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.

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.

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.