Hypotheses presented in a previous paper conceive of an individual's position on an attitude scale as represented by a mean position (status score) and a sigma (dispersion score). The testing of these hypotheses requires a determination of these two scores for each individual. A rationale is presented here for the determination of these two scores for an individual from his forced choice responses to pairs of items of nearly equal scale value.

A battery of pencil-and-paper tests is commonly used for predicting a single criterion. If the score on each test is the number of correct answers, the composite battery score would normally be the sum of the weighted test scores, where the weights are the raw score regression weights. Knowing the reliability of each test, it is possible to alter the lengths of the tests in a manner such that the weights will all be equal. The composite battery score would then simply be the total number of items answered correctly and scoring would be greatly simplified. Such simplification is particularly desirable where the volume of testing is large. Section I of the article outlines the procedure for altering the lengths of the tests, and Section II gives a proof of the method.

Two current methods of deriving common-factor scores from tests are briefly examined and rejected. One of these estimates a score from a multiple-regression equation with as many terms as there are tests in the battery. The other limits the equation to a few tests heavily saturated with the desired factor, with or without tests used to suppress the undesired factors. In the proposed methods, the single best test for each common factor is the starting point. Such a test ordinarily has a very few undesired factors to be suppressed, frequently only one. The suppression test should be univocal, or nearly so. Fortunately, there are relatively univocal tests for factors that commonly require suppression. Equations are offered by which the desired-factor test and a single suppression test can be weighted in order to achieve one or more objectives. Among the objectives are (1) maximizing the desired factor variance, (2) minimizing the undesired factor variance, (3) a compromise, in which the undesired variance is materially reduced without loss in desired variance, and (4) a change to any selected ratio of desired to undesired variance. A more generalized solution is also suggested. The methods can be extended in part to the suppression of more than one factor. Equations are derived for the suppression of two factors.

This is a continuation of a paper in the previous issue of Psychometrika (Sept., 1948, pp. 147-174) in which some historical facts about the First World War were expressed in mathematical form. The present introduction is intended to make the sequel readable by a person who has never seen Part I. This can perhaps just be done; although for ease, clarity, and conviction a prior study of Part I is strongly to be recommended. The independent variable is t, the time.

On the basis of the assumption that distributions of scores on psychological functions are generated by a finite number of equally probable factors, a method is presented which yields the number of factors and their probability p. The statistics β1 and β2 are used for this purpose. An experiment utilizing a code-transcription test is described in which the method was employed to analyze performance at several stages in the learning process. n was found to be 10 for the first 2 minutes of practice and 19 for the second 2 minutes. For the third, fourth, and fifth 2-minute periods, no value could be obtained owing to the pronounced leptokurtosis of the distributions. The first 3 periods of practice, when lumped together, gave an n of 33. It is suggested that the method offers a means of comparing the variability and “complexity” of otherwise non-comparable psychological functions. The use of the method as an instrument of investigation in the field of factor analysis is described.

A method of exhaustion has been described for calculating regression coefficients. This method dispenses with the solution of simultaneous equations but utilizes a process of successive extraction in obtaining β's, where each successive β is maximized. This procedure permits the worker to discard as he goes along those weights which are deemed unsatisfactory for purposes of prediction. The β coefficients and the R in a problem involving a criterion and six independent variables were calculated in sixty minutes. The R's obtained by this method are smaller than those yielded by the Doolittle technique, but in problems which have been considered this discrepancy has not exceeded .05.

A table to aid in testing the statistical significance of the difference between correlated percentages is presented. Percentage points which range from .10 to .001 are included.

A test of the significance of a row or column agent in an analysis of variance may be expressed in the form of correlation between the agent and the variate. A test of the significance of interaction variance may be expressed in the form of correlation between the agents. These expressions are principally of theoretical interest in that the degree of significance in an F test or the value of a correlation coefficient may be controlled at will, or inadvertently, within certain limits.

A person can be openly friendly, or hostile, or war-weary. Depth-psychology has shown that the underlying mood is also important. In the present theory the independent variable is time, and the dependent variables are the numbers of persons in different warmoods in two opposing nations. The rate of conversion of persons from one mood to another is taken to be proportional both to the number of susceptible persons and to the number of influencing persons, as in Kermack and McKendrick's theory of epidemics of cholera (15). This formulation leads to a set of differential equations nearly but not quite like those whereby Volterra (41) described the interaction between predator and prey among fish. Here the constants are fitted to the history of the First World War. The gregarious motive to follow a fashion of either peace or war is formulated as an amendment.

From the factorial analysis of a battery of nineteen individual tests seven factors were obtained, six of them being interpreted. Some of these factors are probably related to similar ones isolated in different batteries and by different authors. The test Progressive Matrices here analyzed is loaded in some of the factors, as was expected according to the nature of its problems and to previous experience with the test. The loadings of the tests and of the factors in the second-order factors are given. It is expected that this extension of factor theory will prove useful for the isolation of more fundamental psychological parameters, although at present it is difficult to evaluate precisely the theoretical and practical implications of this aspect of factor analysis.

A method of computing ΣX, ΣX2, ΣXY and higher moments on IBM equipment is described. The basic method is that of successively summary punching, collating a variable number of blank cards behind these summary cards, gang-punching the data from the summary cards into the blank cards, and totalling the entries on these summary cards. The method appears to have several advantages over those previously described, especially if coded data are used.

Professor Godfrey Thomson, in a communication to this laboratory, has drawn our attention to certain computational and editorial errors in a recent publication on the effects of selection in factor analysis. The purpose of this note is to rectify these errors. In so doing, we wish to express our gratitude to Professor Thomson for preventing their prepetuation.

Since the particular tables under consideration appear also in Chapter XIX, “The Effects of Selection,” of and are identical with those of, the material presented in this note is equally applicable to both publications. These tables were numbered identically in both and may be referred to non-differentially. In order to facilitate the exposition in this note, the numbers assigned to each of the corrected tables and figure correspond to the analogous tables and figure of the original publications.

Outlined is the method used at present by the Educational Testing Service for computing intercorrelations from basic summations. This procedure is adapted to the use of high speed calculators in performing the calculations, and much of its value lies in the complete system of checks that is a part of the method. Besides the correlations that are the object of the procedure, covariances, means, standard deviations, and the number of cases are also recorded on the completed form to be available for further statistical steps.

Data originally analyzed by Charles H. Goodman on the MacQuarrie Test for Mechanical Ability are subjected to the principal axes factoring method. The maximum variance was extracted with three factors. Rotation to an oblique simple structure yielded a factor pattern which satisfies the simple structure concept more adequately than the orthogonal factor matrix, thus leading to greater clarity of interpretation of the factors.

There appears to be a gap in published computational techniques inasmuch as nowhere in the literature nor in textbooks can one find a model to be followed in computing the numerous zero-order correlation coefficients for a correlation matrix. The purpose of this paper is to present, by means of an illustration, such a model. The model consists of two computational matrices, matrix one being the Summation Matrix and matrix two the Computational Matrix. The entries on these matrices are arranged so as to facilitate the future computations.

Ignorance concerning the standard error of individual factor loadings and their differences has been a major obstacle to the proper interpretation of factorial results. The effects of three types of experimental error (selection of variables, selection of subjects and selection of scores) are here reported. In order to handle the errors of rotation systematically, it has been necessary to introduce a new semi-analytical criterion for the attainment of simple structure. Variability in results which may theoretically be eliminated is discussed under the heading of non-essential error.

Factorial analysis begins with an n × n correlation matrix R, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated that R will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data in R. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.