This is a very simple explanation of factor analysis primarily for the “factorial layman.” The interpretation of factors and the comparisons of various factor solutions are illustrated with a hypothetical example so designed that the reader can visualize all the relationships in a two-dimensional graph.

A type of interaction of two active groups is considered, in which the opposition of each group increases as the success of the other increases. Some possible applications of this situation are discussed.

The reliability coefficient is unlike other measures of correlation in that it is a quantitative statement of an act of judgment,—usually the test maker's, — that the things correlated are similar measures. Attempts to divorce it from this act of judgment are misdirected, just as would be an attempt to eliminate judgment of sameness of function of items when a test is originally drawn up. A “coefficient of cohesion,” entirely devoid of judgment, measuring the singleness of test function is proposed as an essential datum with reference to a test, but not as a substitute for the similar-form reliability co-efficient.

A comparison of the “test-retest” and “internal consistency” estimates of reliability coefficients is given, and it is shown that the two methods give different results. Application of the analysis of variance and covariance method reveals that there is not just one but a number of reliability coefficients involved, and that an estimate of each of these may be obtained. The analysis shows that in using the test-retest method the error or remainder effects are not independent on the two trials, possibly because the individuals remember the items and their responses to them on the previous trial.

The graphic item-counter is described and its use as a statistical device is explained. Procedures are presented for obtaining Pearson product-moment correlations by means of the graphic item-counter.

This paper is concerned particularly with certain regression effects which appear whenever matched groups are drawn from populations which differ with regard to the characteristics being studied. It is shown that regression will produce systematic differences between specifically matched. The size and direction of these differences depends upon the differences between the parent populations both in the matching and in the experimental variables and upon the correlation between the matching and experimental variables. Formulas are presented for estimating the expected regression effect. Several alternative procedures are suggested for avoiding the erroneous conclusions which the regression effect is likely to suggest.

A proposed criterion for the number of factors is developed on the basis of the similarity between a factorial residual and the partial correlation coefficient; something is known concerning the sampling error of the latter. Instead of computing the residuals as partials, a formula is presented for adjusting the standard deviation of the distribution of residuals so as to approximate the S.D. of the residuals as partial correlations. The criterion requires that factors be extracted until the adjusted S.D. reaches or falls below 1/√N. When tried out on six samples drawn from six universes of known factorial description, the criterion indicated the correct number of factors each time. The requisites of situations adequate for such empirical checks are discussed.

This paper relates the constant process used in psychophysics to the problem of item selection. Each test item may be described in terms of a limen, which is an index of the point at which an item discriminates, and the standard deviation of the limen, which is an index of the “goodness” of discrimination. The method developed may be related not only to the description of items but also to the description of persons. Thus a person's ability may be described in terms of a limen and its standard deviation.

In this note are presented facilitating tables for the estimation of the standard error of a tetrachoric r and also tables providing significant and very significant tetrachoric coefficients for various sizes of samples and various combinations of proportions in the dichotomized distributions.

In this article are presented a number of unusual achievement test exercises of both the essay and the objective types. These exercises may suggest to others engaged in the construction of achievement tests certain forms which they may find useful either as models or as points of departure in the invention of new forms. The article also calls attention to certain problems which must be solved if achievement testing is to have a sound, scientific basis.

A simple test of multiple serial response is proposed which makes use of materials that are inexpensive and easily prepared, and that do not get out of order. The test employs twenty-four colored blocks placed in two rows on a table. The total equipment needed consists of the blocks and a watch.

Previously derived equations for the control of the behavior of one social group by another are developed further. Cases of interaction of three social classes are studied, and the variation of such an interaction with time is investigated. A previously derived equation for the ratio of urban to rural population is generalized, and a theory of interaction of industrial and agricultural classes is outlined. Some of the theoretically derived equations are compared with available sociological data and found in fair agreement.

Several sets of learning data furnished by I. Krechevsky have been analyzed in terms of meaningful parameters of the learning curve, and the changes in the frequency distributions of these parameters with changes in the experimental conditions have been studied. One of the parameters represents the animal's initial preference for the light or dark, the other represents learning ability. The analysis shows that destruction of about ten or fifteen per cent. of the cortex, increases the animal's preference for the light and decreases the learning ability slightly. By ordinary methods of analysis, it is not possible to discover that both initial preference and learning ability have been changed by any given factor.

Tests of auditory function in the fields of pitch, loudness, quality (timbre), and time, auditory analysis, synthesis, and memory, together with age, intelligence, and four tests of visual memory, were studied factorially. The subjects were 200 high-school students. The intercorrelations were factored to nine factors by a modification of the centroid technique and rotated to an oblique simple structure. No general auditory factor appeared. Instead there appeared group factors tentatively identified as pitch-quality discrimination, loudness discrimination, “auditory integral for perceptual mass,” auditory resistance (synthesis and analysis), speed of closure, auditory span formation, memory span (auditory and visual), memory or incidental closure and an unidentifiable residual plane. The average intercorrelation among the primary vectors was low, only one intercorrelation being greater than .34. A number of queries are answered by the interpretation of the results.

Allport's J-curve hypothesis of conforming behavior and its attendant treatment of appropriate data are criticized on the following points: (1) narrowness of application; (2) flexibility of interpretation of results; (3) arbitrary selection of a criterion of conformity; (4) lack of a means by which the extent of conformity in one situation can be compared with that in another; (5) inequality of “telic” units. As an alternative treatment of such data, the method of higher moments is suggested and rationalized. Data from Allport and Solomon are reworked by this method and results compared.

The Lexis Ratio is discussed in its application to distributions of test scores where the items of the test can be assumed to be of equal difficulty. The ratio indicates the extent to which inter-individual variation operates as a source of the variance. The concept is related to the Lexis, Bernoulli, and Poisson distributions and illustrated by urn schemata. The Ratio is applied to the scores of 560 university freshmen on the Robinson Reading Test. The relation of the Lexis Ratio to the Kuder-Richardson estimation of reliability is also discussed and the latter authors' case IV is rewritten explicitly in terms of the Ratio.

General solutions of the analysis of variance in the case of unequal numbers of observations in the subclasses are presented. If we have k criteria for the classification, there will be k! solutions in making a complete analysis and 2k−1 answers, bearing different meanings, for the sum of squares between subclasses of each criterion. The sum of squares for the interaction of any order, however, will be identical in different solutions of same problem.

In this article the authors examine the effect of including alternate test forms in a factor matrix upon the validity of the resultant factor loadings, finding that in this particular instance the effect is negligible. Comparisons of the factor loadings derived from matrices in which only one of the alternate test forms is included with those in which both forms are included reveal practically no difference in the magnitude of either the original or rotated factor loadings, or in that of the computed communalities.

The purpose of this study is to investigate, by the method of paired comparisons, a possible scaling of individuals who have made certain test scores, such that the additive property will be satisfied and such that a stability in scaling will be maintained,—in other words, a scaling such that the scaled score of an individual will remain relatively the same regardless of the grouping of individuals in which he may be placed. The results show that it is possible to utilize psychophysical methods in psychological and educational test situations. Among the major findings are that Case V of the Law of Comparative Judgment is applicable to the data in this problem, the method of dividing the intermediate category equally between the greater and the less was the best of three possible methods, internal consistency was satisfied, and, finally, when a new test of stability was applied, it was found that the distances between the hypothetical individuals remain the same.