This paper presents a new method of determining the minimum rank in factor analysis, appropriate to the principal axes solution. The new method is compared with a former method which, with some adjustment, is more convenient for the centroid approach. Both methods are applied to two familiar examples.

In seeking a suitable topic for the address at this annual event of the psychometricians, I tried to keep in mind the motto of our official publication, Psychometrika, to promote “the development of psychology as a quantitative, rational science.” The best that I have been able to do falls short of this ideal in two important respects. Although the material that I am to bring before you pretends to be quantitative, in the sense that measurements have been made, the problem has not been rationalized and the data have thus far yielded only to graphical treatment.

The table of intercorrelations published by Dr. R. L. Thorndike in the Genetic Psychology Monographs, 1935, 17, No. 1, has been reanalyzed into ten factors. Rotation of the reference vectors resulted in a configuration which, for all practical purposes, may be said to show a simple structure and (except for one variable on one factor) a positive manifold. Five of the factors can be interpreted with some confidence, one only with considerable caution; three factors are specific to the apparatus employed, and for the one remaining factor, no interpretation is attempted; it appears to be a residual plane.

A stimulus (or stimulus-complex) is pictured as giving rise to a random series of sensory nerve “pulses,” which manifest themselves in contractions of individual muscle fibers. Assuming the expected time-frequency of these pulses to be proportional to the intensity of the stimulus, probability distributions are computed representing the cumulative effect of these pulses on the state of the organism, that is, on its degree of awareness of the stimulus. Preliminary results suggest a modification of the Weber-Fechner formula for intensity discrimination for certain types of stimuli: the psychological scale to be measured by I1/2 instead of log I.

The initial problem of factor analysis is described as a search for clustering of the test vectors. Curves are developed which give a visual picture of the clustering tendency, and an index of clustering is derived which provides a simple estimate for the number of factors.

An analysis by Thurstone's centroid method of the intercorrelations of fifty-two tests was carried to ten factors. Included were tests of social intelligence, Philip's attention tests, and Seashore's tests of musical ability. After rotation of axes, the most important factors appeared to pertain to operations conventionally alluded to by the following terms: verbal facility; spatial ability; numerical ability; attention; musical ability; and memory (or memory span). The social intelligence tests proved to be mainly tests of the verbal factor. A factorial sex difference was indicated by the superiority of men in tests of spatial ability.

Conditions are determined under which a single self-exciting neuron can reach a state of permanent excitation, in the case that the neuron develops both an excitatory and an inhibitory substance as well as in the case that it develops only a single, excitatory substance. In case both substances are developed it is possible to have periodic solutions, a possibility which does not arise when the circuit consists of neurons, or of a neuron, developing only an excitatory substance.

A technique for determining a set of weights for the linear combination of a number of measures is applied to a concrete problem. The set of weights meets the criterion of maximum separation of the total scores of two different occupational groups.

Correlation coefficients derived from an hypothetical simple structure for twenty tests and four factors were “loaded” with chance error components. Centroid analyses and rotations to give least square determinations of the hypothetical simple structure were made for several conditions. It is concluded that the experimental situation of inaccurate coefficients and estimated communalities permits accurate determination of primary trait loadings provided that the rank of the centroid matrix is equal to or greater than that of the underlying primary trait matrix. Of several criteria for the completeness of factorization which were tested, none was wholly satisfactory.

From the original Stanford-Binet scale, those items passed by between 10 and 90 per cent of a group of ten-year-old children were analyzed by the centroid method. Upon rotation, there appeared a common factor, for which two explanatory hypotheses are offered, the more tenable being that it is an effect of maturation. Primary factors tentatively identified are Number, Space, Imagery, Verbal Relations and Induction. A sixth factor apparently involves a reasoning ability and a seventh can not be interpreted.

A shortened method of finding the regression for estimation of the scores on mental factors is presented. If the correlations among n tests are accounted for by r factors, the regression equation involves the inverse of an r × r matrix instead of the inverse of an n × n matrix. When r is much less than n, there is considerable saving in computational labor. A numerical example is presented to illustrate the method.

The article discusses the general principles involved in factorizing correlations between persons, and compares the special techniques put forward by Burt and Stephenson respectively. The chief points of agreement and disagreement are summarized. In statistical procedure the two writers prove to be largely in accord; but they differ over the psychological and the methodological implications, the chief source of divergence being the so-called “reciprocity principle.”

In continuation of previous studies, an abstract mathematical theory is developed for the interaction of several social classes, of which one influences and controls the behavior of the others. In some cases such interaction results in the existence of two configurations of equilibrium for the social structure, characterized by different types of behavior. Each configuration corresponds to one definite type of behavior. The transition from one configuration to the other, in other words, the transition from one behavior to another, occurs rather rapidly. Equations governing these transitions are given. In other cases, namely when the efforts of the individuals to influence others into a given behavior lessen as the success of this influence increases, there is only one stable configuration characterized by a mixed behavior. Due to the dissimilarity of parents and progeny, the composition of each social class changes with time, as generations change. This results in the appearance of instabilities of the social structure and in relatively sudden social changes. Possibilities of quasi-periodic alterations of different social structures are discussed. Finally, the developed equations are applied to the case of physical conflicts between groups of individuals, such as riots, wars, etc. Possible factors in addition to mere physical force which may determine the outcome of such conflicts are investigated.

In continuation of previous studies, relations are studied between two classes of population, of which one is characterized by a much greater ability to organize and supervise the productive activities of the other. Under some special and rather simple assumptions, it is shown that this kind of interreaction results first in an increase of the ratio of the cumulative results of productive activities for the two classes in favor of the first. With time, however, this ratio reaches a maximum and declines. An expression for the “life span” of such organized classes is obtained.

In the second section, a study is made of possible non-uniformities of spatial distribution of the population. Some possible applications are made to the theory of the variation of the ratio of urban to rural population, showing how that ratio may increase with increasing total population.

A mechanism for localizing different stimulus intensities within the nervous system, essentially as described by Rashevsky, is considered in first approximation. The mechanism is central, and therefore general. There results a one-parameter family of theoretical curves, graphs of the Weber (discriminal) ratio against stimulus intensity. Comparisons are made with experimental visual, acoustical, and tactual data.

The results of J. P. Guilford's card-sorting experiment by the method of equal appearing intervals are analyzed mathematically. The constants of the psychometric functions are obtained by processes analogous to the constant process. When these constants are determined (a) independent of the supposition that the series of groups into which the cards were sorted is quantitative in character and (b) under this supposition, good agreement is found between them. Guilford's results agree with Weber's and Fechner's laws in both cases.

Certain assumptions and procedures basic to factor analysis are examined from the point of view of the mathematician. It is demonstrated that the Hotelling method does not yield meaningful traits, and an example from the theory of gas mixtures with convertible components is cited as evidence. The justification of current methods for determining the adequacy of the reproduction of a correlation matrix by a factorial matrix is questioned, and a x2 criterion, practical only for a small matrix, is proposed. By means of a hypothetical example from geometry, it is shown that results of a Hotelling analysis are necessarily relative to the population at hand. The factorial effects of the adjunction of a “total test” to a group of tests are considered. Some of the general considerations and questions raised are pertinent to types of analysis other than the Hotelling.

In a study of color preferences, Winch, using the order of merit technique, adopted as a measure of group preference an index equivalent to the average rank assigned by N individuals. He awarded to the specimen in first place in each array a value of 1, to second place specimens a value of 2, and so on, summed the values awarded to each item for the N arrays and used their relative magnitudes as a measure of group preference. These values divided by N would give the average rank of the specimens.

In the computation of scale values from ranked items by the method developed by Thurstone, the first step is the determination of the frequency with which each item is preferred to every other for the N arrays. In this method it is assumed that ranked data yield comparative judgments and that they can be treated in the same manner as data from paired comparisons. The first specimen in a ranked order is considered preferred to all specimens following it, and similarly for other specimens.

With the mechanism of common elements we designed numerous sets of variates correlated with each other in a known manner and also correlated with the primary and specific factors in the same predetermined fashion. To the correlations from theoretical populations, and also from experimental samples, Thurstone's centroid method of factoring was applied. The resulting centroid co-ordinates were rotated to yield the test vectors. These vectors were close approximations to the theoretical and sample correlations.