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OU_1 64071 > OSMANIA UNIVERSITY LIBRARY OUP 557 13-7-71 3,000. OSMANIA UNlVESr| LIBRARf f<:' Call No. 3>\\- VE Accession No. ^ S" Title K*-\WJU s This booked be returned on or before the dat: bht marked below. 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- DATIONS OF THE WAR PRODUCTION BOARD HAVE BEEN OBSERVED FOR THE CONSERVATION OF PAPER AND OTHER IMPORTANT WAR MATERIALS. THE CONTENT REMAINS COMPLETE AND UNABRIDGED. COPYRIGHT, 1939 BT CYRIL H. GOULDEN Att Rights Reserved This book or any part thereof must not be reproduced in any form without the written permission of the publisher. 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 ooooo 1050000 xhc^ OO Oi rH O I s "" IS* oooooo O O CO IO -i O - OS <N l>- O ooooooo 22 do 33, S rHO>r-( CS| COrH F-t i-t rH rH OS O5 IO CO<NOr-r-l OOOO 1 J ooooo oooooo 1 1 i o LO ^o t* co 01 oo S^CO^O'H^^ O d rH CO 00 W ooooooo oooo 1 i Si oooo 1 i ooooo ! 1 1 <^ CO GO O> "# iO rH Cl TH rH O O rH^5o<NTfJoi CO r-t O O -*H in oooooo II I 1 HOOOOO* 1 1 1 1 Oi H CO I s " w <o -* 5*<l> OO o oooo t^ot^q 8 ^t>coc lOcOQe COiOOC >r-,-iO*COC JOiQ^COC 52|gsi oooo I i ooooo MM r-10000^ I I coco 0*0 I CO I s - O5 (MQCD rH OOO O -co oooo i 1 CO CO p-4 do I S! 00 o o I <N co * iO co r^ : =5 * o : : -"S^ s*^ N . a *^8a N 3,.g o g -43^-0 ?3S'S x .^ CNCOCOOW d d o* o o II II \\ II II II H oSoiox^oa 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 s I I, !! S I! CO g d g? 5c? 1 I! tOQ ss 8 1 OiOO CO 00-^ %vt co eo * Sis Nd^ 8S ss too to 3S3& cocfco 8iOO ^^ iOO i-iOC OtOtO COdCO top to tOOOQ CO CO tO u s I 3 if 2 | to B r-l g 3 if tv I s * t 00 CO tOO H i~4 eo*2o cococo Cl^C OOO t to too *co"co" tO tO tO loSd to to to S.S.8 CifcOCO oo ss si ^ 1 * 8^.S I o'cojo 1 1 CO tO tO CO-HtO -f 1 + 00 OOO COCO'-' esicoto sa ""I 00 tO OOOr-l i- fHd *J I CO CO 3J I j ft j tO CO 36 fcsa r-t CO CO feSSS 88^' >OdO tOdC ^dco 8 ^dc. H ^ X M J rt ' J j 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 H << 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 + o I ^ 1 1 gfi pa ZS COI>kO co epos COCOO IQTt^ -iQ OOrH OO 1C ^ Tf" 328 CM"CNT>O > CO CM CO rHt^.rH g ^ rH 3 c? IO rH Sg fe rH 1 s 00 CM 00 CO "I 1 r-l-^COO Gi 00 O V* Gb 00 i I I^IOO^^H IOCOOOO) o-o 1 + + i g o ' Tt< tffod" CO CM CO CO O CO *O CO 00 + 1 1 CMrH , N OF NORMAL ] 1 ** 00 rH if rHlblO- oo-icO 33^ 1 Sg ir^t>^ CO^CM"" iO *O rn'6 +7 +1665.0445 (0.003528) 2 o o S7 +0.412,519 rH 5 X ^ si 3 s rH(N co^toco t^OOO>O-i <NCO^ IOCO t* ^HCMCO^ 04 3 98J9Aa^[ CN rt< t>. rH IO>O IO kO *O <O CM CO >O rH T"H Qi ^< OOi-id + 1 ++ II II II II 8G8 1 < t> X N co 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 I I t> 3 Q G5 00 rf o I S S w < H * ^ w 2 w S^ &, ^ o o BCD W -^ 5 PH ^ 1.1 I O I s oo M CQ O s PQ u s m "5 M B S PQ CO M PQ 1 O O c 'S O r-tO^jOOO^OOl^t^" CNCNC^CNCOCOiNCSI - CO CO CO CO CO CO a '3 O a ' O a "S O -S "S cS O CQ 81O C^l O5 CO CO r-4 O o as a> o co t- w Jt^t^cOcOcOcOcOcO S COCOCO(MCOCOCOCO CO CO CO CO 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 *. 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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