 129.9 2 /4 71.73 
" 0=1721.15- 82.7 2 /4= 11.33 

Total within = 183.34 

Actually it was not necessary to calculate the last sum of squares as we could have 
obtained it by subtracting the sum of squares for between varieties from the total. 
Thus: 

Total Between Within 
476.23 -292.89 - 183.34 

However, when possible it furnishes a very easy check on the calculations to obtain 
the error sum of squares directly and indirectly. 



SELECTING A VALID ERROR 



127 



The third step is to set up the analysis of variance and make the tests of sig- 
nificance. This is performed in Table 30. 

TABLE 30 
ANALYSIS OF VARIANCE 





Sum of 
Squares 


Degrees 
Freedom 


Variance 


F 


Jlog.F 


Between varieties 
Error 


292 89 
183 34 


2 
9 


146 4 
20 37 


7.19 


0.9863 














Total 


476 23 


11 





















In Fisher's tables we look up the 5% point of z for n\ = 2 and nz ~ 9. The value 
is 0.7242, so that the variety differences here are quite significant. Using Snedecor's 
tables of F (Table 96) we find that the 5% point for F is 4.26, and we of course reach 
exactly the same conclusion. 

Example 30. Twofold Classification of Variates. In a swine-feeding experiment 
Dunlop (2) obtained the results given in Table 31. The three rations, A, B, and C 
differed in the substances providing the vitamins. The animals were in 4 groups of 
3 each, the grouping being on the basis of litter and initial weight. For our purpose 
we shall assume that the grouping is merely a matter of replication. 

TABLE 31 
GAINS IN WEIGHT OF SWINE FED ON RATIONS A, B, C 

I II III IV Totals 



Ration 



29.5 
The form of the analysis is 



A 


7.0 


16 


10.5 


13.5 


47 




B 


14 


15.5 


15.0 


21.0 


65.5 




C 


8 5 


16.5 


9 5 


13 5 


48 





48.0 



35 



48.0 



160 5 



Sum of Squares DF 

Rations 2 

Groups 3 

Error 6 



Total 

Calculating the sums of squares we have 



11 



2316.75 - (160.5) 2 /12 - 2316.75 - 2146.6875 - 170.0625 

- 54.1250 

+ 48 . 0*)/3 - 2146 . 6875 - 87 . 7292 

remainder - 28.2083 



Total 

Rations - (47. 0* + 66.5 1 + 48.0)/4 - 2146.6875 

Groups - (29.5*4- 

Error 



128 THE ANALYSIS OF VARIANCE 

This givee us an analysis of variance as follows: 





Sum of 
Squares 


Degrees 
Freedom 


Variance 


F 


5% Point 


Rations . . . 
Groups 
Error 


54.1250 
87.7292 
28.2083 


2 
3 
6 


27.06 
29.24 
4.701 


5,76 
6.22 


5.14 
4.76 














Total... 


170.0625 


11 









The variance for rations is just significant. The meaning of the significance of 
the variance for groups depends on the manner in which the classification into groups 
has been made. We have assumed here that the groups are merely replications, in 
which case the error variance is a result of variations within groups not due to the 
rations. It is therefore valid to consider this variance as an error variance with 
which the others can be compared. The group variance, since it results from the 
plan of the experiment, is an expression of error control. If the arrangement had 
been other than in groups we would have had a simple classification into within and 
between rations. The variance for within rations would have been much larger than 
it is according to the present arrangement, and consequently the experiment would 
have been less precise. 

Example 31. Selecting a Valid Error. A series of 5 wheat varieties were grown 
at 4 stations and baking tests made on the flour. A sample of each variety was taken 
from each station and milled into flour. Two loaves were baked from each sample. 
The error of determination was given, therefore, by the differences between the loaf 
volumes of the duplicate loaves. These data were supplied by courtesy of the 
Associate Committee on Grain Research of the National Research Council dfcCanada. 

TABLE 32 

DUPLICATE LOAF VOLUMES FOB 5 VAEIETIES OF WHEAT GROWN AT 4 STATIONS 
(Loaf volumes in cc. 500)/10 



Stations 
II III 



IV 



Totals 



Varieties 



Totals 



1 
2 
3 
4 
5 



7.5 4.5 


15.5 14.0 


16.5 14.5 


19.0 18.6 


110.0 


12.5 13.2 


20.0 18.6 


15.0 14.0 


23.8 24.4 


141.4 


7.0 1.0 


10.0 8.0 


15.5 14.0 


17.8 18.5 


91.8 


1.5 2.0 


13.0 15.0 


8.5 9.0 


14.8 16.6 


80.4 


28.0 29.0 


19.5 16.0 


10.5 12.0 


22.0 24.8 


161.8 



106.2 



149.5 



129.5 



200.3 



585.5 



On examining the form that the analysis of variance will take, we note first that 
we must have a station variance represented by 3 degrees of freedom, and a variety 
variance represented by 4 degrees of freedom. There must also be an interaction 



SELECTING A VALID ERROR 



129 



effect which may be regarded as the differential response of the varieties at the 
different stations. The rule for finding the degrees of freedom for an interaction is to 
multiply the degrees of freedom for the interacting factors. The interaction variance 
must therefore be represented by 3 X 4 12 degrees of freedom. There is a total 
of 40 -determinations, so that there is a total of 39 degrees of freedom. The remaining 
20 degrees of freedom must represent the error of duplicate determinations, and we 
have a check on this because there are 20 pairs of loaves and since each pair gives us 1 
degree of freedom there must be 20 in all. The final form of the analysis is: 

Variance DF 

Stations 3 

Varieties 4 

Interaction 12 

Error 20 



Total. 



To obtain the sums of squares another table as given below is required. This 
table gives the values of (x y) and (x + y), where x and y are taken to represent 
the paired values. 



1 
2 

fe-ir) 3 
4 
5 



II 



III IV 



3.0 


1.5 


2,0 


0.4 


0.7 


1.5 


1.0 


0.6 


6.0 


2.0 


1.5 


0.7 


0.5 


2.0 


0.5 


1.8 


1.0 


3.5 


1.5 


2.8 



fe+y) 



Totals 



1 
2 
3 
4 
5 



II 



III 



IV 



106.2 149.5 129.5 200.3 



Totals 



12.0 


29.5 


31.0 


37.6 


110.1 


25.7 


38.5 


29.0 


48.2 


141.4 


8.0 


18.0 


29.5 


36.3 


91.8 


3.5 


28.0 


17.5 


31.4 


80.4 


57.0 


35.5 


22.5 


46.8 


161.8 



585.5 



The first half of this table may be used fo> calculating the error sum of squares. A 
general rule for the sum of squares for differences within paired values is to use the 
identity 

Total minus between pairs |S(a; y) 2 



The two expressions on the left are SGc 2 ) - Tf/N and S(x -f yf/2 - Tf/N* On 
subtracting and simplifying we obtain \ S(a? - y)\ The calculations give 

Within pairs (error) - (93.33) - 46.66 



16V 



TJtiJU ANALYSIS U* VAJtUAAIUJii 



From the second half of the calculation table we determine 

20566.13 585. 5 



Between pairs 



2 40 

10283.065 - 8570.256 - 1712.81 



Diauons 
Varieties 


10 
73186.61 


- oo/u.^oo = *oi.oo 

ft^*7fl O^ift J'7ft (Y7 


8 




Interaction 


= Remainder 


- 653.09 



This procedure gives us a general rule for the calculation of interaction sums of 
squares. In the table considered we find the total and subtract the sum of squares 
for the two interacting factors. The remainder is the interaction. 
The analysis of variance is as follows 





Sum of 
Squares 


DF 


Variance 


Stations 


481 . 65 


3 


160 5 


Varieties 


578.07 


4 


144.5 


Interaction 


653.09 


12 


54 42 


Error 


46 66 


20 


2 333 










Total 


1759.47 















We now have to decide whether we should use the variance from the duplicate 
loaf volumes or the interaction variance to test the significance of the differences 
between stations and varieties. If the purpose of the experiment is to determine 
which of the varieties will give the highest loaf volume over the whole area that the 
stations sample, it will be necessary to use the interaction variance because in this 
light the stations are merely replications of the experiment. The error from dupli- 
cate loaf volumes will give an indication merely of the accuracy of the laboratory 
technique. If it is large it will reduce the significance of the differences, because it 
raises the value of the interaction variance. 

On comparing the variety variance with the interaction variance we get an F 
value of 2.66; and since the 5% point is 3.26, we must conclude that, considering the 
whole ares being sampled, the differences in loaf volume are not significant. In other 
words, the variation in the order of the mean loaf volumes of the varieties, from 
station to station, is so great that the differences between the means for the whole 
area may easily be accounted for by this variation. 

The interaction variance is very much higher than that arising from differences 
between duplicate loaf volumes. This means that the laboratory error is not an 
appreciable factor affecting the precision of the results in this experiment. 

Since variety tests are conducted in replicated plots at each station, it follows 
that if loaf volume determinations had been made on each plot another measure of 
error could have been obtained. This error would have measured the variation due 
to soil heterogeneity; and, if the variety variance for the whole area was significant 
when compared to the pooled error due to soil heterogeneity, this would indicate that 
in general at each station the differences between the means of the varieties were 



SELECTING A VALID ERROR 



131 



greater than could be accounted for by such sampling variation. This would not, 
however, alter our conclusion based on the test using the interaction as an error. 

Example 32. Threefold Classification of Variates. In testing out a machine 
for molding the dough in experimental baking, Geddes, et al. (5), used 3 adjustments 
of the machine, designated A, B, and C, and tried them out on a series of 5 flours 
baked according to 2 formulae. The loaf volume data are given in Table 33. 



TABLE 33 

LOAF VOLUME RESULTS IN A TEST OF A MACHINE FOR MOLDING THE DOUGH 
(Loaf volume in cc. - 500) /10 



Formula 


Machine 
Setting 


Flours 


Totals 


1 


2 


3 


4 


5 




A 


9.4 


2.6 


12.3 


4.6 


13.5 


42.4 


Simple 


B 


9 6 


3.1 


13.0 


4.3 


13.8 


43.8 




C 


9.6 


2.7 


12.4 


1.8 


13.0 


39.5 




Flour 
















subtotals 


28.6 


8.4 


37.7 


10.7 


40.3 


125.7 




A 


13.7 


21.6 


19 4 


13.5 


24.5 


92.7 


Bromate 


B 


12.7 


22.6 


20 6 


10.4 


24.3 


90.6 




C 


12.6 


21.8 


20.9 


6.8 


23.2 


85.3 




Flour 
















subtotals 


39.0 


66.0 


60.9 


30.7 


72.0 


268.6 




Flour 
















totals 


67.6 


74.4 


98.6 


41.4 




394 3 














112.3 





On working out the form of the analysis we find that there is an additional com- 
plication here as compared to those that have been worked out previously. The 
6 rows in Table 33 represent 2 classifications, but for the present we shall consider 
them as 6 classes giving us a simple twofold classification. The form of the analysis 
is then: 

Flours 4 DF 

Classes 5 DF 

Interaction (a) 20 DF 



Total 29 DF 



But the 5 degrees of freedom for classes must be split up into: 

Machine settings ABC 2 DF 

Formulae SB 1 DF 

Interaction ABC X SB 2 DF 



132 



THE ANALYSIS OF VARIANCE 



Hence interaction (a) in the first analysis is an interaction of the above three factors 
with the flours. Realizing this, we can then write out the form of the analysis in full: 

Flours (1 ... 5) 4 DP 

Machine settings (ABC) 2 DF 

Formulae (SB) 1 DF 

Interaction (ABC X SB) 2 DF 

(1... 5 X ABC) SDF 

" (1 ... 5 X SB) 4DF 

" (I ...5 X ABC X SB)... SDF 

Total 29 DF 

The last interaction is known as a triple interaction. In this case it represents the 
degree to which the interaction of (ABC X SB) is different for the different flours. 
If the interaction (ABC X SB) is the same for each flour, the triple interaction will 
be zero. 

To determine the sums of squares for the components set out above it is necessary 
to set up 3 calculation tables as below: 



<*r t 


Flours 




Machine 
Settings 


1 


2 


3 


4 


5 


Totals 


A 


23.1 


24.2 


31.7 


18.1 


38.0 


135.1 


B 


22.3 


25.7 


33.6 


14.7 


38.1 


134.4 


C 


22.2 


24.5 


33.3 


8.6 


36.2 


124.8 


Totals... 


67.6 


74.4 


98.6 


41.4 


112.3 


394.3 





Flours 




Formulae 












Totals 




1 


2 


3 


4 


5 




S 


28.6 


8.4 


37.7 


10.7 


40.3 


125.7 


B 


39.0 


66.0 


60.9 


30.7 


72.0 


268.6 


S + B 


67.6 


74.4 


98.6 


41.4 


112.3 


394.3 


S -B 


10.4 


57.6 


23.2 


20.0 


31.7 





Machine Settings 


Formulae 


A 


B 


C 


Totals 


S 


42.4 


43.8 


39.5 


125.7 


B 


92.7 


90.6 


85.3 


268.6 


S + B 


135.1 


134.4 


124.8 


394.3 


S-B 


50.3 


46.8 


45.8 


142.9 



SELECTING A VALID ERROR 



133 



The calculations are 1 
Total 

Flours (1- -5) 
Settings (ABC) 
Formulae SB 



- 6618.43 - 



394.3 2 
30 



- 6618.43 - 5182.42 



34,152.33/6 - 5182.42 
51,890.41/10-5182.42 - 
(268.6 - 125.7) 2 /30 



- 1436.01 
509.64 



680.68 



Interaction (ABCXSB) m S(S--B) 2 /10 - 680.68 - 6817.97/10 - 680.68 - 1.12 

Interaction (1 - -5) X (ABC) 

Total for table - 11,436.57/2 - 5182.42 - 535.86 

Flours (I--- 5) - 509.64 

Settings (ABC) 6.62 

Remainder (1 -5) X (ABC) - 19.60 

Interaction (1 -5)X (SB) - S(S - #) 2 /6 - 680.68 - 5369.05/6 - 680.68 - 214.16 
Interaction (1 - -5 X ABC X SB) remainder 4.19 

The analysis of variance when set up in detail is as follows: 





Suras of 
Squares 


DF 


Variance 


F 


5% 
Point 


Flours (1 5) 


509 64 


4 


127 4 


243 1 


3 84 


Formulae (SB) 


680.68 


1 


680.7 


1299.0 


5.32 


Interaction (1 -5 X SB) 


214.16 


4 


63.54 


102.2 


3.84 


Settings (ABC) 


6 62 


2 


3.31 


6.31 


4.46 


Interaction (ABC X SB) 


1.12 


2 


0.560 


1.07 


4.46 


" (1"-5X ABC) 


19 60 


8 


2.450 


4.68 


3.44 


(1---5X ABCX SB) 


4.19 


8 


0.524 






Total 


1436.01 


29 





















It is of interest to make a detailed study of Example 32 from the 
standpoint of the selection of a valid error. We note first that the 
determinations were not made in duplicate so that we have no real 
measure of the error in the technique; and, if such an error is the one 
that should be used throughout for tests of significance, we shall have 
to select one of the other variances that gives us a close approximation 
of what the error of duplicate loaf volumes would be. In the second 
place it must be remembered that the primary object of the experiment 
is to study the differences in the loaf volumes due to the different settings 
of the machine and the differential responses due to these same settings. 
For this reason the analysis of variance has been separated into two 

1 Note the method used to calculate interactions for a series of paired values. 
This will be explained in more detail in the next example. 



134 THE ANALYSIS OF VARIANCE 

portions. The three effects in the first group are of no particular in- 
terest, as previous experience would have enabled the cereal chemists 
to predict that just such results would be obtained. The separation 
of these three effects into one group is not a result of the data obtained in 
the experiment, but was preconceived, and it was decided before the 
experiment was operated that this would be done. 

Considering the variance due to the settings, the first question to be 
asked is whether or not it should be tested against a variance representing 
purely laboratory error or against the interaction of the settings with 
the flours. The answer follows from the fact that we are concerned 
not so much with the interaction of the settings with the flours as with 
attempting to find out the best single setting of the machine for all 
purposes; and therefore we do not anticipate that, in differentiating a 
set of flours, all the settings that have been tried here will be used. 
Actually our measure of significance in this experiment must be based 
on the usual experimental error of the laboratory, because, if the machine 
settings cause differences significantly greater than those resulting from 
experimental error, it is obvious that before the machine is used for 
general purposes the most desirable setting must be worked out. In 
other words we ought to see to it that the machine does not introduce a 
greater error into the determinations than already exists as the result of 
the ordinary procedures of the laboratory. 

On this basis it follows that the triple interaction is the most logical 
error to use, as it is the least likely to represent a significant effect and 
is not likely to be lower than the error due to differences between dupli- 
cate loaf volumes. The latter statement is the same as saying that, if 
there is no actual triple interaction effect, the variance will be equal to 
the error that would have resulted from using duplicate determinations. 

The F values with their 5% points are given in the analysis, and with 
their aid the results may be summarized very quickly. The flour and 
formula differences as well as the interaction between them are very 
large in comparison to the experimental error and may be dismissed 
with that statement. The primary interest in the experiment is in the 
settings of the machine and the interaction of the settings with the other 
factors. The settings are significant in relation to experimental error, 
and glancing at the totals we note that this must be due to the fact that 
the C setting gives a somewhat lower loaf volume than A or B. The 
interaction of ABC with the formulae (SB) is not significant, indicating 
that the differences between thfc settings are reasonably consistent for 
both methods of baking. The interaction of the flours with the settings 
is significant, and we can conclude that the results with the flours are 
to a certain extent changed by the machine settings. From an inspec- 



METHODS OF CALCULATING SUMS OF SQUARES 135 

tion of the results this would seem to be due to flour 4, as for this flour 
the B and C settings depress the loaf volume to a greater extent than 
for the others. 

7. Summary of Methods of Calculating Sums of Squares. After 
the form of analysis has been worked out, the greatest difficulty that 
confronts the student of the methods of this chapter is the calculation 
0f the sums of squares. Most of the methods have been dealt with in 
the above examples, but it would seem to be desirable to summarize 
them under one heading. 

(a) Total for a set of n single variates. xi, #2, # 



Tl 

2(* - )* - Z(a) - 
i i n 



We square each value and summate, then subtract the square of the 
total divided by the number of variates. 

(6) For a set of k groups when each group is made up of n variates. 
It there are k groups we can represent the totals for the groups as T\ t 
T%, 2Y T*; and the means for the groups by 1, fe, x - - *. 






i n kn 

We square each total, summate, and then divide by the number of 
variates entering into each total. From this we subtract the square of 
the grand total divided by the number of variates. 

(c) For a set of k groups when the number of variates is not the 
same for each group. If we represent a particular series with the corre- 
sponding number of variates in each group as follows : 



We calcfulate: 



Group totals Ti t T 2 , T 3 , T* 

Numbers a, 6, c, d 



Tf , If Tl Tl T* 

I _ j_ _j ... 



abed (a + 6 + c + d) 

In this case we square each total and divide by the number entering into 
it. The quotients are summated, and from this sum we subtract the 
square of the grand total divided by the total number of variates. 

(d) For within and between pairs. If a set of paired values are 
represented as follows: 



136 



THE ANALYSIS OF VARIANCE 

1 2 3 4---n Totals 



XI 

Vi 


X2 
1/2 


Xz 
2/3 


X4" 'X n 
1/4- "2/n 


T x 

T v 



The sum of squares for between pairs is: 

Z(s + y) 2 

2 
And for within pairs it is: 



2n 



If each x and y value represents k variates we have : 

%( x + y)* T 2 

Between = -- 



Within 



(e) For two groups only. The totals for the groups may be T s and 
as above in (d) . The sum of squares is : 



where N is the total number of variates. 

(/) Simple interaction in a 2 X n table. The table is as in (d), in 
which each value of x and y represents k variates. The interaction 
(1, 2, 3, n) X (x2/) is given by: 



S(z - 



2V) 2 



2kn 



(g) Simple interaction for a 2 X 2 table. The following is a 2 X 2 
table in which each value of x is a total for k variates. 

A B 




METHODS OF CALCULATING SUMS OF SQUARES 137 

The interaction (AB X I II) is given by 



4k 

(h) Simple interaction for a k X n fold table. A table of this type is 
illustrated in Section 5 above, and equation (2) shows how the sums of 
squares and degrees of freedom are broken up. The sum of squares 
for within groups and classes is the same as for the interaction and can 
be calculated by subtracting the two terms on the right from the total. 
The procedure therefore is as follows: 

Total - S(z 2 ) - Tl/kn 

For n classes - Z(T?)/* - Tl/kn 
For k groups = S(Tj)/n - Tl/kn 

Interaction = Difference 

(i) Triple interaction. In more complex analyses it is sometimes 
necessary to calculate triple interactions. We shall illustrate the method 
for the simple case of 2 X 2 tables: 1 






The interaction to be calculated is (XYZ X I II X AB). Assume each 
value to be made up of k variates; then for each of the above tables we 
have : 

For X (I II X AB) = (xi + * 3 - x 2 - x*) 2 /4k 



Y (I II X AB) = 
Z (I II X AB} 



+ x 3 - x 2 - 
+ z 3 ~ x 2 - 



Summating these gives us the sum of the interactions of (I II X AB) 9 
taking each X, Y y and Z group separately. Next we find (I II X AB) 
for X, Y t and Z combined, having set up another 2X2 table. 

1 If the three factors have only two levels the triple interaction is also represented 
by only one degree of freedom and may therefore be calculated from a difference 
between two correctly chosen totals. The method of building up these totals will 
be clear after a study of the methods of the following chapter. 



138 



THE ANALYSIS OF VARIANCE 

X + Y + Z 
A B 



II 



XI 



For X, Y, and Z, (I II X AB) = (xi + x 3 - x* - x*) 2 /12k, which, 
when subtracted from the sum obtained for the three tables above, 
gives the triple interaction (XYZ X I II X AB). 

According to the same principle, triple interactions may be calculated 
for any three factors. Note that there are three different ways in which 
the calculations may be carried out, as repeated calculations of any one 
of the three simple interactions will finally give the triple interaction. 
Always examine the three possible methods and decide which one will 
require the least amount of labor. 

8. Exercises. 

1. Table 34 taken from data by Crampton and Hopkins (1) gives the gains in 
weight of pigs in a comparative feeding trial. The 5 lots of pigs represent 5 different 
treatments, and there were 10 pigs in each lot. Make an analysis of variance for 
the data, and test the significance of the treatment differences. 

TABLE 34 
GAINS OF PIGS IN A COMPARATIVE FEEDING TRIAL 



Replicate 


Lot I 


Lot II 


Lot III 


Lot IV 


LotV 


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 



The error variance in this experiment works out to 84S.6. 

2. In a study of hog prices in Iowa, Schultz and Black (9) have given prices by 
months, years, and districts. The districts are obtained by dividing the state into 4. 
A portion of the data is given in Table 35. After completing the analysis of variance 
for these data, devise graphical means of illustrating the interaction of months with 



EXERCISES 



139 



years. It is not necessary in this exercise to make tests of significance of the results, 
as it is being used here merely to show how the technique of the analysis of variance 
can be used to separate out the various effects in a set of data. 

Sum of squares for months X years =* 83.3418. 

3. In agronomic trials of varieties of cereal crops it is desirable to conduct the 
trials at various points in the area under consideration and to carry them on for a 
period of 2 or more years. Immer, et al. (8), have given data on barley yields at 
several stations in Minnesota over a period of 2 years. Table 36 gives the yields at 3 
of the stations for 2 years for 6 varieties. Analyze the results. 

Note that the blocks are numbered 1, 2, and 3, but this does not mean that block 
1 at University Farm has any relation to block 1 at Waseca or any other station. 
Consequently the sum of squares and degrees of freedom for blocks are worked out at 
each station and lumped together in the final analysis. A common error that 
beginners make in sorting out the degrees of freedom for an experiment of this kind 
is to regard the blocks as a factor occurring at three levels and thus they have such 
expressions in their analysis as these: 

Blocks X Stations 
" X Years 
" X Stations X Years 
etc. 

These expressions obviously have no meaning as the block numbers do not represent 
definite levels that are uniform at all stations. The correct procedure is therefore 
to calculate the block sum of squares for each experiment and add all these sums of 
squares together in order to show them in the final analysis. 

The following values for the sums of squares will assist in checking the calculations. 

Total .............................. 11,504.61 

Varieties ........................... 1J5MM 

Varieties X Stations X Years ......... 



TABLE 35 
HOG PRICES PAID TO PRODUCERS IN IOWA 1928-29 TO 1930-31 





1928-29 


1929-30 


1930-31 




Districts 


Districts 


Districts 




A 


B 


C 


D 


A 


B 


C 


D 


A 


B 


C 


D 


Oct. 


9.48 


9.46 


9.47 


9.56 


8.79 


8.98 


8.90 


9.15 


8.84 


8.83 


8.80 


8.86 


Nov. 


8.41 


8.13 


8.43 


8.44 


8.32 


8.30 


8.34 


8.53 


8.04 


8.23 


8.17 


8.45 


Dec. 


7.91 


7.85 


7.79 


7.96 


8.58 


8.50 


8.44 


8.54 


7.39 


7.31 


7.32 


7.34 


Jan. 


8.14 


8.28 


8.12 


8.24 


8.79 


8.69 


8.71 


9.02 


7.06 


7.10 


7.11 


7.17 


Feb. 


9.14 


9.03 


9,00 


9.10 


9.84 


9.59 


9.63 


9.82 


6.44 


6.62 


6.63 


6.65 


Mar. 


10.57 


10.61 


10.44 


10.49 


9.75 


9.81 


9.78 


10.10 


6.80 


6.87 


6.84 


6.88 


Apr. 


10.65 


10.53 


10.56 


10.67 


9.20 


9.22 


9.26 


9.42 


6.78 


6.86 


6.92 


6.92 


May 


10.36 


10.20 


10.07 


10.14 


9.12 


9.10 


9.06 


9.26 


6.03 


6.10 


6.06 


6.30 


June 


9.95 


9.86 


9.97 


10.03 


9.16 


9.17 


9.14 


9.33 


5.40 


5.39 


5.57 


5.60 


July 


10.64 


10.47 


10.70 


10.54 


8.24 


8.11 


8.31 


8.60 


6.00 


5.85 


6.16 


6.24 


Aug. 


10.35 


10.34 


10.34 


10.56 


8.22 


8.52 


8.68 


8.75 


5.91 


5.66 


6.24 


6.36 


Sept. 


9.37 


9.46 


9.40 


9.51 


9.52 


9.52 


9.64 


9.73 


6.07 


5.20 


5.26 


5.38 



140 



THE ANALYSIS OF VARIANCE 



TABLE 36 

YIELDS IN BUSHELS PER ACRE OF 6 VARIETIES OF BARLEY GROWN AT 3 STATIONS 

IN EACH OF 2 YEARS 



Block 
No. 


Man- 
churia 


Gla- 
bron 


Svan- 
sota 


Velvet 


Trebi 


Peat- 
land 


Station 


Year 


1 


29.2 


44.6 


33.9 


36.7 


41.2 


38.5 






2 


25.0 


39.1 


39.4 


41.0 


31.9 


29.6 


University Farm 


1931 


3 


26.8 


45.5 


32.1 


42.0 


36.6 


30.2 






1 


19.7 


28.6 


20.1 


20.3 


19.3 


22.3 






2 


31.4 


38.3 


30.8 


27.5 


22.4 


30.8 


University Farm 


1932 


3 


29.6 


43 5 


31.4 


32.6 


45.5 


31.1 






1 


47.5 


55.4 


44.5 


56.9 


63.9 


41.2 






2 


52.2 


53.4 


46.0 


40.6 


63.8 


51.5 


Waseca 


1931 


3 


46.9 


56.8 


51.5 


53.2 


63.8 


53.0 






1 


40.8 


44.4 


41.0 


44.6 


53.5 


39.8 






2 


29.4 


34.9 


41.1 


41.4 


44.2 


39.2 


Waseca 


1932 


3 


30.2 


33.9 


33.4 


26.2 


50.0 


29.1 






1 


24.0 


27.5 


26.5 


27.2 


42.1 


24.7 






2 


24.7 


25.5 


21.5 


28.0 


42.5 


29.5 


Morris 


1931 


3 


33.6 


33.3 


29.3 


23.2 


46.7 


35.4 






1 


29.6 


36 6 


27.1 


35.9 


40.0 


35.7 






2 


34.1 


34.3 


35.7 


33.9 


46.9 


41.9 


Morris 


1932 


3 


39.4 


34.5 


42.3 


46.7 


53.0 


52.0 







4. Find the 5% points of F for the following values of n\ and n 2 : 



ni 

3 

6 

4 

12 

7 

11 

16 

18 

17 

36 

28 



51 
43 
92 

195 
36 
64 
39 

215 
19 
28 

154 
42 



REFERENCES 141 



5, Prove: (1) That S(s*) - T%/n - S(a? - S) 2 . 
i i 

(2) That the interaction for a 2 X 2 table is given by (x\ + a* - a* - 

xt)*/kn. See Section 7(0). 

(3) That the sum of squares for the two subtotals T a and 7*6 is given 

by (T a - Ttf/N. See Section 7(e). 

(4) That in a series of pairs the sum of squares for within pairs is 
given by 1 2(x - y)*. See Section 7(d). 



REFERENCES 

1. E. W. CBA^FTON and J. W. HOPKINS. Jour. Nutrition, 8: 329-339, 1934. 

2. G. DUNLO*. Jour. Agr. Sci. t 25: 445-459, 1935. 

3. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter VII. 

4. R. A. FISHER. The Design of Experiments. Oliver and Boyd, London, 1937. 

Reading: Chapter X, Section 65. 

5. W. F. GBDDBS, et al. Can. /. Research, 4: 421-482, 1931. 

6. C. H. GOULDEN. Modern Methods of Field Experimentation. Sri. Agric., 11: 

681-701, 1931. 

7. C. H. GOULDEN. Application of the Variance Analysis to Experiments in Cereal 

Chemistry. Cereal Chem., 9: 239-260, 1932. 

8. F. R. IMMEB, et al. Jour. Am. Soc. Agr on., 26: 403-419, 1934. 

9. T. W. SCHULTZ and A. G. BLACK. Research Bull. 161, Iowa State Agric. Exp. 

Station, 1933. 

10. G. W. SNEDECOR. Calculation and Interpretation of the Analysis of Variance 

and Co variance. Collegiate Press, Inc., Ames, Iowa, 1934. Reading: Parts I, 
II, and III. 

11. G. W. SNEDECOR. Statistical Methods. Collegiate Press, Inc., Ames, Iowa, 

1937. Reading: Chapters X and XL 

12. L. H. C. TIPPBTT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter VI. 



CHAPTER XII 
THE FIELD PLOT TEST 

GENERAL PRINCIPLES AND STANDARD DESIGNS 

1. Soil Heterogeneity. The fact of soil heterogeneity as it affects 
the yields of crops has been commented on by various writers. In the 
agronomic test it is the chief source of error in comparing varieties, soil 
and fertilizer treatments, and factors of a similar type. If soil hetero- 
geneity was practically non-existent a single pair of plots would be suffi- 
cient to make a comparison of two varieties, but even then it is doubtful 
whether that condition would be highly desirable. By a sufficient 
expenditure we might render a piece of soil completely homogeneous, 
but by doing so we would partly defeat the purpose of the test which 
is to determine the behavior of varieties and treatments under a limited 
range of conditions. We would have selected one particular soil type 
for our experiment and therefore restricted the area to which our results 
would apply. The ideal agronomic test is one conducted on a piece of 
land in which the range in soil type, etc., is the same as that in the dis- 
trict to which the results are to be applied. Usually agronomic tests are 
on soil that is much less subject to variation than the surrounding dis- 
trict so that in general the results from them are considered as applicable 
over too wide an area. This is not to argue that more variable soils 
should be selected, for that might again defeat the purpose of the test 
by rendering the results insignificant, but rather to point out the limita- 
tions of the tests as ordinarily conducted and that the ideal cannot be 
reached by any method of increasing the uniformity of the soil. 

2. Replication. In order to obtain greater accuracy in field experi- 
ments, the most effective method is to increase the number of replica- 
tions. Increasing the plot size is also effective, but increasing replication 
is much more so. In previous pages it has been pointed out that the 
standard error of a mean is given by s/\/n, where s is the standard error 
of a single determination and n is the number of determinations averaged. 
It follows, therefore, that, in replicating field plots, the decrease in the 
standard error of the mean of one variety or treatment is proportional 
to the square root of the number of replications. This rule applies only if 
the variation due to the replicates themselves is removed from the error, 
but, as will be pointed out below, this follows naturally from the plan of 
the test and the use of the analysis of variance. 

142 



RANDOMIZATION 143 

A most important consideration in the use of replications is that they 
furnish an estimate of the error of the experiment, and this estimate can 
be obtained in no other way. The error of the experiment arises from 
the differences between plots of the same variety or treatment that are 
not due to the average differences between the replicates. From this it 
is clear that, if there is only one complete set of plots of all the varieties 
or treatments, there is no possibility of obtaining a measure of random 
soil variability that can be used as an error in tests of significance. In 
terms of the theory which has been emphasized repeatedly in the previous 
pages, the variance of the variety or treatment means is subject to test- 
ing on the hypothesis that it has arisen purely from random variations 
in the fertility of the field. Since the only way in which we can form a 
reliable estimate of these random variations is to replicate the experi- 
ment, it follows that without replication there is positively no method 
of making a test of the significance of the variety or treatment differ- 
ences. 

3. Randomization. As pointed out above, the estimate of error is 
taken from differences between plots that are treated alike. R. A. 
Fisher states that " an estimate of error so derived will only be valid 
for its purpose if we make sure that in the plot arrangement, pairs of 
plots treated alike are not nearer together, or further apart than, or in 
any other relevant way, distinguished from pairs of plots treated differ- 
ently." This point is obvious if we consider a simple replicated experi- 
ment containing, say, 4 varieties, that we shall designate as A, B> C, and 
D. Suppose, merely for purposes of argument, that the plots are square 
and the arrangement of the plots in the field is as follows: 

Replicate 1 A B C D 

Replicate 2 A B C D 

ReplicateS A B C D 

Replicate 4 A B C D 

The form of the analysis will be: 

DF Variance 

Replicates 3 r 

Varieties 3 v 

Error 9 e 



Total 15 



and now, if there are no variety differences it can be expected that on 
the average the variance v will be equal to the error e, and unless our 
experiment is designed to make this true it is unbalanced, or in the 



usual terminology it is subject to a bias. On this basis it is possible to 
picture the situation with respect to bias in this simple experiment, on 
varying the location of the plots with respect to distances between plots 
of the same variety and plots with different varieties. In the first place, 
suppose that the replicates are only 1 foot apart so that there is for ex- 
ample only a space of 1 foot between the plot of A in the first replicate 
and the plot of A in the second replicate. Then between the plots of 
different varieties there are 6-foot buffer plots of some other crop. This 
situation presents a very obvious bias in that the plots of different 
varieties are farther apart than plots of the same variety. The result is 
that, if there are no differences between the varieties, the variance v will 
on the average be larger than e. This very proposition was recognized 
by agronomists at an early stage in the development of field plot tests, 
and as a remedy for it suggestions were made as to the distribution of the 
plots in a systematic manner over the whole field. These suggestions, 
however, did not take into consideration the possibility of a bias in the 
opposite direction to that of the design outlined above. That such a 
bias is a distinct possibility has been shown by Tedin (10), in an exten- 
sive study of data from uniformity trials. A bias in the direction that 
tends to make the error too large, and the variety or treatment variance 
too small, is in effect just as disastrous as the opposite type of bias, as 
it means that, on the average, certain significant effects will be over- 
looked. 

A systematic type of distribution of the plots might be as follows: 

A B C D 

C D A B 

A B C D 

C D A B 

and it will be noted that the plots of the same variety are scattered 
widely over the field. This is the type of arrangement that is likely to 
result in an error that is too large, but, disregarding that point, there 
is another type of bias common to all systematic arrangements. This 
may be referred to as an intravarietal bias, in that comparisons between 
different pairs of varieties are not of equal precision. For example, in 
both of the systematic arrangements that we have outlined above, the 
varieties A and B occur on adjacent plots in every replication while 
the varieties A and D are on the average farther apart. This is a very 
undesirable feature of such experiments, for if a single error is used for 
the whole experiment it means that real differences between the varieties 
that are close together may be overlooked and other differences that 
actually do not exist may be judged significant. 



EBROR CONTROL 145 

From the above discussion it may appear to the reader that the 
field plot test is extremely complicated and difficult to set up in such a 
way that there is no bias. Actually, all these difficulties may be very 
easily overcome by the simple process of arranging the varieties at ran- 
dom in each replication. Thus, instead of either of the arrangements 
that have been outlined, we would make up one as follows, in which the 
positions of the varieties are determined entirely at random. 

D C A B 

C B A D 

B C D A 

A D B C 

Then, regardless of the size or shape of the plots, it can be proved either 
mathematically or by actual trial that, in a series of such tests, using a 
different random arrangement each time, the variance v will on the aver- 
age be equal to the variance e. Details of the methods used for randomi- 
zation are given in Chapter XVI. 

4. Error Control. In replicated experiments, the differences between 
the plots of any one treatment are due in part to experimental error and 
in part to the average differences between the replicates. The latter is 
not relevant to the comparisons we wish to make, as each treatment is 
represented by one plot in each replicate or block. The variance due 
to blocks is therefore removed from the error, and, the larger the propor- 
tion of the total variability that is removed, the more accurate the experiment. 
This has a very important bearing on the plan of an experiment, espe- 
cially in relation to the shape of the blocks and of the plots. The differ- 
ences between long narrow plots, when they are placed side by side, are 
usually less than those between square plots, and similarly for blocks, 
and since we want the differences between plots as small as possible and 
the differences between blocks as large as possible, the ideal plan is one 
which combines long narrow plots with square blocks. Practical con- 
siderations limit the shape of the plots, however, and consequently limit 
also the shape of the blocks; but, if we keep this fundamental principle 
in mind in drawing up experiments, the greatest possible efficiency will 
be obtained. 

The arrangements for error control by means of replication differ 
according to the plan of the experiment. There are two fundamental 
plans, randomized blocks, and the Latin square. Others that will be 
described later may be referred to as special types in that they are to a 
certain extent modifications of the fundamental types, and especially 
adapted to certain purposes. 



148 



THE FIELD PLOT TEST 



. Randomized Blocks. This plan is the simplest of all the types in 
which any measure of error control is obtained. It is illustrated in the 
following diagram, which represents an experiment with 8 treatments 
in 4 blocks. 



II 



G 


A 


H 


D 


E 


D 


H 


A 


F 


B 


C 


E 


G 


C 


F 


B 


B 


H 


D 


F 


G 


F 


C 


A 


C 


E 


A 


G 


E 


D 


B 


H 



III 



TV 



In the general case let k represent the number of blocks and n the number 
of treatments. Then the equation for sums of squares is: 

(D (2) (3) (4) 



n 



(1) 



where Xb is the mean of a block and x v is the mean of a treatment. The 

n* 

last term on the right is actually S(x Xb v + x) 2 , but is abbre- 

viated for convenience. The corresponding equation for degrees of 
freedom is: 

(D (2) (3) (4) 

tit - 1 = (t - 1) + (n - 1) + (n - 1)(* - 1) (2) 



In calculating the sums of squares the following formulae are the most 
convenient. 



(1) Total 2(s - x) 2 = S(x 2 ) - T 2 /nk 

(2) Blocks n 2(#& - x) 2 = 2(T?)/ - !T 2 M 

i i 



(3) Treatments k 2(fc- x) 2 Z(T?)/t - T 2 /nk 
i i 



(4) Error 



T = grand total 
for all plots 

T b = total for 
one block 

T v = total for 
one treat- 
ment 

Subtract 
blocks and 
treatments 
from total. 



THE LATIN SQUARE 



147 



The analysis of variance is set up in the usual way. 
The standard error of the experiment is given by 



and for the mean of one treatment 



(3) 



(4) 



6. The Latin Square. The following diagram illustrates a 5 X 5 
Latin square where the letters represent the treatments. 



E B C D A 

A C D E B 

D E B A C 

C D A B E 

B A E C D 



Note that the plots are arranged in 5 rows and 5 columns, and that there 
must be the same number of treatments as rows and columns. The 
treatments are placed at random, subject to the restriction that a treat- 
ment can occur only once in any row or column. 

Let n represent the number of rows, columns, and treatments, and 
the equations for the sums of squares and degrees of freedom are as 
follows: 



25(s - 
1 



r - x) 2 + nZ(*c - x) 2 + n S(z, - x) 2 + 
11 11 



(5) 



where x r and x c represent the means of rows and columns respectively. 

(n 2 - 1) = (n - 1) + (n - 1) + (n - 1) + (n - 2) (n - 1) (6) 
The calculations for sums of squares are: 
(1) Total 25(* - z) 2 = S(z 2 ) - T 2 /n 2 



T = grand total of 
all plots 



(2) Rows nS(x r - x) 2 = S(!T 2 r )/n - T 2 /n* T r 



(3) Columns S(x c - x) 2 - 2(T*)/n - T 2 /n 2 T c 



total for one 
row 

total for one 
column 



148 



THE FIELD PLOT TEST 



(4) Treatments n S( , - ) 2 = Z(T?)/n - T 2 /n 2 T v = total for one 



treatment 



(5) Error 



S(d 2 ) = (1) - (2) - (3) - (4) Subtract, rows, 
1 columns, and 

treatments 
from the 
total. 
The standard error in a Latin square is given by 



n 



' (n - 2) (n - 1) 
And for the mean of one treatment 



(7) 



(8) 



The Latin square gives error control in two directions across the field, 
so that soil gradients are always taken care of. For a few treatments it 
is a very efficient type of experiment, and it is very doubtful that a 
better one can be devised. When the number of treatments are more 
than 8 the Latin square is cumbersome and a point is soon reached 
where the increase in accuracy does not warrant the added labor. 
Moreover, as the number of treatments are increased the rows and col- 
umns become longer in proportion to their width and a point is reached 
finally where further accuracy through error control is not obtained. 

Example 33. Randomized Blocks. Table 37 gives the yields of 6 wheat varieties 
obtained in an experiment consisting of 4 randomized blocks. The marginal totals 
are given in the table so as to facilitate calculation. 

TABLE 37 

YIELDS IN BUSHELS PER ACRE BY BLOCKS 
OF 6 WHEAT VARIETIES 



Blocks Variety 


1234 Totals 


A 


27.8 


27.3 


28.5 


38.5 


122.1 


B 


30.6 


28.8 


31.0 


39.5 


129.9 


C 


27.7 


22.7 


34.9 


36.8 


122.1 


Varieties D 


16.2 


15.0 


14.1 


19.6 


64.9 


E 


16.2 


17 


17.7 


15.4 


66.3 


F 


24.9 


22.5 


22.7 


26.3 


96.4 


Block Totals. 143.4 133.3 148.9 176.1 


601.7 



THE LATIN SQUARE 



149 



Calculating the sums of squares we have: 

Total SO& 2 ) - T*/nk - 16,460.05 - 15,085.12 - 1374.93 

Blocks S(n 2 )/n- T*/nk - 15,252.48 - 15,085.12 - 167.36 

Varieties *L(Tf)lk - T 2 /nfc - 16,147.87 - 15,085.12 - 1062.75 

Error - 1,374.93 - 167.36 - 1062.75 - 144.82 

The analysis of variance is then as follows: 





Sum of 
Squares 


DP 


Variance 


F 


5% Point 
ofF 


Blocks 


167 36 


3 


55.79 


5.78 


3.29 


Varieties 


1062 75 


5 


212.50 


22.0 


2.90 


Error. . . . 


144 82 


15 


9 655 


















Total 


1374.93 


23 





















The block and variety differences are seen to be significant, and if we wish to compare 
any two varieties we make use of the standard error. 

3.122 



V9.655 - 3.122 



1.561 



The standard error of a difference between the means of any 2 varieties is then 
1.561 X \/2 2.21. Now suppose that we wished to compare varieties D and F 
for which the means are 16.2 and 24.1 respectively. The difference is 7.9 and we have 

-=- 

From Table 94 we note that for 15 degrees of freedom t 2.95 at the 1% point, 
so that the difference between the 2 varieties is very significant. We take t for 
15 degrees of freedom corresponding to the number of degrees of freedom available 
for estimating the error variance. Unless the degrees of freedom are decidedly 
limited a short cut can be employed for testing significance. From Table 94 we 
note that t at the 5% point is approximately 2. Therefore a significant difference 
will be 2 X \/2 X s m * 2.82 m . Roughly a significant difference is 3 s m . 

Example 34. The Latin Square. The following is a plan of a Latin square 
which was used to test the efficiency of different methods of dusting with sulphur 
in order to control stem rust of wheat. The key to the treatments is given with the 
plan- 
Columns KEY TO TREATMENTS 



12345 



I 

II 

Rows III 

IV 

V 



B 
C 
D 
E 
A 


D 

A 
C 
B 

S 


E 
B 
A 
C 
D 


A 
E 
B 
D 
C 


C 
D 
E 
A 
B 



A = Dusted before rains. 
B Dusted after rains. 
C Dusted once each week. 
D Drifting once each week. 
E - Check (undusted). 



150 



THE FIELD PLOT TEST 



All applications were 30 pounds to the acre at each treatment. Drifting means 
that the dust was allowed to settle down over the plants from above. In the ordinary 
procedure the sulphur is forced down among the plants by a blast of air. 

The plot yields in bushels per acre are given in Table 38. The figures in the table 
correspond with the position of the plots in the above plan. 



TABLE 38 
PLOT YIELDS IN BUSHELS PER ACRE 



I 

II 

Rows III 

IV 

V 

Column 
Totals 



Columns Row 
12346 Totals 


4.9 


6.4 


3.3 


9.5 


11.8 


35.9 


9.3 


4.0 


6.2 


5.1 


5.4 


30.0 


7.6 


15.4 


6.5 


6.0 


4.6 


40.1 


6.3 


7.6 


13.2 


8.6 


4.9 


39.6 


9.3 


6.3 


11.8 


15.9 


7.6 


50.9 



TREATMENT 
TOTALS 

A 34.2 

B 32.3 

C 65.6 

D 39.8 

E 24.6 



36.4 39.7 41.0 45.1 34.3 196.5 



In order to obtain the treatment totals we must select the yields according to 
the position of the treatments in the plan. Thus for treatment B we have 4.9 + 7.6 
+ 6.2 4- 6.0 4- 7.6 - 32.3. Finally we have all the treatment totals as given in 
Table 38. 

The calculations are as given below: 

(1) Total Sfce 2 ) - T*/n z - 1829.83 - 1544.49 - 285.34 

(2) Rows 

(3) Columns 



S(T 2 )/n - T*/n 2 - 1591.16 - 1544.49 - 46.67 
i 

S(!T 2 )/n - TVn 2 - 1558.51 - 1544.49 - 14.02 



(4) Treatments 2J(T!)/n - TVn 1 - 1741.10 - 1544.49 - 196.61 
i 



(5) Error - d)-(2)- (3)~(4) 

Then the analysis of variance is: 



28.04 





Sum of 
Squares 


DF 


Variance 


F 


5% Point 
of F 


Rows 


46.67 


4 


11.67 


4.99 


3.26 


Columns 


- 14.02 


4 


3.50 


1.50 


3.26 


Treatments 


196.61 


4 


48.62 


20.8 


3.26 


Error 


28.04 


12 


2.34 


















Total 


285.34 























SPLIT PLOT EXPERIMENTS 151 

7. Factorial Experiments. As the name denotes, in factorial experi- 
ments, an attempt is made to study the various treatment factors. Thus 
an experiment designed to study, at the same time, rate and depth of 
seeding of a cereal crop would be a factorial experiment in which the 2 
factors, rate and depth of seeding, are represented at 2 or more levels. 
We may use, for example, 3 rates and 3 depths, giving us in all 9 treat- 
ment combinations. Usually, there are more than 2 factors, as it is 
easily seen that the greater the number of factors the greater the scope, 
and inductive value of the experiment. The experiment on rates and 
depths, for example, might well be conducted with more than 1 variety, 
AS it is conceivable that results obtained with 1 variety might not apply 
to others. In factorial experimentation, therefore, the study of the 
interactions is a very important consideration and, until the advent of 
the development of a suitable technique, was very frequently completely 
overlooked. 

The introduction of factors is of course limited by space and the cost 
of experimentation, and, in addition, it is easy to add so many factors 
that the analysis becomes rather complex. If we have to study all the 
possible combinations in an experiment with 4 factors at 3 levels each, 
we must have 81 different combinations. The addition of another factor 
at 3 levels would increase the number of combinations to 243, at which 
point the experiment would become extremely unwieldy, and since the 
blocks would be very large, error control would not be highly efficient. 

If all the factors are of equal importance, the obvious method is to 
make up the total number of combinations and randomize them indis- 
criminately in each block. We shall see later that with this plan con- 
siderable increases in precision can be obtained by a process of splitting 
up the replicates into smaller units and confounding with these smaller 
blocks certain relatively unimportant degrees of freedom. In many 
cases the factors are not of equal importance and very efficient use can 
be made of the split plot design, in which more than one error variance 
is obtained, each one appropriate for testing certain comparisons. 

8. Split Plot Experiments. An experiment was conducted in 1932 on 
the experimental field of the Dominion Rust Research Laboratory, which 
is a good example of the split plot type. This particular study was de- 
signed to determine the effect on the incidence of root rot, of variety of 
wheat, kinds of dust for seed treatment, method of application of the 
dust, and efficacy of soil inoculation with the root-rot organism. 

The plan of the experiment with the key to the treatments is given 
below and is sufficient to indicate hotf the experiment was worked out. 
Two varieties of wheat, Marquis and Mindum, were used. These vari- 
eties were planted in 4 blocks, half of each block being sown to one variety 



152 



THE FIELD PLOT TEST 



and half to the other. The strips were then divided into 10 plots each. 
With 5 different kinds of dust and 2 methods of application, dry and wet, 
there were 10 different treatments, and one of these was assigned at 
random to each plot in each strip. The plots were then divided length- 
wise and on one half the seed was sown with inoculated soil and on the 
other half with uninoculated soil. The final result was as shown in the 
plan of the experiment. It will be noted that the disposition of varieties, 
dust treatments, and soil treatments is purely at random throughout 
the experiment. 

In order to analyze this experiment it is necessary to sort out the 
degrees of freedom corresponding to the various components of the test. 
In the first place, for the 160 plots there is a total of 159 degrees of free- 
dom. The 160 plots are in pairs, one of each pair being inoculated (I), 
and one uninoculated (U). A convenient initial classification of the 
degrees of freedom (DF) is to consider the field as made up of 80 pairs 
of plots, and since there is one DF within each pair, we have 



Between 80 pairs 
Within 




(9) 



Total 159 DF 



Then, proceeding to the splitting up of the DF of these two components, 
and dealing first with the 79 DF for between pairs, we note that the units 
now are plots exactly twice the size of the original plots, and the DF can 
be analyzed out without any reference whatsoever to the fact that the 
plots are divided into I and U portions. If the experiment is considered 
first as a test of 10 treatments replicated 8 times, the analysis would be as 
follows: 

Blocks 7DF 

Treatments 9 DF (10) 

Error 63 DF 

But the experiment is not actually replicated 8 times, as 4 of these blocks 



PLAN OF A SPLIT PLOT EXPERIMENT 
2345678 



10 



Marquis 



Mindum 



5 


4 


3 


8 


7 


1 


9 


10 


2 


6 


UI 


UI 


UI 


UI 


IU 


UI 


IU 


UI 


IU 


IU 


6 


2 


8 


10 


3 


4 


1 


7 


5 


9 


IU 


UI 


UI 


IU 


UI 


IU 


IU 


UI 


IU 


UI 



SPLIT PLOT EXPERIMENTS 



153 



PLAN or A SPLIT PLOT EXPERIMENT Continued 
1 23456 789 10 



Marquis 
Mindum 

Mindum 
Marquis 

Mindum 
Marquis 



Key to Treatments 

I SB Inoculated soil. 
U = Uninoculated soil. 



1. Dry, Ceresan. 
3. " Semesan. 
5. " DuBay. 
7. " Check. 
9. " CaCo 8 . 



2. Wet, Ceresan. 
4. " Semesan. 
6. " DuBay. 
8. " Check. 
10. " CaCo 8 . 



9 


10 


4 


2 


1 


5 


7 


6 


3 


8 


IU 


IU 


IU 


IU 


UI 


IU 


IU 


UI 


IU 


IU 


6 


9 


2 


5 


1 


8 


10 


4 


3 


7 


TJI 


UI 


IU 


IU 


UI 


IU 


UI 


UI 


IU 


UI 




10 


6 


9 


1 


w 

i 


5 


2 


3 


4 


8 


IU 


IU 


UI 


UI 


UI 


UI 


UI 


IU 


UI 


UI 


4 


5 


1 


2 


3 


6 


7 


9 


10 


8 


UI 


IU 


IU 


UI 


IU 


IU 


UI 


UI 


UI 


IU 




4 


8 


9 


5 


1 


3 


10 


2 


6 


7 


UI 


UI 


IU 


UI 


IU 


IU 


UI 


IU 


IU 


UI 


8 


10 


3 


2 


4 


5 


7 


1 


6 


9 


UI 


UI 


IU 


IU 


UI 


IU 


UI 


IU 


Ut 


UI 



II 



III 



IV 



are sown to Marquis wheat and 4 to Mindum wheat. The 7 DF for 
blocks contain, therefore, 1 DF for varieties and 3 DF for the interaction 
of varieties with blocks, where the blocks consist now of two sets of all 
the treatments, one set with Marquis wheat and one set with Mindum 
wheat. The 3 DF for the interaction of varieties with blocks obviously 
represent the error for determining the significance of the differences 
between the varieties. The final disposition of the 7 DF as given in 
(10) is therefore: 

Blocks 3 DF 

Varieties 1 DF (11) 

Error (1) 3 DF 

We take next the 9 DF as given in (10) for treatments. The key to 
treatments shows that there are 4 different dusts and 1 check, so that 



154 



THE FIELD PLOT TEST 



we have 4 DF for treatments. Then each dust is applied dry (D) and 
applied wet (W), so that we must have 1 DF for D W. The remaining 
4 DF represent the interaction of dusts with D W, so that the 9 DF are 
finally split up as follows: 



Dusts 

DW 

Interaction 



4DF 
IDF 
4DF 



(12) 



The effect of the varieties (V) on the factors given in (12) must also be 
considered; therefore we must have in the 63 DF for error given in 

(10): 

V X Dusts 4 DF 

V X D W IDF (13) 

V X Dusts X D W 4 DF 

The 9 DF represented in (13) must obviously come out of the 63 DF for 
error as given in (10), so that there are actually only 54 DF representing 
true error. Finally the complete disposition of the 79 DF for between 
pairs of plots can be shown as follows: 

Blocks 3 DF 

Varieties 1 DF Group (1) 

Error (1) 3 DF\ 

Dusts 4 DF] 

DW IDF 

Dusts X D W 4 DF 

V X Dusts 4 DF \ Group (2) 

V X D W 1 DF 

V X Dusts X D W 4 DF 

Error (2) 54 DF 

Total 79 DF 

Error (2) is applicable to all the factors in the second group. 

TABLE 39 

PLOT YIELDS IN A SPLIT PLOT EXPERIMENT 
123456789 10 



64 


68 
69 


68 


71 


62 


73 


56 


67 


78 


69 


67 
73 


66 
67 


71 


64 
70 


64 


75 


70 


66 


67 


65 


76 


66 


72 


75 


81 


72 


71 


70 


72 


72 


72 


85 


76 


70 


71 


74 



SPLIT PLOT EXPERIMENTS 



155 



PLOT YIELDS IN A SPLIT PLOT EXPERIMENT Continued 
23456789 



10 



66 


63 


63 


51 
72 


58 


64 


60 


57 


55 


60 


53 


61 


73 


56 


59 


55 


47 


58 


64 
73 


55 


54 

I 


74 


73 


73 


64 


79 


68 


68 


72 


76 


69 


66 


78 


67 


63 


69 


74 


76 



83 


73 


68 


60 


82 


79 


73 


81 


84 


94 


77 


76 


74 


77 


76 


73 


69 


70 


75 


88 


51 


59 


57 


57 


63 


60 


57 


61 


63 


65 


64 


61 


60 


65 


56 


67 


61 


74 


73 


55 




63 


72 


72 


83 


78 


69 


70 


66 


60 


66 


68 


70 


63 


68 


64 


61 


59 


63 


65 


77 


60 


69 


60 


67 


67 


52 


61 


56 


61 


69 


58 


62 


60 


72 


57 


54 


58 


58 


64 


65 



Considering now the 80 DF for within pairs, the first point to note is 
that, since these 80 DF represent only differences between members of 
pairs of adjacent plots, they do not contain any direct effects due to 
blocks, varieties, or dust treatments. The differences between such 
plots do represent, however, the effect of I and U corresponding to 1 DF. 
The first split up of the 80 DF is therefore : 



IU 
Remainder 



Total 



IDF 
79 DF 

80 



(14) 



The 79 DF for the remainder must contain the DF representing the inter- 
action of I U with all the other factors as given in Groups (1) and (2) ; 
hence we can set these down in order. 



I U X V 1 DF 

I U X Dusts 4 DF 

I U X D W 1 DF 

I U X Dusts X D W 4 DF 

I U X V X Dusts 4 DF 

ITJXVXDW IDF 

Total 15 DF 



(15) 



Note that we have left out (I U X Blocks) and the quadruple interaction 
(I U X V X Dusts X D W). The former belongs to error, and the latter 
is very unlikely to be significant, and even if it might turn out significant, 
its interpretation would probably be too complex to have any practical 



156 THE FIELD PLOT TEST 

bearing on the use of the treatments. The final analysis of the 80 DF 
for within pairs can now be written down : 

I U 1 DF 

I U X V IDF 

I U X Dusts 4 DF 

I U X D W IDF 

I U X Dusts X D W 4 DF 

I U X V X Dusts 4 DF 

IUXVXDW IDF 

Error (3) 64 DF 



(Group (3) 



Total 80 DF 

The three groups may be placed together as one complete analysis or 
dealt with separately. It will usually be found most convenient in 
checking calculations to consider the three groups together in one com- 
plete analysis. 

After completing the sorting out of the DF the next step is to draw 
up the tables from the actual data that are necessary for calculation of 
the sums of squares. In the first place a table such as Table 39 is 
required, giving the data for the individual plots in a plan corresponding 
to the plan of the experiment. Comparing the table and the plan we 
can then draw up Table 40, which is a series of small tables required for 
calculating the sums of squares. 

The following is an outline of the analysis of variance for the whole 
experiment, with figures in the fifth column indicating the calculation 
tables from which the corresponding sums of squares are obtained. 

From Table 39 we calculate the total sum of squares for all the plots. 
Then from the calculation Table 12, for the differences within pairs of 
plots, we determine the sum of squares for the 80 DF representing within 
pairs. Subtracting this from the total sum of squares gives the sum of 
squares for 79 DF representing Groups (1) and (2). 

We proceed next to calculate, from the tables, the sums of squares as 
indicated in the outline of the analysis of variance, leaving items error 
(2) and error (3) to the last. From the sum of squares representing 
within pairs for 80 DF, we subtract the first seven items in Group (3). 
The remainder is the sum of squares for error (3). From the sum of 
squares for between pairs (79 DF) we subtract the total for group (1) 
and the first six items in Group (2). The remainder is the sum of 
squares for error (2). 

The method of calculation of triple interactions has been described in 
a previous chapter. 



SPLIT PLOT EXPERIMENTS 



157 





Sums of 
Squares 


DF 


Variance 
or Mean 
Square 


Calculation 
Table 


Blocks 


989.6 


3 


329.9 


1 


Varieties (V) 


3638.6 


1 


3638 6 


1 


Error (1) 


647.6 


3 


215.9 


1 


Dusts 


987.6 


4 


246.9 


2 


Dry vs. Wet (D W) 


117.3 


1 


117.3 


2 


Dusts X D W 


46.2 


4 


11.6 


2 


V X Dusts 


146.7 


4 


36 7 


3 


VX D W .. 


91.5 


1 


91 5 


4 


V X Dusts X D W 


148.1 


4 


37 


5 


Error (2) 


1059.1 


54 


19.6 




Inoculated vs. Uninoculated (I U) . . 
I UX V 


965.3 
0.3 


1 
1 


965.3 
0.3 


6 
6 


I U X Dusts ... .... 


379.8 


4 


95.0 


7 


I UX D W 


68.9 


1 


68.9 


8 


I U X Dusts X D W 


25.8 


4 


6.4 


9 


I U X V X Dusts 


119.4 


4 


29.8 


10 


IUXVXDW 


3.9 


1 


3.9 


11 


Error (3) 


931.1 


64 


14 5 














Total 


10,366.8 


159 

















Number 



(D 



TABLE 40 
SERIES OF SUBTABLEB FOB CALCULATING SUMS OF SQUARES 

Blocks 
I II III IV 



(2) 



(3) 



Ma 

Mi 


1351 
1454 


1178 
1408 


1229 




10,739 
Ca 


Ma 
Ma 

D 
W 


f Mi 
- Mi 


2805 2586 
-103 -230 

Ce Se Du Ch 


1044 
1039 


1059 
1045 


1062 






D + W 
D- W 


2083 2104 
5 14 

Ce Se Du Ch Ca 


Ma 
Mi 




970 
1113 


988 
1116 


967 






Ma 
Ma 


+ Mi 2083 2104 
- Mi -143 -128 



10,739 



10,739 



158 



THE FIELD PLOT TEST 



Number 



(4) 



(5) 



(6) 



(7) 



(8) 



(9) 



TABLE 40 (Continued) 
SERIES OF SUBTABLES FOB CALCULATING Sous OF SQUARES 

Blocks 
D W 



Ma 
Mi 


2498 
2940 




10,739 
Du Ch Ca 


Ma + Mi 

Ma D 
W 


5438 
Ce 


Se 


482 
488 


487 
501 


480 


D + W 
D- W 

LMi D 
W 


970 988 
-6 -14 


562 
551 


572 

544 


582 


D + W 1113 1116 
D - W 11 28 



4,988 



5,751 



Ma 



Mi 



I 
u 

1+ U 

I 
u 

I+U 
I -U 

I 
u 
i + u 


2594 
2394 




10,739 
Du Ch Ca 


4988 
Ce Se 


1069 
1014 


1070 
1034 


1054 


2083 2104 
55 36 

D W 


10,739 


2791 
2647 




5438 



10,739 



Ce 



Se 



Du 



Ch 



Ca 



D I 
U 
I+U 
I-U 

IW I 
U 
I+U 
I -U 


531 
513 


535 
524 


533 






1044 1059 
18 11 


538 
501 


535 
510 


521 






1039 1045 
37 25 



5,438 



5,301 



SPLIT PLOT EXPERIMENTS 

TABLE 40 (Continued) 
SERIES OF SUBTABLES FOR CALCULATING SUMS OF SQUARES 



159 



Number 

'] 
(10) . 

.] 

(11) 
(12) 

Dusts: 

Ce(O 

Se(Se 
Du(E 
Ch(C 
Ca(O 


Ma I 
U 
I + U 
I -U 

Mi I 

U 
I+U 

I -U 

I 

U 
I + U 


Ce 


Blocks 
Se Du Ch Ca 


498 
472 


507 

481 


480 


4,988 


970 
26 


988 
26 


571 
542 


563 
553 


574 


5,751 


1113 1116 
29 10 

Ma 
D W 


Mi 
D W 


1289 
1209 




1502 
1438 






2498 4,988 2940 
DIFFERENCES BETWEEN PAIRS OF PLOTS 


5,751 


4 3 
7 6 


11 
6 


11 9 
1 2 


1 7 11 4 
6 2 13 6 


2 
3 






3 12 
20 1 


6 
9 


3 5 

11 4 


8 17 4 11 


9 










eresan) 
mesan) 
hiBay) 
heck) 
alciujp carbonate) 


ABBREVIATIONS 

Varieties Soil Treatment 
Ma (Marquis) I (Inoculated) 
Mi (Minclum) U (Un inoculated) 

Method of Applying Dust 
D(Dry) 
W (Wet) 



160 THE FIELD PLOT TEST 

CONFOUNDING IN FACTORIAL EXPERIMENTS 

9. Orthogonality and Confounding. F. Yates (16) has given the 
following definition of orthogonality. It is " that property of the design 
which ensures that the different classes of effects shall be capable of 
direct and separate estimation without any entanglement/' Thus, in 
a randomized block experiment, the treatments are orthogonal with 
blocks in that the effects of each are capable of direct and separate esti- 
mation. This orthogonality is accomplished in the design by seeing to 
it that each block contains the same kind and number of treatments. 
If by any chance some of the plots in one or more of the blocks are lost, 
non-orthogonality is introduced, and special methods may be required 
in order to separate the treatment and block effects. These methods, 
which have been worked out and described in some detail by Yates, 
require additional computation, and sometimes the whole procedure may 
be rather laborious. Consequently in designing an experiment we make 
every effort to keep within the requirements of orthogonality. In simple 
experiments this presents no difficulty, but in more complex ones for 
which a new design is being worked out it is quite easy unwittingly to 
introduce an element of non-orthogonality. New designs, therefore, 
require careful scrutiny before they are put into practice. 

In factorial experiments involving a fairly large number of combina- 
tions, non-orthogonality is sometimes introduced deliberately, and this 
process is now referred to as confounding. The purpose of confounding 
in general, as we shall see later in more detail, is to increase the accuracy 
of the more important comparisons at the expense of the comparisons 
of lesser importance. In many instances, however, although a certain 
portion of the information concerning the comparisons of lesser impor- 
tance is sacrificed, the precision with which all the effects are estimated 
is increased to a point such that even the partially confounded compari- 
sons are more accurately estimated. 

The student should at this point make quite certain of the meaning 
of confounding, and a few elementary illustrations may be of assistance. 
Suppose that three fertilizers AT, P, and K are being compared at 2 levels 
of each, so that we have 8 different combinations that we shall designate 
by Wotfo, NoP*Ki, Wi^o, # iPoXo, NoPiKi, NiPoKi, N^Ko, 
and NiPiKij where the subscript numbers refer to the amounts or dosage 
of each kind of fertilizer. Since NoPoKo means that no fertilizer is 
applied, and NoPoKi means that only K is applied, these terms may be 
abbreviated to 0, K, P, N, PK, NK, NP, and NPK. In these 8 com- 
binations it will be noted that we have 4 without N and 4 with N. If 



ORTHOGONALITY AND CONFOUNDING 161 

now we divide the blocks ordinarily containing 8 plots into halves such 
that one half contains the treatments 0, K, P, PK and the other half 
N> NK, NP, NPK, then the effect of N which may be represented 
algebraically by (Ni Wo) is completely confounded with block effects. 
The other main effects are still orthogonal with the blocks. For example, 
in earh block we have 2 plots containing P and 2 plots that do not 
contain P. We would not consider a design of this type in actual 
practice, as it defeats what is obviously one of the main purposes of 
the experiment. Assuming, however, that accuracy can be gained by 
reducing the size of the blocks, it may be worth while to examine all the 
comparisons to see whether certain of these may be deemed sufficiently 
unimportant to be sacrificed in order to increase the precision of the re- 
maining comparisons. 

The treatment effects may be set out as follow with the correspond- 
ing degrees of freedom. 

N IDF 

P 1 DF Main effects, 3 DF 

K IDF 

NXP IDF 

NX K I DF Simple interactions, 3 DF 

PX K 1 DF 

NX PX K I DF Triple interaction, 1 DF 

To the best of our judgment the triple interaction N X P X K would 
seem to be the least important. At least, even if significant in effect it is 
the most difficult to interpret in terms of actual fertilizer practice. We 
shall decide, therefore, to confound this one degree of freedom with 
blocks, and it remains only to determine the distribution of the treat- 
ments ih the blocks in a manner which will confound this one comparison 
and leave all the others intact. Algebraically, all the treatment effects 
can be represented as follows 

N = (Ni - No) (Ki + Ko) (Pi + Po) 
P = (Ni + No) (Ki + K ) (P l - Po) 
K = (Ni + No) (K, - Ko) (Pi + Po) 



N X P = (Ni - No) (K, + Ko) (Pi - Po) 
N X K - (N! - No) (Ki - Ko) (Pi + Po) 
P X K = (Ni + No) (^ - Ko) (Pi - Po) 

N X P X K = (Ni - tfq) (Ki - Ko) (Pi - Po) 



162 THE FIELD PLOT TEST 

and on expanding the last expression we have 

+^ p o#i + tfiPitfi + NoPiKo + 



N X P X K | - jv Po#o - NiPiKo - NoPiKi - 

\+K + NPK + P + N 
01 1-0- NP - PK - NK 

This means simply that, if we let the symbols represent the actual yields 
from the corresponding plots, the sum of squares for the triple inter- 
action will be given by 

-~ [(N + P + K + NPK) - (0 + NP + PK + NK)] 2 

where k is the number of plots represented in each total such as (0 + NP 
+PK + NK). Now if we divide each replication into 2 blocks and in 
one of these put the treatments 0, NP, PK, NK, and in the other, 
N t P, K, NPK, then the above sum of squares will contain not only the 
triple interaction effect but also the effect of the blocks. The 1 degree of 
freedom for triple interaction will have been completely confounded 
with blocks. The analysis of variance for the experiment, assuming 
4 replications, will be of the form 

Blocks 7DF 

Main effects 3 DF 

Simple interactions 3 DF 

Error 18 DF 



Total : 31 DF 

Since 7 DF have been utilized for error control instead of 3 as in an 
ordinary randomized block experiment, with a moderate degree of soil 
heterogeneity, it may be expected that the remaining effects will be 
estimated more accurately by the confounded experiment than by the 
randomized blocks. 

10. Partial Confounding and Recovery of Information. The pro- 
cedure illustrated above resulted in the complete sacrifice of the infoi^ 
mation on the triple interaction, and it may be argued that, regardless 
of the apparent unimportance of the information sacrificed, this is not 
good experimental procedure in that the experimenter is taking too much 
for granted in attempting to forecast a result on which he has no previous 
information, and using this as a basis for the experimental design. The 
difficulty can be overcome by a process known as partial confounding, 
which amounts to confounding different degrees of freedom in different 
replications and using the results TfronTtEe blocks in which the particular 



PARTIAL CONFOUNDING AND RECOVERY OF INFORMATION 163 

effects are not confounded to recover a portion of the information de- 
sired. In order to partially confound the experiment described above 
and at the same time recover a portion of the information on all the com- 
parisons, we shall require at least 4 replications. In each replication 
we can conf ound with blocks a degree of freedom from one of the inter- 
actions. The method of laying out the treatments in the blocks is ob- 
vious if we expand algebraically each of the expressions for the inter- 
actions. Thus 

^^ NK) 



NK+ P+NPK\ 
III (NXK) -(Ni-Nt)(Pi+Pd(Ki-K*)- N K-NP- PR) 

/4-0-f PK+ N+NPK\ 

iv (PXK) -(Ni+NMPi-Pd(Ki-Kti-{+p^ K I N pL NK) 

Then in the first replication we can confound the triple interaction and 
conserve it in all the remaining replications. In the second replication 
we can confound the simple interaction N X P and conserve it in all 
the remaining replications. With 4 replications we can confound each 
interaction in 1 replication and conserve it in all the others. 

In recovering information with respect to the interactions it will, of 
course, be necessary to make the desired comparisons only in those 
replications in which the particular interaction is not confounded. Thus 
if we are computing the sum of squares forNXP we omit replication 
II entirely and make up our totals from the other three. The final 
analysis will be of the form : 

Blocks ........................ 7 DF 

Main effect* ................... 3 DF 

Simple interactions ............. 3 DF 

Triple interactions .............. I DF 

Error ......................... 17 DF 

Total ..................... 31 DF 

The result of this procedure is to sacrifice one-quarter of the information 
on each interaction, but the main effects and that portion of the informa- 
tion with respect to the interactions that is recovered may be estimated 
with greater accuracy. 

Using a set of figures from uniformity data the procedure for designing 
and analyzing a partially confounded 2X2X2 experiment is illus- 
trated in Example 35. 



164 THE FIELD PLOT TEST 

Example 36. Partial Confounding in a 2 X 2 X 2 Experiment. 



TABLE 41 

PLAN OF FIELD SHOWING LOCATION OF TREATMENTS AND CORRESPONDING YIELDS, 
FOR A PARTIALLY CONFOUNDED 2X2X2 EXPERIMENT 



Replication 
No. 


Treat- 
ment 


Yield 


Treat- 
ment 


Yield 


Treat- 
ment 


Yield 


Treat- 
ment 


Yield 






NK 


159 


P 


153 





145 


K 


189 









179 


NPK 


202 


PK 


191 


P 


272 




I 


PK 


135 


N 


153 


NK 


300 


N 


160 






NP 


130 


K 


182 


NP 


240 


NPK 


305 








603 




690 




876 




926 


3,095 




NPK 


155 


N 


191 





226 


P 


266 






NP 


129 


NK 


138 


K 


159 


NK 


300 




II 


K 


151 


P 


188 


NPK 


240 


PK 


233 









159 


PK 


210 


NP 


182 


N 


278 








594 




727 




807 




1077 


3,205 




P 


154 


K 


143 


P 


186 


N 


209 






NK 


77 


NP 


119 


NPK 


173 


K 


93 




III 





92 


N 


115 





170 


PK 


224 






NPK 


128 


PK 


179 


NK 


213 


NP 


245 








451 




556 




742 




771 


2,520 




N 


113 


P 


136 


PK 


182 


K 


293 






PK 


127 


NK 


197 





175 


NK 


226 




IV 


NPK 


185 


K 


182 


NPK 


156 


NP 


248 









148 


NP 


212 


N 


183 


P 


269 








573 




727 




696 




1036 


3,032 




















11,852 



Table 41 gives the location of the treatments in the field and the corresponding 
yields. The latter were taken from uniformity data as the results from an actual 
experiment were not available. Note that the replicate numbers (actually two 
replicates) correspond with the numbers given opposite the expansion of the inter- 
actions on page 163. Thus in replicate I the triple interaction NXPXK is con- 
founded with blocks, and so forth for the other interactions in the remaining replica- 
tions. Within each block the treatments are assigned to the plots at random. 

In Table 42 the treatment totals are arranged in a convenient form for the 
calculation of sums of squares. For example, in calculating the triple interaction 



PARTIAL CONFOUNDING AND RECOVERY OF INFORMATION 165 



TABLE 42 

TREATMENT TOTALS REQUIRED TOR CALCULATION 
OF SUMS OF SQUARES 







Minus 


Minus 


Minus 


Minus 


Treatment 


Replications 


Replication 
I 


Replication 
II 


Replication 
III 


Replication 
IV 





1294 


970 


909 


1032 


971 


N 


1402 


1089* 


933 


1078 


1106 


P 


1624 


1199 


1170 


1284 


1219 


K 


1392 


1021 v 


1082 


1156 


917 


NP 


1505 


1135 


1194 


1141 


1045 


NK 


1610 


1151 


1172 


1320 


1187 


PK 


1481 


1155 


1038 


1078 


1172 


NPK 


1544 


1037* 


1149 


1243 


1203 



N X P X K we must use the totals from the replicates in which this interaction ia 
not confounded. These are given in the third column, and we find 

NXPXK** (1021 + 1037 + 1199 + 1089 - 970 - 1135 - 1155 - 1151) 2 /48 - 88 
Similarly the interaction P X K is calculated from the totals in the sixth column 
P X K - (1219 + 917 + 1045 + 1187 - 971 - 1172 - 1106 - 1203) 2 /48 147 

The main effects are of course calculated from all the replicates, so we make use of 
the totals in the second column. 

TABLE 43 

COMPLETE ANALYSIS FOR PARTIALLY CONFOUNDED 
2X2X2 EXPERIMENT 





Sums of 
Squares 


DF 


Mean 
Square 


Blocks 


112,462 


15 


7,497 


N 


1,139 


1 


1,139 


p 


3,249 


1 


3,249 


K 


638 


1 


638 


N X P 


9 


1 


9 


N X K 


3,781 


1 


3,781 


P X K . ... 


147 


1 


147 


NX PX K 


88 


1 


88 


Error 


61,895 


41 


1,510 










Total 


183,408 


63 













166 THE FIELD PLOT TEST 

11. Splitting up Degrees of Freedom into Orthogonal Components. 

Before considering the problem of confounding in experiments of a more 
complex type, the student should acquaint himself with the methods 
of separating effects representing more than 1 degree of freedom into 
component parts that are mutually independent and therefore may be 
separately estimated from the data. Thus if we have 3 levels of nitrogen 
in a fertilizer experiment, there are 2 degrees of freedom representing 
the effect of nitrogen. These 2 degrees of freedom may be separated 
with their appropriate sums of squares in an infinite number of ways, 
but unless the separation is a purely formal one we will probably wish 
to separate them in some way such that they will represent definite facts 
relative to the interpretation of the experiment. In the case of the 3 
levels of nitrogen Ni, N%, and N$, the 2 degrees of freedom can be 
expressed by 

(a) #3 - Ni 

(b) 2N* -Nt-N* 

and in this form (a) represents the linear effect of N on yield, and (6) the 
quadratic effect. If the yields are represented graphically, (6) will be 
zero if the 3 points lie exactly on a straight line. These two expressions 
merely bring out the fact that any 2 points can be fitted by a straight line 
function, and any 3 points by a quadratic function. Any other division 
of the degrees of freedom that we might make would probably not have 
as valuable a meaning as this one, although if one felt quite certain that 
#3 was a decided overdose of nitrogen one might wish to measure the 
linear effect by N% N\ 9 and the quadratic effect by 2ATa Ni #2. 
In general, however, the expressions such as (a) and (b) are the most 
useful. 

If we have 4 levels of nitrogen the 3 degrees of freedom may be 
divided: 

(c) 3N* + N* - N 2 - 3Ni Linear term 

(d) Nt - ^ r 3 - N 9 + Ni Quadratic term 

(e) N* - 3N 3 + 3# 2 - Ni Cubic term 

The rule for writing out the expressions for the division of degrees of 
freedom is to see to it that in each expression the sum of the coefficients is 
zero, and for any pair of expressions the sum of the products of the 



SPLITTING UP DEGREES OF FREEDOM INTO COMPONENTS 167 

corresponding coefficients is zero. Thus, in the set immediately above, 
the sums of the coefficients are 

(c) 3 + 1-1-3 = 

(d) 1-1-1 + 1 = 

(e) 1-3 + 3-1 = 
Then multiplying the coefficients: 

(cXd) 3-1 + 1-3 = 

(cXe) 3-3-3 + 3 = 

(dXe) 1+3-3-1 = 

We must remember, however, that if we wish to write the polynomial 
expressions as has been done here there is only one set that can be 
written. 

The sum of squares for any one of the above expressions may be cal- 
culated by means of a simple rule. For example, if we have the expres- 
sions (a) and (6) the sums of squares are 

(a) i (N* - Ntf (b) ^ (2N 2 -Ni- Ntf 

where the numerical portion of the divisor comes from summing the 
squares of the coefficients within the bracket. The value of k comes 
from the number of units entering into each subtotal. For example, in 
(6), NI, N2, and N$ may represent subtotals from 8 plots, whence the 
complete divisor is 48. 

An actual example of the division of 3 degrees of freedom according 
to the scheme outlined above is given by Yates (17). The figures are 
for response to nitrogen, and the results of the analysis are reproduced 
below: 

DF SS 

Linear term 1 19,536.4 

Quadratic term 1 480.5 

Cubic term 1 3.6 

Total 3 20,020.5 

When compared with the error of the experiment, the quadratic term 
turned out to be insignificant, and the cubic term was below expecta- 
tion. Undoubtedly, this type of result is quite usual in agricultural 
experiments, and since we can separate out not only main effects in the 



168 



THE FIELD PLOT TEST 



above manner but also interaction effects, it follows that if a portion of 
the degrees of freedom for an interaction effect is to be sacrificed by 
confounding it is desirable in general to sacrifice that portion that is least 
likely to be significant. At any rate, it may be wise to ensure that at 
least the interaction between linear effects may be partially recovered 
from the confounded experiment. 

If the interaction between nitrogen at 2 levels and potash at 2 levels 
can be represented by (N2 Ni) (K% K\) it follows that, if there are 
3 levels of nitrogen, the interaction N X K can be broken up into two 
parts: 

(K* - Ki) (N 3 - Ni) and (K 2 - KI) (2N 2 - NI - N*) 

where the second expression represents the interaction of the quadratic 
effect of nitrogen with potash. This point may be more obvious if we 
consider (2A^ NI Nz) as representing deviations from linear 
regression instead of the quadratic response, and hence the interaction 
may be written as K regression X N deviation or K T X N d . Now if we 
have 3 levels of potash as well as 3 levels of nitrogen the 4 degrees of 
freedom for the interaction N X K may be broken up as follows: 



NrXK r 

N r XK d 

N d X Kd 



(#3~ 

(2Nz~- 



-K$ 
-Ki) 
- N Z )(2K* -Ki- JRT 8 ) 



IDF 
IDF 
IDF 
IDF 



NXK 



4DF 



and it may be of interest to do this from the standpoint of obtaining 
complete information with respect to the interaction. Yates (17) has 
given a useful table for calculating the sums of squares, which is repro- 
duced below in Table 44. 

TABLE 44 



GUIDE FOR CALCULATING SUMS OF SQUARES FOR THE 
INTERACTIONS IN A 3 X 3 TABLE 



N r XK d 



NdXKr 



N d XKd 




Divisor 4Jc 



12k 12k 

k Number of units in each cell. 



36A- 



SPLITTING UP DEGREES OP FREEDOM INTO COMPONENTS 169 

To use the table it is necessary to set up a table of subtotals in the same 
form as the above squares. The subtotals are added or subtracted 
according to the signs in the appropriate table. Thus if the subtotals 
are represented by 

X\ X2 3 

y\ 2/2 2/3 

Z\ Z2 Z3 



we get the sum of squares for N d X K r by 

1 
12k (22/I " 



-f 



In certain cases it may not be necessary to divide up the degrees of 
freedom into orthogonal components that have any definite meaning, in 
which case we refer to the division as a purely formal one. A 3 X 3 
table, for example, may be represented as follows: 

Pi P 2 P 3 




and from knowledge that has been derived from a study of the properties 
of the Latin square, Fisher, (2), it can be shown that the 4 degrees of 
freedom representing the interaction N X P can be split up into two 
orthogonal components by making up totals from the diagonals of the 
above square. Thus 2 degrees of freedom of the interaction is repre- 
sented by the differences between the totals (11 + 22 + 33), (21 + 32 + 
13), (31 + 12 + 23), and the other 2 by the differences between the 
totals (11 + 32 + 23), (21 + 12 + 33), (31 + 22 + 13). As a matter of 
fact this provides a very useful method of calculating the interaction in 
a 3 X 3 table as it is a direct method and the total sum of squares cal- 
culated independently from the same table may be used to obtain a 
perfect check on all the calculations. 1 The division of the 4 degrees of 
freedom is, however, purely formal. In other words, we would expect 
that on the average the two components would give us equal estimates 
of the interaction variance. 

1 Note that the second set of totals can be obtained most easily by setting up the 
numbers in the first three totals in the form of another square, and taking from this 
square the same diagonals as were used in the first instance. 



170 THE FIELD PLOT TEST 

12. Confounding in a 3 X 3 X 3 Experiment. We shall now con- 
sider the possibilities of confounding in a 3 X 3 X 3 experiment. The 
3 main factors can be represented by JV, K, and P, and since each 
of these occurs at 3 levels there are 27 different combinations. The 26 
degrees of freedom for treatments can be subdivided at first as follows: 

N 2DF] 

K 2 DF [ Main effects 6 DF 

P 2 DF\ 

NXK 4DF] 

NX P 4 DF [Simple interactions 12 DF 

KXP 4 DF\ 

NXKXP 8 DF Triple interaction 8 DF 

Now if we wish to conserve the main effects and the simple interactions 
we must have at least 9 plots in each block. That is, the 3 levels of 
each fertilizer must each be represented by 3 plots, and the 9 combina- 
tions of each pair of fertilizers must each be represented by 1 plot. The 
required combinations to fulfill these conditions are given by a 3 X 3 
Latin square in which the rows may be taken to represent the 3 levels 
of nitrogen, the columns the 3 levels of potash, and the Latin letters 
(here replaced by numbers) the 3 levels of phosphate. R. A. Fisher, 
in introducing this solution, points out that there are only 12 solutions 
of this 3X3 square and that these 12 fall into 4 sets such that in any one 
set the other 2 may be generated by cyclic substitution of the numbers in 
the square. The entire 12 solutions are reproduced below. 




132 

II 3 2 1 

2 1 3 

1 2 3 

III 3 1 2 

2 3 1 

1 3 2 

IV 2 1 3 

3 2 1 



To make the meaning of these squares perfectly clear, suppose that we 
consider the treatments represented by the square I (a). These are, 

^ etc. In any one replica- 



PARTIAL CONFOUNDING IN A 3 X 3 X 3 EXPERIMENT 171 

tion we must have all the treatments of one complete set such as I, II, 
III, or IV, and within the replication the division of the treatments into 
blocks is according to the division of the sets into (a), (6), and (c). In a 
single replication we have 2 degrees of freedom for blocks, and these 
must represent 2 degrees of freedom of the triple interaction that have 
been confounded, as we have seen to it that the main effects and the 
simple interactions have all been conserved. It follows also that it is 
impossible, if the main effects and the simple interactions are conserved, 
to confound more than 2 out of the 8 degrees of freedom of the triple 
interaction. Such being the case, we shall still have, after confounding, 
6 degrees of freedom for the triple interaction, which we may use to test 
the significance of the residual portion of this effect. 

The actual procedure of confounding in an experiment of this kind is 
to set up the treatments and divide them into blocks according to one of 
the cyclic sets. The same division of the treatments into blocks is 
retained throughout the remaining replications. In analyzing the 
results, if set I has been used for confounding, then sets II, III, and IV 
are used to build up the treatment totals from which the sum of squares 
for the triple interaction is calculated. The details of this are given in 
Example 36. 

13. Partial Confounding in a 3X3X3 Experiment. By the 
methods described above we are able to divide the 8 degrees of freedom 
for the triple interaction into 4 sets of 2 that are mutually independent 
and therefore may be separately estimated from the data. But these 
sets represent purely formal differences, and although we confound only 
2 of them and conserve 6, we are not able to separate out particular 
effects such as that represented by N r X K r X P r for particular study. 
To do this we must adopt the method of partial confounding which 
results from using each of the cyclic sets once, one for each replication. 
We require therefore a minimum of 4 replications. Space is inadequate 
here to go into detail regarding the method of separating out the particu- 
lar components, but the student interested in these further aspects of 
confounding will be able to obtain further information from R. A. 
Fisher's "The Design of Experiments," and from the monograph by 
F. Yates, "Factorial Experimentation." 

Example 36. A Confounded 3X3X3 Experiment. In the preparation of this 
example, data from a uniformity trial have been used. It serves therefore merely 
to show the technique of setting up and analyzing a 3 X 3 X 3 experiment in .which 
2 degrees of freedom from the triple interaction have been confounded with blocks. 

As indicated in Table 45 giving the treatment numbers and the corresponding 
yields, the distribution of the treatments into the 3 blocks of each replication is 
according to cyclic set I as described above. In order to abbreviate, only the sub- 
script numbers of the treatments are given, it being assumed that the three ingredient* 



172 THE FIELD PLOT TEST 

such as NKP are in the same order in each case. Within the blocks the treatments 
are, of course, randomized. 

Table 46 is obtained by collecting the plot yields from Table 45. It is used for 
calculating the main effects and the simple interactions. At the foot of this table are 
given the treatment totals from which the sum of squares for the 6 degrees of freedom 
for the triple interaction is calculated. These treatment totals may be obtained very 
quickly by the combined use of the cyclic sets as given on page 170 and the 3X3 
tables for N and K, one for each level of P. Knowing that set I has been used for 
confounding, we obtain our treatment totals for calculating the triple interaction, 
from the application of sets II, III, and IV, to the data given in Table 46. For 
example, taking set II we note that the 1's in group (a) correspond in Table 46 (a) 
with 1604, 1523, and 1912; the 2's correspond inJTable 46 (6) with 1893, 2030, and 
1845; and the 3's in Table 46 (c) with 1741, 1838, and 1917. Adding all these values, 
we obtain 16,303. Then to obtain the next total the same process is repeated, using 
the square indicated by II (5), and finally the third square, II (c), gives the third 
total. The sets III and IV are then used in a similar manner to obtain the remaining 
totals. The sum of squares is calculated for each set of 3 totals and these are added 
to give the sum of squares for the 6 degrees of freedom of the triple interaction. 

Mainly as an exercise, the sums of squares for the individual degrees of freedom 
as represented by the regression and deviation from regression effects have all been 
calculated and are shown in the analysis of variance Table 47. These calculations 
are very simple if one makes use of Yates's diagram as given on page 168. A few of 
the calculations are reproduced below for further guidance: 

N, (15,393 - 16,900) 2 /144 = 15,771.17 

N r X K r (5403 + 5706 - 5376 - 4894) 2 /96 = 7,332.51 

Nd X P r (2 X 5244 + 5057 + 5596 - 4812 - 5667 - 2 X 5297) 2 /288 = 16.06 
N d X K d (5403 -f 5376 + 4 X 5436 + 4894 + 5706 - 2 X 5520 

- 2 X 5096 - 2 X 5818 - 2 X 5074) 2 /864 - 13.25 



METHODS FOR TESTING A LARGE NUMBER OF VARIETIES 

14. General Principles. In factorial experiments, when the total 
number of combinations is fairly large, we have seen that greater accu- 
racy can be obtained by confounding with blocks certain of the degrees 
of freedom for the higher-order interactions. In variety experiments the 
numbers are frequently quite large and we again meet with the problem 
of insufficient accuracy owing to the large size of the blocks. In order 
to overcome this difficulty Yates has developed methods that, by a pro- 
cedure analogous to confounding in factorial experiments, enables us to 
divide up the replications into much smaller blocks, and these are used 
as error control units. Since the small blocks contain only a fraction 
of the total number of varieties, they are referred to as incomplete blocks. 
Yates (20) in a preliminary examination of uniformity data concluded 
that incomplete block experiments would give increases in efficiency over 
randomized blocks of 20 to 50%. Goulden (6) arrived at practically 
the same conclusion after a fairly extensive study. 



GENERAL PRINCIPLES 



173 



TABLE 45 

TREATMENT NUMBERS AND CORRESPONDING PLOT YIELDS FOR 3X3X3 EXPERI- 
MENT. THE SAME Two DEGREES OP FREEDOM FROM THE TRIPLE INTERACTION 
CONFOUNDED IN ALL REPLICATES 



Treat- 




Treat- 




Treat- 




Treat- 




Treat- 




Treat- 




ment 


Yield 


ment 


Yield 


ment 


Yield 


ment 


Yield 


ment 


Yield 


ment 


Yield 


No. 




No. 




No. 




No. 




No. 




No. 




212 


159 


131 


153 


121 


145 


Ill 


189 


232 


153 


233 


210 


321 


179 


311 


202 


233 


191 


122 


272 


112 


226 


132 


225 


231 


135 


232 


153 


323 


300 


212 


160 


131 


122 


121 


290 


133 


130 


333 


182 


331 


240 


313 


305 


311 


281 


312 


262 


313 


155 


213 


191 


211 


226 


223 


266 


213 


334 


211 


369 


111 


129 


112 


138 


312 


159 


321 


300 


333 


208 


331 


150 


223 


151 


221 


188 


222 


240 


231 


233 


221 


276 


113 


267 


332 


159 


123 


210 


113 


182 


332 


278 


322 


355 


222 


338 


122 


154 


322 


143 


132 


186 


133 


209 


123 


247 


323 


323 


Block 
























totals 


1351 




1560 




1869 




2212 




2202 




2434 


122 


77 


213 


119 


233 


173 


133 


93 


131 


303 


132 


271 


313 


92 


112 


115 


132 


170 


321 


224 


221 


221 


331 


251 


133 


128 


333 


179 


113 


213 


332 


245 


213 


216 


211 


199 


321 


113 


123 


136 


323 


182 


313 


293 


232 


319 


121 


129 


231 


127 


311 


197 


312 


175 


122 


226 


311 


269 


323 


237 


111 


185 


131 


182 


121 


156 


231 


248 


123 


221 


113 


282 


212 


148 


*22 


212 


211 


183 


223 


269 


333 


359 


312 


303 


332 


215 


221 


120 


222 


138 


111 


228 


322 


319 


233 


300 


223 


132 


232 


162 


331 


192 


212 


326 


112 


314 


222 


247 


Block 
























totals 


1217 




1422 




1582 




2152 




2541 




2219 


231 


102 


131 


105 


121 


154 


223 


154 


112 


233 


132 


269 


212 


143 


333 


227 


233 


145 


133 


197 


311 


297 


323 


283 


313 


171 


123 


218 


312 


214 


332 


197 


232 


318 


312 


179 


321 


190 


112 


180 


331 


219 


321 


227 


131 


250 


233 


191 


111 


159 


213 


165 


211 


186 


212 


222 


221 


227 


121 


137 


122 


279 


232 


173 


132 


187 


111 


230 


123 


197 


331 


179 


223 


125 


311 


156 


323 


148 


231 


204 


322 


161 


211 


162 


332 


150 


322 


212 


222 


309 


122 


201 


213 


228 


113 


202 


133 


104 


221 


234 


113 


246 


313 


251 


333 


247 


222 


250 


Block 
























totals 


1423 




1670 




1808 




1883 




2158 




1852 


313 


124 


213 


276 


323 


276 


212 


228 


221 


260 


233 


388 


133 


136 


333 


269 


132 


225 


313 


295 


311 


166 


222 


228 


122 


297 


131 


255 


331 


343 


111 


304 


213 


309 


323 


244 


212 


265 


221 


164 


233 


145 


231 


212 


333 


246 


121 


324 


223 


209 


112 


264 


113 


258 


122 


320 


131 


309 


211 


421 


111 


180 


311 


277 


121 


194 


133 


325 


322 


252 


113 


344 


332 


259 


232 


283 


222 


280 


321 


464 


112 


335 


331 


336 


321 


215 


322 


259 


312 


304 


332 


376 


232 


247 


312 


249 


231 


262 


123 


243 


211 


285 


223 


410 


123 


269 


132 


360 


Block 
























totals 


1947 




2290 




2310 




2934 




2393 




2804 



174 



THE FIELD PLOT TEST 



TABLE 46 

TREATMENT TOTALS COLLECTED FROM TABLE 45 FOB CALCULATION 
OF SUMS OF SQUARES 



(a) Pi 



(6) 



tfi 
N t 
AT, 



(e) 



Pi 

N 3 ) P 2 



K,) 



AT, 



+ JVo + 



K* 



(a) 
(*>) 
to 



1,604 
2,031 
1,845 


1,529 
1,690 
1,912 


1,679 
1,523 
1,910 


4,812 
5,244 
5,667 


k 8 


5,480 


5,131 


5,112 


15,723 



KI KZ KZ 


5,524 
5,489 
5,637 


k - 8 
k -8 

k ** 24 
k =24 

k - 24 
k =24 


1,805 
1,651 
1,845 


1,826 
2,030 
1,913 


1,893 
1,808 
1,879 


5,301 


5,769 

#2 


5,580 

#3 


16,650 

5,057 
5,297 
5,596 


1,994 
1,838 
1,686 


1,741 
1,716 
1,993 


1,322 
1,743 
1,917 


5,518 


5,450 
K* 


4,982 

#3 


15,950 

15,393 
16,030 
16,900 


5,403 
5,520 
5,376 


5,096 
5,436 
5,818 


4,894 
5,074 
5,706 


16,299 


16,350 

#2 


15,674 

#3 


48,323 

15,723 
16,650 
15,950 


5,480 
5,301 
5,518 


5,131 
5,769 
5,450 


5,112 
5,580 
4,982 


16,299 
Pi 


16,350 


15,674 


48,323 

15,393 
16,030 
16,900 


4,812 
5,244 
5,667 


5,524 
5,489 
5,637 


5,057 
5,297 
5,596 


15,723 
Pi 


16,650 


15,950 

P3 


48,323 

16,299 
16,350 
15,674 


5,480 
5,131 
5,112 


5,301 
5,769 
5 t 580 


5,518 
5,450 
4,982 


15,723 

II 
16,303 
15,720 
16,300 


16,650 

III 
15,836 
16,506 
15,981 


15,950 

IV 

15,831 
15,764 
16,728 


48,323 

k - 72 



48,323 48,323 48,323 



INCOMPLETE BLOCK EXPERIMENTS 



175 



TABLE 47 

ANALYSIS or VARIANCE FOB 3X3X3 EXPERIMENT SHOWING SUMS OF SQUARES 
FOR INDIVIDUAL TREATMENT DEGREES OF FREEDOM 





DF 


SS 


SS 


DF 


MS 


F 


5% 
Point 


Blocks 


23 


548,407 


548,407 


23 


23,844 


8.83 


1.59 


N r .. . 


1 


15 771 1 












N d 


1 


126) 


15,897 


2 


7,948 


2.94 


3.05 


P r 


1 


358] 
i 












Pd 


1 


6,128] 


6,485 


2 


3,243 


1.20 


3.05 


K r 


1 


2,713) 












K d 


1 


1,223] 


3,936 


2 


1,968 


0.73 


3.05 


N r X P r . 


1 


1,040 












N r X Pd . . . 


1 


4,737 












N d X P r 


1 


16 


5,909 


4 


1,477 


55 


2.43 


N d X Pd 


1 


116 












N r X K r 


1 


7,332 












N r X K d 


1 


1,508 












NdX K r . . . - 


1 


1,765 


10,619 


4 


2,655 


0.98 


2.43 


N d XKd 

P r X K r 


1 
1 


13 
294 












P r X K d 


1 


1,850 












P d XK r 


1 


7,422 


11,357 


4 


2,839 


1.05 


2.43 


PdX Kd 


1 


1,791 












NXPX K... 
Error 


I 2 

2 
(2 

168 


3,131 
3,452 
8,048 
453,509 


14,631 
453,509 


6 

168 


2,438 
2,699 


90 


2.15 




















215 


1,070,750 













16. Incomplete Block Experiments. There are a number of different 
types of incomplete block experiments, and only those are described here 
that would seem to be of the greatest practical value in agronomic tests. 
The type which can probably be regarded as the most elementary is 
known as the two-dimensional quasi-factorial with two groups of sets. By 
extending this type to three groups of sets we have a somewhat greater 
degree of complexity, and this complexity continues to increase with the 
number of groups of sets until we reach the point of using all possible 
groups of sets, wherein the entire process of analysis suddenly becomes 
very much simplified. The latter type may be referred to as a symmet- 
rical incomplete block experiment. Quasi-factorial experiments of the 



176 THE FIELD FWT TUiST 

three-dimensional type are also possible, and one of the simplest of these 
will be described. 

In discussing the general principles involved in incomplete block ex- 
periments we shall consider an hypothetical experiment with only 9 
varieties. With such a small number of varieties it would probably not 
be worth while to use these methods, but a small example of this kind 
will be quite sufficient to illustrate the general principles. First, we 
take 9 numbers to represent the varieties and write them down in the 
form of a square. These are two-figure numbers, the first figure rep- 
resenting the row and the second the column of the square. 

11 12 13 
21 22 23 
31 32 33 

If we suppose now that this square represents, instead of 9 different 
varieties, 9 combinations of 2 factors at 3 levels as in a simple 3X3 
factorial experiment, the degrees of freedom can be divided as follows: 

A (factor for which levels are indicated by first figure of two-figure numbers) 2 DF 
B (factor for which levels are indicated by second figure of two-figure numbers) 2 DF 
A X B (interaction) 4 DF 

Furthermore, since the 4 DF for the interaction can be separated into 
two orthogonal components, each represented by 2 DF, the total of 
8 DF can be split up into 4 pairs. Then if the 9 combinations making 
up a complete replication are divided into 3 blocks, either one of the 
above pairs of degrees of freedom may be confounded with blocks. If 
we should decide to confound the A factor with blocks, the degrees of 
freedom for one replication will be apportioned as follows: 

Blocks 2 DF 

B 2DF 

AXB 4DF 

and the method of confounding would be merely to put the treatments 
together in the same block that occur in the rows of the square given 
above. Similarly the B factor may be confounded by putting the treat- 
ments in the same block that occur in the columns of the square. Then 
from our knowledge of the properties of a Latin square it is clear that 
if the interaction A X B is to be confounded it is only necessary to put 
the treatments together in the same block that occur in the diagonals 
of the square. In one replication we can confound only 2 out of the 4 



INCOMPLETE BLOCK EXPERIMENTS 177 

degrees of freedom. For example, in one replication the arrangement 
of the treatments in the blocks might be as follows: 

Block 1 11 22 33 

Block 2 21 32 13 

Block3 31 12 23 

and the degrees of freedom will be divided in the following manner: 

Blocks 2 DF 

A 2DF 

B 2 DF 

A X B 2DF 

Alternative to the above scheme 2 degrees of freedom from the inter- 
action may be confounded with blocks by this arrangement: 

Block 1 11 32 23 

Block 2 21 12 33 

Blocks 31 22 13 

Finally, it works out that in each replication a different pair of degrees 
of freedom may be confounded with blocks, in which case the analysis 
of variance will be of the following form: 

Blocks 11 DF 

A 2DF 

B 2DF 

AX B 4DF 

Error 16 DF 

By a process of partial confounding all the degrees of freedom for the 
9 treatment combinations can be recovered, and at the same time error 
control has been improved by the use of smaller blocks. The loss of 
information due to partial confounding is seen to be exactly ^, since 
each pair of degrees of freedom has been confounded in 1 replication and 
conserved in 3. In other words, both the main factors and the inter- 
action are determined with % of the precision that would have resulted 
if there had not been any confounding. The presumption, of course, 
is that the error will be sufficiently reduced by confounding to more 
than make up for the loss in precision. 

Returning now to the testing of 9 different varieties, it should be 
obvious that, if the varieties are designated by numbers and arranged 
in a square as above, we can go through the same procedure of partial 
confounding as has been outlined above for a 3 X 3 factorial experiment, 
and theoretically the same increase in accuracy due to confounding will 
be obtained. The method of analysis will also be clear from these con- 



178 THE FIELD PLOT TEST 

siderations, as we work it out in the first place as though it is a factorial 
experiment and, after finding the sums of squares for the imaginary 
factors and their interaction, we combine these to form the variety sum 
of squares. 

The fact that the variety numbers are first arranged in the form of a 
square simulating a two-factor experiment is the basis of the term " two- 
dimensional." The number of groups of sets is based on the number of 
groups of degrees of freedom that are confounded with blocks. In the 
quasi-factorial 3X3 experiment, for example, the 8 DF for the 9 treat- 
ments can be divided orthogonally into 4 pairs, and if we confound only 
2 of these pairs, the experiment is said to consist of " two groups of sets." 
With 9 varieties we have seen that 4 pairs of degrees of freedom can 
be confounded, in which case we might refer to the experiment as one 
with "four groups of sets," but as pointed out above it is usual to refer 
to experiments of this type as symmetrical incomplete block experiments. 

In a quasi-factorial experiment with only two groups of sets it will 
be obvious that all comparisons are not made with the same precision. 
Suppose, for example, that the blocks are made up out of the rows and 
columns of the square, in which case the analogous factorial experiment 
would be outlined as follows: 

Blocks 5 DF (assuming 2 replicates only) 

A 2DF 

B 2DF 

AX B 4 DF 

Error 4 DF 

In which the imaginary factors A and B are confounded in one replicate 
and conserved in the other, while the interaction A X -B is conserved 
in both replicates. The main factors A and B are determined with 
only ^ the precision with which the interaction is determined, and 
transferring these ideas to a variety experiment it becomes clear that the 
varieties that occur in the same row and in the same column are compared 
more accurately than those that do not occur at all in the same block. 
Another point that we should note here is that in estimating the result 
for any one treatment combination of the partially confounded factorial 
experiment, or of one variety in the quasi-factorial experiment, it will 
be necessary to make a correction for the blocks in which they occur. 
The actual totals are partially confounded with blocks. One variety 
may occur mainly in low-yielding blocks and another one in high- 
yielding blocks, and therefore the actual yield of the first variety must 
be increased and the yield of the second variety lowered, in order to 
make the two variety yields comparable. The details of this method 
of correction are given below. 



QUASI-FACTORIALS WITH TWO GROUPS OF SETS 179 

16. Two-Dimensional Quasi-Factorials with Two Groups of Sets. 
Assuming that only 9 varieties are to be tested, the first step is to take 
9 numbers to represent the varieties, as pointed out above, and arrange 
them in the form of a square. The next step is to arrange the varieties 
in sets according to the rows and columns of the square. These are 
given below and the first group of sets is referred to as group X and the 
second group of sets as group F. 

Group X Group Y 



11 12 13 11 21 31 



21 22 23 12 22 32 



31 ,32 33 13 23 33 



The varieties in the sets are those that are assigned to the incomplete 
blocks, and each group makes up a complete replication. The varieties 
occurring in the same block are, of course, those that are between the 
same set of parallel lines in the above figure. The groups can now be 
repeated as many times as we wish in order to bring up the replicates 
to the required number. The varieties are randomized within each 
block, but the blocks themselves may be placed in any order. 1 

Figure 11 illustrates diagrammatically the set up of the experiment 
assuming 4 complete replications. The yields may be arranged in a 
form somewhat similar to this for convenience in calculation. After 
setting up the original yields they must be combined for each group and 
then for both groups. The marginal totals are then obtained for each 
group and for both groups combined, and we are ready to proceed with 
the calculation of the sums of squares and the corrected variety means. 

The calculation of the variety sum of squares follows from the analogy 
to a factorial experiment. 

DF 

In Group Y A = 2(Yl)/np - Y 2 ./np* p~l 

In Group X B~ Z(X? f )/np - X 2 ./np* p-1 

Group X + Group Y(A X B) - 2(T2.)/2n - 2(Tl)/2np 

- Z(T? t )/2np + T?../2np* 



1 In certain cases the experimenter may decide, even after conducting the experi- 
ment as a quasi-factorial, to use the actual yields or some other character of the 
varieties, without correction. For example, he may wish to make quality or other 
testa on composite samples made up from ail the replicates. For this purpose it is 
somewhat better to have the incomplete blocks randomized within each replication. 



180 



THE FIELD PLOT TEST 



Group X 



11 12 13 



21 22 23 



31 32 33 



11 12 13 



21 22 23 



31 32 33 



Group Y 



11 


12 


13 




11 


12 


13 


21 


22 


23 




21 


22 


23 


31 


32 


33 




31 


32 


33 



X. v 



#11 12 13 


x, 


#21 #22 ^23 


*2. 


Xsi XS2 #33 


X 3 . 


y vr "\r- 

.A.I A .2 -A .3 


X.. 



Y. v 



yn 


^12 2/13 


Fi. 


2/21 


#22 2/23 


F 2 . 


2/31 


2/32 2/33 


Fj. 


F.! F.2 F. 3 


F.. 



T.. 

FIG. 11. Representation of a miniature example of a two-dimensional quasi-factorial 
experiment with two groups of sets. 

where p is the number of varieties in one set and n is the number of 
repetitions of each group. 

Yates (20) gives a direct method of calculating the sum of squares for 
varieties which is probably quicker than the one used above. Yates's 
formula is 

Varieties (SS) = Z(Tl)/2n+I,(X u . - F..) 2 /2np+S(Z., - F..) 2 /2np 
(X.. Y..) 2 /2np 2 [S(X.) + ^/(Y^^/np 

We next calculate the total sum of squares for all the plots and for the 
blocks, and obtain the error sum of squares by subtraction. The sum- 
marized analysis is of the form Df , 

Blocks 2np 1 

Varieties p 2 1 

Error (p l)(2np p - 1) 



QUA8I-FACTORIALS WITH TWO GROUPS OF SETS 181 

Just as in the factorial experiments that have been confounded all 
comparisons must be made within blocks. This means that to compare 
2 varieties directly we cannot use the actual variety totals but must 
prepare for these varieties ratings based on their behavior as compared to 
other varieties in the same blocks. The least squares method gives us 
as the best rating for any variety uv, the following expression which we 
shall refer to as a corrected variety mean. 

^ - 5= + ~ (X.. - F. v ) + -i- (F u . - X*.) 
2n 2np 2np 

If a large table of yields is to be corrected it may save time to set up the 
corresponding portions of the correction in the margins of the table. If 

we let C. v = ~ (X. v - F.,) and C. = ~ (Y v . - X u .), then C.i will be 
2np 2np 

the portion to be added to all the variety means in the first column, and 
Ci. will be the portion to be added to all the variety means in the first 
row. 

In this as in all other quasi-factorial arrangements the error variance 
must be multiplied by a factor depending on the type of experiment, to 
give the variance for comparing 2 varieties by their corrected means. 
If s 2 is the error variance, the variance of the difference between the 
corrected means of 2 varieties that occur in the same set is 



For 2 varieties not having a set in common the variance of the difference 



The mean variance of all comparisons is 



and when p is not too small we may use the latter variance for all com- 
parisons without appreciable error. 

Example 37. Two-Dimensional Quasi-Factorial with Two Groups of Sets. 
Using uniformity data and assuming a test of 25 varieties in 4 replications this 



182 



THE FIELD PLOT TEST 



example has been worked through in detail in order to show the methods of calcula- 
tion. Setting up first the specifications of the test: 

Varieties in each set (p) 5 

Varieties (v) - p 2 = 25 

Sets () * 2p =10 

Replications of each group (n) * 2 

Replications (r) 2n = 4 

Blocks (6) = 2np =20 

Total number of plots (N) 2wp 2 =100 

The variety numbers are first written down in the form of a square: 



11 
21 
31 
41 
51 



12 
22 
32 
42 
52 



13 
23 
33 
43 
53 



14 
24 
34 
44 
54 



15 
25 
35 
45 
55 



and the 10 sets in 2 groups of 5 taken from the rows and columns of the square. The 
varieties in these sets are then randomized in the blocks as indicated in Table 49. 
Here the groups are repeated twice so that (n 2) and (r ** 4), and the groups are 
separated in the field. It might be wise if there is a marked difference in variability 
in different parts of the field to randomize the blocks over the whole field instead of 
keeping them together as complete replications, but in general this would seem to be 
unnecessary and it is a decided convenience from the standpoint of making observa- 
tions on the plots to have all the plots in one replication together. 

After obtaining the block totals and the grand total the next step is to set up 
Table 50, the construction of which should present no difficulty. Note that the 
marginal totals X Vf and Y, v are those in which variety and block effects are con- 
founded. 

By the shortest method the sum of squares for varieties is calculated as follows 

Z(Ti)/2n - 1,961,637.50 

X(X U . - F v .) 2 /2np - 81,162.50 

2(X. V - Y. 9 )*/2np - 117,817.50 

-(Z-. - F..) 2 /2np 2 -- 61,076.50 

+ S(F.J)} /np - -2,058,800.00 (Groups -f Sets -f Mean) 



Total = Varieties (SS) . 



50,741.50 



The total sum of squares for all plots is 630,266.00 and for blocks is 467,586.00. 
Having obtained these, we can set up the analysis of variance. 

TABLE 48 

ANALYSIS OF VARIANCE 
Two DIMENSIONAL QUASI-FACTORIAL Two GROUPS OF SETS 





SS 


DF 


MS 


F 


5% Point 


Blocks 


467,586.00 


19 


24,609.8 


12.3 


1.78 


Varieties ....'... 


50,741.50 


24 


2,114.2 


1.06 


1.72 


Error 


111,938.50 


56 


1,998.9 


















Total 


630,266.00 


99 





















QUASI-FACTORIALS WITH TWO GROUPS OF SETS 183 

In order to obtain the corrected variety yields we calculate 

C. v - ^ (JT.. - Y. v ) for 9 - 1, 2, 3, 4, 5 



C. - ~ (F. - X u .) for u - 1, 2, 3, 4, 5 

These are entered in the margins of a (5 X 5) table as in Table 51 and added to the 
actual means of corresponding cells in the table. 

To obtain a further check on the sums of squares for varieties we can now calculate 
it in another way using the formula 

Varieties (SS) **2(tu V -T uv ) - S(f w .-Z.) - Sfl.,, F.,) 

where ti. t for example, is the mean of all the l uv values in the first row of Table 51 
and LI is the mean of the first column. 

To make comparisons between the corrected means we may if we wish to be exact 
take into consideration whether or not the varieties being compared occur in the same 
set. To compare varieties 21 and 22, for example, we calculate the variance accord- 
ing to the formula 



F<fc-W-;^;-^y-X=)-lM 



SEfa - IK) = V 1199.3 = 34.63 

_ 161.50 - 123.75 
' 34.63 - 1 ' 99 

To compare varieties 11 and 54 we would have 

38.Q 7\ 

- 1399.23 




'1399.23 =37.41 

135.25 - 170.25 ^ 
37.41 



0.94 



We would obviously not be very far wrong, even with a p value as low as 5, to use 
for all comparisons the mean variance for the difference between 2 varieties. This 
would be 



- 1332.6 
n 



5^ m - Vl332.6 - 36.50 

* The t used here is, of course, the statistic defined by R. A. Fisher in "Statistical 
Methods for Research Workers. 11 



184 



THE FIELD PLOT TEST 



TABLE 49 

POSITION OF VARIETIES IN THE FIELD AND CORRESPONDING PLOT YIELDS. 
TWO-DIMENSIONAL QUASI-FACTORIAL EXPERIMENT 
WITH Two GROUPS OF SETS 



Set 
No. 


Vari- 
ety 
No. 


Yield 


Vari- 
ety 
No. 


Yield 


Vari- 
ety 
No. 


Yield 


Vari- 
ety 
No. 


Yield 


Vari- 
ety 
No. 


Yield 


Block 
Totals 


iy 


31 


215 


21 


300 


51 


255 


41 


185 


11 


145 


1,100 


2y 


22 


150 


12 


50 


52 


45 


32 


105 


42 


155 


505 


5y 


55 


125 


35 


30 


15 


65 


25 


130 


45 


55 


405 


4y 


14 


85 


34 


55 


54 


110 


24 


130 


44 


40 


420 


3y 


53 


45 


43 


45 


13 


60 


23 


15 


33 


-5 


160 


iy 


11 


210 


21 


290 


41 


325 


31 


230 


51 


220 


1,275 


2y 


12 


310 


32 


230 


22 


155 


52 


195 


42 


245 


1,135 


% 


15 


315 


45 


215 


55 


160 


25 


285 


35 


230 


1,205 


3y 


63 


185 


43 


220 


33 


175 


13 


275 


23 


185 


1,040 


4y 


14 


130 


24 


190 


34 


160 


44 


110 


54 


155 


745 


lx 


14 


140 


15 


165 


11 


265 


13 


150 


12 


180 


900 


4x 


41 


190 


42 


135 


45 


100 


43 


145 


44 


205 


775 


3x 


33 


250 


31 


150 


35 


150 


34 


195 


32 


155 


900 


2x 


22 


75 


21 


105 


25 


130 


23 


180 


24 


90 


580 


5x 


55 


40 


54 


155 


53 


65 


52 


60 


51 


40 


360 


5x 


55 


115 


54 


185 


53 


240 


51 


120 


52 


125 


785 


lx 


11 


145 


13 


105 


14 


50 


15 


130 


12 


135 


565 


3x 


32 


150 


33 


115 


34 


60 


35 


110 


31 


25 


460 


2x 


21 


5 


24 


65 


25 


70 


23 


60 


22 


20 


220 


4x 


41 


30 


42 


50 


43 


35 


45 


20 


44 


50 


185 


















Grand T 

II 


otal - 


13,720 



TABLE 50 

YIELDS OF VARIETIES BY GROUPS, AND TOTAL YIELDS FOR BOTH GROUPS 

Values of x* v 



Group X 



\ v 
u \ 


1 


2 


3 


4 


5 


Xu. 


1 

2 
3 
4 
5 


410 
110 
175 
220 
160 


315 
95 
305 
185 
J85 


255 
240 
365 
180 
305 


190 
155 
255 
255 
340 


295 
200 
260 
120 
155 


1,465 
800 
1,360 
960 
1,145 


X. v 1075 


1085 


1345 


1195 


1030 


5,730 - 



QUASr-FACTORIALS WITH THREE GROUPS OF SETS 



185 



TABLE 50 Continued 
Values of y uv 



Group 7 



X 

u \ 


1 


2 


3 


4 


5 


Y n 


1 

2 
3 
4 
5 


355 
590 
445 
510 
475 


360 
305 
335 
400 
240 


335 
200 
170 
265 
230 


215 
320 
215 
150 
265 


380 
415 
260 
270 
285 


1,645 
1,830 
1,425 
1,595 
1,495 



Y. v 2375 1640 1200 1165 1610 | 7,990 - F., 
Values of T w 



Group X 

+ 
Group Y 



X 

u \ 


1 


2 


3 


4 


5 


T*. 


1 

2 
3 
4 
5 


765 
700 
620 
730 
635 


675 
400 
640 
585 
425 


590 
440 
535 
445 
535 


405 
475 
470 
405 
605 


675 
615 
520 
390 
440 


3,110 
2,630 
2,785 
2,555 
2,640 


T. 3450 


2725 


2545 


2360 


2640 


13,720 - 



1 

2 
3 
4 
5 



X. v - Y. v 

-1300 

- 555 
145 

30 

- 580 



~X U . 



180 

1030 

65 

635 

350 



(X.. - F..) -2260 



(Y.. - X..) - 2260 



TABLE 51 
CALCULATION OF CORRECTED VARIETY MEANS (*) 



\ 

tt\ 


1 


2 


3 


4 


5 


C tt . 


1 


135.25 


150.00 


163.75 


111.75 


148.75 


9.00 


2 


161.50 


123.75 


168.75 


171.75 


176.25 


51.50 


3 


93.25 


135.50 


144.25 


122.25 


104.25 


3.25 


4 


149.25 


150.25 


150.25 


134.50 


100.25 


31.75 


5 


111.25 


96.00 


168.50 


170.25 


98.50 


17.50 



C. v -65.00 -27.75 7.25 1.50 -29.00 

C.i - -1300/20 - -65.00 
Ci. - 180/20 - 9.00 

17. Two-Dimensional Quasi-Factorials with Three Groups of Sets. 

A possible criticism of the quasi-factorial method with two groups of 
sets as described above is that there is too great a discrepancy between 
the estimates of the error variance for comparing varieties in the same 
and in different sets. This can be partly overcome by increasing the 



186 THE FIELD PLOT TEST 

number of groups, and hence the type with three groups of sets is theo- 
retically an improvement over the previous type. It requires, however, 
more computation, and the number of replications must be a multiple 
of 3. Details for setting up and analyzing such experiments may be 
found in the reference of Yates (20). 

18. Three-Dimensional Quasi-Factorial^ with Three Groups of Sets. 
In the two-dimensional types the varieties were represented by two- 
figure numbers corresponding to the two dimensions of a square. In 
the three-dimensional types the varieties are represented by three-figure 
numbers (uvw) corresponding to the three dimensions of a cube. Thus 
in a cube with p numbers on a side we can represent jfl varieties, and 
taking these numbers in sets of p by slicing in three directions we can 
make up 3p 2 sets. There will be three groups of p 2 sets, each one cor- 
responding to a direction in which the cube is sliced. At this point 
the student should draw up a cube, put in the numbers, and practice 
writing out the sets. It will then be noted that the sets can be written 
out directly for any value of p by expanding the sets given below for 
p = 3. 

When the number of varieties is very large, say 216 or more, there 
are decided advantages in using this type of experiment, as with any 
other type the blocks would still be rather large. 

The details of setting up and analyzing a three-dimensional experi- 
ment may be obtained from Example 38. 

Example 38. Three-Dimensional Quasi-Factorial Experiment with Three Groups 
of Sets. The specifications are: 

Varieties (v) p* = 27 

Sets () - 3p* - 27 

Replications of each group (n) 2 

Complete replications (r) = 3n 6 

Total number of blocks (6) = 3np* - 64 
Total number of plots (N) - 3np - 162 

After forming the (3X3X3) cube we can write out the sets as follows: 

Group X ( vw) Group Y (u - w) Group Z (uv - ) 

Set No. Set No. Set No. 

1 111 211 311 1 111 121 131 1 111 112 113 

2 112 212 312 2 211 221 231 2 121 122 123 

3 113 213 313 3 311 321 331 3 131 132 133 



4 


121 


221 


321 


4 


112 


122 


132 


4 


211 


212 


213 


5 


122 


222 


322 


5 


212 


222 


232 


5 


221 


222 


223 


6 


123 


223 


323 


6 


312 


322 


332 


6 


231 


232 


233 


7 


131 


231 


331 


7 


113 


123 


133 


7 


311 


312 


313 


g 


132 


232 


332 


8 


213 


223 


233 


8 


321 


322 


323 


9 


133 


233 


333 


9 


313 


323 


333 


9 


331 


332 


333 



QUASI-FACTORIALS WITH THREE GROUPS OF SETS 187 

After the distribution of the blocks over the field and the randomization of the 
varieties within the blocks we have such an arrangement as is shown in Table 53, 
in which the individual plot yields corresponding to the varieties are given. In this 
case the blocks are distributed at random over the whole field, but it would have been 
more convenient to keep them together in complete replications. 

The calculations are carried out in tabular form in Table 54. The data are first 
collected by groups so that the yield of any one variety in one group will be a total 
of n plots. The marginal totals are obtained as indicated in three directions, and it 
will be noted that X. w , F M . W , and Z uv . represent the totals for the sets. The complete 
variety totals represented by T uv are entered next and all the marginal totals of these 
obtained. 

For calculating the corrected variety means ft,**) the most convenient formula is 

T 

tuvw ** *7| h C'-tw "I" Cv'W + Cuv 
on 

where 

- T. v . -{- 3F. V .) 



(pT u *v> 3pF w .tt> T. . w -f- 3Z. . 



wip~ 
Thus 

C.u - (3 X 2735 - 9 X 340 - 9875 -f 3 X 3635) - 57.176 

M - -;L ( 3 X 3330 - 9 X 1385 - 9645 + 3 X 3105) - -25.972 
108 

Cn. = ~~ (3 X 3305 - 9 X 1185 - 9470 + 3 X 3180) = - 6.296 

Having obtained all the correction terms, we check by obtaining the total, which in 
this case comes to +0.001. This is a sufficiently close check. 

The corrected means are obtained by adding the corresponding correction terms 
to the actual means. For example, ui- 151.667 +57.176 -25.972- 6.296 176.575. 

To obtain the sum of squares for varieties we first average the corrected means in 
three directions to give t.w, t v . w , and *. To illustrate this: 

<.ii - | (176.575 + 190.001 + 164.723) - 177.100 
h-i - I (176.575 + 192.222 + 224.028) - 197.608 
hi- - $ (176.575 + 180.556 + 197.917) - 185.016 
The sum of squares for varieties is then given by 

\rn<M*s>4t*\<M f Q Qf\ -L ^F*/f T* % ^PfY 4 \ *$* fV 4 \ V/5P l \ 

Varieties (p>) * (t u vw -luvw) Zt(JL. V v't"w>) ~~ ^i^u-wht-tp; (i uv .-t uv .) 

which in this case is 

5,847,432.06 - 5,754,971.44 - 92,460.62 



188 



THE FIELD PLOT TEST 



Then after calculating the total and block sum of squares from Table 53, we can set 
up the analysis of variance. 

TABLE 52 
ANALYSIS OF VARIANCE 

THREE-DIMENSIONAL QUASI-FACTORIAL EXPERIMENT 
WITH THREE GROUPS OF SETS 





SS 


DF 


MS 


F 


5% Point 


Blocks 


1,154,025 


53 








Varieties . 


92,461 


26 


3556 


1 23 


1 62 


Error 


236,872 


82 


2889 


















Total 


1,483,358 


161 





















The variances and standard errors for comparing the varieties are as follows. 
It will be noted that such comparisons now fall into three groups that can be de- 
termined from the variety numbers. 



2s 2 

5 

.2 
_ 



2X2889 



X 13 = 1391 



2889 


2889 



X 31 =1658 



** 37.30 
= 40.72 
~ 42.02 



-, (2p*+3p+6) - r X 33 - 1766 
onp 64 

And the mean variance of all comparisons is 

^X^-1630 



19. Symmetrical Incomplete Block Experiments. It will be remem- 
bered from the discussion of Section 15 that, if all the possible groups of 
degrees of freedom are not confounded, certain of the comparisons are 
determined with less precision than others. For this reason in using 
the quasi-factorials we have two or more standard errors depending on 
the " dimensions " of the experiment. This difficulty can be overcome 
by confounding all the possible groups of degrees of freedom or in other 
words by using all the possible groups of sets. We then have a design 
that is perfectly symmetrical and not only do we have equal precision 
for all comparisons but also the calculations are considerably simplified. 

The chief problem in setting up the design of a symmetrical experi- 
ment is in writing out the sets. For this purpose we can conveniently 



SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 189 



TABLE 63 

POSITION OP VARIETIES IN THE FIELD AND CORRESPONDING PLOT YIELDS THREE- 
DIMENSIONAL QUASI-FACTORIAL EXPERIMENT WITH THREE GROUPS OF SETS 



Set 
No. 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Block 
Totals 


Set 
No. 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Block 
Totals 


2x 


212 


315 


312 


370 


112 


360 


1045 


4y 


122 


195 


112 


310 


132 


315 


820 


5y 


222 


265 


232 


355 


212 


345 


965 


6s 


233 


215 


231 


330 


232 


270 


815 


6y 


322 


245 


312 


185 


332 


160 


590 


9y 


333 


290 


313 


95 


323 


140 


525 


8y 


223 


285 


233 


355 


213 


240 


880 


7x 


231 


330 


131 


410 


331 


235 


975 


2y 


211 


325 


221 


315 


231 


300 


940 


6y 


312 


255 


322 


375 


332 


305 


935 


5z 


122 


240 


322 * 


220 


222 


350 


810 


4x 


121 


255 


321 


235 


221 


230 


720 


2x 


212 


360 


312 


230 


112 


225 


815 


5y 


232 


275 


222 


245 


212 


140 


660 


3y 


331 


270 


311 


255 


321 


170 


695 


5s 


223 


270 


222 


230 


221 


135 


635 


Ox 


323 


175 


123 


290 


223 


330 


795 


5s 


222 


95 


221 


245 


223 


330 


670 


6x 


323 


180 


123 


275 


223 


290 


745 


9s 


332 


215 


333 


300 


331 


255 


770 


3y 


321 


155 


331 


180 


311 


160 


495 


3x 


213 


185 


313 


145 


113 


150 


480 


9y 


323 


120 


313 


70 


333 


100 


290 


9x 


333 


50 


133 


45 


233 


105 


200 


7y 


113 


100 


123 


170 


133 


65 


335 


8s 


322 


155 


323 


125 


321 


30 


310 


to 


233 


55 


333 


145 


133 


40 


240 


7x 


131 


65 


331 


130 


231 


55 


250 


Ix 


111 


35 


311 


45 


211 


55 


135 


2s 


122 


85 


123 


55 


121 


110 


250 


Tjr 


123 


140 


133 


45 


113 


15 


200 


9s 


331 


130 


332 


40 


333 


45 


215 


ti 


111 


85 


211 


65 


311 


55 


205 


3s 


131 


45 


132 


60 


133 


15 


120 


IB 


112 


80 


111 


115 


113 


165 


360 


3x 


313 





213 


70 


113 


65 


135 


ly 


121 


180 


111 


265 


131 


290 


725 


8y 


223 


285 


213 


270 


233 


185 


740 


fix 


222 


150 


122 


55 


322 


50 


255 


ly 


111 


210 


131 


265 


121 


185 


660 


8x 


332 


130 


132 


215 


232 


155 


500 


2y 


211 


95 


221 


95 


231 


155 


345 


4x 


121 


210 


221 


90 


321 


95 


395 


4s 


213 


160 


212 


140 


211 


125 


42ft 


7s 


311 


140 


312 


195 


313 


310 


645 


Is 


111 


210 


112 


290 


113 


325 


82ft 


4y 


132 


230 


122 


220 


112 


310 


760 


4s 


211 


230 


213 


155 


212 


195 


580 


3s 


132 


245 


133 


315 


131 


215 


775 


8x 


132 


160 


232 


285 


332 


230 


675 


6s 


232 


185 


233 


220 


231 


175 


580 


7s 


311 


275 


313 


185 


312 


130 


590 


2s 


121 


190 


122 


160 


123 


110 


460 


8s 


323 


155 


321 


150 


322 


240 


545 



divide such experiments into two types: (1) where the number of vari- 
eties (v) = p 2 ; and (2) where v = p 2 p + 1. There are, of course, 
other types, but the two mentioned are likely to be of the most value in 
field experiments. Considering the first type, (v = p 2 ), it is obvious 
that the variety numbers can be written in the form of a square. Sup- 
pose that we have 9 varieties; then the square is 



11 
21 
31 



12 
22 
32 



13 
23 
33 



The first two groups of sets are written as for a two-dimensional quasi- 
factorial, from the rows and Columns of the square. Two more groups 
may then be written from the diagonals of the above square. These are 



11 
21 
31 



22 
32 
12 



33 
13 
23 



11 
21 
31 



32 
12 
22 



23 
33 
13 



190 



THE FIELD PLOT TEST 



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SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 191 

the second one being written from the diagonals of the first. This must 
be all the groups, as we know from a study of the degrees of freedom in 
a Latin square, and also from the fact that, if we repeat the process on 
the last square written, the original square is regenerated. The maxi- 
mum number of groups that can be written is always p + 1. On exam- 
ining these sets we note that each variety occurs once and once only in 
the same set with any other variety. Taking variety 11 the sets in 
which it occurs are 

(11 12 13), (11 21 31), (11 22 33), (11 32 23), 

and in these four sets all the other varieties have occurred once. 

If p is a prime number the above method of writing out the sets will 
work for the type (v = p 2 ). If p is not a prime number we must mak^ 
use of a completely orthogonalized square, if such a square can be pre- 
pared. For p = 6 the orthogonalized square is impossible, so that we 
cannot write more than three groups of sets. This is the same as saying 
that a Latin square is possible for any number of rows and columns, 
but Graeco-Latin squares are impossible for certain numbers, Fisher (2). 
A completely orthogonalized 4X4 square is given below, and further 
squares are given in R. A. Fisher's "Design of Experiments," 1937. 

Completely Orthogonalized 4X4 Square 
111 234 342 423 
222 143 431 314 
333 412 124 241 
444 321 213 132 

This square may be used to show how the sets for 16 varieties can be 
made up. 

The first two groups of sets are obtained from the rows and columns 
of the square of variety numbers in the usual way, and the orthogonalized 
square is used to write out the remaining groups. Assuming that the 
square of variety numbers is as follows : 

11 12 13 14 

21 22 23 24 

31 32 33 34 

41 42 43 44 

and is superimposed on the orthogonalized square, we note, considering 
the first of the three-digit numbers only, that 1 corresponds with the 
variety numbers 11, 22, 33, 44; 2 with the numbers 21, 12, 43, 34; 3 
with 31, 42, 13, 24; and 4 with 41, 32, 23, 14. These are the sets for the 
third group, and we make up two more groups by using the second and 
third figures of the orthogonalized square. 



192 THE FIELD PLOT TEST 

To write the sets for the type v = p 2 p + 1, it is only necessary 
to modify the above procedure. Suppose that t; = 13; then p = 4 and 
p 1 = 3. A convenient method of designating the varieties is as 
follows: 

01 02 03 04 
11 12 13 
21 22 23 
31 32 33, 

and if the sets are written for the 9 numbers in the square, the sets for the 
13 varieties are obtained by making one set out of 01, 02, 03, 04, and the 
remaining sets by adding one of these to the sets of each group formed 
by the other 9 numbers. The sets finally are as follows : 



01 02 03 04 
01 11 12 13 
01 21 22 23 
01 SI 32 33 


02 11 21 31 
02 12 22 32 
02 13 23 33 


03 11 22 33 
03 21 32 13 
03 31 12 23 


04 11 32 23 
04 21 12 33 
04 31 22 13 



If the number of varieties is 21, the numbers would be written out as 
below: 

01 02 03 04 05 

11 12 13 14 

21 22 23 24 

31 32 33 34 

41 42 43 44 

and we would have to use a completely orthogonalized 4X4 square in 
order to make up the 20 sets for the 16 numbers in the square, to which 
the remaining numbers would be added as described above. 

Special mention should be made of the fact that, as the sets are 
written out by the methods described above for the v = p 2 p + 1 
type, the blocks cannot be arranged so that they form complete replica- 
tions. There is a method of making up the sets (Youden's square) by 
means of which all the blocks are placed side by side and all the plots in a 
single row from one end of the field to the other would form a complete 
replication. This method is likely to be of considerable value in labora- 
tory experiments, but in field plot experiments it is not likely that the 
long narrow strips one plot wide would be of any value in error control. 

Example 39. A Symmetrical Incomplete Block Experiment for 25 Varieties and 
6 Replications* The sets have been written out by the method described above, and 
those for each group have been kept together to form complete replications. This 
will be obvious from Table 55, and it will be noted also that no attempt has been 
made to randomize the blocks* All the randomization is of the varieties within 
blocks. It is convenient to enter on the plan of the field the individual yields and 



SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 193 

the block totals. The variety totals are obtained by collecting the individual yields 
as in Table 56. These are denoted by T uv . The figures in the column headed 
are obtained by adding for any one variety the totals for all the blocks in which that 
variety occurs. Thus from Table 55 for variety 11 we have 

Sn - 257 + 181 4- 177 + 265 + 271 + 303 - 1454 

The second last column is obtained as indicated, and this can be checked by adding, 
as the total for all the (pT uv - 2 ttt) ) values is zero. The last column gives the cor- 
rected variety means (*) which are given by the formula 



where m is the general mean of the whole experiment and v is the number of varieties. 
The sum of squares for varieties is given simply by 

Varieties <SS) - 



The analysis of variance can then be set up as at the foot of Table 56. The 
method is also given for calculating the variance of a difference between two cor- 
rected means. The general formula is 

V m - (-^ ) 
r \ p / 

where r is the number of replications. 



194 



THE FIELD PLOT TEST 



TABLE 55 

LOCATION OF THE VARIETIES IN THE FIELD AND CORRESPONDING YIELDS. SYM- 
METRICAL INCOMPLETE BLOCK EXPERIMENT FOR 31 VARIETIES AND 6 REPLICATIONS 

Replicate VI 



Plot No. 
Variety 
Yields 


12345 
52 34 25 11 43 
57 52 38 60 50 


6 7 8 9 10 
12 44 35 21 53 
31 31 28 32 24 


11 12 13 14 15 
31 13 54 22 45 
24 40 19 20 30 


16 17 IS 19 20 
41 14 32 23 55 
40 35 32 19 36 


21 22 23 24 25 
33 15 51 42 24 
36 44 46 57 68 


Block 
totals 


257 


146 


133 


162 


251 



Replicate V 



Replicate total = 949 



Plot No. 
Variety 
Yields 


12346 
35 23 42 54 11 
54 39 28 40 20 


6 7 8 9 10 
33 52 14 21 45 
14 11 10 24 19 


11 12 13 14 15 
55 12 43 24 31 
30 42 32 28 30 


16 37 18 19 20 
41 53 15 34 22 
32 38 26 16 20 


21 22 23 24 25 
25 44 32 13 51 
19 24 8 12 26 


Block 
totals 


181 


78 


162 


132 


89 



Replicate IV 



Replicate total = 642 



Plot No. 
Variety 
Yields 
Block 
totals 



12345 


6 7 8 9 10 


11 12 13 14 15 


16 17 18 19 20 


21 22 23 24 25 


32 24 45 53 11 


34 13 21 42 55 


52 23 15 31 44 


54 33 12 41 25 


35 22 51 43 14 


57 39 25 32 24 


7 23 18 24 24 


30 42 16 16 23 


25 39 35 18 21 


20 23 15 16 27 


177 


96 


127 


138 


101 



Replicate III 



Replicate total = 639 



Plot No. 


12345 


6 7 8 9 10 


11 12 13 14 15 


16 17 18 19 20 


21 22 23 24 25 


Variety 


33 44 22 55 11 


15 32 43 54 21 


14 53 31 25 42 


13 35 24 52 41 


12 23 34 51 45 


Yields 


74 57 34 49 51 


45 43 31 44 40 


41 36 28 8 16 


19 30 23 22 35 


25 31 19 23 27 


Block 












totals 


265 


203 


129 


129 


125 



Replicate II 



Replicate total = 851 



Plot No. 
Variety 
Yields 
Block 
totals 



12345 


6 7 8 9 10 


11 12 13 14 15 


16 17 18 19 20 


21 22 23 24 25 


11 31 51 21 41 


22 12 42 52 32 


13 53 43 33 23 


14 44 54 24 34 


45 55 35 15 25 


52 57 40 79 43 


36 33 24 44 32 


22 27 11 18 32 


37 29 24 37 32 


22 19 21 29 26 


271 


169 


110 


159 


117 



Replicate I 



Replicate total - 826 



Plot No. 


12345 


6 7 8 9 10 


11 12 13 14 15 


16 17 18 19 20 


21 22 23 24 25 


Variety 


12 14 15 13 11 


23 22 21 24 25 


34 35 32 31 33 


43 44 41 45 42 


53 51 52 55 54 


Yields 


74 65 54 66 44 


48 57 37 44 55 


33 35 37 38 30 


46 72 57 62 89 


76 54 55 75 84 


Block 












totals 


303 


241 


173 


326 


344 



Replicate total - 1387 
Grand total = 5294 



SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 195 



TABLE 56 

YIELDS OF SINGLE PLOTS BY VARIETIES, VARIETY TOTALS, VALUES OF 2 wt , AND THE 

CORRECTED MEANS ftn>). SYMMETRICAL INCOMPLETE BLOCK EXPERIMENT FOR 25 

VARIETIES AND 6 REPLICATIONS 



Vari- 


VI 


V 


IV 


III 


II 


I 


T w 


2r 


pT UV 2,1,; 


*.- 
pT uv 2 . 


ties 




















V 


11 


60 


20 


24 


51 


52 


44 


251 


1,454 


-199 


27.33 


12 


31 


42 


35 


25 


33 


74 


240 


1,043 


157 


41.57 


13 


40 


12 


23 


19 


22 


66 


182 


860 


50 


37.79 


14 


35 


10 


27 


41 


37 


65 


215 


932 


143 


41.01 


15 


44 


26 


16 


45 


29 


54 


214 


1,133 


-63 


32.77 


21 


32 


24 


18 


40 


79 


37 


230 


1,035 


115 


39.89 


22 


20 


20 


23 


34 


36 


57 


190 


1,041 


-91 


31.65 


23 


19 


39 


42 


31 


32 


48 


211 


946 


109 


39 65 


24 


68 


28 


39 


23 


37 


44 


239 


1,119 


76 


38.33 


25 


38 


19 


21 


8 


26 


55 


167 


971 


-136 


29.85 


31 


24 


30 


16 


28 


57 


38 


193 


995 


-30 


34.09 


32 


32 


8 


57 


43 


32 


37 


209 


973 


72 


38.17 


33 


36 


14 


39 


74 


18 


30 


211 


1,015 


40 


36 89 


34 


52 


16 


7 


19 


32 


33 


159 


942 


-147 


29 41 


35 


28 


54 


20 


30 


21 


35 


188 


847 


93 


39 01 


41 


40 


32 


18 


35 


43 


57 


225 


1,158 


-33 


33.97 


42 


57 


28 


24 


16 


24 


89 


238 


,152 


38 


36.81 


43 


50 


32 


16 


31 


11 


46 


186 


,159 


-229 


26.13 


44 


31 


24 


23 


57 


29 


72 


236 


,112 


68 


38.01 


45 


30 


19 


25 


27 


22 


62 


185 


956 


-31 


34 05 


51 


46 


11 


15 


23 


40 


54 


189 


,181 


-236 


25.85 


52 


57 


26 


30 


22 


44 


55 


234 


,104 


66 


37 93 


53 


24 


38 


32 


36 


27 


76 


233 


,038 


127 


40 37 


54 


19 


40 


25 


44 


24 


84 


236 


,158 


22 


36.17 


55 


36 


30 


24 


49 


19 


75 


233 


,146 


19 


36 05 


Totals 


949 


642 


639 


851 


826 


387 


5294 


26,470 








pv 



324,354 
125 



= 2594.83 



Replications 2 (a: 2 ) 

I 83,531 

II 32,228 

III 34,039 

IV 19,029 
V 19,568 

VI 40,367 

Total = 228,762.00 
CT 186,842.91 



5294 
150 



S(7V)/5 
CT 

Blocks 



35.29 



1,088,496/5 



217,699.20 
186,842.91 

30,856 29 



= 41,919.09 



Analysis of Variance 





SS 


DF 


MS 


F 


5% PL 


Blocks 
Varieties. . . 
Error 


30,856.29 
2,594.83 
8 467 97 


29 
24 
96 


108.12 
88 21 


1.22 


1.63 














Total.... 


41,919.09 


149 









196 



THE FIELD PLOT TEST 
2 X 88.21 



Vmd - 



X() -36.28 



5.94 



I 



Example 40. A Symmetrical Incomplete Block Experiment for 31 Varieties in 
6 Replications. The sets were written out by setting up the variety numbers as 
follows: 

01 02 03 04 05 06 



11 


12- 


13 


14 


15 


21 


22 


23 


24 


25 


31 


32 


33 


34 


35 


41 


42 


43 


44 


45 


51 


52 


53 


54 


55, 



writing out the 6 groups of sets for the 5X5 square and adding, to each, one of the 
numbers in the first row. An additional set was then made up from the numbers in 
the first row, giving 31 sets in all. The blocks were arranged as indicated in Table 
58, after randomizing the varieties within the blocks. The variety totals are collected 
as in Table 59, and it is convenient for this purpose and for obtaining the values of 
Z vv to make up a table similar to Table 60 giving the sets with their corresponding 
numbers and block totals. Then, to collect the yields of, say, variety 23, we can 
locate it in each group, note the numbers of the sets, and then proceed from the 
table of individual yields to obtain the total. Similarly to obtain ^23 we add the block 
totals in the same line as 23 throughout the table. 

From this point the calculations are exactly as in Example 39 for 25 varieties, 
except that, since this experiment is of the v = p 2 p + I type, the variance for 
the difference between two corrected variety means is 



V m 1 



r \P*-P 
The analysis of variance is given in Table 57. 



TABLE 57 

ANALYSIS OP VARIANCB 
INCOMPLETE BLOCK EXPERIMENT FOR 31 VARIETIES IN 6 REPLICATIONS 





SS 


DF 


MS 


F 


5% Point 


Blocks 


1,083,491 


30 


36,116 


10.5 


1 53 


Varieties 


103,977 


30 


3,466 


1.01 


1.53 


Error 


429,756 


125 


3,438 


















Total :... 


1,617,224 


185 





















SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 197 



TABLE 58 

LOCATION OF THE VARIETIES IN THE FIELD, CORRESPONDING PLOT YIELDS, AND 
BLOCK TOTALS. SYMMETRICAL INCOMPLETE BLOCK EXPERIMENT WITH 31 VARIETIES 

AND 6 REPLICATIONS 



Set 
No. 


Vari 
ety 


Yield 


Vari 
ety 


Yield 


Vari 
ety 


Yield 


Vari 
ety 


Yield 


Vari 
ety 


Yield 


Vari 
ety 


Yield 


Block 
Totals 


1 


11 


315 


13 


370 


01 


360 


14 


265 


12 


355 


15 


345 


2,010 


2 


23 


245 


22 


185 


21 


160 


01 


285 


24 


355 


25 


240 


1,470 


3 


01 


325 


33 


315 


32 


300 


35 


240 


31 


220 


34 


350 


1,750 


4 


45 


360 


43 


230 


42 


225 


01 


270 


41 


255 


44 


170 


1,510 


5 


01 


175 


53 


290 


51 


330 


54 


220 


52 


220 


55 


265 


1,500 


6 


31 


195 


11 


310 


21 


315 


02 


215 


41 


330 


51 


270 


1,635 


7 


22 


290 


52 


95 


02 


140 


32 


330 


12 


410 


42 


235 


1,500 


8 


13 


255 


23 


375 


43 


305 


33 


255 


02 


235 


53 


230 


1,665 





54 


275 


44 


245 


34 


140 


24 


270 


14 


230 


02 


135 


1,295 


10 


45 


95 


35 


245 


02 


330 


25 


235 


15 


200 


55 


285 


1,390 


11 


44 


180 


11 


275 


33 


290 


55 


155 


03 


180 


22 


160 


1,240 


12 


03 


120 


32 


70 


21 


100 


15 


100 


43 


170 


54 


65 


625 


13 


53 


55 


42 


145 


31 


40 


25 


35 


03 


45 


14 


55 


375 


14 


24 


140 


13 


45 


35 


15 


03 


85 


62 


65 


41 


55 


405 


15 


45 


80 


23 


115 


34 


165 


03 


85 


51 


55 


12 


120 


620 


16 


32 


215 


11 


300 


45 


255 


24 


185 


01 


145 


63 


150 


1,250 


17 


13 


50 


34 


45 


55 


105 


42 


155 


21 


125 


04 


30 


510 


18 


23 


65 


15 


130 


44 


55 


31 


85 


04 


55 


52 


110 


500 


19 


25 


130 


33 


40 


41 


45 


12 


45 


54 


60 


04 


15 


335 


20 


35 


-5 


04 


70 


22 


65 


43 


35 


14 


255 


51 


80 


500 


21 


05 


180 


11 


255 


23 


290 


42 


285 


35 


270 


54 


185 


1,465 


22 


21 


150 


52 


55 


14 


50 


45 


210 


33 


265 


05 


185 


915 


23 


55 


130 


24 


215 


12 


155 


31 


95 


05 


95 


43 


155 


845 


24 


15 


210 


41 


90 


63 


95 


22 


160 


05 


140 


34 


125 


820 


25 


32 


140 


05 


195 


13 


310 


51 


195 


25 


130 


44 


285 


1,255 


26 


11 


210 


34 


290 


43 


325 


25 


230 


52 


220 


06 


310 


1,585 


27 


12 


230 


44 


155 


35 


195 


53 


245 


06 


315 


21 


215 


1,355 


28 


13 


160 


31 


285 


54 


230 


22 


185 


45 


220 


06 


175 


1,255 


29 


14 


275 


55 


185 


06 


130 


32 


190 


41 


160 


23 


110 


1,050 


30 


15 


155 


42 


150 


24 


240 


06 


130 


33 


145 


51 


125 


945 


31 


01 


220 


05 


215 


06 


195 


03 


240 


02 


295 


04 


230 


I t 395 




























34,960 



198 



THE FIELD PLOT TEST 



TABLE 59 

YIELDS OF SINGLE PLOTS BY VARIETIES, VARIETY TOTALS, VALUES OF 2 r , AND THE 

CORRECTED MEANS i*,. SYMMETRICAL INCOMPLETE BLOCK EXPERIMENT WITH 31 

VARIETIES AND 6 REPLICATIONS 



Vari- 












ety 


Single Plot Yields 


T uv 


s ww 


P^-Z* 


tr 


No. 












01 


360 


285 


325 


270 


175 


220 


1,635 


9,635 


175 


193.6 


02 


215 


140 


235 


135 


330 


295 


1,350 


8,870 


-770 


163.2 


03 


180 


120 


45 


85 


85 


240 


755 


4,660 


-130 


183.8 


04 


145 


30 


55 


15 


70 


230 


545 


4,490 


-1220 


148.6 


05 


180 


185 


95 


140 


195 


215 


1,010 


6,695 


-635 


167.5 


06 


310 


315 


175 


130 


130 


195 


1,255 


7,585 


-55 


186.2 


11 


315 


310 


275 


300 


255 


210 


1,665 


9,185 


805 


214.0 


12 


355 


410 


120 


45 


155 


230 


1,315 


6,665 


1225 


227.5 


13 


370 


255 


45 


50 


310 


160 


1,190 


7,090 


50 


189.6 


14 


265 


230 


55 


255 


50 


275 


1,130 


6,145 


635 


208.5 


15 


345 


200 


100 


130 


210 


155 


1,140 


6,290 


550 


205.7 


21 


160 


315 


100 


125 


150 


215 


1,065 


6,510 


-120 


184.1 


22 


185 


290 


160 


65 


160 


185 


1,045 


6,785 


-515 


171.4 


23 


245 


375 


115 


65 


290 


110 


1,200 


6,760 


440 


202.2 


24 


355 


270 


140 


185 


215 


240 


1,405 


6,210 


2220 


259.6 


25 


240 


235 


35 


130 


130 


230 


1,000 


6,410 


-410 


174.8 


31 


220 


195 


40 


85 


9S 


285 


920 


6,360 


-840 


160.9 


32 


300 


330 


70 


215 


140 


190 


1,245 


7,430 


40 


189.3 


33 


315 


255 


290 


40 


265 


145 


1,310 


6,840 


1020 


220.9 


34 


350 


140 


165 


45 


125 


290 


1,115 


6,580 


110 


191.9 


35 


240 


245 


15 


-5 


270 


195 


960 


6,865 


-1105 


152.4 


41 


255 


330 


55 


45 


90 


160 


935 


5,755 


-145 


183.3 


42 


225 


235 


145 


155 


285 


150 


1,195 


6,305 


865 


215.9 


43 


230 


305 


170 


35 


155 


325 


1,220 


6,720 


600 


207.4 


44 


170 


245 


180 


55 


285 


155 


1,090 


7,155 


-615 


168.2 


45 


360 


95 


80 


255 


210 


220 


1,220 


6,940 


380 


200.3 


51 


330 


270 


55 


80 


195 


125 


1,055 


6,455 


-125 


184.0 


52 


220 


95 


65 


110 


55 


220 


765 


6,405 


-1815 


129.4 


53 


290 


230 


55 


150 


95 


245 


1,065 


6,955 


-565 


169.8 


54 


220 


275 


65 


60 


185 


230 


1,035 


6,475 


-265 


179.4 


55 


265 


285 


155 


105 


130 


185 


1,125 


6,535 


215 


194.9 




215 


7495 


3680 


3555 


5490 


6525 


34,960 


09,760 











m-* 


960 


188.0 














m ~ 186 ~ 



SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS 199 



TABLE 60 

SETS ARRANGED IN ORDER OF NUMBERS WITH CORRESPONDING BLOCK TOTALS. 
INCOMPLETE RANDOMIZED BLOCK EXPERIMENT 



Set 
No. 


Block 
Totals 


1 


01 


11 


12 


13 


14 


15 


2010 


2 


01 


21 


22 


23 


24 


25 


1470 


3 


01 


31 


32 


33 


34 


35 


1750 


4 


01 


41 


42 


43 


44 


45 


1510 


5 


01 


51 


52 


53 


54 


55 


1500 



7 

8 

9 

10 



11 
12 
13 
14 
15 



02 11 21 31 41 51 1635 

02 12 22 32 42 52 1500 

02 13 23 33 43 53 1655 

02 14 24 34 44 54 1295 

02 15 25 35 45 55 1390 



03 11 22 33 44 55 1240 

03 21 32 43 54 15 625 

03 31 42 53 14 25 375 

03 41 52 13 24 35 405 

03 51 12 23 34 45 620 



Set 
No. 


Block 
Totals 


16 


04 


11 


32 


53 


24 


45 


1250 


17 


04 


21 


42 


13 


34 


55 


510 


18 


04 


31 


52 


23 


44 


15 


500 


19 


04 


41 


12 


33 


54 


25 


335 


20 


04 


51 


22 


43 


14 


35 


500 



21 
22 
23 
24 
25 



26 
27 
28 
29 
30 
31 



05 11 42 23 54 35 1465 

05 21 52 33 14 45 915 

05 31 12 43 24 55 845 

05 41 22 53 34 15 820 

05 51 32 13 44 25 1255 



06 11 52 43 34 25 1585 

06 21 12 53 44 35 1355 

06 31 22 13 54 45 1255 

06 41 32 23 14 55 1050 

06 51 42 33 24 15 945 

06 01 02 03 04 05 1395 



Grand Total 34,960 



200 THE FIELD PLOT TEST 

20. Choosing the Best Type of Incomplete Block Experiment for a 
Given Test After a study of the various incomplete block experiments 
it will be noted that each has certain limitations. On account of general 
simplicity the symmetrical incomplete blocks are to be preferred to the 
quasi-factorials, and in addition all comparisons are made with equal 
precision. However, for the symmetrical types we must have, when 
v = p 2 , p -f- 1 replications, and when v = p 2 p + l,p replications. 
For a test of 121 or 133 varieties we require 12 replications, and if the 
number of varieties is greater than this it is obvious that in general the 
test will be more expensive than is usually warranted in such cases. At 
a certain point, therefore, it would seem that the quasi-factorials should 
be extremely useful. On account of its relative simplicity the two- 
dimensional quasi-factorial with two groups of sets is preferable to the 
three-dimensional type, but the latter will probably be the most efficient 
if the number of varieties is quite large. These points can now be used 
as a basis for setting up a general schedule as to the type of experiment 
best suited to a given number of varieties. For this purpose Table 61 
has been prepared, taking as a basis the number of varieties that can be 
tested by at least one of three types. 

In Table 61 the dotted lines indicate the range through which the 
methods are generally recommended. The two-dimensional quasi- 
factorial can be used at the point where the number of replications for 
the symmetrical type becomes too large. For very large numbers the 
three-dimensional quasi-factorial is probably the most efficient, but, 
since it can be applied easily only to numbers that are cubes, the two- 
dimensional type must be extended to include fairly high numbers. 

A possible objection to incomplete block experiments in general may 
be that certain numbers of varieties cannot be tested and hence the 
experimenter may feel that it is still necessary to use randomized blocks. 
However, it would seem to be desirable where possible to suit the num- 
ber of varieties to the experiment even if it involves using "dummy" 
varieties. Also, for those who wish definitely to use other numbers than 
those listed here, Yates (20), has developed methods for laying out and 
analyzing quasi-factorials in which the dimensions are not equal. Thus 
instead of a 12 X 12 quasi-factorial for 144 varieties we might use a 
12 X 11 for 132 varieties. These modifications, however, require addi- 
tional computations and will be avoided if possible. 



INCOMPLETE BLOCK EXPERIMENT FOR A GIVEN TEST 201 



TABLE 61 

VALUES OF p AND r REQUIRED FOR DIFFERENT NUMBERS OF VARIETIES 

AND RANGES THROUGH WHICH THE THREE GENERAL TYPES OF 

INCOMPLETE BLOCK EXPERIMENTS ARE RECOMMENDED 



Symmetrical Incomplete 


No. of Blocks 




Varieties p* 


r 


13 


4 


4 | 


16 


4 


5 


21 


5 


5 


25 


5 


6 


27 






31 

QA 


6 


6 


OU 

49 


7 


8 


57 


8 


8 


64 


8 


9 


73 9 


9 


81 9 


iot 


91 10 


10 


100 10 


11 


111 11 


11 


121 11 


12 


125 




133 12 


12 


144 12 


13 


157 13 


13 


169 13 


14 


183 14 


14 


196 14 


15 


211 15 


15 


216 




225 15 


16 



etc. 



Two-Dimensionai 

Quasi-Factorial 

p r 



4 
5 


2n 
2n 


6 


2n 


7 


2n 


8 


2n 


9 


2n 


10 


2n 


11 


2n 






12 


2n 


13 


2n 


14 


2n 



15 



2n 



Three-Dimensional 
Quasi-Factorial 

p r 

2 3n 



3n 



3n 



3n 



3n 



etc. 



* p mm number of plots in one block. 

r *> number of replications. 

f Completely orthogonaliased squares greater than (9 X 9) have not yet been written, and 
therefore we cannot if we wished go beyond this point at the present time. 



202 



THE FIELD PLOT TEST 



TABLE 62 

YIELDS OF OAT VARIETIES IN AN EXPERIMENT ON THE EFFECT 
OF SOIL INOCULATION WITH A ROOT ROT ORGANISM 



Variety 


Soil 
Treatment 


Replicates 


1 


2 


3 


4 


I 


/ 
U 


24.1 
65.4 


16.1 
49.3 


31.6 
39.8 


28.9 

48.4 


II 


I 

U 


30.6 

51.8 


51.7 

74.8 


51.7 
76.5 


42.5 
56.6 


III 


I 

U 


39.1 
68.7 


47.4 
42.0 


36.9 
81.6 


28.9 
57.3 


IV 


I 

U 


120.1 
112.2 


69.5 
88.6 


96.2 
102.8 


69.7 
85.0 


V 


I 

U 


118.7 
58.5 


24.1 
68.0 


45.9 

77.7 


10.4 
54.7 


VI 


I 

U 


76.2 
109.1 


66.3 
91.5 


77.7 
124.1 


65.3 
96.9 


VII 


I 

U 


57.8 
112.2 


45.9 
95.9 


29.7 
91.1 


56.4 
77.3 


VIII 


I 

U 


58.0 
127.3 


40.1 
66.3 


47.6 
77.0 


38.4 
63.4 


IX 


I 

U 


81.8 
100 3 


23.6 
73.8 


31.6 
81.4 


32.1 
52.7 


X 


I 

U 


85.3 
81.6 


78.2 
94.3 


99.4 
96.4 


85.0 

77.2 



EXERCISES 203 

21. Exercises. 

1. The results of a randomized block experiment are given in Table 62. Ten 
varieties of oats were tested for their reaction to root rot. The plots were arranged 
in pairs of which one plot was inoculated with the root-rotting organism and one 
plot uninoculated. Analyze the results. State in words the meaning of a significant 
interaction between varieties and the soil inoculation. 

DF MS 

Replicates 3 2,042.08 

Varieties 9 2,654.19 

Error (1 ) 27 270.54 

Treatments 1 12,226.51 

Varieties X Treatments 9 401.32 

Error (2) 30 232.30 

2. In a fertilizer experiment conducted In an 8 X 8 Latin square, the yields of 
wheat given in Table 63 were obtained. The fertilizer combinations are designated 
N, P, K, NP, NR, NPK, 0. In the table the yields are in the exact position of the 
plots in the field, and above each yield figure is the fertilizer treatment which the 
plot received. Work out the analysis of variance for this experiment, and, by means 
of the standard error, compare: 

(a) Yields for plots receiving N with those receiving no N. 
(6) Yields for plots receiving K with those receiving no K. 
(c) Yields for plots receiving P with those receiving no P. 

The results for the sums of squares are given below to provide a check on the work, 
but the sum of squares for the treatments must be split up to correspond to individual 
degrees of freedom. 

SS DF 

Rows , 102.20 7 

Columns 84.24 7 

Treatments 513.79 7 

Error 91.99 42 

3. Complete the analysis of the split plot experiment described in Section 8, 
above. Assume that the plan of this experiment is to be rearranged so that the 
most accurate comparison is to be between D and W, and make the plan accordingly. 

The sums of squares for the three errors as given below will provide a com- 
plete check on the calculations. 

Error (1) 647.6 Error (2) 1059.1 Error (3) 931.1 

4. Assuming that the following sets of figures represent the response to fertilizer 
at 4 levels, for each set work out the sums of squares for the total and then for the 
linear, quadratic, and cubic responses. Graph the actual yield results as given below, 
and then point out the relation between the shape of these graphs and the results 
obtained for the sums of squares. 



(a) 
(b) 
(c) 


HI 


n2 


n s 


fU 


22 
19 
24 


65 
61 
58 


54 

58 
13 


78 
27 
41 



204 THE FIELD PLOT TEST 

The sums of squares are 



Quadratic . 
Cubic 



(a) 


(W 


W 


1232.45 
90.25 
396.05 


22.05 
1332.25 
14.45 


1.80 
9.00 
1155.20 



6. Table 64 gives the plan of a field for a 3 X 3 X 3 confounded experiment, 
with treatment numbers and plot yields. The numbers such as 123 and 321 represent 
NiKzPs and NtKJPi. Cyclic set II was used to confound 2 degrees of the triple 
interaction N X K X P with blocks. Work out the complete analysis of variance 
for this experiment giving the results for treatment effects by individual degrees of 
freedom. 

The following excerpts from the results for the sums of squares will assist in 

checking the calculations. 

Total for treatments. . .2,434.93 

N T ..' 9.46 

N r X K d 4.73 

K d XPr 438.90 

N X K X P 149.98 (for one pair of DF) 

Error 5,770.81 

6. Table 65 gives the plan of the field with variety numbers and corresponding 
plot yields for a two-dimensional quasi-factorial experiment with two groups of sets. 
Make a complete analysis of the results. 

The variety sum of squares is 253,538. 

7. Table 66 gives the plan of the field with variety numbers and corresponding 
plot yields for an incomplete block experiment with 21 varieties. Analyze the results, 
and make a test of the significance of the mean difference between the varieties 
01 and 04. 

8. Prepare plans for the layout of: 

(a) Two-dimensional quasi-factorial experiment to test 36 varieties. 

(b) Symmetrical incomplete block experiment to test 31 varieties. 

(c) Three-dimensional quasi-factorial experiment to test 125 varieties. 



EXERCISES 



205 



TABLE 63 
YIELDS OF WHEAT IN AN 8 X 8 LATIN SQUARE FERTILIZER EXPERIMENT 



p 

18 8 


N 
12.2 


NP 

18.3 


K 

15.8 


NK 
11.4 



11.5 


MPtf 
19.4 


PK 
18.9 


N 
12.9 


NK 
7.3 


PK 

17.4 


NPK 

17.2 


P 
19.7 


K 
12.0 


NP 
19.0 


O 
15.6 


NK 
10.7 


NP 
17.5 


N 
10.4 


P 
18.0 


O 

9.8 


NPK 
16.6 


PK 
17.5 


K 

14.3 


PK 
18.3 


K 
12.6 


NPK 

14.2 


O 
12.2 


N 
11.4 


NP 
14 5 


P 
16.9 


NK 

16.1 


NP 
17 9 


O 

12.8 


NK 
13.3 


N 
11 3 


PK 
16.5 


P 
15 6 


K 
10.9 


NPK 

16.7 


K 
14.9 


PK 

18.2 


O 

12.8 


NP 
17.1 


NPK 
15.8 


N 
9.5 


NK 
8.9 


P 
20.6 


NPK 
19.0 


P 

18 9 


K 

11.2 


PK 
17.1 


NP 

17.9 


NK 

8.6 



10 2 


AT 
14 5 


O 
17.5 


NPK 
20.4 


P 
20.8 


NK 
16.4 


j 
16.8 


PK 
18.5 


AT 
13.6 


NP 
23.0 



206 



THE FIELD PLOT TEST 



TABLE 64 
PLAN OF FIELD AND PLOT YIELDS FOE A (3 X 3 X 3) CONFOUNDED EXPERIMENT 



Variety 


Yield 


Variety 


Yield 


Variety 


Yield 


111 


465 


112 


364 


113 


549 


123 


395 


121 


348 


122 


348 


132 


556 


133 


421 


131 


463 


213 


343 


211 


455 


212 


346 


222 


413 


223 


374 


221 


394 


231 


408 


232 


507 


233 


363 


312 


337 


313 


421 


311 


449 


321 


421 


322 


374 


323 


217 


333 


308 


331 


334 


332 


355 


333 


353 


121 


381 


332 


244 


312 


486 


133 


403 


323 


246 


213 


219 


313 


75 


113 


82 


321 


544 


211 


325 


122 


280 


123 


478 


331 


141 


221 


195 


231 


391 


112 


259 


131 


196 


222 


311 


223 


254 


311 


178 


111 


302 


322 


259 


212 


222 


132 


542 


232 


398 


233 


309 


222 


374 


133 


299 


311 


196 


321 


358 


331 


273 


131 


259 


213 


468 


232 


437 


122 


361 


231 


316 


322 


485 


233 


345 


111 


307 


121 


311 


221 


207 


333 


570 


313 


343 


113 


16 


312 


427 


211 


353 


323 


199 


123 


380 


112 


454 


212 


114 


132 


400 


223 


251 


332 


240 


132 


611 


121 


403 


113 


302 


123 


444 


331 


338 


323 


256 


312 


550 


322 


405 


311 


367 


333 


573 


223 


331 


131 


268 


213 


706 


313 


522 


221 


400 


111 


423 


211 


319 


332 


446 


321 


749 


232 


383 


233 


515 


231 


529 


133 


292 


212 


420 


222 


424 


112 


554 


122 


384 



EXERCISES 



207 



TABLE 65 

PLAN OP A FIELD WITH VARIETY NUMBERS AND CORRESPONDING PLOT YIELDS FOR 
A TWO-DIMENSIONAL QUASI-FACTORIAL EXPERIMENT WITH 49 VARIETIES 





Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 




12 


189 


15 


284 


13 


218 


14 


392 


11 


211 


17 


304 


16 


182 




26 


280 


25 


342 


27 


345 


23 


214 


21 


327 


22 


270 


24 


320 


Repli- 


32 


300- 


34 


357 


36 


298 


31 


366 


37 


356 


35 


283 


33 


292 


cate I 


45 


132 


44 


250 


46 


292 


43 


384 


42 


279 


41 


197 


47 


214 


Group X 


56 


50 


52 


42 


51 


339 


54 


283 


53 


126 


57 


82 


55 


37 




65 


153 


66 


310 


63 


306 


64 


303 


61 


182 


67 


121 


62 


197 




72 


214 


71 


380 


77 


345 


74 


363 


75 


274 


76 


330 


73 


242 




71 


234 


11 


283 


51 


125 


41 


336 


31 


233 


61 


339 


21 


269 




32 


280 


62 


367 


72 


305 


12 


309 


52 


148 


22 


252 


42 


147 


Repli- 


73 


414 


23 


399 


33 


381 


43 


184 


13 


162 


63 


191 


53 


62 


cate II 


44 


217 


54 


331 


24 


295 


64 


277 


34 


273 


14 


307 


74 


287 


Group Y 


35 


144 


15 


202 


25 


196 


45 


375 


55 


329 


65 


221 


75 


141 




26 


161 


16 


204 


56 


214 


46 


450 


66 


165 


76 


203 


36 


197 




27 


278 


47 


291 


67 


214 


17 


316 


57 


243 


37 


134 


77 


169 




15 


263 


16 


111 


12 


255 


14 


201 


17 


150 


11 


95 


13 


259 




22 


129 


21 


156 


24 


192 


26 


173 


25 


133 


27 


371 


23 


255 


Repli- 


31 


284 


34 


240 


32 


214 


35 


326 


33 


149 


37 


254 


36 


19i 


cate III 


42 


130 


47 


206 


45 


234 


43 


290 


44 


211 


41 


225 


46 


358 


Group X 


54 


165 


57 


93 


52 


267 


51 


242 


53 


158 


55 


102 


56 


339 




65 


259 


66 


11 


64 


285 


62 


312 


67 


165 


61 


196 


63 


301 




73 


307 


74 


168 


71 


245 


75 


301 


76 


223 


72 


265 


77 


361 




71 


139 


61 


169 


31 


268 


41 


173 


51 


188 


11 


29 


21 


79 




72 


126 


52 


142 


42 


180 


12 


-16 


22 


-8 


32 


-63 


62 


62 


Repli- 


63 


187 


23 


254 


53 


100 


43 


-29 


33 


65 


73 


10 


13 


199 


cate IV 


44 


257 


64 


159 


74 


118 


24 


209 


34 


174 


14 


112 


54 


108 


Group Y 


15 


254 


35 


289 


65 


244 


55 


191 


25 


142 


75 


395 


45 


265 




36 


249 


76 


201 


16 


140 


26 


248 


56 


235 


46 


235 


66 


176 




67 


216 


27 


186 


77 


209 


37 


336 


57 


233 


17 


105 


47 


27 



208 



THE FIELD PLOT TEST 



TABLE 66 

PLAN OF FIELD WITH VARIETY NUMBERS AND CORRESPONDING PLOT YIELDS FOR 
A SYMMETRICAL INCOMPLETE BLOCK EXPERIMENT WITH 21 VARIETIES 



Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Vari- 
ety 


Yield 


Block 
Totals 


13 


465 


11 


393 


14 


556 


01 


343 


12 


413 


2170 


22 


408 


21 


337 


24 


421 


23 


308 


01 


353 


1827 


31 


486 


01 


219 


34 


544 


32 


478 


33 


391 


2118 


01 


311 


44 


302 


43 


542 


42 


374 


41 


358 


1887 


41 


468 


21 


316 


02 


307 


11 


570 


31 


427 


2088 


02 


380 


32 


400 


42 


611 


12 


444 


22 


550 


2385 


43 


573 


13 


706 


23 


423 


02 


749 


33 


529 


2980 


44 


424 


02 


638 


24 


736 


14 


488 


34 


758 


3044 


22 


364 


11 


348 


44 


421 


33 


455 


03 


374 


1962 


12 


507 


34 


421 


43 


374 


21 


334 


03 


381 


2017 


24 


403 


42 


75 


13 


325 


31 


141 


03 


259 


1203 


32 


254 


23 


259 


03 


398 


14 


299 


41 


273 


1483 


24 


437 


11 


485 


32 


311 


04 


343 


43 


353 


1929 


14 


454 


33 


251 


21 


403 


42 


338 


04 


405 


1851 


31 


331 


04 


522 


44 


319 


12 


383 


23 


292 


1847 


22 


554 


04 


626 


41 


753 


34 


505 


13 


668 


3106 


42 


549 


34 


348 


05 


463 


11 


346 


23 


394 


2100 


05 


363 


21 


449 


13 


217 


44 


355 


32 


244 


1628 


43 


246 


31 


82 


14 


280 


22 


195 


05 


196 


999 


41 


178 


,05 


222 


24 


309 


12 


196 


33 


259 


1164 


02 


361 


04 


345 


01 


207 


05 


16 


03 


199 


1128 



REFERENCES 209 



REFERENCES 

1. L. D. BATCHKLOR and H. S. REED. Jour. Agr. Research, 12: 245-283, 1918. 

2. R. A. FISHER. The Design of Experiments. Oliver and Boyd, London, 1937. 

Reading: Chapters V, VI, VII, and VIII. 

3. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter VIII, Sections 48 and 49. 

4. R. A. FISHER and J. WISH ART. The Arrangement of Field Experiments and the 

Statistical Reduction of the Results. Imp. Bur. Soil Science, Tech. Com., 10. 

5. C. H. GOULDEN. Sci. Agr., 11: 681-701, 1931. 

6. C. H. GOULDEN. Can. Jour. Research, C, 15: 231-241, 1937. 

7. C. H. GOULDEN. Modern Methods for Testing a Large Number of Varieties. 

Dom. of Canada, Tech. Bull. 9. 1937. 

8. F. R. IMMER, H. K. HAYES, and LEROY POWERS. Jour. Amer. Soc. Agron., 26: 

403-419, 1934. 

9. W. SAYER, M. VAIDYANATHAN, and S. S. IYER. Indian Jour. Agr. Sci. t 6: 

684-714, 1936. 

10. O. TEDIN. Jour. Agr. Sci., 21: 191-208, 1931. 

11. G. A. WIEBE. Jour. Agr. Research, 50: 331-357, 1935. 

12. J. WISHART. The Analysis of Variance Illustrated in its Application to a 

Complex Agricultural Experiment on Sugar Beets. Archiv. Pflanzen, 5:4, 1931 . 

13. J. WISHART. Jour. Roy. Stat. Soc., 1: 26-61, 1934. 

14. J. WISHART. Jour. Roy. Stat. Soc., 1: 94-106, 1934. 

15. J. WISHART and H. G. SANDERS. Principles and Practices of Field Experimenta- 

tion. Empire Cotton Growing Corp., London, 1936. 

16. F. YATES. The Principles of Orthogonality and Confounding in Replicated 

Experiments. Jour. Agri. Sci., 23: 108-145, 1933. 

17. F. YATES. Complex Experiments. Supp. Journ. Roy. Stat. Soc., II: 181-247, 

1935. 

18. F. YATES. Incomplete Randomized Blocks. Ann. Eugen., 7: 121-140, 1936. 

19. F. YATES. The Design and Analysis of Factorial Experiments. Imp. Bur. Soil 

Sci., Tech. Comm. 35, 1937. 

20. F. YATES. A New Method of Arranging Variety Trials Involving a Large 

Number of Varieties Jour. Agr. Sci., 26: 424-455, 1936. 



CHAPTER XIII 

THE ANALYSIS OF VARIANCE APPLIED TO LINEAR 
REGRESSION FORMULAE 

1. Significance of the Regression Function. If, in a series of paired 
values, y is the dependent and x is the independent variable, the regres- 
sion of y on x is represented by the linear equation Y = y + b(x ), 
where b is the regression coefficient and Yi is a value of y estimated from 
the equation for x = rr t . Now if the equation is used to estimate each 
value of y from the corresponding values of x, it can be shown that 

- F) 2 



And since S(y #) 2 = (1 r 2 )2(y $) 2 + r 2 2(^/ ) 2 , it is obvious 
that, if the total sum of squares for the dependent variable is broken up 
into two parts, one part 2(t/ F) 2 , representing deviations from the 
regression function, and another part S(F ) 2 , representing that 
portion of the total variability that is accounted for by the regression 
function, these two parts are proportional to (1 r 2 ) and r 2 , respectively. 
It should be clear that S(y F) 2 represents deviations from the regres- 
sion function because for each value of y we are taking the square of the 
deviation of that value from the corresponding F value on the regression 
line. Similarly 2(F y) 2 represents the regression function itself 
because for each value of y we take the square of the difference between 
y and the corresponding point on the regression line. As the slope of 
the regression line increases, 2(F y) 2 must increase also, and as the 
y values approach more closely to the regression line the value of 
2(y F) 2 decreases correspondingly. 

The direct relation between 2(F y} 2 and the regression equation 
may be shown by equating it to 

2(F - y) 2 = 2{y + b(x - x) - y} 2 = b 2 Z(x - x) 2 (2) 

In the expression on the right S(x x) 2 is obviously independent of the 
correlation so that any variations in S(F y) 2 are due entirely to b. 
This is an important concept as it shows that, since the value of 

210 



SIGNIFICANCE OF THE REGRESSION FUNCTION 



211 



2(F ij) 2 for any given distribution of y is dependent on a single 
statistic 6, it must represent only 1 degree of freedom. Hence the 
analysis of variance corresponding to equation (1) will be: 





"Sum of Squares 


DF 


Mean Square 


Regression function 


b 2 2(x x) 2 


L 


b*2(x - ) 2 


Deviations from regression 
function 


2(v - F) 2 


ri - 2 


^(y - F) 2 /n' - 2 










Total 


S(j/ - i/) 2 


n' - 1 













where n' is the number of pairs of values of x and y. 

In calculating the sum of squares b 2 S(x x) 2 it is frequently con- 
venient to make use of the equality 



(3) 



If b has already been obtained it is of course just as convenient to mul- 
tiply S(z - x) 2 by b 2 . 

If the correlation coefficient has been determined, a short method 
of determining the significance of r xy which is exactly comparable 
to determining the significance of b vx arises from the substitution of 
(1 - r 2 ) S(i/ - y) 2 for 2(y - F) 2 , and r^(y - y) 2 for Z> 2 S(z - z) 2 , in 
the sum of squares column of the analysis of variance. Then F works out 
to r 2 (n r 2)/l r 2 , and this is all the calculation necessary. In other 
words, for a total correlation or a regression coefficient, F = 2 , and 
tables either of F or of t may be used to test their significance. Refer 
here to Chapter VII, equation (11), and note that F = Vb/v e . 

2. Test for Non-Linearity. When correlation data are set up in the 
form of a correlation table the total sum of squares may be split up into 
two portions, one part representing differences between the means of 
arrays and the other representing differences between values within 
arrays. The equation is 



- y) 2 = 



,(& - y) 2 

Between 



Within 



where n p is the number in an array and y r is the mean of an array. The 
second summation in the term on the right means that the sums of 
squares are first computed for each array and these are summated. 



212 VARIANCE APPLIED TO LINEAR REGRESSION FORMULAE 
The equation for the corresponding degrees of freedom is as follows: 

n 1 - 1 = (q - 1) + (*' - q) (5) 

where q is the number of arrays in the table. 

If we picture the sum of squares for between arrays as being due to a 
set of means running diagonally across the table following in general the 
regression straight line, it is obvious that the sum of squares for between 
arrays includes the sum of squares 6 2 S(x x) 2 , worked out above for 
deviations due to the regression function, and that the remainder will be 
due to deviations of the means of arrays from the regression line. The 
equation is 

Sn,(fr - y) 2 = 2n,(k - F) 2 



(6) 



Between 



Deviations 
of means of 
arrays from 
regression line 



Due to linear 
regression 



If the means of arrays fall directly on the regression line, ^,n p (y p F) 2 
will be zero, and correspondingly its value will increase as the trend 
of the mean values gets farther away from the trend of the straight 
regression line. Then since the sum of squares for within arrays 
measures the random variability in the values of y a comparison of the 
estimates of variance obtained from Sn P (y P F) 2 and 22 (y y p )' 2 
should provide a measure of the linearity of regression, or the goodness 
of fit of the regression straight line to the data in question. 

The equation for the degrees of freedom corresponding to equation 
(6) will be (q - 1) = (q - 2) + 1. 

The complete analysis of variance may be represented as follows: 





Sum of Squares 


DF 


Sum of Squares 


D/? 


Between arrays 


Sn p (# p #) 2 


< Linear 
regression 
Deviations, 


.-. 


1 






means of arrays 


2>n p (y p - F) 2 


ff-2 






from regression 










line 






Within arrays 


ZZ(y - ft,) 2 


n'-g 






Total 


z&r-tf 


n'-l 







TEST FOR NON-LINEARITY 213 

For the purpose of testing linearity, however, it suffices to set up; 





Sum of Squares 


DF 


Variance 


Deviations, means of arrays 
from regression line. 


2n n (ti n - F) 2 


(7-2 


Sntet, - F) 2 /0 - 2 


Within arrays 


2S(w - L,) 2 


n' -q 


S2(y - v) 2 /n' - ff 










Total 


S(y - F) 2 


n' -2 








* 





There are various methods of obtaining the sums of squares for the 
above analysis, but one of the most convenient and direct is first to 
calculate Sn,,(j?p t/) 2 , making use of the identity 



(7) 



y) 



We square the total of each array and divide by the number in the 
array. These are summated, and from the sum we subtract the square 
of the y total divided by the number of paired values. Then we calcu- 
late b 2 2(x x) 2 and, S(y t/) 2 being known, the two sums of squares 
required can be obtained by subtraction. The procedure is obvious by 
reference to the outline of the analysis of variance above. 

Example 41. Significance of a Regression Function. In Chapter VII, Ex- 
ample 13, we determined the correlation coefficient for the yields of adjacent barley 
plots and in Chapter VI, Example 11, ^e determined the regression line. Using the 
same data and the analysis of variance to test the significance of the regression 
function we should get a similar result. The sums of squares are 

2(* - *) 2 3952 - 850 2 /200 339.60 

- *)* - 0.4492 2 X 339.60 - 68.60 

- ) 2 - 8180 - 1246 2 /200 417.42 

- K) 2 - 417.42 - 68.60 - 348.92 



Then the analysis of variance is as follows: 





Sum of 
Squares 


DF 


Variance 


F 


1% Point 


Regression function 


68.60 


1 


68.60 


38.9 


6.76 


Deviations from regression 


348.92 


198 


1.762 






Total 


417.42 


199 





















The F value is well beyond its 6% point, indicating a high degree of significance. 



214 VARIANCE APPLIED TO LINEAR REGRESSION FORMULAE 



Example 42. The Test for Non-Linearity. We shall again use the data of 
Chapter VI, Table 12, for this test. Since we already have 2(y - #) 2 (Example 41, 
above) the first step is to calculate ^n p (y p #) 2 . In Chapter VI, Table 13, the 
totals for the y arrays are given, so we proceed as follows: 



Between arrays ....... 20 2 /4 - 

Linear regression ..... 6 2 2(x - ) 

Deviations from regres- 

2n p (y p F) 2 



-f 60 2 /13 + - - + 42 2 /6 - 1246 2 /60 78.70 
- fV ~ 0.4492 2 X 339.50 - 68-/50 



sion 



Difference 



Total &\y y)~ 

Between arrays 2n p (y p - , 

Within arrays S2(t/ y p ) 2 = Difference 

Setting up the analysis of variance, we have: 



68.50 

- 10.20 

= 417.42 
= 78,70 

= 338.72 





Sum of 
Squares 


DF 


Variance 


F 


5% Point 


Deviation means of arrays 
from regression line 


10 20 


5 


2 040 


1 16 


2.26 


Within arrays 


338.72 


193 


1.755 



















The F value does not approach its 5% point, so we conclude that there is no evidence 
of non-linear regression. 

3. Significance of Multiple Correlations. In multiple correlation 
where x\ represents the dependent variable and x 2 and x$ two independ- 
ent variables the regression equation is 



bi 2 (x 2 



(8) 



and this may of course be extended for any number of variates. The 
normal equations corresponding to (8) are 



Zzi(a; 2 - x 2 ) = bnZfa - x 2 ) 2 
Sxi(x 3 - ft) = bi22x 2 (x3 - ft) 
and from these we can derive the solution 



x 3 ) 2 



- i) 2 = 



- Xi) 2 + 6i 2 Sxi(x 2 - ft) + 



(9) 



- ft) (10) 



This equation corresponds to (1) above where the first term on the right 
represents the portion of the sum of squares for x\ that is independent 
of x 2 and #3. The other two terms on the right represent the portion of 
the sum of squares for xi that is dependent on x 2 and X&. These terms 
may of course be written b\ 2 TZ(x 2 x 2 ) 2 and 6132(0:3 ft) 2 , in which 



SPECIAL APPLICATIONS 



215 



form they correspond to 6 2 S(x x) 2 as above in equation (2), Equa- 
tion (10) may also be written 

(11) 



where R is the multiple correlation coefficient. Also 

- Zi) 2 , and 



= bi22xi(x2 fe) + 6132x1(0:3 a). 

It follows from (10) and (11) that a multiple regression can be 
expressed as an analysis of variance as follows : 





Sum of Squares 


D F 


Variance 


F 


Regression 
function 


<-*> 


P 


-z ta -*./, 


(x> \ / ' i\ 
* \/n p 1 "-!* 
If,,. ri, I 


i - z A P y 


Deviations 
from 










regression 
function 


(1 - R*)2(xi - *i) 2 


n' p 1 


(1 - fl2)Z(a:i - >i) 2 




n' - p - 1 


Total 


xfa - ** 


n' I 







where p is the number of independent variables. To test the significance 
of a multiple correlation therefore it is only necessary to find 



(12) 



and look up the 5% point of F corresponding to n\ = p and 
n> = n 1 p 1. 

Example 43. The Significance of a Multiple Correlation. Let /? 1.2345 = 0.6457, 
and it has been obtained from a series of 84 values of x\, xi, x$, #4, und x&. We have 



_ /a416928\/79\ 
\0.583072/\4/ 



For p = 4 and n' p 1 = 79, the 1% point of F is 3.56, so that the multiple 
correlation is highly significant. 

4. Special Applications. The analysis of variance can be used to 
determine the significance of the additional information obtained in cal- 
culating multiple correlation coefficients. This method was used by 
Geddes and Goulden (2) in a practical problem in cereal chemistry. 
Correlations were first determined between loaf volume of wheat flour 
and the percentage of protein. In later studies the protein was sep- 
arated into two portions, peptized and non-peptized, and using these two 



216 VARIANCE APPLIED TO LINEAR REGRESSION FORMULAE 



portions as variables the multiple correlation for their combined effect on 
loaf volume was calculated. If the proportions of the two kinds of pro- 
tein have an important effect on loaf volume the multiple correlation 
should be significantly higher than the simple correlation for total pro- 
tein and loaf volume. A method of comparing the two correlations 
would determine therefore the practical significance, for purposes of 
predicting flour quality, of knowing the amounts of peptized and non- 
peptized protein in addition to the total protein. 

If we let x\ represent loaf volume, #2 the peptized protein, x& the 
non-peptized protein, and x p the total protein, the corresponding simple 
and multiple correlation coefficients are r\ p and Ri.23- The total pro- 
tein is of course (0:2 + 3), the sum of the two fractions. 

Assuming these correlations to be determined from 20 pairs of values, 
the sums of squares representing deviations from the regression function 
are proportional to (1 r\ p ) and (1 #1.23)) respectively, and the 
corresponding degrees of freedom are 18 and 17. The effect of using 
more variables to estimate x\ as in the case of multiple regression is to 
decrease the sum of squares due to deviations from the regression func- 
tion, but for each additional variable introduced 1 degree of freedom is 
lost and unless the reduction of the sum of squares is more than propor- 
tional to the loss in degrees of freedom there is no gain in precision. An 
analysis may therefore be set up as follows: 





Sum of Squares 


DF 


Variance 


Deviations from regression of 
x-n on xi 


1 _rf_ 


18 




Deviations from regression of 
#2 and XB on x\ 


1 - #?23 


17 


(1) 


Additional degree of freedom 


(1 - rj,) - (1 - R\ M ) 


1 


(2) 



Applying the z test to the mean squares (1) and (2), using (1) as an 
error, we can determine the significance of the gain in information due to 
the addition of another variable. 

In one actual experiment for a series of 20 flours from No. 2 Northern 
wheat r\p 0.511 and #1.23 = 0.732. The analysis gives: 



Sum of Squares 


DF 


Variance 


F 


1% Point 


1 - rf 0.738879 
1 - #1.23 0.464176 


18 
17 


0.02730 






Difference 0.274703 


1 


0.2747 


10.06 


8.40 



EXERCISES 



217 



In this case there was a decided gain in information owing to the 
separation of the protein into two components. 

In the general case to which this method may be applied note that 
(1 r 2 ) represents (n f 2) degrees of freedom and (1 R 2 ), 
(n f p 1) degrees of freedom. The difference between the two 
sums of squares will be represented therefore by (ri 2) 
(n' p 1) s (p l) degrees of freedom. 

6. Exercises. 

1. For the data in Chapter VI, Table 15, determine the significance of the re- 
gression function by means of the analysis of variance, where the flour carotene is 
taken as the dependent variable. F 159.5. 

2. For the same data as in Exercise 1 above, test for linearity of regression. 

F - 3.21. 

3. Apply the test for non-linearity to the data in Table 67 for the relation between 
loaf volume according to a standard baking formula and the percentage protein of 
wheat flour. If there is evidence of non-linearity calculate the regression equation 
and make a graph showing the regression line and the means of the arrays. 

TABLE 67 
CORRELATION SURFACE TOR RELATION BETWEEN PROTEIN AND LOAF VOLUME 

Protein in Percentage 
11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 



950 
900 
850 
800 

Loaf 750 
volume 
in cc. 700 

650 
600 
550 
500 

















1 






1 
7 
22 
45 
50 
13 
11 
12 
2 
1 














2 


5 














5 


2 


5 


6 


3 


1 








6 


15 


7 


3 


12 


2 








9 


12 


14 


5 


6 


2 


1 


1 




1 


7 


3 


1 


1 












4 


5 


2 














4 


6 


2 


















2 


















1 




















5 13 23 23 35 15 16 26 6 2 


164 



218 VARIANCE APPLIED TO LINEAR REGRESSION FORMULAE 

4. For n' = 40, determine the multiple correlation #1.234 that is just significant. 

6. Determine the significance of the gain in information through the calculation 
of multiple correlations in the examples given below. For each comparison, state 
your conclusion in words. 

n ' 40 ri 2 0.7643 # L23 4 - 0.8031 

n 1 = 62 r 12 = 0.8744 #1.2345 - 0.9664 

n ' = 20 n 2 = 0.7621 #03 - 0.7635 

n' = 20 na = 0.7316 #1.23456 - 0.7329 

REFERENCES 

L R. A. FISHBB. Statistical Methods for Research Workers. Oliver and Boyd, 
London, 1936. Reading: Chapter VIII, Sections 44, 46, 47. 

2. W. F. GEDDES and C. H. GOULDEN. Cereal Chem., 1: 527-556, 1930. 

3. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, London, 

1931. Reading: Chapter VII, Sections 7.23, 7.33; Chapter IX, Sections 9.3; 
Chapter XI, Sections 11.6, 11.72, 11.63. 



CHAPTER XIV 
NON-LINEAR REGRESSION 

1. An Example of Non-Linear Regression. In Chapter XIII, 
Section 5, Exercise 3, a test for non-linearity was applied to a correlation 
surface for the relation between protein and loaf volume of wheat flour 
in a baking experiment. The non-linearity is significant, and on plotting 
the means of the arrays we find that with increasing protein there is at 
first a very rapid increase in the loaf volume, but with higher protein 
flours the increase in loaf volume is slower and finally there are indica- 
tions that the loaf volume is actually decreasing. Here we have a 
typical example of non-linearity, and it is obvious that, in such cases, 
methods for the prediction of values of the dependent variable from 
specific values of the independent variable cannot be based on a straight- 
line equation. 

2. The Correlation Ratio. In cases of non-linear regression the 
correlation ratio (1) is sometimes used to represent the relation between 
the two variables. The correlation ratio is defined by 



y) 2 



and its relation to the correlation coefficient will be obvious from the out- 
line of the analysis of variance of Chapter XIII, Section 2. The corre- 
lation coefficient may be defined as follows if we take into account its 
numerical value only: 

- y) 2 

(2) 



y) 2 

and it is clear that in the correlation ratio the numerator contains the 
sum of squares S(F y) 2 plus the sum of squares due to deviations of 
means of arrays from the regression line. Hence rj 2 is always greater than 
r 2 unless the means of the arrays fall exactly on the regression line. The 
correlation ratio measures the total variability of the means of arrays, 
and this may be due in part either to a linear relation between the vari- 
ables or to some other type of relation. It does not, however, represent 
a relation that can be expressed by a mathematical equation, either 

219 



220 NON-LINEAR REGRESSION 

linear or curvilinear. The correlation ratio is therefore not a very satis- 
factory statistic as it cannot be used to predict one variable from another. 
Its use must be confined to a measurement of the significance of the 
total variability of the means of the arrays and in this respect must be 
interpreted in terms of the analysis of variance. Thus in Chapter 
XIII, Section 2, the analysis of variance test will involve a comparison 
of the variance between arrays with the variance within arrays. 

The popularity of the correlation ratio was occasioned partly by 
the use of Blakeman's criterion (r/ 2 r 2 ) as a test for linearity (1). 
R. A. Fisher (3) has shown that this test is not satisfactory and that the 
analysis of variance can be used as described in Chapter XIII to provide 
an accurate test. The correlation ratio as such is therefore not much 
used at the present time. It may frequently be necessary to apply a 
test of significance to the variance for the means of arrays in a correlation 
surface, but this does not necessitate the actual calculation of the corre- 
lation ratio. Elaborate methods have been developed for testing the 
significance of the correlation ratio, but these are now unnecessary as 
the problem has been completely solved by Fisher's z distribution and 
the analysis of variance. The test, as we have noted in the previous 
chapter, is now quite simple. 

3. Types of Regression Equations. The procedure in making a 
critical study of the relation between two variables when this relation is 
non-linear is to endeavor to find some type of mathematical equation 
that will give a good fit. This is obviously not always a simple problem 
as there are a number of types of equations to choose from and in each 
case the method of making an accurate test of the goodness of fit must 
be considered. The first step is to examine the trend of the values in the 
regression graph and from its general characteristics decide as to the 
type of equation to be used. After the type has been selected the 
actual equation must be determined by direct methods. 

The simple straight-line equation that we have dealt with previously 



Y = y + b^(x - *) fl - 
and since y b vx is a constant we can write this equation in the form 

Y = GO + cix 

where CQ = y b va and c\ = &, the regression coefficient. This is a 
convenient form with which to represent the various kinds of regression 
equations, which in general are of two types: (1) polynomials, and (2) 
logarithmic. Typical examples are as follows: 



A GENERAL METHOD OF FITTING POLYNOMIALS 221 

POLYNOMIALS LOGARITHMIC 

Y *= CD + cix Y co + ci log x 

Y = co + cix + C2Z 2 log F *= co + ciz 

Y = co + cix -f C2X 2 + cjar 8 log F -= co -f ci log x 
etc. etc. 

Of the polynomials the first is the simple straight-line equation, the 
second is the simple parabola or quadratic, and the third is the cubic. 
The simple parabola has only one maximum or minimum point, and 
there are no points of inflection. The cubic has both a maximum and a 
minimum point and one point of inflection. Curves of higher degree 
have more maximum and minimum points and tend to twist oftener and 
more rapidly. A most interesting characteristic of the polynomial 
equations is one that has already been noted in Chapter XII, in dealing 
with the separation of sums of squares corresponding to individual 
degrees of freedom. The effects represented by the polynomials of 
different degree are independent, and we refer to them as the orthogonal 
polynomials. This property is of particular value in curve fitting as it 
simplifies materially the problem of testing the goodness of fit at each 
stage of fitting. 

Logarithmic curves may be regarded as modifications of the other 
types. Thus the straight-line equation Y = CQ + c\x may be changed 
to a logarithmic equation by replacing x by log x. The result of this 
change is a crowding together of the x ordinates farthest away from zero. 
A straight line with a positive slope is changed therefore to a curved line 
which has a very decided slope at the origin but changes rapidly as x 
increases and reaches a point finally where the slope is fairly constant 
but much less than that of the original straight line. Logarithmic 
curves, in addition, cannot be used to represent negative values, and in 
this respect are therefore much more limited in their application than 
the polynomials. 

The characteristics of the different types of equations are most easily 
learned by working out the F values for some imaginary equations and 
plotting the curves on graph paper. 

4. A General Method of Fitting Polynomials. With the data such 
as those of Table 67, Chapter XIII, before us in the form of a correla- 
tion surface, we may inquire as to the possibility of expressing the rela- 
tion between protein and loaf volume by some simple mathematical 
equation, the end result of our inquiry being to obtain the best method 
available for predicting the loaf volume that will be obtained from the 
flours of a given protein content. The selection of the best type of 
equation is fairly easy in this case. First we prepare a graph of the 
means of the y arrays as in Fig. 12, connecting the points with a dotted 



222 



NON-LINEAR REGRESSION 



line. The general trend of the points seems to follow fairly closely 
the first half of the second degree parabola, or of the portion of a third- 
degree curve up to the maximum point. There is very little resemblance 
to a logarithmic curve as the first portion of it is nearly straight and with 



10 
9 
8 
7 
6 

Y 5 
4 
3 
2 



3456789 10 



FIG. 12. Graph of means of y arrays; data of Table 67. 

a greater curvature towards the end. Of course polynomials of higher 
degree may give a better fit than those of the second degree or third 
degree, and the problem resolves itself therefore into the selection of a 
polynomial that will give the greatest degree of precision in predicting y 
from particular values of x. 



SELECTION OF EQUATION GIVING THE BEST FIT 

The problem of selecting an equation of the degree that gives the 
greatest precision for prediction purposes is of paramount importance 
in curve fitting and one which may easily be overlooked in a maze of 
technical details leading to the fitting of curves of a high order. Unless 
we can be sure that a curve fits better than a straight line it would be 
better not to use the curve. In certain cases the improvement in fit due 
to one equation over another is clearly visible by inspection, but this is 
certainly not generally true. For example, in comparing second and 
third-degree curves, the latter often appear to fit better than the former, 
but a critical test may show that the situation is definitely otherwise. 



A GENERAL METHOD OF FITTING POLYNOMIALS 223 

In the methods of curve fitting described below, particular attention 
is given to the problem of determining goodness of fit. We begin by 
fitting a straight line or a curve of low degree and follow up with addi- 
tional stages of fitting. At each stage one degree of freedom is utilized 
in fitting, and the variance represented by this degree of freedom is tested 
against the error of regression. As a general rule, when a curve has 
been obtained that passes reasonably well through the points, and if in 
making use of an additional degree of freedom there is no gain in preci- 
sion, the curve of lower degree fitted previously is taken as giving the 
best fit. 

METHOD 

The fitting of polynomials is an application of the method of least 
squares. Where Y represents the values of y estimated from the regres- 
sion equation for given values of z, the type regression equation is as 
follows : 

Y = co + ax + c 2 x 2 + + c m x m (3) 

and consequently the error of estimation is given by 

- F) = S(y - Co - cix - c 2 z 2 - - c m x) 2 (4) 



The best values for substitution in the equation for co, ci, 02, *c m are 
taken as those that give a minimum value to S(y F) 2 . Minimizing 
the expression on the right in (4) we obtain a set of m + 1 simultaneous 
equations, where m + 1 is the number of unknowns and m is the highest 
power of x in the polynomial equation to be derived. These simultane- 
ous equations are known as the normal equations, owing to the sym- 
metrical nature of the coefficients. For the general case they are as 
follows, where x and y are measured from their means: 



nco 
Z(*)co 

S(z 2 )co + Z(x*)ci + S(:r 4 )c2 + - - - + S(^+ 2 )c w = 2(x 2 ?/) (5) 



-f + %(x* m )c m = 2(x m y) 

The symmetrical nature of the coefficients allows for a method of 
solution commonly known as the Doolittle method wherein the total 
amount of calculation involved is very considerably reduced as com- 
pared with the ordinary method of solving a set of simultaneous equa- 
tions. After Co, Ci, C2, c m have been solved for, the setting up of the 



224 



NON-LINEAR REGRESSION 



regression equation is merely a matter of substituting the values of these 
statistics in equation (3) 



TESTING THE GOODNESS OF FIT 

The method of testing the significance of the variance corresponding 
to each degree of freedom used in fitting is merely an extension of the 
method described in Chapter XIII for testing the significance of a 
straight-line regression function. 

Let flo = Zfo - #) 2 , Ri = S(J/ - Fi) 2 , and S(Fi - #) 2 is the 
sum of squares due to the regression function for one degree of fitting. 
The analysis is of the form: 

88 DF 



Regression function 


2(Fi - #) 2 


1 


First residual 


Ri - 2(y - Fi) 2 


ri - 2 








Total 


Ro = 2(y - #) 2 


ri - 1 









If a second statistic is fitted the residual R\ will be reduced by an amount 
equal to the difference between the sums of squares for the two regres- 
sion functions, i.e., by S(F2 y) 2 S(Fi t/) 2 , which for conve- 
nience we will put equal to S(Fi F2) 2 . The new residual may be 
represented by #2, and the analysis will be: 



88 



DF 



Difference, regression func- 
tions 


2(7i - y 2 )2 


1 


Residual 


jR 2 


n' - 3 








First residual 


Ri 


n'~2 



Obviously this process can be continued indefinitely, providing at each 
stage a test of the significance of the additional statistic fitted in the 
regression equation. Isserliss has shown how the sums of squares for 
each regression coefficient can be obtained simultaneously with the so- 
lution of the equations for the unknowns. His method involves solving 
for the regression coefficients CQ, ci, Cm, by means of algebraical 
formulae, and since this method appears to be somewhat laborious, the 
work in the following examples is performed in tables by a technique 



A GENERAL METHOD OF FITTING POLYNOMIALS 225 

similar to that used in solving the equations for partial regression and 
correlation coefficients. It is shown also how the sums of squares 
required for the tests of significance may be obtained directly from 
these tables. 

The analysis of variance test as used here should not be confused 
with the test for non-linearity as described in Chapter XIII. The 
regression straight line may not be a good fit, but, if it is a better fit than 
the horizontal line representing the mean of y, the test we use here will 
show it to be significant. At the same time, the test for non-linearity 
will indicate significant deviations of the means of the y arrays from the 
regression line. As a matter of fact, after fitting a straight line it is 
desirable to apply the test for linearity. If there is no evidence of 
non-linearity there is no object in proceeding to the fitting of a curve of 
higher degree. 

Example 44. For this example we shall use the data of Table 67 and fit poly- 
nomials by successive stages up to the third degree. 

The first step in the procedure of fitting regression lines is to obtain the values of 
the coefficients for the normal equations. These are best obtained as in Table 68, 
which is divided into sections, each section representing the data necessary for cal- 
culating one additional constant. Thus Section A is necessary for fitting a straight 
line; if we wish to fit a second-degree curve we proceed with Section B, and so forth. 
This is continued until it is obvious that further fitting is unnecessary. In actual 
practice we will probably not have to go beyond fitting to the third degree. 

Note that the actual classes for both y and x are replaced by 1, 2, 3, .. .9. This 
reduces the labor a great deal, and, when the Y values have finally been calculated 
for drawing the curve, they may be converted to actual values by the method de- 
scribed in Chapter II, Section 8, for converting means; or the whole equation may be 
converted to actual values by methods similar to those described in Chapter VI, 
Section 5. 

The easiest method for calculating the sum of the powers of x is by continuous 
multiplication. First, N xv x is calculated for each array, and to obtain the figures in 
Nx-ifi? 1 we simply multiply each of the N xv x values by x. When we reach the last 
column of one section it is good practice to check this column using a table of powers 
of x. This checks all the previous calculations of the powers of x. 

Having carried out the calculations as in Table 68, Section A, we write the normal 
equations for fitting a straight line. For the general case these are 



(6) 



and substituting the actual coefficients we have 

164co + 851ci - 1014 
851 ro + 5181ci - 5695 



226 



NON-LINEAR REGRESSION 



TABLE 68 

CALCULATION OF COEFFICIENTS FOB FITTING A POLYNOMIAL UP TO THE 

THIRD DEGREE 



Section A 





Frequency 


Totals 


Means 




Frequency 










y 


of y for x 
Arrays 


for y 
Arrays 


for y 
Arrays 


X 


of x for y 
Arrays 


N xv x 


N xv x z 


XTy Z ~ 

N xy xy x 


T* X /N XV 




Ny X 


T* 


V* 




N xv 










i 


I 


13 


2.6000 


1 


5 


5 


5 


13 


33 8000 


2 


2 


43 


3.3077 


2 


13 


26 


52 


86 


142.2308 


3 


12 


115 


5 0000 


3 


23 


69 


207 


345 


575.0000 


4 


11 


137 


5 9565 


4 


23 


92 


368 


548 


816.0435 


5 


13 


234 


6.6857 


5 


35 


175 


875 


1170 


1564.4571 


6 


50 


100 


6 6667 


6 


15 


90 


540 


600 


666.6667 


7 


45 


115 


7.1875 


7 


16 


112 


784 


805 


826.5625 


8 


22 


199 


7.6538 


8 


26 


208 


1664 


1592 


1523.1154 


9 


7 


44 


7.3333 


9 


6 


54 


486 


396 


322.6667 


10 


1 


11 


7.0000 


10 


2 


20 


200 


140 


98.0000 




164 


1014 






164 


851 


5181 


5695 


6568.5427 



Section B 


Section C 


N xv x* 


N xy x* 


7- 2 T - 
X I y X 

N xy x 2 y x 


N xy x* 


N xv x 


x*T yl = 
N xv x% 


5 


5 


13 


5 


5 


13 


104 


208 


172 


416 


832 


344 


621 


1,863 


1,035 


5,589 


16,767 


3,105 


1,472 


5,888 


2,192 


23,552 


94,208 


8,768 


4,375 


21,875 


5,850 


109,375 


546,875 


29,250 


3,240 


19,440 


3,600 


116,640 


699,840 


21,600 


5,488 


38,416 


5,635 


268,912 


1,882,384 


39,445 


13,312 


106,496 


12,736 


851,968 


6,815,744 


101,888 


4,374 


39,366 


3,564 


354,294 


3,188,646 


32,076 


2,000 


20,000 


1,400 


200,000 


2,000,000 


14,000 


34,991 


253,557 


36,197 


1,930,751 


15,245,301 


250,489 



A GENERAL METHOD OF FITTING POLYNOMIALS 227 



SUMMARY OF COEFFICIENTS 



Section A 


Section B 


Section C 


n' = 164 


Z(z 3 ) - 34,991 


S(;c 5 ) - 1,930,751 


2(x) = 851 


S(x 4 ) = 253,557 


S(z 6 ) - 15,245,301 


2(2/) = 1,014 


Z(afy) - 36,197 


Sfccfy) = 250,489 


SCr 2 ) = 5,181 






I,(xy) 5,695 






^(T^/Ar^) 6,568.54 






S(y - y) 2 = 428.512 







The solution of these equations is carried out as in Table 69, the method being 
identical with that described in Chapter VIII for partial regression and correlation 
coefficients. Note the "check sum" column, which is used for checking the calcula- 
tions as you proceed, and in addition the "check line" just below the "reverse," that 
gives a complete check on all the calculations including those in the reverse. In 
Table 69 the check line is obtained as follows: 

164 X 3.244,175 + 851 X 0.566,340 = 1014 

It is merely a substitution of the statistics CQ and ci in the first equation of (6). 

At the foot of Table 69 we have the analysis of variance for testing the significance 



of the degree of freedom due to the regression straight line. 
obtained from Table 68, using the equality 



2(2, - 



RQ S(y y) 2 is 



jO 2 is then obtained from the solution of the normal equations by multiplying 
the figure in line 5, column 1 (5,1), by the square of the figure in line 6, column 
A'(6,A r ) 2 . The difference is the sum of squares 2(y Fi) 2 = RI, and may be taken 
to represent the error of regression and is therefore appropriate for testing the sig- 
nificance of the variance due to the regression line. In the example, we find that 
the regression is decidedly significant but we proceed to the second stage in order to 
determine whether or not greater accuracy can be obtained. 

Proceeding to the fitting of a polynomial of the form Y = Co + c\x -f czx 2 , we 
write the normal equations 



n'co-f 2(*)ci 
S(:r)co -f 2(x 2 )ci 
-f S( 



(7) 



-f 



and the necessary data for solving the equations are obtained as in section B of Table 
68. The solution of the equations is performed according to Table 70, and note that in 
this table columns (0) and (1) can be copied directly from Table 69, and column K 
can be copied as far as line 6. The reverse and the check line are calculated in the 
usual way. For the analysis of variance RI is brought forward from Table 69, and 
S(Fi - F2) 2 is calculated by multiplying (10,2) by (ll,^) 2 , where the numbers in 



228 



NON-LINEAR REGRESSION 



TABLE 69 

SOLUTION OF NORMAL EQUATIONS FOR FITTING A STRAIGHT LINK 
Line 1 K Sum 



1 

2 


164 
-1.0000 


851 
-5.189,024 


1014 
-6.182,927 


2029 
-12.37,195 


3 
4 
5 
6 




5181 
-4415.8594 
765.1406 
-1.0000 


5695 
5261.6703 
433.3297 
-0.566,340 


11,727 
-10,528.530 
1,198.470 
-1.566,340 


i 
ci= +0.566,340 ! 

co = +3.244,175 i 

P 


3 1 

: 2 


+3.244,175 


+0.566,340 
-2.938,752 


+0.566,340 
+6.182,927 




Check 


532.0447 


+481.9553 


= 1014 







S (sq.) 


DF 


Variance 


F 


5% Point of F 


JK - 2(y - 2/) 2 


428.512 


163 








(5,1) X (6,tf) 2 


245,412 


1 


245.4 


225 


3.90 


^=S(y-Fi) 2 


183.100 


162 


1.130 







the brackets correspond to line and column respectively. The difference between 
the two sums of squares is R%, which can now be taken to represent the error of 
regression. In the example we find that the variance due to the additional degree 
of freedom used in calculating the second-degree curve is quite significant, so we can 
conclude that a real gam in precision has been made. 

If the method of procedure up to this point has been thoroughly understood it 
will be found that the fitting of additional statistics can be carried forward without 
difficulty. The work involved in fitting to the third degree in the present example 
has been performed in Table 71. Note that the columns 0, 1, and 2, can be copied 
directly from previous calculations and that column K can be copied as far as line 11. 

The analysis of variance indicates that the variance due to the additional degree 
of freedom used in fitting a polynomial of the third degree is insignificant. It is, in 
fact, less than the variance due to error of regression. The conclusion is that the 
third-degree curve, although it fits the data satisfactorily, is less useful for predicting 
loaf volume from protein than the second-degree curve. In making use of another 
degree of freedom to determine a new regression function, precision has actually been 
lost. 



A GENERAL METHOD OF FITTING POLYNOMIALS 



229 





rH 


rH 


SS$- 






g 


s3 


OO C^ ^O <N 

rH rH CO rH 


ifc-jg! 








^ 


^Sco * 


isr 






, 


IN. 

S 


8I 


rH 
O 00 OO l> 

rh <NcOO 


O cO OO 




s 


rH 


CO CO *O 




rH IO rH 


O 


g* 


*<* CO 
rH I 

o 


gssi'f 


isl 


OOCO 


rH 
O 


g 

PW 


rH 


""T 


CO'of-^ I 

, 




rH 
11 


1 





CO !> lO 
COCOCJ 
OOCD4O 


CO rHuSS 


fcC'io'co"' 
O CO OO 


<N 


1 N 


00 CO 


silf 


IO cO OO O5 ' 


?++ 


1 


CQ 


to" 


CO s ! 


CO CO O CO 

kOcOOO 




l 


< 




1 


C^ rH | 






O 












1 


^. 






Sco 




fo 


o 








rH 


o 


8" 

rH 






S'c^ 
CO 
I--OO 




"- 


if 


ooSg^y 




rHOO 


i 


o 




tQ '<g< 






rH 


H 




1 






4" 


! 












1 








CM 




1 








co^ 


S 


" O 








co 


rH 


o 


2rf 






o 


1 


i 












s 












& 


rH<N 


m*ot><D 


t-OOCROrH 


rHC^CO 




3 






rH rH 


^AOH 












2oco 












IW 












OrHO 


1 










II II II 





I 



$s 
^ 



X 

R 

' 



230 



NON-LINEAR REGRESSION 





CM 

CO 
(N 

CO 


o ^ 


2 

00 

CMCOWD WD 


>co*^oo 






s 

p! 









rH IN. 

ss 


W3N CO,,* 
t^OOOiSS 
CNOi(N2 


JN. r~l O5 CO CO 

b C3i O5 00 "-H 

5O r_< C\J rH 


illla 






S 

W3 








3 1 


00 rH I 
i i 00 ' 
Cs| 


sssss 


slll^ 
















1 


cf-r i 
1 


tCoTtCr-T 
^111 












.j 

< 


182927 


CO l> CO 


2 

OOOOCI> 
^ CM COO 

t^^QOr-4 


TfQCMOOQ 
00 CO 00 COO 


CM O5 ^h CM 

luQ 1 CO Oi 

CO'^CO'CM" 
oocpoo 

O ""l 5 rH 


I 


fc. 






E POLYNOM 
K 


THO 

oj 


*r-(COO 
.89 i 

10 Q 

1 


|>CO il>O 
O5COOXN 

^HO0^ 

CO*~CM"^~ 1 

COCO | 


O5COOXN OO 
00 ^ CO *-4 OO l 

^cot-t^ i 

S2S^ 

CMCM , 


OOOCD 

+ 1 ++ 


Tt< 

o 

II 


Variance 


o 

rH 

8 

O 


1 

O 


S 

o 


00 


00 

Tf< 


o 




CM ScOCO 
UjCOOil- 


CM 








w 

p 


g 

CO 


2Igg 


Oi 
0000 "t 00 
t- CO 00 CD 


oo^^^o 


iD'rH S CM~ 
iO CO iQ 


00 








Q 

tt 

a co 


rnCO 
0i I-H 
05 CM 


t>- O3 t^ ^ 

IO CO 00 O5 
O O Os I 


T-HCO -H CM W5 

UDr-l T-N ^ 

t> ^t 1 f* CO i 


O I> "* C35 O> i 

COCOOi-^ rH 1 


OOOO 
+ 111 


s 


6s 

r^ 


rH rH 

co 


s 


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<< 


S 1 


w^^r ' 

OCX)I^ 
(M I-H 
| 


icTcM (N 
O CO CD 
14 t~ 


OOCMCMxh 

"^ CO t^ 00 CM 
CM^t-^Ol 

io^t>Tco"" 1 




+ 








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^ 1 1 












gfi 

pa 
ZS 






COI>kO 

co epos 

COCOO 


IQTt^ -iQ 
OOrH OO 




1C ^ Tf" 

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CM"CNT>O > 

CO CM CO 

rHt^.rH 


g 

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rH 


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c? 


IO rH 

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r-l-^COO 

Gi 00 O V* 

Gb 00 i I 


I^IOO^^H 

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1 + + 


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g 


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Tt< tffod" 

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CO CO O CO 
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+ 








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tfc 


? 



FITTING LOGARITHMIC CURVES 



231 



5. Fitting Logarithmic Curves. The procedure is best illustrated by 
means of an example. 

Example 45. The data given in Table 72, and presented graphically in Fig. 13, 
were obtained in a study by Geddes (4), of the effect of time of heating on the baking 
quality of wheat flour. 



100 



80 



60 



1 40 



012345678 
TIME IN HOURS (X) 

FIG. 13. Relation between time of heating and baking quality of wheat flour. 

From an examination of Fig. 13 it is obvious that a straight line cannot give a good 
fit to the results. It is also obvious by inspection that a polynomial cannot be 
expected to give a good fit as the curve tends to flatten out and run parallel to the 



TABLE 72 

INFLUENCE OF THE TIME OF HEATING AT 170 F. ON THE BAKING QUALITY OF 

STRAIGHT GRADE FLOUR 



Time in Hours 
0.25 
50 
0.75 
1.0 
1.5 
2.0 
3.0 
4.0 
6.0 
8.0 



Baking Quality 
Single Feature Estimate 
93 
71 
63 
54 
43 
38 
29 
26 
22 
20 



zero axes at both ends. From x to x 4, the curve might be fitted fairly well 
by a second-degree polynomial, but as x increases from that point, the curve flattens 
out and runs almost parallel to the x axis. This is typical of logarithmic curves and 



232 



NON-LINEAR REGRESSION 



decidedly not typical of polynomials. We decide therefore that a logarithmic curve 
will give the best fit. 

The next step is to examine the three principal types of logarithmic curves, as 
given on page 221 , and make a preliminary determination of their goodness of fit to 
the results by plotting the three pairs of variables, y and log y, log y and x, log y and 
log x, against each other in a rough graph and noting which of the three give points 
that fall most nearly in a straight line. As illustrated in Fig. 14, the set of points 
falling most nearly in a straight line are those given by log y and log x, so we proceed 
to fit a curve of the type log Y CQ + c\ log x. 

The calculations, using log y and log x as variables, are exactly the same as 
in fitting a straight line. These are given in Tables 73 and 74, together with the 
analysis of variance to determine the significance of the fit of the regression line. 



l_VU 1 1 

20 






0-1- Y AND LOG X 




\ \ LOG Y AND X 


19 


\ \ + + IDG Y AND LOf, X 


100- 


\ \ 


18 90- 


^b ^N- 


80- 
17 
70- 


x \ \ 

- Vv \ 


16 60- 


L \ \ \ 


50- 


\ --- x 


15 


\ *v \ 


40- 




14 30- 


^-o^\. 


20- 


^"~-o-_ -C^c 


13 


~~^0 


10- 





X 

LOG X 



-6-4-20 2 4 .6 8 

FIQ, 14. Result of preliminary test to determine the logarithmic equation giving 
the best fit to the data of Table 72. 

Note that the goodness of fit is determined on the basis of the logarithms of y and Y, 
and not on the basis of the actual values. Thus the error of regression is given by 
Sflog y log F) 2 . This can be taken as a general rule, i. e., that when the regression 
equation gives logarithmic values, the test of goodness of fit must be in terms of the 
logarithms estimated. It arises from the fact that logarithms express the relative 
differences between numbers and not their absolute differences. With two numbers 
such as a and 6, their absolute difference is a 6, but log a log 6 is log a/6, and if a 
and 6 are variables and a given percentage increase in a results in a similar percentage 
increase in 6, log a/6 is constant and the relation between the logarithms can be ex- 
pressed by a straight-line equation. To test this fact it is essential that we deal with 
logarithms throughout and not with actual values. 

For graphical purposes it is suitable to express the results of fitting a logarithmic 
equation as in Fig. 15, where the actual values of x are plotted against the anti- 
logarithms of log F, and a smooth curve drawn through the points. The small 
circles in Fig. 15 represent the original values of y and x. 



FITTING LOGARITHMIC CURVES 



233 




I 2345678 

TIME IN HOURS 

FIG. 15. Logarithmic curve for the equation 

log Y = co + ci log x, 
fitted to the data of Table 72. 



TABLE 73 
CALCULATION OF COEFFICIENTS FOR THE CURVE LOG Y co + ci log x 



X 


y 










Time 
in Hours 


Baking 
Quality 


Xl 

= Log a; 


2/1 
= Log^ 


Logy 


y 


0,25 


93 


-0.6021 


1.9685 


1.9937 


98.6 


0.50 


71 


-0.3010 


1.8513 


1.8528 


71 2 


0,75 


63 


-0.1249 


1.7993 


1.7704 


58 9 


1.0 


54 


0.0000 


1.7324 


1.7120 


51 5 


1.5 


43 


0.1761 


1.6355 


1.6296 


42.6 


2.0 


38 


0.3010 


1.5798 


1.5711 


37.2 


3.0 


29 


0.4771 


1.4624 


1.4887 


30.8 


4.0 


26 


0.6021 


1.4150 


1.4302 


26.9 


6.0 


22 


0.7782 


1.3424 


1.3478 


22 3 


8.0 


20 


0.9031 


1.3010 


1.2893 


19.5 



2.209,600 
2.601,671 
6.085,600 
0.355,642,8 



4.169,362 
= 3.703,453 



Z(y - )2 = 0.465,909 = RQ 



i o<. led by subtracting 1 . 



234 



NON-LINEAR REGRESSION 



TABLE 74 

CALCULATION OF STATISTICS AND TEST OF GOODNESS OF FIT FOR THE CURVE 

Log Y - co + c\ log x 



Line 



K 



S 





1 
2 


10 
-1.0 


2.2096 
-0.22096 


6.0856 
-0.60856 


18.2952 
-1.82962 




3 
4 
5 
6 




2.60167 
-0.48823 
2.11344 
-1.0000 


0.355,643 
-1.344,674 
-0.989,031 
0.46797 


5.16691 
-4.04251 
1.12440 


ci -0.46797 
co - +0.71196 


1 
2 


+0.71196 


-0.46797 
+0.10340 


-0.46797 
+0.60856 




Check Line 




7.1196 


-1.0340 


-6.0856 





S(sq.) 


DF 


Variance 


F 


1% Point 


flb 


- 0.465,909 


9 








(5,1) X (6,K) 2 


= 0.462,835 


1 


0.4628 


1205 


11.26 


Ri 


- 0.003,074 


8 


0.000384 







Equation log Y = 1.71196 - 0.46797 log x. 

6. Fisher's Summation Method of Fitting Polynomials. When the 
y values are, or can be assumed to be, of equal weight and are given for 
equal intervals 1 of x, the method of fitting polynomials developed by 
R. A. Fisher provides a very decided short cut from the actual to the 
theoretical polynomial values. The arithmetical labor is likewise easy 
as it consists largely of a process of continuous summation. The pro- 
cedure will be illustrated by an example. 

A summary of formulae for fitting polynomials is given below, and in 
Tables 79, 80, and 81 the constant factors in the formulae have been 
calculated for n = 5 to 20 and r = to 6, where r represents the degree 
of fitting. 

1 Professor Fisher has now developed this method for application to the case 
wherein the y values are of unequal weight. See the references at the end of this 
chapter. 



FISHER'S METHOD OF FITTING POLYNOMIALS 235 



SUMMARY OF FORMULAE FOB FITTING POLYNOMIALS BY THE SUMMATION METHOD 

1. Si, St, 3, 4, 5, 5$, ' ' *S r + i (by summation) 

2. a - Si a' - a 

n 

2 6' - a - 6 



n(n + 1) 



24 

<*' - a - 66 + lOc - 5d 



'n(n + l)---(n+3) 

120 

5 n(n-f l)...(n + 4)' 

720 



a - 106 + 30c - 35d + 14e 



126c - 42/, 



' . where the rule for the formation of the coeffi- 

' " '\ r ~r cients is to multiply successively by 



r 
n(n + l).-.(n + r) r(r + 1) (r - 1) (r + 2) (r - 2) (r + 3) 

1.2 ' 2.3 ' 3.4 

and so on, until the series terminates. 

3. Coefficients 

Yi =+ 1 X (a 1 +W + &' + 7d' + W + llf) 135 7 9 11 

A 

D l Yi - - 7 - - (6 ; + 5c' + 14d' + 30e' + 55f) 1 5 14 30 55 
(n 1) 

60 

c ' + 7d/ + 27e/ + 77 /') 1 7 27 77 



(n - 1) (n - 
840 






Each formula is seen to be composed of two parts that are best cal- 
culated separately. For the component on the right Fisher gives the 
coefficients for fitting curves to the tenth degree. They are reproduced 
here for fitting up to the fifth degree. The factors on the left are of 



236 NON-LINEAR REGRESSION 

alternate positive and negative signs and in generalized form are as 
follows: 

-2.3 3.4.5 __ -4.5 6.7 (r -f 1) (r +2)- -(2r+l) 

n - 1 ' (n - 1) (n - 2) ' (n - 1) (n - 2) (n - 3) '"'' (n - 1) (n - 2)- -(n- 4) 

4. Polynomial values Y\ Yz Fa, etc., by process of summation. 1 

Example 46. The y values in Table 75 represent the percentages of cars of 
smutty wheat graded at Winnipeg, Manitoba, for the years 1925 to 1933 (6). The 
* values are therefore years and can be replaced by the numerals 1 to 9. We shall 
use these data in order to show the procedure of fitting a curve of the fifth degree. 
Such a curve would probably be of very little practical value for analyzing data of 
this kind but it is quite suitable as a numerical example. Summing the y values 
from top to bottom we write down the sum showing on the machine after each value 
is added. This process is repeated in succeeding columns, the sums of the columns 
being designated Si, $2, etc., and if we are fitting a curve to the fifth degree we must 
go as far as S&. At this point the summations must be very carefully checked. 
This is accomplished simply by adding all the columns and noting that the last 
figure in any one column must correspond with the sum of the column on the left. 

The second step is to calculate values that are denoted by the letters a, 6, c, d t e, 
f, and from these obtain a', 6', c', d' t e', and /'. The formulae for these calculations 
are given on page 235. In our example we have 

a - 53.1/9 = 5.900,000 a' - 5.900,000 

6 - 253.3/45 = 5.628,889 V - 0.271,111 

c - 790.8/165 - 4.792,727 c' - - 1.401,213 

d - 2020.8/495 - 4.082,424 d' - 0.358,184 

e = 4577.5/1287 = 3.556,721 e' = 0.302,174 

/ = 9543.6/3003 - 3.178,022 /' - 0.088,117 

The third step is the calculation of Y\ the polynomial value of y corresponding 
to x 9, and five other values known as the first, second, third, fourth, and fifth 
differences. From FI and the differences represented by the symbols 



the polynomial values are built up by a process of summation as illustrated in 
Table 76. For Y and the differences we get 

FI - 1.000,000 X 0.888,833 - 0.888,833 
D l Yi - - 0.750,000 X 2.162,125 - - 1.621,594 

- 1.071,428 X 11.035,206 - 11.823,429 

- - 2.500,000 X 6.238,530 - - 15.596,325 
Z) 4 Fi - 9.000,000 X 1.271,461- 11.443,149 
IfYi - -49.500,000 X 0.088,117 - - 4.361,792 

1 If necessary the actual equation may be written. Details of the calculations are 
given by Snedecor in "Statistical Methods." 



FISHER'S METHOD OF FITTING POLYNOMIALS 



237 



The summation process as illustrated in Table 76 is started in the lower right- 
hand corner. Beginning with D 4 Fi we add successively the value of D 5 Fi. The 
other columns are then built up merely by starting with the first figure at the bottom 
and adding the figures in the same row in the column to the right. The values in 
the last column on the left are the calculated polynomial values of y. Note that in 
the second column only five values are required but we require one more in each 
column as we proceed to the left and also that if only two decimal places are required 
for the polynomial values the number of decimal places are reduced by one for each 
column after the second. A final check on all the work following the calculation 
of Si t $z, -$6 is to add the last column. This should give us S, the total for all the 
values of y. 

The summation method is particularly well adapted to fitting by successive 
stages and to the application of the analysis of variance at each stage. Assuming at 
the outset that fitting will probably be carried to the fifth degree we first calculate 
$1, 2 - -#6 as in Table 75 and the constants a', b', c f , d', e', /'. For each stage of 
fitting we require only Yi and the corresponding differences. If desirable we can 
determine the significance of each degree of freedom used in fitting before we go to 
the trouble of actually calculating the polynomial values and in this way save our- 
selves the labor of calculations that are not going to be of any value. The formulae 
for the sums of squares represented by each additional degree of freedom used in 
fitting are as follows: 



Degree of 

Fitting (r) 





Sum of Squares 
S\/n 



na' 2 (Represents fitting of the mean) 



' + 8 ' > 



, 

(n - 1) (n - 2) (n - 3) 



, 

(n - 1) (n -2)---(n - 4) 



.11 



,, 

(n - 1) (n - 2)- "(n - 5) 



(2r 



(n - 1) (n -2)--.(n - r) 



2 



For the exairple that has already been fitted to the fifth degree the sums of squares 
and corresponding analyses of variance are given in Table 77. \fter fitting to the 
second degree there is no further gain in precision, consequently in actual practise 



238 



NON-LINEAR REGRESSION 



we would proceed direct to the calculation of the polynomial values for a second- 
degree curve. This calculation is given at the foot of Table 77. 



TABLE 75 

CALCULATION OF Si, $2, 3, 4, S$, AND St FOR FITTING A POLYNOMIAL OF THE 
FIFTH DEGREE BY THE SUMMATION METHOD 



1 


2.2 


2.2 


2.2 


2.2 


2.2 


2.2 


2 


1.2 


3.4 


5.6 


7.8 


10.0 


12.2 


3 


2.6 


6.0 


11.6 


19.4 


29.4 


41.6 


4 


5.5 


11.5 


23.1 


42.5 


71.9 


113.5 


5 


16.5 


28.0 


51.1 


93.6 


165.5 


279.0 


6 


17.0 


45.0 


96.1 


189.7 


355.2 


634.2 


7 


6.5 


51.5 


147.6 


337.3 


692.5 


1326.7 


8 


1.1 


52.6 


200.2 


537.6 


1230.0 


2556.7 


9 


0.5 


53.1 


253.3 


790.8 


2020.8 


4577.5 




53.1 


253.3 


790.8 


2020.8 


4577.5 


9543.6 



TABLE 76 
CALCULATION OF POLYNOMIAL VALUES 



1 


2.34 












2 


0.87 


1.467 










3 


2.06 


- 1.190 


2.265,7 








4 


7.91 


- 5.845 


4.655,4 


- 1.998,50 






5 


14.40 


- 6.495 


0.649,9 


4.005,52 


- 6.004,019 




6 


15.90 


- 1,497 


- 4.997,9 


5.647,75 


- 1,642,227 




7 


9.47 


6.429 


- 7.926,1 


2.928,18 


2.719,565 




8 


-0.73 


10.202 


- 3.772,9 


- 4.153,18 


7.081,357 




9 


0.889 


- 1.6216 


11.823,43 


-15.596,325 


11.443,149 


-4.361,792 



Example 47. The whole process of fitting by successive stages may be carried 
out in tabular form as in Table 78. The data are for the relation between pH and 
the activity of the enzyme asparaginase (5). Note that three columns are required 
for fitting to the first degree and thereafter each additional column provides the 
data for fitting one additional constant. Lines 14 and 15 determine the degree to 
which the curve should be fitted. In the example it is obvious that the fitting should 
be carried to the fourth degree; consequently, the remainder of the work applies to 
a fourth-degree curve only. 



FISHER'S METHOD OF FITTING POLYNOMIALS 



239 



TABLE 77 

ANALYSES OF VARIANCE SIGNIFICANCE OF DEGREES OF FREEDOM USED IN 
FITTING TO THE FIFTH DEGREE IN SUCCESSIVE STAGES 



Degree 
of 
Fitting 




Sums of 
Squares 


Degrees 
of 
Freedom 


Variance 


F 


5% Point 


1 


Total 


334.96 


8 










Regression 


2.48 


1 


2.48 








Error 


332 48 


7 


47.50 






2 


Regression 


173.55 


1 


173.6 


6.55 


5 99 




Error 


158.93 


6 


26.49 






3 


Regression 


31.75 


1 


31.75 


1.25 


6 61 




Error 


127.18 


5 


25 44 






4 


Regression 


75 54 


1 


75 54 


5.85 


7.71 




Error 


51 64 


4 


12 91 






5 


Regression 


27 48 


1 


27.48 


3.41 


10 13 




Error 


24 16 


3 


8 053 







Y l = IX (5.900,000 -f 3 X 0.271,111 - 5 X 1.401,213) = -0.292,732 

= -0.75 X (0.271,111 - 5 X 1.401,213) 
D 2 Yi = 1.071,428 X -1.401,213 



x 
1 
2 
3 
4 
5 
6 
7 
8 
9 -1.501,30 



.401,213) 


* 


5.051,216 







-1.501,299 




Polynomial 






Values 






- 1.92 




-5.458 


3.54 




-3.957 


7.50 




-2.455 


9.95 




-0.954 


10.90 




0.547 


10.36 




2.049 


8.31 




3.550 


4.76 




5.0512 


- 0.293 





Total 



53.1 



240 



NON-LINEAR REGRESSION 



I 



& 







rHO3COCOOO 



O*0 CO 

rH rHd 



JOOrH rHiO rH IO 
5 CO rH rH (N rH rH 



.SI 



Is. COO 

COCOrH 

OO^ rH rHCO O 

tot^co^oo o?co ooo 

b- rH O C<l rH rH CO rH rH C 



CO 



g 



CO 

<N ^ csf 



CO C^ rH O O CO 

(N O CO O* O CO CO 
IOOO 1 CO 
OOCp I <M 



5^ ^HIO oooi- 

go>ococp co?o< 
CO l>- O5 O> "* ^l * 



4- 

~ 



rH rH (N (N CO T^ TlH 



QQ Q5 ^Q QJ j ^5 ^ 

o osc^cocooo ooo5 



(N CO 
fvj U 

o* 



00 t>- OO t>- 



O ^ C> G5 00 rH rH Q C^ r- 1 CO 



i 

I 

9 

CO 



o> S? 

06 



25 c 



. 

S 



8 



JD : . .- 

cj .- 



:oo 

:_e 



CQ 5 cP 



EXERCISES 



241 



(1) r +i values entered as columns are summated. 

(2) Divisor for S r +i values, taken from Table 79. 

(3) Division of line 1 by line 2 gives the constants a, b, c t d, 

(4) The constants a', 6', c', d', . . .are calculated from a, 6, c, d, . . .as indicated in 
summary on page 235. 

(5) Squares of a', &', c', d',.... 

(6) Factor taken from Table 80. 

(7) Line 5 multiplied Ijy line. 6 gives the sum of squares S(F r -i F r ) 2 repre- 
sented by 1 DF. For each DF utilized in fitting this is the reduction in the sum of 
squares due to error of regression. 

(8) Enter S(j/) 2 in first column. 

(9) Repeat S(F r -i - F r ) 2 values. 

(10) Subtracting 9 from 8 in the first column gives the remainder in line 10. Then 
subtract the values in line 9 successively, putting down the remainders in line 10. 

(11) The DF for error of regression are entered here. The DF for the sums of 
squares in line 9 is 1 in each case so that they do not need to be entered. 

(12) Line 9 repeated, reducing to 4-figure accuracy. 

(13) Line 10 divided by line 11. 

(14) F - * fa. 

(15) Enter 5% points from Table 96 

(16) Calculate as in section 3 of summary of formulae. 

(17) Enter factors from Table 81. 

(18) Line 16 multiplied by line 17. 



CALCULATION OF POLYNOMIAL VALUES FOB FOURTH-DEGREE CURVE 



1 


0.2748 










2 


0.1691 


0.105,651 








3 


1.6903 


-1.521,180 


1.626,831 






4 


4.0161 


-2.325,746 


0.804,566 


0.822,265 




5 


6.4768 


-2.460,689 


0.134,943 


0.669,623 




6 


8.5554 


-2.078,651 


-0.382,038 


0.516,981 




7 


9.8877 


-1.332,274 


-0.746,377 


0.364,339 




8 


10.2619 


-0.374,200 


-0.958,074 


0.211,697 




9 


9.6190 


0.642,929 


-1.017,129 


0.059,055 




10 


8.0525 


1.566,471 


-0.923,542 


-0.093,587 




11 


5.8087 


2.243,T84 


-0.677,313 


-0.246,229 




12 


3.2865 


2.522,226 


-0.278,442 


-0.398,871 




13 


1.0373 


2.249,155 


0.273,071 


-0.551,513 




14 


-0.234,608 


1.271,929 


0.977,226 


-0.704,155 


0.152,642 



7. Exercises. 

1. Calculate the correlation ^atio for the data of Table 67, Chapter XIII, and by 
means of the analysis of variance test the significance of the variance for the means 
of the arrays. 



242 



NON-LINEAR REGRESSION 



TABLE 79 

FOR USE IN CALCULATION OF o, 6, c, d, e,f- 
Degree of Fitting (r) 



n 





1 


2 


3 


4 


5 


6 


5 


5 


15 


35 


70 


126 


210 


330 


6 


6 


21 


56 


126 


252 


462 


792 


7 


7 


28 


84 


210 


462 


924 


1,716 


8 


8 


36 


120 


330 


792 


1,716 


3,432 


9 


9 


45 


165 


495 


1,287 


3,003 


6,435 


10 


10 


55 


220 


715 


2,002 


5,005 


11,440 


11 


11 


66 


286 


1001 


3,003 


8,008 


19,448 


12 


12 


78 


364 


1365 


4,368 


12,376 


31,824 


13 


13 


91 


455 


1820 


6,188 


18,564 


50,388 


14 


14 


105 


560 


2380 


8,568 


27,132 


77,520 


15 


15 


120 


680 


3060 


11,628 


38,760 


116,280 


16 


16 


136 


816 


3876 


15,504 


54,264 


170,544 


- 17 


17 


153 


969 


4845 


20,349 


74,613 


245,157 


18 


18 


171 


1140 


5985 


26,334 


100,947 


346,104 


19 


19 


190 


1330 


7315 


33,649 


134,596 


480,700 


20 


20 


210 


1540 


8855 


42,504 


177,100 


657,800 



TABLE 80 



- - FOR CALCULATION OF SUMS OF SQUARES 

n - l) ( n -2)---(n - r) J 



Degree of Fitting (r) 



n 





1 


2 


3 


4 


5 


6 


5 


5 


22.5000 


87.5000 


490.000 


5670.000 






6 


6.0 


25.2000 


84.0000 


352.800 


2268.000 


30,492.00 




7 


7 


28.0000 


84.0000 


294.000 


1386.000 


10,164.00 


156,156.00 


8 


8 


30.8571 


85.7143 


264.000 


1018.286 


5,393.14 


44,616.00 


9 


9.0 


33.7500 


88.3928 


247.500 


827.357 


3,539.25 


20,913.75 


10 


10.0 


36.6667 


91.6667 


238.333 


715.000 


2,621.67 


12,393.33 


11 


11 


396000 


95.3333 


233.567 


643.500 


2,097.33 


8,427.47 


12 


12 


42.5454 


99.2727 


231.636 


595.636 


1,768.00 


6,268.36 


13 


13 


45.5000 


103.4091 


231.636 


562.545 


1,547.00 


4,962.45 


14 


14.0 


48.4615 


107.6923 


233.007 


539.245 


1,391.38 


4,110.91 


15 


15 


51.4286 


112.0879 


235.385 


522.737 


1,277.80 


3,523.64 


16 


16 


54.4000 


116.5714 


238.523 


511.121 


1,192.62 


3,100.80 


17 


17.0 


57.3750 


121.1250 


242.250 


503.135 


1,127.39 


2,785.88 


18 


18.0 


60.3529 


125.7353 


246.441 


497.912 


1,076.68 


2,544.88 


19 


19.0 


63.3333 


130.3922 


251.005 


494.838 


1,036.80 


2,356.37 


20 


20 


66.3158 


135.0877 


255.872 


493.467 


1,005.21 


2,206.24 



EXERCISES 



243 



TABLE 81 
FOB CALCULATION OF Y\ AND "DIFFERENCES" 



Degree of Fitting (r) 



n 





1 


2 


3 


4 


5 


6 




+ 


_ 


+ 


_ 


4. 





+ 


5 




1.5 


5.0 


35.0 


630.0 






6 




1.2 


3.0 


14.0 


126.0 


2772.0 




7 




1.0 


2.0 


7.0 


42.0 


462.0 


12,012.0 


8 




0.8571,4286 


1.4285,7143 


4.0 


18.0 


132.0 


1,716.0 


9 




0.75 


1.0714,2857 


2.5 


9.0 


49.5 


429.0 


10 




0.6666,6667 


0.8333,3333 


1.6666,6667 


5.0 


22.0 


143.0 


11 




0.60 


0.6666,6667 


1.1666,6667 


3.0 


11.0 


57.2 


12 




0.5454,5454 


0.5454,5454 


0.8484,8485 


1.9090,9091 


6.0 


26.0 


13 




0.50 


0.4545,4545 


0.6363,6364 


1.2727,2727 


3.5 


13.0 


14 




0.4615,3846 


0.3846,1538 


0,4895,1049 


0.8811,1888 


2.1538,4615 


7.0 


15 




0.4285,7143 


0.3296,7033 


0.3846,1538 


0.6293,7063 


1.3846,1538 


4.0 


16 


1 


0.40 


0.2857,1428 


0.3076,9231 


0.4615,3846 


0.9230,7692 


2.4 


17 


1 


0.375 


0.25 


0.25 


0.3461,5385 


0.6346,1538 


1.5 


18 


1 


0.3529,4118 


0.2205.8824 


0.2058,8235 


0.2647,0588 


0.4479,6380 


0.9705,8824 


19 


1 


0.3333,3333 


0.1960,7843 


0.1715,6863 


0.2058,8235 


0.3235,2941 


0.6470,5882 


20 


1 


0.3157,8947 


0.1754,3860 


0.1444,7884 


0.1625,3870 


0.2383,9009 


0.4427,2446 



2. For the following equations, calculate the values of Y for x 1 to x = 20, 
and plot the curves on graph paper. 



(a) 
(b) 
(c) 
(d) 



Y = 2. 58 -f 0.84 x 
Y = 2.58 + 8. 4 logs 
Log Y = 0. 258 + 0.058* 
Log Y = 0.213 + 0.662 log x 



Describe the effect of the logarithmic transformation of equation (a) into equa- 
tions (b) and (c). 

3. Using the data given in Table 85, determine the type of logarithmic curve that 
should be fitted to the data. Having selected the type of curve proceed with the 
fitting as in Tables 73 and 74. Prepare two graphs, one showing the fit of the 
straight-line logarithmic equation to the logarithms of y, and another showing the 
curve for the actual values of Y estimated from the regression equation. Table 82 
may be used for a similar exercise. 

4. Table 83 gives the values of y, N yx , and T yx from a correlation surface for the 
area and head length of 500 bull spermatozoa, Isa (7). The three columns are 
similar to the first three columns of Table 68 and provide all the data necessary 
for calculating polynomial regression equations. Find the regression equation that 
gives the best fit to the data. Then calculate the Y values and construct a graph 
similar to Fig. 15, showing the means of the arrays and the regression line. 



244 NON-LINEAR REGRESSION 

6. Using the data for x and y given below, determine the goodness of fit of curves 
up to the sixth degree. Select the curve to which the data should be fitted, and 
proceed accordingly to the calculation of the polynomial values. Graph your results. 

# I 2 3 4 5 6 7 8 9 10 11 12 13 14 
y 12.6 13.8 14.1 13.9 12.3 7.2 4.8 2.8 2.4 2.1 3.7 5.3 7.8 8.3 

6. In economic analysis, methods of curve fitting are very frequently utilized 
in order to study secular trend in a time series. Secular trend means the smooth 
long-term movement of a series of statistical values and is entirely distinct from 
seasonal and cyclical fluctuations. Cyclical fluctuations are not as periodical as the 
seasonal ones but as a general rule have sufficient regularity to show definite swings 
above and below the normal through periods of depression and prosperity. Curve 
fitting may, on the one hand, be used to measure the secular trend of a statistical 
series, and, on the other hand, using the fitted curve as a normal, we can plot the 
deviations from the normal in such a way as to bring out the characteristics of 
cyclical fluctuations. 

Take the data given in Table 84 of the bank clearings in New York City for the 
years 1860 to 1923 and combining them in 4 year groups obtain 16 points to which a 
curve may be fitted. Determine the best-fitting polynomial and graph your results 
on a large sheet of graph paper giving the 16 calculated values and the actual bank 
clearings for individual years. Measure off the deviations of the values for individual 
years from a smooth curve drawn through the 16 calculated points, and graph these 
deviations on another sheet showing them as deviations from a straight horizontal 
line. 

TABLE 82 
HBAT or HYDRATION IN CALORIES AND WATER IMBIBED PER GRAM OF FLOUR 

Cc. Heat of 

Water Imbibed Hydration 

0.012 2.3 

0.025 5.7 

0.039 7.4 

0.049 9.2 

0.064 10.7 

0.073 12.4 

0.091 14.6 

0.099 15.1 

0.123 16.8 

0.146 17.8 



EXERCISES 



245 



TABLE 83 

DATA FROM CORRELATION SURFACE FOR AREA (y) AND HEAD LENGTH (x) OF 

600 BULL SPERMS 



y 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 



Frequency 
of y for 
x Arrays 

2 


7 

7 

14 

12 

22 

36 

70 

112 

133 

69 

2 

2 

1 

Total = 500 



Totals for 
y Arrays 

6 

24 

63 
247 
618 
939 
1038 
897 
557 
311 

82 

29 

41 

29 
7 



TABLE 84 

BANK CLEARINGS IN NEW YORK CITY (1860-1923) 
Figures in thousands of millions 



1860 


7.2 


1876 


21.6 


1892 


36.7 


1908 


79.3 


61 


5.9 


77 


23.3 


93 


31.2 


09 


103.6 


62 


6.9 


78 


19.9 


94 


24.4 


10 


97.3 


63 


14.9 


79 


29.2 


95 


29.9 


11 


92.4 


64 


24.1 


80 


38.6 


96 


28.8 


12 


100.7 


65 


26.0 


81 


49.4 


97 


33.4 


13 


94.6 


66 


28.7 


82 


46.9 


98 


42.0 


14 


83.0 


67 


28.7 


83 


37.4 


99 


60.8 


15 


110.6 


68 


28.5 


84 


31.0 


1900 


62.7 


16 


159.6 


69 


37.4 


85 


28.2 


01 


79.4 


17 


177.4 


70 


27.8 


86 


33.7 


02 


76.3 


18 


178.6 


71 


29.3 


87 


33.4 


03 


66.0 


19 


235.8 


72 


33.8 


88 


31.1 


04 


68.6 


20 


243.2 


73 


35.5 


89 


35.9 


05 


93.8 


21 


194.4 


74 


22.9 


90 


37.4 


06 


104.7 


22 


217.9 


75 


25.1 


91 


33.7 


07 


87.2 


23 


214.0 



246 



NON-LINEAR REGRESSION 



TABLE 85 
MOISTURE CONTENT AND HEAT OF HYDRATION OF FIFTH MIDDLINGS FLOUR (6) 



Per Cent 
Moisture 

<y) 

1.7 

2.9 

4.2 

5.6 

6.6 

8.1 

9.0 
10.8 
11.6 
14.0 
16.3 



Heat of 

Hydration 

in Calories 

(*) 

18.3 

16.0 

12.6 

10.9 

9.1 

7.6 

5.9 

3.7 

3.2 

1.5 

0.5 



REFERENCES 

1. J. BLAKEMAN, Biometrika, 4: 332. 1905. 

2. MORDECAI EZEKIEL. Methods of Correlation Analysis. John Wiley & Sons, 

Inc., New York, 1930. Reading: Chapter VI. 

3. R. A. FISHER. Statistical Methods for Research Workers. Oliver and Boyd, 

London, 1936. Reading: Chapter V, Sections 27, 28, 28.1, 29.2. 

4. W. F. GEDDES, Can. Jour. Research, I: 528-558. 1929. 

5. W. F. GEDDES and A. HUNTER, Jour. Biol. Chem., 77: 1928. 

6. W. F. HANNA and W. POPP, Sci. Agric., 11: 200-207, 1930. 

7. J. ISA. Unpublished data from master's degree thesis, University of Manitoba. 

8. G. W. SNEDECOR. Statistical Methods. Collegiate Press Inc., Ames, Iowa, 

1937. Reading: Chapter XIV. 

9. L. H. C. TIPPETT. The Methods of Statistics. Williams and Norgate, Ltd., 

London, 1931. Reading: Chapter IX. 
10. C. A. WINKLER and W. F. GEDDES. Cereal Chem., 8: 455-475, 1931. 



CHAPTER XV 
THE ANALYSIS OF COVARIANCE 

1. The Heterogeneity of Covariation and the Principle of Covariance 
Analysis. We have noted from our study of the analysis of variance 
that for a single variable the variation is frequently heterogeneous and 
may be sorted out into components determined largely by the way in 
which the data are taken. The same is true for the correlated vari- 
ability or covariation of two variables, and the mechanism for sorting 
out the covariance effects is known as the analysis of covariance. In 
order to think in terms of actual values, we may suppose that the two 
variables are yields of grain and straw from cereal plots. The total 
covariance for grain and straw yields is made up in part by the covari- 
ance for the means of the treatments and in part by the covariance within 
the plots of the same variety. The degree of correlation may be differ- 
ent for the two components and hence the total correlation is hetero- 
geneous. In the same way we may consider the covariance for the 
replicate means as another component. In fact the components may 
be taken as exactly equivalent to those according to which the data may 
be classified for an analysis of variance of either variable. 

2. Division of Sums of Products and Degrees of Freedom. Just as 
the analysis of variance arises from the fact that the sums of squares and 
degrees of freedom may be subdivided according to the way in which the 
data are classified, the analysis of covariance arises from the fact that 
the sums of products of the deviations and corresponding degrees of 
freedom can be subdivided in the same manner. 

Representing a set of data for two variables as follows : 



k groups X2iy2 * 222 



in which there are k groups of n pairs of variates of x and y. Then 

(arii ) = On xi) + (x\ x) 

and (y\i y) = (yn yi) + (yi 2/) 

247 



248 THE ANALYSIS OF COVARIANCE 

Multiplying to obtain a single product of the deviations: 

(xn x) (yn j/) = (xn xi) (yn yi) + (xi - x) (y\ y) 

+ (xn xi) (yi y) + (xi - x) (yn - #1) 

On summating for all the pairs in the first group the last two terms dis- 
appear and we have : 

n n 

^?* ( nf & J ( it _ '//) ^ T -T-i it'll 1*7 1 | I W ( *Y -i *T*I I 1*7 1 l"7l 

A/V'*' *'/\fir y) & *' J 'i)\y yi) i /H-*'! ^) \y\ y) 
Then summating over the A: groups: 

nk I f n 1 * 

^iT or*) (-1/ V/i ~~~ T5^l 'Vfl* / T* I i7/ - ')"/ I I I 'M ^V i / y* r '1*1 (fl"7 f T/\ ill 

z/v**- *'/ \y y) ^LZ/V**' *'*/ \y y&> j < i^z^\^g ^j \yg y/ v 1 / 

where x g and ft are group means for x and y. This is the fundamental 
equation for the sums of products on which the analysis of covariance is 
based. If the same data are divided into n classes as well as k groups, 
the equations for sums of products and degrees of freedom are: 

iik nl 

2(:r - x)(y - y) = 2(x - x t - x c + x)(y - ft - y c + J7) 
(nk - 1) (n - l)(fc - 1) 

+ n S(x* - x)(ft - y) + k 2(4, - x)(ft - y) (2) 
+ (fc - 1) + (n - 1) (3) 

The method of calculating the sums of products is not according to 
these formulae but by means of equalities similar to those used for cal- 
culating sums of squares. These equalities are described below under 
Example 48. 

3. Coefficients of Correlation Corresponding to Sums of Products 
and Squares. Considering the simple classification of the pairs of 
variates into k groups of n pairs, we have the sums of products and corre- 
sponding sums of squares of x and y as follows : 



n* n* t 

2 OF -x)(y -y) = 2(z - x g )(y - ft) + n 2(x g - x)(ft - y) 
2 OF - x) 2 = 2(x - x g ) 2 + n S(x g - x) 2 

y*) 2 + n 2(ft - y) 2 



(4) 



COEFFICIENTS OF CORRELATION 



249 



It is now clear that each vertical set represents the factors necessary to 
calculate correlation or regression coefficients. Hence we can write: 



r xv (total) = 



- x)(y - g) 



y) 2 



nt 



-0 



* nfc 

2(x- 
DF = nk - 2 



(within) 



- s f )(# - g f ) 






k(n - 1) 



(between) = 



n2(^ - x)(y g - y) 



(5) 



DF = k - 2 



Note that for each component the degrees of freedom for estimating the 
coefficients are one less than for the corresponding estimates of the 
variance. 

Since it can be proved that the variances and covariances for between 
and within groups are unbiassed estimates of the true values for the 
population sampled, it follows that the corresponding coefficients of 
correlation and regression are also unbiassed estimates of the correla- 
tion and regression parameters of the population. They can be used, 
therefore, to test the significance of the covariance effects represented by 
the various components for which they are calculated. One practical 
application of this principle will be seen at once. Total correlation 
coefficients are obviously incapable of definite interpretation if they 
represent heterogeneous covariance effects, and tests of significance 



250 



THE ANALYSIS OF COVARIANCE 



applied to them cannot give a clear-cut answer. The coefficients 
calculated from each component, however, are capable of definite inter- 
pretation. In the simple case of covariance within and between groups, 
if the total covariance is made up largely of the covariance between the 
means of the groups, the total correlation is often referred to as contain- 
ing a spurious effect. By the covariance method this effect is taken 
care of in the calculation of the covariance between the means and is 
completely removed from the covariance within the groups. Thus the 
so-called spurious effect is not only removed but completely evaluated as 
a distinct component of the total. 

4. Applications of the Covariance Method to the Control of Error. 
One of the most important applications of the analysis of covariance is 
in the control of errors that arise at random throughout the experiment 
and cannot be taken care of by replication. In the case, for example, of 
number of plants per plot for such crops as mangels and sugar beets, 
the variations in number of plants arise at random throughout the 
experiment and, so far as they affect the yields of single plots, add to the 
experimental error. Correction of the yields on the basis that yield is 
directly proportional to the number of plants is a frequent practice, but 
it is not difficult to demonstrate that yield is rarely if ever proportional 
to the number of plants per plot, and that such an adjustment is likely 
to exaggerate the yields of plots in which plants are missing. Correction 
on the basis of the exact relation between yield and number of plots as 
indicated by the data is, however, perfectly justifiable, and the method of 
making such a correction is a natural development of the covariance 
technique. Numerous applications of the same method will undoubt- 
edly occur to workers in other fields. 

In order to demonstrate the control of error by the covariance 
method, wo shall represent a covariance analysis algebraically as follows, 
in which the experiment is presumed to be a randomized block field plot 
test. 





DF 


2(* 2 ) 


2(*y) 


2(2/ 2 ) 


byx 


b y ^L(xy) 


2G/' 2 ) 


DF 


Blocks . . 


P 


Ao 


Bo 


Co 










Treat- 


















ments 


<l 


Ai 


Bi 


Ci 


bi Bi/Ai 


biBi 


Ci - 6A 


q-l 


Error . . . 


n 


A 2 


Bi 


C 2 


62 = Bt/A* 


b 2 B 2 


C 2 - 6 2 B 2 


n - 1 


T + E . 


n -f q 


At 


B t 


c t 


b t = B t /At 


b t B t 


Ci btB t 


n + q- 1 



COVARIANCE METHOD TO THE CONTROL OF ERROR 251 

In the column headings, x is written for (x x), y for (y y), b vs for the 
regression coefficient of y on z/and S(t/' 2 ) indicates a sum of squares for 
y adjusted by the regression coefficient in the same line. 

The calculations are complete in each line of the table. The regres- 
sion coefficient is B/A, and the adjustment in the sum of squares for y is 
bB or B 2 /A. In the last line we are considering only treatments and 
error so that At = AI + A 2 , B t = B\ + B 2 and Ct = Ci + C 2 . 

The second step in the procedure is indicated as follows : 

DF S (sq.) Variance 



T + E 


n + q 


- 1 C t - b t B t 




E 


n 


1 C 2 - 6 2 B 2 


F 2 


T 


q 


Ci + W*2 - btBt 


F! 


T 


q - 


1 Ci - 6iBi 


Fa 



- 6 2 ) 1 biBi + 6 2 B 2 - b t Bt V* 



The first sum of squares for treatments is obtained by differences and, 
since it has not been adjusted by the treatment regression coefficient, is 
still represented by q degrees of freedom. The second treatment sum of 
squares is written down from the first table and is represented by q 1 
degrees of freedom, as it has been adjusted by the treatment coefficient. 
On subtracting the second treatment sum of squares from the first, we 
have a sum of squares given by biBi + 62^2 b t B t , and it is not diffi- 
cult to prove the following equality: 

biBi + b 2 B 2 - btB t = b\Ai + b 2 2 A 2 - b?A t = ~^~ (bi - 6 2 ) 2 (6) 

Ai + A 2 

It follows that when 61 = b 2 this sum of squares is zero, and that a test of 
significance of the corresponding variance (4) is a test of the significance 
of the difference between the error and treatment regression coefficients. 
The test of significance of the treatment differences after adjustment 
for the regression of y on x involves a comparison of the variances V 2 and 
V\. The fact that Fi may contain a significant effect due to (bi fo) 
does not vitiate the meaning of the test, as such an effect is obviously due 
to some factor characteristic of the treatments. In the case of yield and 
number of plants per plot, the variety regression coefficient (61) might 
be higher than (6 2 ), and this will contribute to the significance of Vi, but 
62 represents the regression of yield on number of plants within varieties, 
and may be taken as a true measure of the effect of number of plants on 
yield. If the treatment regression coefficient is higher this probably 
reflects an additional genetic relationship, and one that should contribute 
to the significance of the differences between the varieties. A further 



252 THE ANALYSIS OF COVARIANCE 

test may be applied, however, to Vs, and by a comparison of the signif- 
icance of Vz and V* a complete picture of the variety effects is obtained. 
The value of such an analysis, if, for example, number of roots has a 
significant effect on yield, is that the error variance and variety variance 
will be reduced proportionately with a consequent increase in the signif- 
icance of the variety differences, if such differences exist. If the anal- 
ysis of the unadjusted yields shows significant differences when the 
adjusted yields do not, this simply means that the original differences 
were due to number of roots and not to the yielding characteristics of the 
varieties as measured by average yield per root. 

R. A. Fisher (4) has pointed out that an appropriate scale for measur- 
ing the effectiveness of methods of reducing the error is the inverse of 
the variance. This is sometimes called the invariance and is represented 
by 1/V. In measuring the reduction of error by means of the covariance 
analysis, this scale is particularly useful. Example 48 is a good illus- 
tration of this point. The original error variance is about three times as 
large as the final error variance obtained by adjusting the sums of squares 
for two associated variables. In other words, in the original form with- 
out any adjustment about three times as many replications would be 
required to give the same accuracy as the adjusted values. One should 
not reason from this that the significance of the differences between the 
treatments will be increased accordingly, as it must be remembered that 
at the same time differences between the treatments due to the associ- 
ated variables are also being removed. 

The test of significance having been applied as outlined, the next step 
is to make an actual correction of the variety means. Since the regres- 
sion coefficient in the error line may be considered as representing the 
actual effect of number of roots on yield, this regression coefficient 
should be used for making corrections. The corrected means should 
then be the best possible estimates of what the means would have been 
if they had not been affected by variations in number of roots. The 
regression equation will be of the form : 

Fi = fr - M*i - *) (7) 

where x\ is the mean of x for one variety, y\ is the mean of y for the same 
variety, b vx is the regression of y on x in the error line, and Yi is the 
estimated mean of the variety. 

To compare two corrected means such as Y p and Y q we must use for 
the standard error of the difference between two means 



A TEST OF THE HETEROGENEITY 



253 



where s 2 is the variance in the error line of the analysis of covariance 
table (for example, in Table 87 it will be 7681.3/35 = 219.5), A is the 
sum of squares for x in the same line, r is the number of replications, and 
(x p x q ) is the difference between the two means used in the two 
expressions for calculating Y p and Y q . Thus 

Y P = y p b yx (x p - x) and Y q = y q b yx (x q x) 

In comparing two means corrected for two variables x\ and X2 we 
calculate the standard error of a mean difference as follows 

2 u~B 2uvP + v 2 A 
_ + ^ B _ p2 

where A and B are the sums of squares in the error line for x\ and #2- 
P is the sum of products for x\ and X2 in the error line. 
u = (XI P i0), difference between xi means, 
v = (fep 29), difference between X2 means. 

The method of error control by means of two or more associated 
variables is described in Example 48. 

6. A Test of the Heterogeneity of a Series of Regression Coefficients. 
The analysis of covariance provides a unique technique for testing the 
significance of the differences between two or more regression coefficients. 
tJsing the same symbolism as in the previous section, the procedure is as 
given below. 



Group 


DF 


2(x 2 ) 


2(xy) 


S(V 2 ) 


byx 


b vx 2(xy) 


2(y' 2 ) 


DF 


1 


9 


Ai 


Bi 


Ci 


61 = Bi/Ai 


&A 


Ci - 6iB t 


q~l 


2 


9 


At 


B z 


C 2 


62 = B 2 /A 2 


bzBz 


2 &2$2 


q-l 


3 


9 


A 3 


B 3 


C 3 


63 = Bs/As 


btB* 


C 8 - 6 3 B 3 


q - - 


P 


9 


A p 


B f 


C p 


b p = B P /A P 


b p B p 


C p - bpB p 


q-l 


Total 


P9 


A t 


B t 


c t 


b t = B t /A t 


b t B t 


Ct btBt 


M - 1 



254 



THE ANALYSIS OF COVARIANCE 





DF 


S(y' 2 ) 


Variance 


Total 


T)Q 1 


Ct btBt 




Within groups. . 


p(9 ~ 1) 


2<C - W) 


Vi error variance 


Difference 


(P-D 


JB(bB) - b,B t 


Vz due to differences between 
regression coefficients 



The last sum of squares may be shown to be 

P pC*/ 

^ T i 

- Ot&i Li \ " 



i - b k ) 2 



+ A 2 -h 



+ - 



where 6y and fyt represent all possible pairs of the regression coefficients and Aj 
and Ak all possible pairs of the corresponding sums of squares for x. 

The comparison of V\ and V% by means of the z test furnishes therefore 
the required test of the heterogeneity of the regression coefficient. 

Example 48. For the sake of brevity this one example will be used to demonstrate 
most of the important applications of the covariance technique. Data are given by 
Crampton and Hopkins (1) on weights, gains, and feed consumption in a comparative 
feeding trial. These data are reproduced in Table 86 for initial weight, feed eaten, 
and final weight. The analysis is concerned with expressing the results for final 
weight corrected for variations in initial weight, corrected for variations in feed 
eaten, and corrected for initial weight and feed eaten. The last is an application of 
the method of partial regression which is described in detail in the paper by Crampton 
and Hopkins. In addition a test will be illustrated of the significance of the dif- 
ferences between the regression coefficients for each treatment. 

(1) Effect of Initial Weight on Final Weight. The analysis of covariance is set 
up in the form shown in Table 87. In performing the calculations for such a table, 
it is recommended that the sums of squares, sums of products, and totals be obtained 
by treatments, as it is necessary to keep these separate if certain tests are to be 
employed at a later stage. In obtaining the sums of products it should be noted 
that a procedure may be followed exactly analogous to that for obtaining sums of 
squares. With k replications of n treatments, the sums of products are given as 
follows: 



Total 



- x)(y - y) 



- T X T V /N 



Bet ween means of treatments k 2 (*< - x)(g t - y) * 2(^*7%) A - T x T y /N 

k k 

Between means of replicates n 2(x r - )(&. - g) = SC^rx^rv)/^ - T x T y /N 
Residual or error Total (treatments) (replicates). 

Where T tx and T ty are treatment subtotals for x and y 
and T rx and T rv are replicate subtotals for x and ^/. 



A TEST OF THP; HETEROGENEITY 



255 



H 
o 



HH 

w 



d 
O 



cc 

o 



W o 

3 



co 
fc 



W 
W 
HH 



I 

O 



s 





-^S 
03 



1 ) 
s 







s 

^3 

05 



'3 

HH t 



COCOCOCOCOCOCO<MCO<N 



I-H to t;- T < O OO GO 
f*~> o^ OO O5 ^5 C> ""* 

CM i 1 T"-" T-H O4 CS1 C^-l 



^i-- 
O^> C^i cO <y? O *O !> co CO CO 

co co co CD t>- co co co r co 



Cy>t^.QOOCTst OOOO50O 



^t* CIO i i^H 
tC^COCi 

cO co cO cO 



^OQOCMt^- 
t -COCOCO 

*O cO cO CO 



Oi-HrHCOt^^OOiOOCO 

CO<N<MCO<MM(M<M<M<M 



256 



THE ANALYSIS OF COVARIANCE 



TABLE 87 

ANALYSIS OF COVARIANCE FINAL WEIGHT AND INITIAL WEIGHT 
zs initial weight x\ * final weight 





(1) 
DF 


(2) 
2(4) 


(3) 

2(ziz 8 ) 


(4) 
2(*?) 


(5) 
611 


(6) 
6u2(zisi) 


(7) 
2(2/' 2 ) 


(8) 
DF 


(9) 

ria 


Replicates . 
Treatments 
Error 


9 
4 
36 


454.4 
509.2 
368.4 


752.0 
1,172.2 
1,001.8 


2,487.2 
5,741.7 
10,405.5 


1.6549 
2.3016 
2 7193 


1,244.5 
2,697.9 
2,724 2 


1,242.7 
3,043.8 
7,681.3 


8 
3 
35 


0.7075 
0.6854 
0.5117 






















Treatments 
+ Error 


40 


877.6 


2,173.8 


16,147.2 


2.4770 


5,384.5 


10,762.7 


39 





(1) DF for unadjusted sums of squares. 

(6) 613 = item in col. (3) divided by item in col. (2). 

(6) 6isS(ziz 3 ) - col. (5) X col (3) or col. (3) 2 /col. (2). 

(7) S(y' 2 ) = adjusted sums of squares = col. (4) col. (6). 

(8) DF for adjusted sums of squares. 

(9) Correlation coefficient (unnecessary for tests of significance). 

From Table 87 we can proceed to the test of significance of the treatment dif- 
ferences adjusted for initial weight and of the difference between the treatment and 
error regression coefficients 



DF S (sq.) Variance 



5% Point 



Treatments -f- Error 


39 


10 762 7 








Error 


35 


7,681.3 


219.5 






Difference = Treatments 
Treatments 


4 
3 


3,081.4 
3,043.8 


770.4 
1,014.7 


3 51 


2 64 


Difference * 6, b t . . 


1 


37.6 


37.6 







Since the difference between the* err or and treatment regression coefficients 
(6, bi) is obviously insignificant the tests of significance are not carried any 
further. 

To adjust the means of the treatment final weights for the initial weights we use 
the equation given above which in terms of the symbols now being used will be 



(2) Effect of Feed Eaten on Final Weight. The procedure is exactly the same as 
above so will be given in tabular form only. 



EXAMPLE OF COVARIANCE ANALYSIS 



257 



TABLE 88 

ANALYSIS OF COVARIANCE FEED EATEN AND FINAL WEIGHT 
#2 feed eaten x\ final weight 





DF 


2(* 2 2 ) 


2 (0:1x2) 


2(*5) 


612 


bvSfrixz) 


2(2/' 2 ) 


DF 


r\2 


Replicates 
Treatments 
Error . . . 


9 
4 
36 


35,150.3 
28,404.9 
90,792.3 


8,774.1 
11,596.5 
24,508.7 


2,487.2 
5,741.7 
10,405.5 


0.24962 
40826 
0.26994 


2,190.19 
4,734.39 
6,615.88 


297.0 
1,007.3 
3,789.6 


8 
3 
35 


0.9384 
9080 
0.7974 




Treatments 
+ Error 


40 


119,197.2 


36,105.2 


16,147.2 


30290 


10,936.26 


5,210.9 


39 







DF 


W 2 ) 


Variance 


F 


5% Point 


Treatments -f- Error 


39 


5210 9 








Error 


35 


3789 6 


108 3 


















Difference = Treatments 
Treatments 


4 
3 


1421.3 
1007.3 


355.3 
335 8 


3.28 
3.10 


2.64 
2.87 


Difference (6 b t ) . . . 


1 


414.0 


414.0 


3.82 


4.12 



There is an indication here of a difference between the regression coefficients for 
treatments and error but it is hardly significant. 

(3) Effect of Initial Weight and Feed Eaten on Final Weight. After obtaining the 
separate sums of squares for each variable and the sums of products for the three ways 
in which the variables can be paired the next step is to determine the partial regression 
coefficients. For three variables the sums of squares and products give two simul- 
taneous equations as illustrated in Chapter VIII. These equations contain the 
partial regression coefficients as unknowns and can be most easily solved by the 
normal equation method, also described in Chapter VIII. The remainder of the 
calculations are as in Table 89. 

TABLE 89 



ANALYSIS OF COVARIANCE- 



~EFFECT OF INITIAL WEIGHT AND FEED EATEN 
ON FINAL WEIGHT 





2(0?!) 


DF 


s&isS (xixz) 


26132(0:1x3) 


2(2/' 2 ) 


DF 


Replicates 


2,487 2 


9 










Treatments. . .... 


5741 7 


4 


4002 8 


960.0 


778.9 


2 


Error 


10,405.5 


36 


5910.9 


988.7 


3505.9 


34 
















Treatments -f- Error 


16,147.2 


40 


9411.5 


2264.0 


4471.7 


38 

















258 



THE ANALYSIS OF COVARIANCE 



TABLE 89 Continued 

ANALYSIS OF COVARIANUE EFFECT OF INITIAL WEIGHT AND FEED EATEN 
ON FINAL WEIGHT 





2V 2 ) 


DF 


Variance 


F 


5% Point 


Treatments + Error 


4,471 7 


38 








Error 


3,505 9 


34 


103 1 


















Difference Treatments 
Treatments 


965.8 

778.9 


4 

2 


241 4 
389 4 


2 34 

3 78 


2 64 
3 26 


Difference 


186 9 


2 


93 4 



















The final result is rather unusual in that the treatment variance corrected by 
its own regression coefficient is significant while the treatment variance as obtained 
by differences is insignificant. This seems to be traceable to the relations between 
x\ and 23 where, as will be noted in Table 87, the difference between the regression 
coefficients is much less than would be expected on the basis of random sampling. 

The equation for correcting the mean final weights will now be 



where 3612 and 2^13 are the partial regression coefficients for the error covariance. 

(4) Test of Heterogeneity of Covariation or the Significance of the Differences 
between Regression Coefficients Calculated for Each Group. If for the above example 
we have kept our raw sums of squares and products separate for each treatment 
we can very quickly set up the results as in Table 90, showing the sums of squares 
and products for 21 and 0-3, the regression coefficients for each group, and finally the 
adjusted sums of squares for x\. 

TABLE 90 
TEST OF HETEROGENEITY OF REGRESSION BETWEEN TREATMENTS 





DF 


2(4) 


Sfrizs) 


2(2?) 


613 


6132(21x3) 


20/' 2 ) 


DF 


Lot I . . .-. 


9 


168.9 


458 5 


2,020.5 


2.7146 


1244.6 


775 9 


8 


Lot II .... 


9 


192.1 


102 6 


715 6 


53410 


54.8 


660 8 


8 


Lot III . . 


9 


132.1 


169.4 


2,869.6 


1.2824 


217 2 


2752 4 


8 


Lot IV... 


9 


158.1 


333.7 


1,964 9 


2.1107 


704 3 


1260 6 


8 


LotV... 


9 


191.6, 


689.6 


5,722 1 


3.5992 


2482 


2740.1 


8 


Total... 


45 


842.8 


1753.8 


12,892 7 


2 0809 


3649 5 


9243 2 


44 



REFERENCES 



259 



TABLE 90 Continued 
TEST OF HETEROGENEITY OP REGRESSION BETWEEN TREATMENTS 





DF 


S(2/' 2 ) 


Variance 


F 


5% Point 


Total 


44 


9243 2 








Treatments 


40 


8189 8 


204 7 


1 29 


2 61 














Difference 


4 


1053 4 


263 4 



















For the test of significance we summate the adjusted sums of squares for each 
treatment and subtracting from the total obtain a sum of squares corresponding to 
4 degrees of freedom representing differences between the 5 regression coefficients. 
In this example there is no evidence of significant heterogeneity of regression. 

6. Exercises. 

1. The data given in Table 91 are grain and straw yields given by Eden and Fisher 
(2) for 8 manurial treatments and 8 replicates of each. Calculate the correlation and 
regression coefficients for treatments, replicates, and residual. Test the significance 
of the grain yield differences for the treatments after correction for straw yield. Test 
the significance of the difference between the regression coefficients for treatments 
and residual, and apply the test for heterogeneity to the regression coefficients 
calculated for each treatment. 

REFERENCES 

1. E. M. CRAMPTON and J. W. HOPKINS. J. Nutrition, 8: 329-340, 1934. 

2. T. EDEN and R. A. FISHER. Jour. Agri. Sci., 17: 548-562, 1927. 

X. R. A. FISHER. Statistical Methods for Research Workers. Sixth Edition. 
Oliver and Boyd, London, 1936. Reading: Chapter VIII, 49.1. 

4. R. A. FISHER. The Design of Experiments. Oliver and Boyd, London and 

Edinburgh, 1937. Reading: Chapter IX. 

5. G. W. SNEDECOR. Statistical Methods. Collegiate Press, Inc., Ames, Iowa. 

1937. Reading: Chapter XII. 

6. J. WISHART and H. G. SANDERS. Principles and Practices of Yield Trials. 

Empire Cotton Growing Corporation, London, 1935. 
7 J. WIBHART. Suppl Jour. Roy. Stat. Soc., III. 79-82. 1936. 



260 



ANALYSIS OF COVARIANCE 



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CHAPTER XVI 

MISCELLANEOUS APPLICATIONS 
I. THE ESTIMATION OF MISSING VALUES 

1. Reasons for Estimating Missing Values and Principles of Esti- 
mation. In most experimental work, and especially in field plot studies, 
the results of one or more observations are occasionally lost or distorted 
by some disturbing factor in such a way as to make the particular 
observations useless. In the laboratory it may be possible to repeat a 
portion of the experiment and obtain new values for those that are miss- 
ing, but in field experiments repetition is impossible and one has to make 
the best of the results available. In other biological experiments 
it is frequently impossible to repeat under the identical conditions 
of the original experiment, and methods of estimating missing or 
distorted values are preferable to discarding the whole or a portion of 
the data. 

A method of estimating the yields of missing plots in field experiments 
on a strictly statistical basis was first developed by Allan and Wishart 
(1). Their methods were developed for the estimation of one missing 
yield; but more recently Yates (3) has extended their methods to the 
estimation of the yields of several missing plots. Since the methods 
developed by Yates are of general application, we shall use them through- 
out, although for single missing plots they are identical with those of 
Allan and Wishart. The mathematical basis of the method of estimat- 
ing missing values is the substitution of a value for the one missing that 
will make the*sum of the squares of the deviations from the mean a mini- 
mum. Equations are written for the sum of squares substituting x 
for the missing value; and after minimizing, the equations are solved 
fora:. 

2. Estimation of Missing Yields in Randomized Block Experiments. 
The data are first arranged in a table according to treatments and 
blocks. Table 92 is an example of an experiment with 6 treatments 
in 4 randomized blocks, and 1 plot of treatment B of block II is miss- 
ing. 

261 



262 



MISCELLANEOUS APPLICATIONS 



TABLE 92 

Treatments 



Blocks 


A 


B 


C 


D 


E 


F 


Total 


I... 


18 5 


15.7 


16 2 


14 1 


13.0 


13 6 


91.1 


II... 


11.7 




12.9 


14 4 


16.9 


12 5 


68 4 = Q 


Ill ... 


15.4 


16.6 


15.5 


20 3 


18 4 


21.6 


107 8 


IV. ... 


16 5 


18.6 


12.7 


15.7 


16 5 


18.0 


98.0 


Total . . 


62.1 


50.9 


57.3 


64 5 


64.8 


65.7 


365 3 = T 



In the generalized formula for x, the yield of the missing plot : 
p = number of treatments, 
q = number of blocks, 
P = total of all the plots receiving the same treatment as the 

missing plot, 

Q = total of all the plots in the same block as the missing plot, 
T = total of all plots. 



The formula is: 



x = 



pP + qQ - T 



(i) 



In Table 93 we have the same data as in Table 92 except that now 
three plots are missing. 

TABLE 93 

Treatments 



Blocks 


A 


B 


C 


D 


E 


F 


Total 


I. . . 


18 5 


15.7 


16 2 


14 1 


13 


13 6 


91 1 


II 
III.... 
IV. ... 


11.7 
15.4 
A 


B 

16 6 
18.6 


12.9 
15 5 

12 7 


D 
20.3 
15.7 


16.9 
18.4 
16 5 


12.5 
21.6 
18 


54.0 
107.8 
81 5 


















Total . . 


45.6 


50.9 


57.3 


50.1 


64.8 


65.7 


334.4 



The procedure in such an example where more than one observation 
is missing is first to substitute approximate values for all the missing 
values except the one to be estimated. We then apply the missing- 
plot formula as given above. The same process is in turn applied to all 
the missing plots. The results given are first approximations, and the 



ESTIMATION OF MISSING YIELDS IN A LATIN SQUARE 263 

whole process is repeated until the estimated values become practically 
constant. 

The methods are illustrated below for the estimation of the missing 
values in Table 93. 

FIRST APPROXIMATION 

Average yield = 334.4/11 = 15.9, 
The T = (334.4 + 2 X 15.9) = 366.2. 

Here the average yield of the plots is used as an approximation of the 
yields of two of the three missing plots. 

A. P = 45.6 Q = 81.5 

x = (6 X 45.6 + 4 X 81.5 - 366.2)/ 15 = 15.6 

B. P = 50.9 Q = (54.0 + 15.9) = 69.9 
x = (6 X 50.9 + 4 X 69.9 - 366.2) / 15 = 14.6 

Note that here we have to substitute a value for D and that the mean of 
all the plots is taken as the best approximation. 

D. P = 50.1 Q = (54.0 + 14.6) = 68.6 

x = (6 x 50.1 + 4 X 68.6 - 366.2)/15 - 13.9 

Here we have to substitute a value for , and the previously estimated 
value is taken as the best approximation. 

SECOND APPROXIMATION 

A. T =(333.4 + 14.6 + 13.9) = 362.9; P = 45.6; Q =- 81.5; 
x = (6 x 45.6 + 4 X 81.5 - 362.9)/15 = 15.8. 

In all the approximations after the first a new value for T is worked out 
for the estimate of each plot, using the estimates from the previous 
approximation. To get P and Q it is best to substitute for the missing 
plot values where necessary, the latest values obtained. 

3. Estimation of Missing Yields in a Latin Square. The best 
arrangement of the data is in a table such that the positions of the figures 
correspond with the positions of the plots in the field. The treatments 
should also be indicated on the table in the exact positions that they 
occur. 

The formula for estimating x the yield of a missing plot is : 

P (Pr + Pc + P|) - 2T 

X (p - l)(p - 2) () 



264 



MISCELLANEOUS APPLICATIONS 



where P r = total of row containing the missing plot. 

PC = total of column containing the missing plot. 
Pt = total of treatment containing the missing plot. 
T = total of all plots. 
p = number, of rows, columns, and treatments. 

If more than one plot is missing, we proceed exactly as for randomized 
blocks, substituting approximate values for the plots not being estimated 
and making continuous applications of formula (2). 

4. Correction to Analysis of Variance Due to Estimation of Missing 
Values. The estimation of missing values for a set of results introduces 
a complication in the analysis of variance. In the first place, one DF 
must be removed from the total for each missing value ; and in the sec- 
ond place a correction must be applied to the sums of squares for treat- 
ments or any other component in the analysis, the significance of which 
is to be tested against the error. An exact mathematical solution of this 
problem for all cases has been provided by Yates (3), but except for 
randomized block experiments, and for Latin square experiments with 
only one missing plot, it is rather complicated for general practice. 

In a randomized block experiment as in Table 93, for which three of 
the missing plot yields were estimated, the following scheme for the 
analysis of variance shows how the correction is applied to the treatment 
variance. In this scheme the "original" values refers to those for the 
21 plots as given in Table 93, and the "completed" values refers to those 
in Table 93 with the addition of the three that were estimated. 



DF 



Sum of Squares Calculated from 



Total 


20 


Original yields 


Error 


12 


Completed yields 








Difference = Blocks -f Treatments . . . 
Blocks 


8 
5 


Original yields 








Difference = Treatments 


3 











The procedure for calculation is as follows: 

(a) Obtain the sums of squares for blocks, treatments, and error 
from the completed yields. 

(6) Obtain total sum of squares for original yields. 

(c) Obtain sum of squares for blocks from original yields, noting 
that not all the blocks contain 6 plots. 



CORRECTION OF TREATMENT MEANS AND STANDARD ERRORS 265 

(d) Set up the analysis of variance as above, obtaining the sums of 
squares first for blocks + treatments and then for treatments by 
subtraction from the known quantities. 

For Latin square experiments with only one plot missing the simplest 
method of determining the correction to the treatment sum of squares is 
to use the formula 



1 



- 1)P, - Pr - PC - 



(p - l)*(p - 2) 2 

which gives the correction directly The scheme of analysis using a 
6X6 Latin square would then be ah follows : 



DF 



Sum of Squares Calculated from 



Total 


34 


Original values 


Error 


19 


Completed values 


Difference = Rows Columns 
Treatments 


15 




Treatments Correction 


5 


Calculate from complete 






values and subtract 
correction 



6. Correction of Treatment Means and Standard Errors. The 

treatment means that contain estimated values for missing plots are in 
effect corrected means and further corrections are not required. The 
standard errors of such means, however, require a definite correction, 
and for methods of doing this accurately the reader should refer to the 
paper by Yates (3). For general purposes it is probably sufficient to 
make a correction for the number of plots averaged, i.e., if there are r 
replications and one plot is missing the standard error of the mean of the 
treatment containing the missing plot will be 



H. METHODS OF RANDOMIZATION 

Randomization can be effected by tossing coins, drawing cards out of 
a shuffled deck, throwing dice, etc., but these methods are too slow and 
in genera] too inaccurate for actual practice. The problem has been 



266 MISCELLANEOUS APPLICATIONS 

greatly simplified by the preparation of Tippett's "Random Sampling 
Numbers" (2), and these numbers are now in general use.* 

If we have a series of numbers 1, 2, 3, n, the problem of random- 
ization is to arrange these numbers in such a way that in forming the 
arrangement any one of the numbers has an equal chance with any other 
number of being placed in a given position. A procedure that is fre- 
quently followed in arranging field plot tests may now be described 
briefly. 

Suppose that the numbers representing the varieties are 1, 2, 3, 4, 5, 
6, 7, 8, 9. Turning to page XI of Tippett's "Tables" (the usual prac- 
tice being to open the book more or less at random), we find that begin- 
ning at the upper left-hand corner we can take a series of random two- 
figure numbers as follows, 40, 81, 89, 58, 87, 74, etc. Assume now that 
there are 9 places to be filled up by the numbers 1 to 9, and the first one 
is selected by dividing the first two-figure number by 9 and taking the 
remainder. Thus for 40/9, the remainder is 4, and number 9 is placed 
in the fourth place. The second number to be placed is 8 and we divide 
the second two-figure number by 8; 81/8 gives a remainder of 1, and 8 is 
placed in the first place. The third number is 7, and dividing it into 89 
the remainder is 5, and 7 is placed in the fifth space counting only those 
that are empty. This procedure is followed until all the numbers have 
been placed and we get finally the following arrangement : 

8, 3, 5, 9, 4, 6, 7, 2, 1 

The same procedure can be modified for application to a Latin square, 
but in that case it is only necessary, starting with a given Latin square 
which may be made up systematically, to randomize the rows, columns, 
and treatments. 

REFERENCES 

1. F. E. ALLAN and J. WISHART. A Method of Estimating the Yield of a Missing 

Plot in Field Experimental Work. Jour. Agr. Sci. 20: 39&-4Q6, 1930. 

2. L. H. C. TIPPETT. Random Sampling Numbers. Cambridge University Press, 

1927. 

3. F. YATES. The Analysis of Replicated Experiments When the Field Results are 

Incomplete. Emp. Jour. Exp. Agr. 1: 129-142, 1933. 

* More recently, Fisher and Yates in " Statistical Tables for Biological, Agri- 
cultural and Medical Research " (Oliver and Boyd, London, 1938) have included an 
excellent set of random numbers which have been thoroughly tested for randomiza- 
tion work. 



TABLES 



TABLE 94 
TABLE OP t* 



Degrees of 
Freedom 


Probability 


0.50 


0.10 


0.05 


0.02 


0.01 


1 


1.000 


6.34 


12.71 


31.82 


63.66 


2 


0.816 


2.92 


4.30 


6.96 


9.92 


3 


.765 


2.35 


3 18 


4.54 


5.84 


4 


.741 


2.13 


2.78 


3.75 


4.60 


5 


.727 


2.02 


2 57 


3.36 


4.03 


6 


.718 


1.94 


2 45 


3.14 


3.71 


7 


.711 


1.90 


2.36 


3.00 


3.50 


8 


.706 


1.86 


2.31 


2.90 


3.36 


9 


.703 


1.83 


2 26 


2.82 


3 25 


10 


.700 


1.81 


2 23 


2.76 


3.17 


11 


.697 


1.80 


2.20 


2.72 


3.11 


12 


.695 


1.78 


2.18 


2.68 


3.06 


13 


.694 


1.77 


2.16 


2.65 


3.01 


14 


.692 


1.76 


2.14 


2 62 


2.98 


15 


.691 


1.75 


2 13 


2.60 


2 95 


16 


.690 


1.75 


2.12 


2.58 


2 92 


17 


.689 


1.74 


2.11 


2.57 


2.90 


18 


.688 


1.73 


2.10 


2 55 


2.88 


19 


.688 


1.73 


2 09 


2.54 


2.86 


20 


.687 


1.72 


2 09 


2 53 


2.84 


21 


.686 


1.72 


2.08 


2.52 


2 83 


22 


.686 


1.72 


2 07 


2.51 


2 82 


23 


.685 


1.71 


2.07 


2.50 


2.81 


24 


.685 


1.71 


2 .06 


2.49 


2.80 


25 


.684 


1.71 


2 06 


2.48 


2.79 


26 


.684 


1.71 


2 06 


2.48 


2 78 


27 


.684 


1.70 


2 05 


2 47 


2 77 


28 


.683 


1 70 


2.05 


2.47 


2.76 


29 


.683 


1.70 


2 04 


2 46 


2 76 


30 


.683 


1.70 


2.04 


2 46 


2.75 


35 


.682 


1.69 


2.03 


2.44 


2.72 


40 


.681 


1.68 


2.02 


2 42 


2 71 


45 


.680 


1.68 


2.02 


2 41 


2.69 


50 


.679 


1.68 


2.01 


2.40 


2.68 


60 


.678 


1.67 


2.00 


2.39 


2.66 


70 


.678 


1.67 


2 00 


2.38 


2 65 


80 


.677 


1.66 


1.99 


2.38 


2.64 


90 


.677 


1.66 


1.99 


2.37 


2 63 


100 


.677 


1.66 


1.98 


2.36 


2.63 


125 


.676 


1.66 


1.98 


2.36 


2 62 


150 


.676 


1.66 


1.98 


2.35 


2.61 


200 


.675 


1.65 


1.97 


2.35 


2.60 


300 


.675 


1.65 


1.97 


2.34 


2.59 


400 


.675 


1.65 


1.97 


2.34 


2.59 


500 


.674 


1.65 


1.96 


2.33 


2.59 


1000 


.674 


1.65 


1.96 


2.33 


2.58 


oo 


.674 


1.64 


1.96 


2.33 


2.58 



* The greater portion of this table taken from R. A Fisher's "Statistical Methods for Research 
Workers," with the permission of the author and his publishers, Oliver and Boyd, London 

267 



268 



MISCELLANEOUS APPLICATIONS 



TABLE 95 
TABLE OF x 2 * 



Degrees 
of 
Freedom 


Probability 


0.99 


0.95 


0.50 


0.30 


0.20 


0.10 


0.05 


0.01 


1 


0.0002 


0.004 


0.46 


1.07 


1.64 


2.71 


3.84 


6.64 


2 


0.020 


0.103 


1.39 


2.41 


3.22 


4.60 


5.99 


9.21 


3 


0.115 


0.35 


2.37 


3.66 


4.64 


6.25 


7 82 


11.34 


4 


0.30 


0.71 


3.36 


4.88 


5.99 


7.78 


9.49 


13.28 


5 


0.55 


1.14 


4.35 


6.06 


7.29 


9.24 


11.07 


15.09 


6 


0.87 


1.64 


5.35 


7.23 


8.56 


10.64 


12.59 


16.81 


7 


1.24 


2.17 


6.35 


8.38 


9.80 


12.02 


14.07 


18.48 


8 


1.65 


2.73 


7.34 


9.52 


11.03 


13.36 


15.51 


20.09 


9 


2.09 


3.32 


8.34 


10.66 


12.24 


14.68 


16.92 


21 67 


10 


2.56 


3.94 


9.34 


11.78 


13.44 


15.99 


18.31 


23.21 


11 


3.05 


4.58 


10.34 


12.90 


14.63 


17.28 


19.68 


24.72 


12 


3.57 


5.23 


11.34 


14.01 


15.81 


18.55 


21.03 


26.22 


13 


4.11 


5.89 


12.34 


15.12 


16.98 


19.81 


22.36 


27 69 


14 


4.66 


6.57 


13.34 


16.22 


18.15 


21.06 


23.68 


29 14 


15 


5.23 


7.26 


14.34 


17.32 


19 31 


22.31 


25.00 


30.58 


16 


5.81 


7.96 


15.34 


18.42 


20.46 


23.54 


26.30 


32 00 


17 


6 41 


8.67 


16 34 


19 51 


21.62 


24.77 


27.59 


33 41 


18 


7.02 


9.39 


17.34 


20.60 


22.76 


25.99 


28.87 


34 80 


19 


7.63 


10 12 


18 34 


21.69 


23.90 


27.20 


30.14 


36 19 


20 


8.26 


10 85 


19.34 


22.78 


25 04 


28.41 


31.41 


37 57 


21 


8.90 


11.59 


20.34 


23.86 


26.17 


29.62 


32.67 


38 93 


22 


9.54 


12.34 


2^.34 


24.94 


27.30 


30.81 


33.92 


40 29 


23 


10.20 


13.09 


22.34 


26.02 


28.43 


32.01 


35.17 


41 64 


24 


10.86 


13.85 


23.34 


27.10 


29 55 


33.20 


36.42 


42.98 


25 


11.52 


14.61 


24.34 


28.17 


30 68 


34.38 


37.65 


44 31 


26 


12.20 


15.38 


25.34 


29.25 


31.80 


35.56 


38.88 


45.64 


27 


12.88 


16.15 


26.34 


30.32 


32.91 


36.74 


40.11 


46.96 


28 


13.56 


16.93 


27.34 


31.39 


34.03 


37.92 


41.34 


48.28 


29 


14.26 


17.71 


28.34 


32.46 


35.14 


39.09 


42.56 


49.59 


30 


14.95 


18.49 


29.34 


33.53 


36.25 


40.26 


43.77 


50.89 



* Taken from R. A. Fisher's "Statistical Methods for Research Workers," with the permission 
of the author and the publishers, Oliver and Boyd, London. 



TABLES 



269 



I 

5 

a 

e 





$ * 

a ! 

H 

< w 



O 

S 
o 





Q 

sT 



Sg 83 SS 






* 2: 52 



OJ^l OO CO <rt 
CO* C0fr 0>0* 



28 



5 S3 .8 SS 

OM <*' COIO COIO 



ft* CO-* CN M 
*M CNM CNM 



35 SS 85 88 M S 



28 



t-OO OOIO O 



Sjf <* lO Ok CN W 
V M t- O CO M 

co't* co'io 



OO OIO CNt- 

10 W <*M COM 

CN M CN M 04 M 



700 <**> CO* 

' s ss -s 



< tfO 0^-1 04IO l^t-> COIO 04r4 ^M 
r* COOl Or^ OOlO COHI <O ^ IO CO <* 



28 M ^2 



$o ^3 SS t^S Si* 



r-H- CM 00 OM 



OM coio 

00 O tOlO 



4<io oto coo t^> 

JH i-HM O>W I>1 



58 8 ^3 



^ to cot* 

O80 -^ IO 
CN M O4 M 



28 



O 00 O4 M C3> t" 

CO 10 0)8 ^M 

^"o cot-' co'io 



M O*^ 
OIO t-M 



88 Sg 

<NM CNM 



So* 



10 gg 00 OJJ 



8C 55 



00 t- t^.H r-l^H 



28 



9M COIO OIO 
-H O5<l b- 



8 3 



28 



i-n* OIO OOIO OOO 00 -H 
00 M OtO tH r-tOk COfr- 



0)0 OOgjJ COJj ^O 



?o t*c 

OIO t- 



28 



^oo coio coio coio coio 



- __ o* oot* 

COM COM OOIO 

oj g oo t 



88 5 2 



-< *HCO 
i^M OS 



, 



83 



28 



S ^o 



Of COIO COIO COIO COMB 



28 



cot- coio coio coio coio coio co** 



28 S3 S3 



28 



COIO >O*-I 



cot* coio coio coio coio coio 



3 S3 8 3 



*. COIO COIO COIO COIO COIO 



\l 8- 

ss 



*<M TMO COIO COM OIO OOO COM OO 

rH0k If-IO ^IO CMO r-HlO O>M 00 O> 00 1- 



^t- cot* coio con 



0000 Ojg t^ 



CN IO CN IO O<<JI ff IO >OP> ft 
COM -"O 0)O OOIO I>M O 

00 O> O - CO 



o 

X 

JD 

-2 
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O 

00 

I 

2 



"o 

a 



S 
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1 



id 



270 



MISCELLANEOUS APPLICATIONS 



D 

2 

H 

S < 



w 
J 
CQ 



o 





8 


COO t-f -< COW C*f 00 0> *< <tO 00 *4 O (O CO** - f O* * 













rt* OOO OIK* t^f COO* OiH O'* (MOO Om t-00 fm 0O OU 

-to ooo ot- a>0 O>M on ao-* 0009 COM f M t-M I^<HI r-~* 










8 


eo0 OM -^Q oso "CM .-t^p r^t- -^M -n O>M ot- ^te >MO 







O>H C4- r^o <NO ooao TfQ o r-t <^M <Nt- o t^o> COM 
r-4-i rno ooo OK* a><0 a>3 a> oo< ob^ oo oo t- r- 










ic 
t^. 


<-<- too oe Tfo o-t com c*0 O>IH INO $- csto ow aooo 

(NrH ^0 OOO Of- Ot* 0)<0 0)10 0010 00 <* OO < 00 CO 0009 r- 










S 


4*^-1 oot cop oop ^oo oo coeo cooo '- ooao co** -*o cite 
oi *-o HOi ooo ot* or* a>o O>M> 010 oo9 oo^ oo^ oow 










S 


r-o ^ei co^ 'HM t-w eou> a>o toe* eoao "- Oia r^-io >o*4 

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o 

CO 


rH< ICQ OO ICO '-'*'< t>^> *^K* OM ODE* COM ^00 CM^l OO 

com <Nof co*-t -<o r-icn ooo ot* or* o>0 oo a>io O>M ano 










Tf 

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CO<4< OM CN^H T-HO r-tO i OOO OOO Of OK* OS 10 O> *O Cfc M 


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ci IH eo<o oo 10 CO<D 05 1- ico <N^I a>oo t-co "feo <N^I oo a>o 

COM COM (NM CN*4 rH O I O rH l O 00 O 00 Of Of Of Oi ft 


OQ 






a 

3 


co 


TfM oiao eof of ico -IM ooie ico co^n oe a>io COH iCf 
^p co^ eo CSJM (NH c<i*-i t-<o *-toi i - 4 * "-^w ooo ooo Of 


s 






1 


*f 


T^O cop I>M co 10 oif coe cow Of COM r^f com ^a of 

00 f *M CO^H com (MM CM^ O4*-l CMO "HO <-HA ^ O i 00 "00 







om (Mm cam <Nm (Mm <Mm (Mm (Mm (Mm <MM IMM <MM <MM 


a 
fe 


(N 


coo oof <MIO ooio ^f o oom cf COM Of oom coo* c 
icoo i# to -mo co^ii com com CMM <M-I CNH CNO "-"O *~* o <o 






<Mm NO cim (Mm <NW (Mm (Mm <Mm <Mm (Mm (Mm <MM CMM 


6 
o 




coo r-w 10*4 ^e IN.^I -**o 10 00-41 coco Tt*'* <MO 010 OCM 


1 

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(Mm <Mm <Mm <Nm (Mm (Mm <Mm (Mm (Mm (Mm (Mm <Mm cim 


*o 

CO 





CMCO Dim (Mm <Mm cim csm <Mm cMm csm CMCO (Mm c^m <MW 


$ 

& 


Cft 


icm a>o rt*oo ooo coo COM OM t^p >cio <MO Oio oo-i r-.f 
coo icoo icf >o 3 -^(0 TJ* > ^ ^i co9 com com COM <MM CN- 


n 






sT 


00 


Ol "fO O>O <C -l OOm IC<0 <MH OO OOiH CO<0 "*<C (MO 

^-^ coo icoo cf icf rf0 ^to rj<io ** eo'* com com COM 






(M'* <M^I (Mm <Mm (Mm (Mm om (Mm <Mm <Mm (Mm cim cim 




I*. 


t-ao 0^1 com cim ooio icf CM-I a>o ro ic# coo ^0 Oi 

t^M t-H COO C00 ICOO Of iCf ^0 flO ^*IO ^3 ^^ C0<9 










CO 


S5 28 8 S2 S SS gS; &S S^ SS H55 53 52 










*o 


8 gS a?3 SSS 8 25 S ^S SS SS SS gg S 




Tt* 


-< CDO "Hf COf COOO OO *** ^fe" (N* 4 O<0 OOM CO 00 "**i 




CO 


ctS S3 38 32 SS S3 Set 5 S3 SS S? S 3 










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S SS S8 SS !gg SS S !JS 3B 38 55 85 8 










r- 


S8 25* 9S 53 58 ^5? S c^S c^S S8 S8 Sk SS 








2 


i 


^tiCcDt-OOOJO-<(MCO-^iC<O 
r _^H^,^, r -.r-.CIC4(M(MCN<M<M 



TABLES 



271 



Q 

M 



* 





oS 


oo oo oo 


Cite 


loS 


in oo 


$"* 


^^ 


OiCO 


oo to 


oeo 


ioo 


8 


* t CO 




- 


. 




" 




. . 






ft- 


























1 


oS 


coo oo oo 


ss 


gs 


oo 

1001 


'000 


CO-* 
to 00 


S GO 


OCO 
iO f 


x to 


r-eo 

f t- 


























CM 


S 


OiCO OOO Ot- 
OH OiH OO 


sg 


ss 


Oi"* 
10 Ol 


KS 


000 
"000 


ss 


CMCO 

1000 


ss 


g? 


o 
o 


r-eo 


ffS K5 S3 





** ^l 

oo 


CMO 

oo 


8S 


iO Ol 


10 en 


oeo 

000 


*f tO 

1000 


co-^n 

<O CO 


























O 


OlO 

i>eo 


oeo co 01 CM to 
r<- eo t *H t- ft 


oS 


o o 


oo 


ss 


as 


ss 


Cf CO 


r^-o 

OOl 


oeo 





oeo 
ooeo 


ss sa ss 


to 


S 


S3 


ss 


OlO 

oo 


^s 


COO 

oo 


C1 00 

OOI 




O 
rt* 


38 


i-4io oeo o>cn 
ooeo ooeo t--co 


OlO 


sa 


K5 


t - T-J 


s- 


egg 


oo 


CDO 


S2 



CO 


001- 

00** 


l>- *# 10 H n< 00 


ooeo 


oo 
ooeo 


00 tO 


a 


to 


C0|> 


CM 10 


rHCO 


OrH 


(N 


COIO 
0510 


otio o>^i o?5 


83 


ss 


CM tO 

ooeo 


ss 


05 Ol 


OO (0 


ss 


>0 CO 




























08 
3 CM. 


8 


OO tt- COIO 


ss 


00 * 


3 


oo 

00-* 


OOCO 


CM IO 

ooco 


<r co 


s?g 


o>oo 




rHCOJ 






















2 < 

08 r-4 


oS 


CM-< ooo cr>e 

Ot- 00 0D 


'^S 


O>IO 


ss 


OJ iS 


001 


00^(1 


dO^* 


00-* 


oo 

JO'* 


























Q 
rt 


s 


ooo ot- or- 


gg 


8 to 
to 


ooeo 


001 


C^IO 


a>io 


CM eo 


** 


3 


O CM 

s 


cow 

<-"O 
CMCO* 


CMO Ot- O3ll 

r-l Ol 100 OOO 

CM" c CM eo =M co 


oS 

CM CO* 


iO<O 

oc- 


off 


OtO 


OtO 


^co 


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O5 tD 


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060 

05 


a 2 

o " 


2S 


>O IO -f CO CM O 


OtO 
*-<00 


oco 


oeo 
ot- 

CMC4 


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ot- 

CM'CO 


0? 
CMCO 


CMO 
Ot- 

CNCO.' 


r-l CO 

OtO 
CM CO 


to 
to 

(M CO 


cr. to 


2 


OtO 
CMO 


05 eo ooo oeo 

i-HO -l O i~ 


ss 


r-H OO 


ss 


ss 


I- O 

000 


cot" 

oc- 


OlO 
Ot- 


SS 


CO H 

ot- 


*i 


CMCO 


CM CO CMCO CMCO 


CMCO 


01 CO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


ap 
fl? OS 
CD 

fib 


CMCO" 


rf ^ CMCO 10 
CMiHI CMO CMO 

CM CO CM CO CM CO 


CM' eo 


58 


:J 


CMCO' 


CMCO 


CM'CO' 


r-i 00 

CM'CO 


SCO 
00 

CM CM 


oo o 

oS 

CM' eo 


^ 00 
? 


CMCO 


CMCO CMCO CMCO 


CMCO 




CMCO 


oi 


CMCO 


CM'CO 


CMCO 


CMCO 


CM eo 




coco 


OtO 'O CO if O 


coco 


OH 

coco 


ooeo 


COIO 


CM*4 


CM *^ 


cot- 

CMO 


?3o 


C^S 




CMCO 


CM CO CM CO CM CO 


CMCO 


o^eo 


CMCO 


CMCO 


CM eo 


CM CO 


CM CO 


CMCO 


CMCO 




%% 


*t CO COO CMt- 


^3 


0000 


COIO 


cow 


co co 


CM tO 


rH^H 


oeo 


00 




CMCO 


CMCO CMCO CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 


CMCO 




t-o 


o to t o coo 


r-H tO 


05^ 


0000 


0* 


OlH 


^01 


cote 


CM^JJ 


53 




CMW 


CMW M' CM'CO 


CMCO 


CMCO 


CMCO 


CM W 


CMM 


CMCO 


01 CO 


CM" co 


CM CO 






t^o So oo 


OCA 


loco 

001 


CCOI 

oeo 


CM O 

ooo 


S3 


a>o 

1000 


ooeo 




Jo?! 
CM'CO" 


CO 





05 O G>M OHO 


s 


S9 


ooo 
ooeo 


3 


ooco 


CO 01 


CM tO 
00 CO 


ooeo 


OCO 

OOCO 


CM 


S3 


S3 S?5 SS 


COCO 


00 Ol 

CMCO 


OlO 
CMCO. 


c^So 


roeo 


CMO 

CM *-* 


sa 


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T-H 


8 


O * OOO t *0 


mo 

rH IO 


25 


rHCO 


OlO 


oeo 


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ss 


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00 0* O 


<3 


t 










CM 


^ 


co 


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Z72 




8 


MISCELLANEOUS APPLICATIONS 

?8 32 38 2 33 83 S3 S5 38 28 S3 83 88 














g 


3 32 38 38 8 8 33 3 8S 38 23 23 23 














g 


3? 3? 33 33 SS 85 33 S3 S3 8 38 28 58 

















38 S3 32 3S 32 3 32 3 33 33 S3 8 32 














.0 


8 38 82 3 5 3 33 32 2 8 35 33 S3 














S 


Q** 00 O CO** -4*f COM >O40 0049 'OOO *O (MM GO fr- CO ** O 4 
CO* /S<0> lOOO X .009 -ft- <f rt-5S <^0 ^ O COM COM CO IO 














s 


Sg S3 38 8 8 33 3? 32 5 32 32 35 32 


fe. 

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G 
CO 


S3 2 2 SS 38 33 2 8 38 3? 3t 5 32 


O 









O 


g 


CN 


3 3 3 32 cog 3 22 S3 S3 8 33 3S 3? 


p 


53 






1 




O 
CN 


8 8 8 S3 S3 3 S3 3 3 88; SS S3 5 


1 








06 


s 

fe 


CO 


2 2 8 32 8 3 3 3 3 S3 3 5 32 










1 'm 

9 O, 

H * 





* 


OD 000 COO 0t- ^<O C4M O><9 t9 OO ^ t- CNM OO Ol t- 


f H 








5 M 
O o 

i < 




CN 


2 SS 32 5 S3 $3 2 SE2 32 32 8 2 3 


A 


o 






s 

u 5 


1 


- 


$8 2 2 32 3 3 3 5 3 2 33 3 


3 fi 


M 






1 - 


s 


o 


3 82 3 S3 2 2 SIS 5 S3 3 S 3 2 


g 


R 









Q 


o 


oS off S og o 3 2 2 S2 S2 g3 S3 SB3 


S 


e* 




CNM WM CNM C*M CNM ~ M ^M ^M ^M ^M ^M ^M *-< M 


H 




oo 


CO 40 i-l IO Off OOO l^f O* COO <-4IO OM OOQ COM! >O rf *H 
-00 -t5 f-40 Ofr Ofr* OK* 0*0 00 O0 Ci5 0>M 0)10 0510 


J 








tf 




^ 


SS 28 J52 J28 SS 2S S8 25 S* o? 08 08 o2 


so 






CNOO CNM CNM CNM CNM CNM CNM CNM CNM CNM CNM CNM CNM 







CO 


O>00 t^M tOM ^<O CNt* -i*i O>4 b HO COM <* O CN IO OM O>Q 
M4 C44 CN4 CSO COO CNO - ^ O ^ % r- '-CO OOD 














10 


QH O0t h'4 CO *4 >OO COlO OO AC* t^^fl CO v4 CO O CN *f <~* M 
^^ COOf CO40 COM COM COM COM CNwt CN <H M -4 CNO CNO CNO 














* 


%P 38 38 32 S3 32 3S 35 33 33 32 32 2 














CO 


8 3 2 3 8 3 8 S3 S 8 38 28 32 








C4|i CN<* CN<P CN<* CN 1 * CN*I CNOO CN W CNOO CNOO CNOO CM 40 C400 






CN 


32 ^S 38 22 38 8 88 S8 82 S? SS 88 S3 








CO* COM C0^ CO^II CO* CO* CO* CO* CO* CO* CO* CO* C4* 






f-l 


85 S3 82 23 So S3 32 32 SS SJS ^? 2 32 








<4*ft *O* *0* CQt* CO** CO 40 CO 40 CO 40 CO 40 CO 40 CO 40 CO 40 CO 40 




i 


r 


8 8 8 3 g 8 8 8 S 8 8 | 



INDEX 



Abnormality, chi-square tests for, 27, 

28, 94 

tests for, 28-31 
types of, 28 
Allen, F. E., 261-266 
a, alpha, 23 

Analysis, of covariance, 247-259 
of variance, 114-138 
applied to linear regression formu- 
lae, 210-217 
division of degrees of freedom in, 

116-119 
division of sums of squares in, 

116-119 

interaction effects in, 124-125 
multiple classification of variates 

in, 120-121 
simple classification of variates in, 

125 

tests of significance in, 119-120 
three-fold classification of variates 

in, 131-133 
two-fold classification of variates 

in, 127-128 
of variation, 114-116 
Arithmetic mean, 8 
calculation of, 9, 16 
decoding of, 17 
properties of, 10 
Association tests for, 95 

6, linear regression coefficient, 55 

Batchelor, L. D., 209 

ft, beta values, 81 

Bias, in field plot tests, 43, 144 

in tra varietal, 144 

planning to remove, 45 
Binominal distribution, 21 

probabilities from, 108-109 
Bivariate frequency distributions, 67 
Blakeman, J., 220, 246 
Brandt, A. E., 97 



Chi square, correction for continuity, 

102 

from mXn-fold tables, 96-97 
from 2Xn-fold tables, 97 
from 2 X 2-fold tables, 97 
Chi-square tables, degrees of freedom 

in, 96 

Chi-square tests, 88, 113 
of goodness of fit, 89-94 
of independence and association, 

95 
Classification of variates, multiple, 120- 

121 

simple, 125-126 
three-fold, 131-133 
two-fold, 127-128 
Cm, non-linear regression coefficients, 

220-221 

Class range, 14 

Class value, selection of, 13-14 
Coding, 57 

Coefficient, of contingency, 98 
of correlation, 65-77 
of partial correlation, 78-83 
of partial regression, 80-81 
of variability, 17 

Confounding, in a 2X2X2 experi- 
ment, 160-162 

in a 3X3X3 experiment, 170-171 
in incomplete block experiments, 175, 

178 

partial, 162-165 
Control of error, by covariance method, 

250, 253 
Correction, of treatment means due to 

missing values, 265 
to analysis of variance due to miss- 
ing values, 264-265 
to chi-square for continuity, 102 
Corrections, for grouping, 15 
to means, in covariance analysis, 
252-253 



273 



274 



INDEX 



Corrections to means, in incomplete 

block experiments, 193 
in quasi-factorial experiments, 181, 

183, 187 

Correlation, definition of, 65 
measurement of, 67-72 
partial and multiple, 78 
Correlation coefficient, 65 
calculation of, 73-75 
from correlation table, 75 
from paired values, 74 
interpretation of, 71-72 
relation to regression coefficient, 69 
test of significance of, 72-73 
z transformation for, 73 
Correlation coefficients in co-variance 

analysis, 248-250 
Correlation ratio, 219-220 
Correlation table, 75 
Covariance analysis, 247-259 
corrections to means, 252-253 
division of degrees of freedom, 247- 

248 

division of sums of products, 247-248 
principles of, 247 
with three variables, 254-258 
Covariance method applied to the 

control of error, 250-253 
Covariation, 65 

heterogeneity of, 247 
Crampton, E. W., 138, 141, 254, 259 
cv, covariance, 54-55 

Degrees of freedom, 12, 34 
division of, in analysis of variance, 

116-119 

in covariance analysis, 247-248 
in chi-square tables, 96 
in estimating the variance, 12 
in linear regression, 56 
in non-linear regression, 224 
in partial correlation, 85 
splitting into orthogonal components, 

166-169 
Departure from normality, tests for, 

27, 31 

Design of experiments, 45 
Differences, methods for testing signifi- 
cance of, 40-42 
Discontinuous variables, 13 



Distribution, binominal, 21 

normal, 22 

of F, 120 

of t, 38 

of z, 120 

"Student's," 38 
Distributions, leptokurtic, 28 

platykurtic, 28 
Dot diagram, 66 
Dunlop, G., 127, 141 

Eden, T., 259 

Enumeration data, 89 

Error, control of, 48 

Error control, by confounding, 160 

by covariance technique, 250-253 

in field plot tests, 145 

in incomplete block experiments, 172 
Error variance, 122-125 
Estimating, missing values, 261-265 

the variance, 35 
Estimation, 6 

of missing yields in randomized 
block experiments, 261-263 

of standard deviation, 33-34 
Experiment, hypothetical, 2 
Experimental design, 45-51 
Ezekiel, M., 63, 246 

F, distribution of, 120 

table of, 269-272 
Factorial experiments, 151 

confounding in, 160-165 
Fiducial limits, 39 
Field plot tests, 142-208 
Fitting, of logarithmic curves, 231-234 

of normal curve, 24-25 
Fitting polynomials, 221-230 

by summation method, 234-243 

summary of formulae, 235-236 
Fitting the regression line, 53-55 
Fisher, R. A., 2, 6, 8, 19, 32, 40, 44, 51, 
63, 72, 77, 87, 100, 107, 113, 116, 
120, 141, 143, 170, 171, 191, 209, 
218, 234, 246, 252, 259 
Frequency distribution, binomial, 21-22 

normal, 22-24 
Frequency polygon, 16 
Frequency table, formation of, 15-16 

graphical representation of, 16 



INDEX 



275 



01, measure of symmetry, 29 

gz, measure of kurtosis, 29 

Geddes, W. F., 87, 131, 141, 215- 218, 
231 

Goodness of fit, 89-94 
of polynomial equations, 224 
tests with small samples, 101 

Goulden, C. H., 77, 141, 172, 209 

Graeco-Latin square, 191 

Grant, J. C. B., 105, 113 

Graphical representation of frequency 
table, 16 

Greenwood, M., 100 

Grouping, 
Sheppard's corrections for, 15 

Hanna, W. F., 246 
Heterogeneity, of covariation, 247 

of soil, 142 

of variation, 114-116 
Heterogeneity test for regression co- 
efficients, 253-254 
Histogram, 16 
Hypothesis, null, 6 

i, class interval, 17 

Immer, F. R., 90, 139, 141, 209 

Incomplete block experiments, 175, 

178 

choosing the best type of, 200-201 
symmetrical, 188-193 
Independence, and association, tests 

for, 95 

tests for, with small samples, 101 
Interaction effects in analysis of vari- 
ance, 124-125 
In variance, 252 
Isa, J., 243, 246 

Koltzoff, N. K., Ill, 113 

Kurtosis, 28 

k statistics, 28-29 

Large number of varieties, methods for 

testing, 172-201 
Latin square, 147-149 
estimation of missing yields in, 263- 

264 
Linear regression, 52 



Logarithmic curves, method of fitting, 

231-234 
Logic of statistical methods, 1 

Mainland, D., 112, 113 
Mean, adjusted for associated vari- 
ables, 252-253 
arithmetic, 8 
calculation of, 16-17 
of a population (m), 35 
of a sample (F) , 35 
variance of, 35 
Mean difference, test of significance, 

40, 42 

Mean square (s 2 ), 35 
Methods of randomization, 265-266 
Miscellaneous applications, 261, 266 
Missing values, correction for, in analy- 
sis of variance, 264-265 
in treatment means, 265 
estimation of, 261-265 
Missing yields, estimation of, in a 

Latin square, 263-264 
in randomized block experiments, 

261-263 

Mitchell, H. H., 44 
Multiple classification of variates, 120- 

121 

Multiple correlation coefficient, 78 
calculation of, 85 
test of significance for, 86 
Multiple regression, 78 
significance of, 214-215 

n, degrees of freedom, 35 
n', number in sample, 35 
N, number in sample, 35 
Neatby, K. W., 112-113 
Non-linear regression, 219, 245 
Non-linearity, test for, 211-213 
Normal curve, calculation of prob- 
ability from, 26 
fitting of, 24-25 
Normal distribution, 22 

definition of, 22 

Normal equations, for fitting poly- 
nomials, 223 

for partial and multiple regressions, 
80-81 



276 



INDEX 



Normal frequencies, calculation of, 

24-25 

Null hypothesis, 6 
Numbers, random, 266 

Orthogonal squares, 191-192 
Orthogonality, 160 



Parameter, 6, 34 ^ 
Partial confounding, 162-165 

in a 2X2X2 experiment, 162-165 

in a 3X3X3 experiment, 171-172 
Partial correlation coefficient, 78 

calculation of, 82 

test of significance, 85 
Partial regression, 78 
Partial regression coefficients, calcula- 
tion of, 80-81 
Pearl, Raymond, 32, 77 
Pearson, Karl, 32 

Polynomial equations, testing of good- 
ness of fit of, 224 

Polynomials, method of fitting, 221- 
230 

summary of formulae for fitting, 235- 
236 

tables for fitting, 242-243 

tabular method of fitting, 240-241 
Population, 5 

Probability, calculations from normal 
curve, 26 

from binomial distribution, 108-109 

inverse, 1 

Quasi-factorial experiments, three-di- 
mensional with three groups of 
sets, 186-188 
two-dimensional, with three groups 

of sets, 185-186 
with two groups of sets, 179-185 

r, correlation coefficient, 65 
R, multiple correlation coefficient, 86 
Random numbers, 266 
Randomization, 46 

methods of, 265-266 

of field plot tests, 143-145 
Randomized blocks, 146-148 

estimation of missing yields in, 261- 
263 



Ratio of variances, 132, 137 
Recovery of information, 162-165 
Reduction of data, 7 
Regression, non-linear, 219-245 

partial and multiple, 78 
Regression coefficient, 53 

methods of calculation, 56-58 

properties of, 55 

relation to correlation coefficient, 69 

standard error of, 56 

test of significance, 55-56 
Regression coefficients, test of hetero- 
geneity, 253-254 
Regression equation, 53 

linear, 54 

multiple, 80 

partial, 80 

Regression equations, types of, 220-221 
Regression function, tests of signifi- 
cance of, 210-211 
Regression graphs, 53 
Regression straight line, 53 

fitting of, 53-55 
Replication, 48, 51, 142, 143 

s, standard deviation, estimated from 
a sample, 35 

s b , standard deviation of regression co- 
efficient, 35 

, standard error of estimate, 35 

*, standard error, 35 

Sample, 5 

Sample mean, standard deviation of, 
12 

Savage, A., 32 

Sayer, W., 209 

Schultz, T. W., 138-141 

Scope, experimental designs that 

broaden, 47 
of an experiment, 48 

Selecting a valid error, 122-125 

Sheppard's corrections for grouping, 15 

Sheppard's tables of the probability 
integral, 23 

a, sigma, standard deviation of popula- 
tion, 11, 12, 35 

2, summation, 10 

Significance, of abnormalities in dis- 
tributions, 29 
of chi square, 90 



INDEX 



277 



Significance, of correlation coefficient, 

72-73 

of differences between means, 40-42 
corrected for associated variables, 

252-253 
in incomplete block experiments, 

193 
in quasi-factorial experiments, 181, 

183, 187 

of differences between regression co- 
efficients, 56, 253 
of linear regression coefficient, 56 
of multiple correlation coefficient, 

214-215 

of non-linearity, 211-213 
of partial correlation coefficients, 85 
of regression function, 210-211 
tests of, 38 
Simple classification of variates, 125- 

126 

Simple interaction, method of calcu- 
lation, 136 
Skewness, 28 
Small samples, tests of significance 

with, 33-42 

Snedecor, G. W., 14, 19, 44, 63, 72, 77, 
89, 97, 100, 120, 127, 141, 246, 259 
Snedecor's table of F t 127, 269 
Soil heterogeneity, 142 
Split plot experiments, 151-159 
Standard deviation, 10 
a measure of variability, 10, 11 
calculation of, 11 
estimation of, 33-34 
of a sample mean, 12 
of large samples, 10-12 
of small samples, 10-12 
Standard error, 35 

of regression coefficient, 56 
Statistical analysis, functions of, 7 
Statistical methods, logic of, 1 
Statistical terms, definition of, 5 
Statistics, 6, 34 
"Student," 38, 44 
"Student's" distribution, 38 
Sum of products, as a measure of cor- 
relation, 67-68 



Sum of products, calculation of, 57 
division of in covariance analysis, 

247-248 

Summation method of fitting poly- 
nomials, 234-243 

Sums of squares, division of, 116-119 
methods of calculation, 135 

Tedin, O., 144, 209 

Tippet, L. H. C., 14, 19, 32, 44, 63, 77, 

87, 100, 113, 141, 218, 246, 266 
Treloar A. E., 72, 77 

Valid error, selection of, 122-125 
Variability, coefficient of, 17 
Variance, 35 

analysis of, 114-138 

estimation of, 35 

heterogeneity of, 114-116 

of a mean, 35 

Variation, heterogeneity of, 114-116 
Varieties, methods for testing large 
number of, 172-201 

Wallace, H. A., 87 
Wiebe, G. A., 209 
Winkler, C. A., 246 
Wishart, J., 87, 209, 259 

x, value of a single variate, 9-10 
f mean of a sample, 9-10 

Y, estimated value of dependent vari- 
able, 54-55 

y, individual value of a variate of de- 
pendent variable, 53-54 

Yates, F., 102, 113, 160, 167, 171, 172, 
200, 209, 261, 266 

Yates's correction for continuity, 102 

Youden, W. J., 192 

Youden's square, 192 

Yule, G. Udny, 32, 77, 113 

z, distribution of, 120 
z' t transformed correlation coefficient, 
73