In building a system of mathematical biophysics, as the foundation of which we took the phenomena of cellular multiplication, we have had naturally several occasions to discuss also various physico-mathematical aspects of psychological phenomena. This is not surprising, since the latter forms a prominent branch of general biology. There is, however, still a big difference in our treatment of biophysics of general biology and in the treatment of biophysics of psychology. In the former we started with a rather general concept of a metabolising system and developed therefrom in an almost purely deductive, synthetic way an elaborate abstract theoretical system of biophysics, which incidentally happened to throw interesting light on actual real phenomena of cellular biology. We intentionally underline here the word “incidentally”, to emphasize that, although we consider the development of mathematical biophysics as eventually of greatest importance for the interpretation of empirical biology, we do not consider this “utilitarian” aim as the principal driving motive for our study.

The resolution of a set of n tests or other variates into components γn, each of which accounts for the greatest possible portion γ1, γ2, ..., of the total variance of the tests unaccounted for by the previous components, has been dealt with by the author in a previous paper. Such “factors,” on account of their analogy with the principal axes of a quadric, have been called principal components. The present paper describes a modification of the iterative scheme of calculating principal components there presented, in a fashion that materially accelerates convergence. The application of the iterative process is not confined to statistics, but may be used to obtain the magnitudes and orientations of the principal axes of a quadric or hyperquadric in a manner which will ordinarily be far less laborious than those given in books on geometry. This is true whether the quadrics are ellipsoids or hyperboloids; the proof of convergence given in an earlier paper is applicable to all kinds of central quadrics.

It has been repeatedly demonstrated that, in a learning situation involving a positive and a negative stimulus, the organism is frequently reacting to the relationship between stimuli rather than to the absolute characteristics of the positive or negative stimulus. For example, if in the original training series an animal has learned to react positively to a ten and negatively to a five centimeter circle, he w,ill, in the majority of trials, when confronted with a ten and a twenty centimeter circle, react positively to the twenty centimeter circle, that is, the larger of the two, and negatively to the ten centimeter circle which was formerly the positive stimulus. This is termed transposition of the relationship greater than from the training or original learning situation to the test or transposition situation.

It is well known that at the present time the Doolittle method is one of the most efficient methods of computing the multiple correlation coefficient between a criterion and several independent variables.

This presentation calls attention to an extremely simple modification of the Doolittle method by means of which (a) a single forward solution will supply all the data necessary for multiple correlation coefficients instead of the usual one, or (b) the multiple correlation between each of several criteria and the same set of independent variables may be obtained with only a little more work than is needed to obtain the multiple correlation between these independent variables and a single criterion. It is even possible to compute the multiple correlation between several independent variables and a criterion; and then to regard one of the former independent variables as a second criterion and compute the correlation between the remaining independent variables and the new criterion. In all these cases the number of back solutions is equal to the number of multiple correlation coefficients desired, but a single forward solution suffices.

The purpose of this paper is to describe one of the methods of analyzing a configuration of psychological tests or traits in the problem of isolating primary factors. In factorial analysis a set of traits or tests is represented by a set of radial vectors which are so related to each other that the scalar product of each pair is equal to the corresponding experimentally observed correlation coefficient. Each radial vector represents a trait. The dimensionality of the trait configuration is the rank of the reduced correlational matrix. The reduced correlational matrix has communalities in the diagonal cells.

If there are n trait vectors in the configuration and if the dimensionality is r, then each of the n traits can be described as a linear function, a weighted sum, of any set of r linearly independent vectors. It is assumed here that n > r. These r traits in terms of vchich one may describe or comprehend each of the traits in the configuration have been denoted reference traits.

There is increasing use of item analysis procedures for the improvement of objective examinations. The development of the procedures of item analysis has consisted chiefly of the invention of various forms of an index of association between the test item and the total test score. At least ten indices of item validity have appeared in various articles, which have been chiefly concerned with the relative effectiveness of the indices as devices for the improvement of tests. Since these indices of “item validity” are substitutes for or approximations to the ordinary coefficient of correlation between the item and the total test score, it may be useful to present certain deductions from simple correlational algebra. The present writer is of the opinion that the ingenuity displayed in the invention of new indices has outstripped the critical examination of the logical foundation for item analysis. The subsequent discussion is therefore concerned only with the underlying rationale of item analysis.

For the readers of a journal devoted to the application of mathematics to psychological research it is scarcely necessary to spend much time in answering the question, “Why mathematics?” Science may best be defined as that set of postulates regarding experience to which the universal assent of competent observers may be obtained, plus the organization of such postulates into theories for which universal assent is likewise obtainable. Of all the propositions about nature those concerned with mathematics are most readily given universal assent. From this state of affairs the Kantian aphorism that a discipline is as scientific as it contains mathematics is entirely consequent. Kant himself doubted the applicability of mathematics to psychology and so was led to question the possibility of a scientific psychology. Fechher, as is well-known, thought differently and from Fechner‘s day the application of mathematical procedures has become an increasingly important part of psychological research until to-day we have a journal devoted to such application alone.