The author arrives at a simple rule for ascertaining when a matrix of correlations, with communalities reducing it to minimum rank, cannot be analyzed into factors such that every column of loadings has at least as many zeros as the number of common factors, as required by Thurstone. A more exact but arithmetically tedious rule is also deduced from Ridley Thompson's boundary conditions, and a correction is made to the latter.

A new item selection technique is presented which takes into account the intercorrelations of the items as well as their correlations with the criterion. The technique is regarded as superior to comparable techniques in that it is considered to achieve greater economy of time, greater objectivity of procedure, higher validity, and higher reliability. The mathematical theory underlying the method is developed. An approximate solution of the mathematical equations is suggested. An approximation procedure for the complete item selection technique is presented, based on the mathematical solution, but much simpler in procedure. The clerical operations involved in the approximation procedure are outlined and illustrated on a sample worksheet.

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.

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery of n tests which are describable in terms of r common factors, with orthogonal reference vectors.

1. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors.

2. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors and the n—1 remaining tests.

Corollary: The square of the multiple correlation of a test j with the n—1 remaining tests is equal to or less than the communality of test j. It cannot exceed the communality.

3. The square of the multiple correlation of a test j with the n—1 remaining tests equals the communality of test j if the group of tests contains r statistically independent ests teach with a communality of unity.

4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a test j equals the square of the multiple correlation of test j with the n—1 remaining tests.

5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a test j with the n—1 remaining tests approaches the communality of test j as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.

A variation of the centroid method is described and illustrated. By the application of new rules for reflecting signs, it may be possible to reduce to insignificance the factor loadings of tests showing insignificant correlation (original or residual) with clusters of tests having relatively high intercorrelations. As a result, a factor common to any one of these clusters may be revealed by the centroid method itself with little or no need for rotation of axes or further calculations.

This paper is a mathematical supplement to the preceding paper by Professor Godfrey H. Thomson. It gives rigorous proofs of theorems enunciated by him and by Dr. J. Ridley Thompson, and extends them. Its basic theorem is that if a matrix of correlations is to be factorized without the aid of higher factors than s-factors (with n-s zero loadings), then the largest latent root of the matrix must not exceed the sum of the s largest communalities on the diagonal.

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.

The topological concepts allow us to assign individuals and goals to certain spatial regions. They allow us to designate what locomotions are possible for an individual and what regions must be traversed in attaining a definite goal. But the topological concepts alone tell us nothing of the actual locomotions performed in psychological activities. So far we have seen that the position of the individual at the start of the psychological activity may be defined in reference to this. Psychological activities may be ordered to locomotions. The individual in this sense has the character of a thing. The space through which the locomotion occurs has the properties of a medium. (Cf. Heider (10). In physical problems where the field construct is used, one may make this same distinction. Bodies falling in the earth’s gravitational field have the properties of things, while the atmosphere is to be characterized as a medium. Similarly in the electro-static field, the isolated conductors have the properties of things and the field has those of a medium.

Given s fallible tests t1,t2, …ts, the problem is to express their intercorrelations in terms of the average correlations between a varying number of parallel forms contained within each test. A new correlation determinant Δ′ is derived containing dii instead of unity as an element on the principal diagonal, where

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This is an experimental study of the isolation, by factor methods, of primary abilities from a battery of tests given to 240 students. The range and nature of the fifty-six tests is briefly described. Tentative interpretations of the twelve orthogonal primary factors are given.

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.

The evaluation of the level of difficulty of a test item is ordinarily derived from the proportion of a specified population passing or failing the item. With items that have a limited number of alternative responses there must be a correction in this proportion to make allowance for chance success. A table of corrected proportions is given for different numbers of alternatives varying from two to eight.

In continuation of a previous paper, some consequences of the fundamental equations established there are studied. For some simple hypothetical cases it is shown how some of the parameters which enter in the equations governing the structure of the social group can be determined by means of those equations from actually observable data. Furthermore some general properties of the variation with respect to time of the fundamental distribution function, which enters in the equations, are derived.

In continuation of a previous paper, a mechanism of delayed reflexes is considered more in detail. Equations governing such a mechanism are established and approximately solved. The formulae thus obtained describe the phenomenon of “concentration” of a conditioned reflex around a definite time interval after stimulation. Applied along the lines discussed in the previous paper to some simple combinations of stimuli and responses the formulae lead to a description of the elimination of errors by trial. They give a relation between the number of repetitions, necessary to eliminate a wrong act, and other constants, describing the situation.

The problem of weighting separate criterion variates is solved by minimizing the differences among the standard scores of the individual upon the various measures. The method is compared with Horst's procedure of maximizing the inter-individual differences. An application is made to personnel data.

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 content unreliability of an essay test is the error due to the items used or the content of the test. The reader unreliability is due to variation in judgment of the persons who read and score the essay test. The content reliability of an essay test is accordingly defined as being independent of the reader reliability. Formulae are derived for the reader reliability and for the content reliability. The content reliability is found to be equal to the geometric mean of the test reliabilities computed from the scores assigned by the two readers, divided by the reader reliability.

Using scores of 1200 students on a long test as a criterion, each of five subtests of different difficulty has maximum correlation with the criterion when the criterion is dichotomized at a value appropriate to the difficulty of the subtest. A 50-item test element is scored on an all-or-none basis with different standards for passing, and the percentage of passes for successive points on the criterion variable is computed. The Constant Method is applied to this relationship. The limen thus computed is a measure of difficulty, the dispersion is a measure of average (or total) validity, and the slope of the curve is a measure of differential validity. The difficulty of a test element is thus directly related to the maximum differential validity.

The implications contained in Richardson's article on item analysis in March 1936 issue of Psychometrika are examined in the light of multiple factor theory. It is shown that item analysis is a necessary, but not a sufficient condition for the construction of a test which shall measure a single trait. The intercorrelations of certain items selected by a method of item analysis are examined, found to contain many zero and some negative correlations. Multiple factor analysis showed that eight traits were measured by the items which had been asserted to measure only one.