Beginning students of analysis of variance frequently find that the explanations concerning numbers of degrees of freedom are rather more mystical than explanatory, and that the dependencies and independencies among parameters and linear restraints are not systematically covered or explicitly presented. It is the purpose of this paper to set forth the simple matrix representation of these models in a way which has been found useful for teaching purposes, and may be found of some value in the setting up of mathematical models for experimental designs calling for the analysis of variance or the analysis of covariance.

The system I shall portray is a more or less natural outgrowth of suggestions made by Dr. R. C. Bose, with whom I have discussed some of the features very slightly. I am also indebted to Professor Frederick Mosteller for assistance with the mechanics of some of the models. Any deficiencies which the system may have are not to be laid to the doors of these gentlemen, however, as they have not had any opportunity to go over the material at any length.

If the lengths of the tests in a battery are altered, their intercorrelations and their validities or correlations with a criterion are also altered. Consequently, the multiple correlation of the battery with the criterion will also be altered. These changes are a function of the reliabilities of the tests. Suppose we have given from a set of experimental data (1) the time allowed for each test in the battery, (2) the reliability of each test, (3) the intercorrelations, and (4) the validities of all the tests. If we specify the over-all testing time we are willing to allow for the test in the future, we can determine the amount by which each test must be altered in order to give the maximum multiple correlation with the criterion. The method is presented, together with numerical examples and the mathematical proof.

The original aim of this investigation was to discover what relations exist between race attitudes and certain personality traits. For that purpose, a standard neurotic inventory and an attitude scale were applied to three dissimilar groups of subjects, and the results, obtained by multiple-factor analysis carried out independently for each of the three groups, compared with one another. The components of neurotic tendency, as displayed by each of the three groups, show a high measure of agreement and appear to be reconcilable with other interpretations of neurotic personality. So far as the original aim of the study is concerned, there appears to be a slight but significant tendency for the neurotic factor of hypersensitiveness to affect the race attitudes of Europeans towards the native in South Africa in a negative or unfavorable direction.

A coefficient of selective efficiency is proposed which can be usefully applied to selection problems involving the evaluation of the validity of (1) dichotomous predictors and (2) continuous predictors at a particular or at successive points of cut. Previously the author has shown that the product-moment correlation can be interpreted as a direct index of selective efficiency if the distribution forms of the criterion and the predictor are similar and the regression of the criterion on the predictor is linear. The coefficient proposed in the present article may be employed to evaluate selective efficiency of a continuous predictor at particular points of cut even when these assumptions are not tenable or are not applicable. It also is demonstrated that the proposed coefficient of selective efficiency may—with somewhat simpler and more generally applicable assumptions than those required in deriving the conventional formula—be employed as a substitute for the biserial correlation coefficient.

In certain problems dealing with percentage differences, it frequently happens that one is interested in a critical value different from zero rather than in the usual hypothesis of no population difference. Moreover, one may be concerned with differences greater than (or less than) a given magnitude, so that only one tail of the distribution of chance values is used. Tables are here presented to facilitate evaluating results of such problems. One per cent and 5 per cent values can be read from the tables, such values corresponding to the more usual 2 per cent and 10 per cent confidence limits.

A pattern consists of fundaments and relations. Its similarity to other patterns can be expressed either in terms of relations alone—in which case a modified correlation coefficient, rs, suffices—or in terms of fundaments and relations, where new coefficients, rp and r′p are required. The derivation of a coefficient of pattern similarity, rp, is presented and its properties are investigated, both deductively and in terms of actual data on culture patterns.

In certain situations it is important to obtain as many measures as possible, all presumably measuring the same function, for each of a group of persons. In general the number and source of the measures may vary from one member of the group to another. We take the mean of the measures for each person as the best estimate of the function for that person. The conventional formulas can not be used to determine the reliability of a set of means so obtained. A formula is developed which provides a unique estimate of the reliability of such a set of means. The formula is more general than some of the well-known reliability formulas, so that these formulas are shown to be special cases of the more general formula.

A simplified method of computing the inverse of a triangular matrix is presented. It is useful with the multiple-group method of factoring the correlation matrix as well as with other factor-analysis and multiple-correlation problems.

As usually interpreted, the standard error of measurement is assumed to be constant throughout the test-score range. In this investigation the standard error of measurement was assumed to be not higher than a second-degree function of the test score. By conceiving a test score to be made up of the scores on two parallel tests, an equation was derived for predicting the standard error of measurement from the test score. In the derivation the corresponding first four moments of the score distributions for the parallel tests were assumed to be identical, and certain errors of estimate involved in predicting the second test score from the first were assumed to be uncorrelated with powers of the score on the first test. An empirical verification was carried out, using nine synthetic tests and a 1000-case sample, and showed good agreement between predicted and observed results. The findings indicated that the standard error of measurement was constant only for a symmetrical, mesokurtic distribution of scores.

Standardization scores on the 1937 Revision of the Stanford-Binet are analyzed by the Thurstone centroid factor technique. Correlation matrices at age levels seven, nine, eleven, and thirteen, each based on two hundred subjects, are analyzed. The factors after rotation are interpreted in terms of the items exhibiting high factor saturations. In contrast to the findings of McNemar (1942) no evidence for a general factor in the Stanford-Binet battery is found in the present study. Rather, I.Q. variance can be explained in terms of psychologically meaningful group factors at the various age levels. The results indicate a need for cautious interpretation and use of single score indices of intelligence such as the Stanford-Binet I.Q.

This is an experimental study of the application of factor analysis to a new domain—the formation of job families. Correlations between jobs are computed from the formula based on the number of common elements between two variables and the job analyses provide the basic data on the presence or absence of the elements. A first-order general factor and four common factors are obtained in a small sample of twenty occupations. Tentative interpretations are made and implications for job analysis and the formation of job families are pointed out.

Matrix methods may be applied to the analysis of experimental data concerning group structure when these data indicate relationships which can be depicted by line diagrams such as sociograms. One may introduce two concepts, n-chain and clique, which have simple relationships to the powers of certain matrices. Using them it is possible to determine the group structure by methods which are both faster and more certain than less systematic methods. This paper describes such a matrix method and applies it to the analysis of practical examples. At several points some unsolved problems in this field are indicated.

The Kuder-Richardson formula (20) is rewritten to be identical with the simplest formula, (21), except for the addition of a term involving the standard deviation, σp, of the item p's. If σp can be estimated, a rapid and superior estimate of test reliability is possible in contrast to the simpler formula (21) used when the number of items and mean and standard deviation of test scores are known.

This note directs attention to the basic similarity between a factor analysis method described by Holzinger in 1944 and what Thurstone has called the multiple group method. With minor modifications and the application of Holzinger's method in several successive cycles until the residuals vanish, the methods are essentially the same.

A new method for correcting erroneous communality estimates is applicable to any completed orthogonal factor solution. It seeks, by direct correction of factor loadings, to make the residuals conform to the chance error criteria of zero mean and zero skewness for each row separately. Two numerical examples, with one and two factors, respectively, are presented. The method can be used as a short cut for Dwyer's extension in adding variables to a matrix. It can also be used as a short cut in cross-validation factor studies. Successful use on problems with many variables and numerous factors is claimed. Factors can be made oblique, after correction, if desired.

A battery of 20 tests originally analyzed by Alexander (1) was reworked according to the principle of simple structure. His results were sustained in general. Both analyses yielded five factors in the first-order domain. Of these, three factors in the re-analysis (v, X and F) have almost exactly the same loadings as the corresponding factors in the original work, and were interpreted in the same way. The loading pattern of a fourth factor, Z, left uninterpreted in the original study, happened to be more clear in the re-analysis, and an interpretation was attempted. It appears to be a factor of perceptual synthesis, and seems to play an important role in intellectual processes. A fifth factor, not present in Alexander's results, appeared in the new analysis: the reasoning factor, involved in inductive and deductive thinking. All four cognitive factors are related to a general factor that can be thought of as representing abstraction and education of relations and correlates, these processes being, therefore, the essential feature underlying intellectual behavior, at least in that sector surveyed by the tests of the present battery.

A simplified method of obtaining sums of squares and sums of cross products by the use of punch card equipment is described. Application of the method has revealed several advantages, which are noted.

Data resulting from the administration of the USAFI Tests of General Educational Development to more than 1000 junior Air Force officers have been statistically analyzed to indicate the reliability of these tests, their correlation with school achievement, the comparability of their intercorrelations with intercorrelations among grades in school subjects, their capacity for differential diagnosis, their factorial composition, and their average item-test correlations. In the light of these findings and the finding that there is a low gradient between achievement on these tests and amount of formal education among these officers, the tests have been evaluated as possessing the practical validity suggested by their face validity for selection of young Air Force officers for assignment to study at civilian colleges and universities.

This paper discusses the properties of distributions of test scores and advances the view that the properties of the distribution should depend on the function which the test is intended to perform. A theory of test discrimination is developed which defines discriminatory capacity in terms of the number of relations of difference established by the operation of administering a test of k items to a sample of n individuals. A simple proof is presented which indicates that maximum discrimination between individuals is achieved when tests are constructed to yield distributions of the rectangular form. A coefficient of test discrimination is developed. The problem of obtaining in practice distributions approximating to the rectangular form is briefly discussed.

A modification of the Wherry-Doolittle test selection method is presented by which tests are included in a multiple correlation (obtained for a given battery of tests) in the sequence in which the rate of return in validity per unit of testing time is greatest, rather than in the order of the size of their contribution to the multiple correlation. It is proposed that the modified method can be utilized profitably when there are economic or practical limits on the time available for test administration.