Methods are developed for factoring an arbitrary rectangular matrix S of rank r into the form FP, where F has r columns and P has r rows. For the statistical problem of factor analysis, S may be the score matrix of a population of individuals on a battery of tests. Then F is a matrix of factor loadings, P is a matrix of factor scores, and r is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrix G of rank r into the form FF′, where F has r columns and F′ denotes F transposed. For the statistical problem of factor analysis, G may be the matrix of intercorrelations, R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal. R is proportional to SS′, and it is shown that S can be factored by factoring R. This may usually be the most economical procedure in practice; it should not be overlooked, however, that S can be factored directly. The general methods build up an F (and P) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.

The introduction of psychological tests for personnel selection in the British forces has given rise to several novel problems in statistical procedure. The solutions proposed are in the main extensions of devices already familiar in educational psychology. The more important are: (i) where the criterion yields a threefold classification only, a method of triserial correlation or of biserial correlation assuming point-distributions for the extremes; (ii) where the data on which validation has to be based are drawn from a selected sample, a simplified form of Pearson's equations to correct for selection; (iii) where the best line of demarcation has to be deduced from theoretical rather than practical considerations, a formula based on the principle of minimal discrepancy.

A procedure is presented for determining the successive principal components of a correlation matrix where it is not necessary to compute the successive tables of residual correlations. The original correlation matrix is bordered with a new row and column for each principal component that is determined.

Second-order factors are defined and illustrated in terms of a literal notation, a physical example, a diagrammatic representation, a geometrical example, and the matrix equations relating the first-order and second-order domains. Both kinds of factors are discussed as parameters which may be not only descriptive of the individual objects in a statistical population but also descriptive of the restrictive conditions under which the objects were generated or selected. Second-order factors may be of significance in reconciling the several theories of intelligence. This paper is concerned with test configurations that show simple structure. If such a structure is not revealed, then the second-order domain is indeterminate.

The general procedure and detailed steps for attaining complete factor analyses of scores are presented. Both orthogonal and oblique factors are considered. It is shown that a single average by conventional procedure gives an incomplete summarization of the data when the rank exceeds one. There should be as many averages as there are common factors.

A technique has been developed which permits the weighting of responses of test items on the I. B. M. scoring machine on the initial scoring, heretofore impossible. This is done by making the length of the response lines on the answer sheet longer or shorter as weights are needed. It is anticipated that this method will prove useful wherever differential weighting serves to increase the validity of tests.

A dilemma was created for factor analysts by Ferguson (Psychometrika, 1941, 6, 323–329) when he demonstrated that test items or sub-tests of varying difficulty will yield a correlation matrix of rank greater than 1, even though the material from which the items or sub-tests are drawn is homogeneous, although homogeneity of such material had been defined operationally by factor analysts as having a correlation matrix of rank 1. This dilemma has been resolved as a case of ambiguity, which lay in (1) failure to specify whether homogeneity was to apply to content, difficulty, or both, and (2) failure to state explicitly the kind of correlation to be used in obtaining the matrix. It is demonstrated that (1) if the material is homogeneous in both respects, the type of coefficient is immaterial, but (2) if content is homogeneous but difficulty is not, the homogeneity of the content can be demonstrated only by using the tetra chorie correlation coefficient in deriving the matrix; and that the use of the phi-coefficient (Pearsonian r) will disclose only the nonhomogeneity of the difficulty and lead to a series of constant error factors as contrasted with content factors. Since varying difficulty of items (and possibly of sub-tests) is desirable as well as practically unavoidable, it is recommended that all factor analysis problems be carried out with tetrachoric correlations. While no one would want to obtain the constant error factors by factor analysis (difficulty being more easily obtained by counting passes), their importance for test construction is pointed out.

For an amount-limit test homogeneous as to content and varied as to difficulty it is established that an individual's number-right score and his limen score as estimated by the constant process are mathematically related. The experimental and the theoretic relationship between normal deviate and limen score are shown to be in good agreement. It is also found that the two methods of evaluating individual test performance yield equally reliable sets of scores for the procedures used. Accordingly where the assumptions basic to the relationship obtain, the more conveniently computed raw score may be considered to be as valid and reliable an index of individual test performance as the limen score. The concept of the dispersion parameter of the individual as a measure of change or error in test score found no experimental verification. Estimates of individual variability are unrelated to differences in score on equivalent forms.

The application of the Müller-Urban constant process to item selection, as considered in a recent paper in this journal, is shown to be closely analogous to a method now in general use for the analysis of insecticidal and other toxicological tests. This method of probit analysis gives the maximum likelihood estimates of the unknown parameters and, with the aid of published tables, the necessary computations can be rapidly performed. The present paper contains an outline of the method and an illustration of the most convenient form of computations for use in analyses of psychometric data.

Four methods of determining the clusters in a correlation matrix are described and compared. The choice of method has to be made according to the size of the matrix and the type of cluster sought. The relativity of clusters is emphasized and a distinction is drawn between phenomenal clusters and nuclear clusters. The relative utility of clusters and factors is briefly commented upon.

The predictive efficiency of a test used to select personnel is defined in terms of total effectiveness of the group thus selected, as compared with chance selection. The formula developed requires the use of an estimate of the ratio of average effectiveness of men selected to the average effectiveness of men not selected by the test. The predictive efficiency of the test varies directly with the magnitude of this ratio and also directly with the percentage rejected.

This paper discusses the application of the principles of factorial design to an experiment in psychology. For the purpose of illustrating the principles, a simple experiment was designed dealing with the determination of the differential limen values of subjects for weights increasing at constant rates. The factorial design was of the type: 4 rates × 7 weights × 2 sexes × 2 sights × 2 dates. The appropriate statistical analysis for this type of design is the analysis of variance. The mathematical formulation of the problem was specified and the appropriate solution for the specific problem was obtained. Greater precision results from this type of design, in comparison with the traditional psychological experiment dealing with a single factor, in that measures are obtained of the effect of each of a number of factors together with their interactions.

Mathematical expressions are derived for such concepts as stimulation of student by instructor, student-instructor rapport, and driving power of instructor, in terms of the student's and the instructor's foci of attention, their strength of concentration, and the intensity of the presentation and of the reception of details of subject matter. Under the assumption of normal distribution, the mathematical methods of combination and integration yield conclusions on summary integral effects of interrelations within the educational team. The psychological interpretation of the mathematical results thus obtained conforms with common sense. The main emphasis of the article is the exposition of how the mathematical method of combination and integration can be used to estimate the resultant effect of various independent combined simple factors acting independently within the individuals forming the educational team. No claim is made as to the absolute truthfulness and reliability of the psychological postulates used at the beginning stage of the mathematical analysis.

A survey of the literature was made to determine the skills involved in reading comprehension that are deemed most important by authorities. Multiple-choice test items were constructed to measure each of nine skills thus identified as basic. The intercorrelations of the nine skill scores were factored, each skill being weighted in the initial matrix roughly in proportion to its importance in reading comprehension, as judged by authorities. The principal components were rather readily interpretable in terms of the initial variables. Individual scores in components I and II are sufficiently reliable to warrant their use for practical purposes, and useful measures of other components could be provided by constructing the required number of additional items. The results also indicate need for workbooks to aid in improving students' use of basic reading skills. The study provides more detailed information regarding the skills measured by the Cooperative Reading Comprehension Tests than has heretofore been provided regarding the skills actually measured by any other widely used reading test. Statistical techniques for estimating the reliability coefficients of individual scores in principal-axes components, for determining whether component variances are greater than would be yielded by chance, and for calculating the significance of the differences between successive component variances are illustrated.

A factorial rotational method is presented which represents a compromise between the use of subjective judgment characteristic of graphical methods and the routine application of analytical methods. At present the analytical methods seem to be inadequate for the discovery of a simple structure, while graphical methods require more subjective judgment. The method herein presented locates the axes for subgroups of tests by an analytical method. The judgments used in the selection of subgroups are based on graphic data concerning interrelation of the factors.

A simple method for extracting correlated factors simultaneously is described. The method is based on the idea that the centroid pattern coefficients for the sections of unit rank of the complete matrix may be interpreted as structure values for the entire matrix. Only the routine centroid average process is required.