In much research in social psychology it is impractical to get quantitative measure of the degree of effectiveness of certain social behaviors, yet associates can sense that effectiveness sufficiently well to detect those who manifest the behavior in very high or in very low degree. This paper develops a technique of biserial correlation from wide-spread classes to deal statistically with such situations, develops standard error formulas for it, and points to a wide range of usefulness for this type of technique.

A factor analysis of the ten sub-tests of the Seashore test of pitch discrimination revealed that more than one ability is involved. One factor, which accounted for the greater share of the variances, had loadings that decreased systematically with increasing difficulty. A second factor had strongest loadings among the more difficult items, particularly those with frequency differences of 2 to 5 cycles per second. A third had strongest loadings at differences of 5 to 12 cycles per second. No explanation for the three factors is apparent, but the hypothesis is accepted that they represent distinct abilities. In tests so homogeneous as to content and form, where a single common factor might well have been expected, the appearance of additional common factors emphasizes the importance of considering the difficulty level of test items, both in the attempt to interpret new factors and in the practice of testing. The same kind of item may measure different abilities according as it is easy or difficult for the individuals to whom it is applied.

The results of three empirical studies on the sampling fluctuation of centroid factor loadings are reported. The first study is based on data which happened to be available on 8 variables for 700 cases and which were factored to three factors for subsamples. The second study is based on fictitious data for 2500 cases which provided separate analyses on 25 samples for each of three situations: 5 variables, one factor; 5 variables, two factors; and 6 variables, three factors. The third study, based on real data for 9 variables and 7000 cases, involves separate factorization for 35 samples of 200 cases. The three studies agree in showing that the sampling behavior of first centroid factor loadings is much like that of correlation coefficients, whereas the sampling fluctuations for loadings beyond the first are disturbingly large.

A multiple-factor analysis was made of a battery of 42 tests of verbal abilities administered to 119 college adults. Where necessary, the distributions of test scores were normalized before the inter-test correlations were computed. Thurstone's M (Memory or Rote Learning) factor has been confirmed, but his V (Verbal Relations) factor seems to have been split into two or possibly three factors, C, J, and G; and his W (Word Fluency) factor has been split into two factors, A and E. The C factor seems to represent the richness of the individual's stock of linguistic responses, and the J factor seems to involve the ability to handle semantic relationships. No satisfactory interpretation can as yet be made of the G factor. The A factor seems to correspond to the speed of association for common words where there is a high degree of restriction as to appropriate responses. The E factor is described as an associational facility with verbal material where the only restriction is that the responses must be syntactically coherent. The new factors are: F, facility and fluency in oral speech; H, facility in attaching appropriate names or symbols to stimuli; and D, speed of articulatory movements.

Bayes' theorem of inverse probability is made the basis of a general equation for scoring objective examinations. The equation so obtained is evaluated by assuming a binomial distribution of examinee knowledge and guessing tendency. A graphical illustration of the application of this equation to a hypothetical test situation is presented. The limitations inherent in the use of Bayes' theorem make it inadvisable to recommend the practical use of the equation unless future experimental evidence indicates an increase in scoring validity which more than compensates for the increase in scoring difficulty.

This article describes the application of three psychometric methods to the problem of measurement of the friendly or hostile relations among states of the world today. To secure judgments, schedules were sent to students of international affairs at several times during the last five years. The method of equal-appearing intervals was used to determine the relative probability of war for 88 pairs of states in January, 1937; the method of “triadic combinations” to determine relative friendliness among the Great Powers in November, 1938; and the method of “multidimensional” or group rank order to measure the attitudes of important states toward the Great Powers in March and April, 1939; June, 1940; and June, 1941. A chart of scale values for the pairs of Great Powers shows the changing trends since 1937. The last two methods were used to depict the Great Powers in multidimensional space according to their mutual friendliness, thus permitting the application of a type of factor analysis. The reliability of the methods employed was high, and various types of evidence support the general validity of the results.

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student “t” and the Fisher “z” tests to a wider range of problems.

In a recent number of this Journal (June, 1941), Professor Guilford has interestingly called into question the assertion of several authors that Thurstone’s method of analysis (by maximal frequency of zero loadings).

To settle the point, Guilford has submitted to the said analysis a fictitious factor matrix which was expressly devised to contain g. The result was that this factor — in opposition to the assertion mentioned above — did duly make its appearance. Thereupon he fairly enough proceeds to conclude that:

Not so acceptable, however, is his wide generalization that:

The most commonly used method of factoring a matrix of intercorrelations is the centroid method developed by L. L. Thurstone. It is, however, necessary to transform the centroid matrix of factor loadings into a simple structure matrix in order to facilitate the interpretation of the factor loadings. Current methods for effecting this transformation are chiefly graphical and require considerable experience and personal judgment. This paper presents a new method for transforming an arbitrary factor matrix into a simple structure matrix by methods almost completely objective. The theory underlying the method is developed and approximation procedures are derived. The method is applied to a matrix of factor loadings previously analyzed by Thurstone.

Two new methods of item analysis are described. One involves the computation of the ϕ coefficient (correlation of a fourfold point distribution) and the other involves chi square. The only data required are the proportions of passing individuals in the upper and lower criterion groups, for the determination of ϕ, and in addition, N, for the determination of chi square. Abacs are presented for graphic solution of the two indices of validity, and tests of significance are provided.

In this paper it is demonstrated that the analysis of variance techniques yield results equivalent to the calculation of t by means of expressions based on the short or the long formula. It is also shown that the covariance technique gives results equivalent to those obtained by means of the formula for t which should be used with matched groups. The conditions necessary for equivalent results are such that the conventional formulas for t would normally be used rather than the variance or covariance techniques. However, a knowledge of the relationships described in this paper should contribute to one's understanding of the variance and covariance techniques.

A formula for estimating the reliability of a test, based on the analysis of variance theory, is developed and illustrated. The data needed for the required computation are the number of correct responses to each item and the score for each subject. The results obtained from this formula are identical with those from one of the special cases of the Kuder-Richardson formulation. The relationships of the new procedure to other approaches to the problem are indicated.

Adaptations of the micro-film projection method to the administration of both individual and group psychological tests are described, together with newly designed auxiliary equipment. The author cites examples of the application of this technique, which can be adapted to a variety of testing situations and which seems to be of unusual promise because of the attention value of visual projection.

A previous mathematical study of a situation, in which the behavior of a larger group of individuals is controlled by a smaller group, is generalized for the case when the “activity” of the individuals in the group is continuously graded. The existence of two possible social configurations and of sudden transitions from one configuration to another are found in this case also.

In order to investigate certain hypotheses concerning the nature of number ability, and, secondarily, the nature of perceptual speed, a battery of thirty-four tests was given to 223 Chicago high school seniors and the data were factored by the centroid method. Seven primary factors were identifiable upon rotation. Several deductions are made relative to the interpretation of the factors and relative to the consistency of the data with the hypotheses which were to be tested.

This presentation deals with the evaluation and transformation of linear forms. Especial emphasis is given to implicit methods in which it is not necessary to find the explicit values, xi. The relation of the Aitken triple product matrix CA−1B to the result of a linear transformation of linear forms is noted, and the numerical computation of this triple product matrix is indicated with the use of the simple Abbreviated Doolittle solution. Application is also made to the evaluation of A−1 and of A−1C.

This paper attempts to show in a non-mathematical way the influence of grouping on standard deviations and correlations, and advances empirical evidence to illustrate with what accuracy values corrected for grouping by Sheppard's correction approximate to values obtained from ungrouped data when the distributions are continuous. This enquiry gained its initial stimulus from the observation that many standard deviations and correlations reported by students of psychology and education are uncorrected for grouping and that frequently errors attributable to the grouping of data are not small when compared with errors of sampling.

This paper is an attempt to integrate the various methods which have been developed for the numerical solution of simultaneous linear equations. It is demonstrated that many of the common methods, including the Doolittle Method, are variations of the method of “single division.” The most useful variation of this method, in case symmetry is present, appears to be the Abbreviated Doolittle method. The method of multiplication and subtraction likewise can be abbreviated in various ways of which the most satisfactory form appears to be the new Compact method. These methods are then applied to such problems as the solution of related equations, the solution of groups of equations, and the evaluation of the inverse of a matrix.

The general problem of estimating correlated or uncorrelated factors is treated. It is specifically indicated wherein the prediction of oblique factors differs from that of orthogonal factors. A shortened method of estimation of correlated factors is developed.

It is shown that the L-method is basic to test-building and to all combining of scores where the several sub-parts of a composite are weighted according to their respective standard deviations, i.e., with “equal gross score weights” or by “adding the several sub-scores.” With chiefly a listing adding machine, a few celluloid strips, and a master matrix table of L's, one may, in a fraction of the time and with equally good or even better practical results, easily duplicate, with test-building material most of the feats obtainable by multiple regression equations.