The resolution of a set of variables into principal components may be performed upon variables having arbitrary variances, or upon variables obtained from these by reduction of all variances to unity. The two procedures do not yield the same results even if the principal components are reduced to unit variance in both cases, except when the original variances are all equal.

Four methods of factorizing the fundamental matrices used in factor analysis are described and illustrated. The first is represented by the techniques already developed. The second is the obverse factor technique. The third and fourth methods are variants of the first and second. The implications of each method for different schools of psychology are pointed out. The methods are complementary, not competitive.

By the use of an algebraic variant of the ordinary formula for bi-serial correlation, tables, and graphic devices, a time-saving systematic procedure for the computation of bi-serial correlation co-efficients is outlined for application to the evaluation of items of a test. A table of for arguments of p=.000 to p=.999 is given.

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.

The widespread use of bi-serial r in item analysis makes a quick and accurate method of evaluating the formula desirable. Such a method is provided by homographs. Dunlap and Kurtz in their Nomograph No. 47 give a method of solving the better known statement of the formula, namely (1)

A derivation of equations fundamental to the technique of path coefficients is given. Suggestions are made with respect to the calculations required in the use of the technique. The relations of the technique to those of partial correlation, semi-partial correlation, part correlation, multiple correlation, and factor analysis are discussed. Some consideration is given to the merits and limitations of the technique of path coefficients.

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.

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution.

A hypothetical interpretation of the canonic components of a score matrix is discussed.

Formulas are derived for simplified computation of partial and multiple correlation coefficients, and generalized ton variables. Time required for computation is compared with other methods.

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.