The validity of a univocal multiple-choice test is determined for varying distributions of item difficulty and varying degrees of item precision. Validity is a function of σd2 + σy2, where σd measures item unreliability and σy measures the spread of item difficulties. When this variance is very small, validity is high for one optimum cutting score, but the test gives relatively little valid information for other cutting scores. As this variance increases, eta increases up to a certain point, and then begins to decrease. Screening validity at the optimum cutting score declines as this variance increases, but the test becomes much more flexible, maintaining the same validity for a wide range of cutting scores. For items of the type ordinarily used in psychological tests, the test with uniform item difficulty gives greater over-all validity, and superior validity for most cutting scores, compared to a test with a range of item difficulties. When a multiple-choice test is intended to reject the poorest F per cent of the men tested, items should on the average be located at or above the threshold for men whose true ability is at the Fth percentile.

Finite Markov processes are reviewed and considered for their usefulness in the description of behavioral data. The various alternative responses in an experimental situation define a vector space, and changes in the probabilities of these alternatives are represented by movements in this space. Methods of fitting the theory to experimental data are considered.

The simplest process, with a constant matrix of transitional probabilities that is applied repeatedly to represent the effect of successive trials, seems inadequate for most learning data. A matrix function that may be useful for learning theory is presented.

The method of successive intervals is a psychological scaling procedure in which stimuli are classified into successive intervals according to the degree of some defined attribute which they are judged to possess. A psychological continuum is defined and the scale values are then taken as the medians of the distributions of judgments on the psychological continuum. It is assumed that the distributions of judgments for each stimulus are normal on the psychological continuum as defined.

An internal consistency check indicates that the cumulative distributions of empirical judgments for the various stimuli can be reproduced by means of a limited number of parameters with an average error that compares favorably with that usually reported for paired comparison data. Furthermore, the scale values obtained by successive interval scaling, for the data reported, are shown to be linearly related to those obtained by the method of paired comparisons.

Under certain assumptions an expression, in terms of item difficulties and intercorrelations, is derived for the curvilinear correlation of test score on the “ability underlying the test,” this ability being defined as the common factor of the item tetrachoric intercorrelations corrected for guessing. It is shown that this curvilinear correlation is equal to the square root of the test reliability. Numerical values for these curvilinear correlations are presented for a number of hypothetical tests, defined in terms of their item parameters. These numerical results indicate that the reliability and the curvilinear correlation will be maximized by (1) minimizing the variability of item difficulty and (2) making the level of item difficulty somewhat easier than the halfway point between a chance percentage of correct answers and 100 per cent correct answers.

The use of a matrix to represent a relationship between the members of a group is well known in sociometry. If this matrix is raised to a certain power, the elements appearing give the total number of connecting paths between each pair of members. In general, some of these paths will be redundant. Methods of finding the number of such redundant paths have been developed for three- and four-step chains by Luce and Perry (3) and Katz (2), respectively. We have derived formulas for the number of redundant paths of five and six steps; and in addition, an algorithm for determining the number of redundant paths of any given length.

In a previous paper (1) were developed three basic theorems which were shown to provide numerical routines, as well as algebraic proof, for existing common-factor methods. New “multiple” routines were also indicated. The first theorem showed how to extract as many common factors as one wished from the correlation matrix in one operation. The second theorem showed how to do the same from the score matrix. The third proved that factoring the correlation matrix was equivalent to factoring the score matrix. A particular application of these theorems is the multiple group factoring method, which the writer first used in practice on some Army attitude scores during World War II. The present paper explains the basic theorems in more detail with special reference to group factoring. Computations are outlined as consisting of five simple matric operations. The meaning of commonfactor analysis is given in terms of the basic theorems, as well as the relationship to “inverted” factor theory.

A technique is outlined which may facilitate the rotation of factor axes to a meaningful position. It is based on certain relationships between the results of test and person factor analysis, and consists essentially of supplementing the test factor space with tests which are the test-equivalents of persons or groups of persons. These persons may be, for instance, well-known “types” in the domain being investigated, or even “freaks.” The ways in which these persons may be selected and used to determine the final rotated position of the factor axes is discussed.

The wiring of the plugboard of the IBM type 405 machine for the computation of sums of squares and cross products of positive and negative numbers is described. The method makes greater demands on the X distributor and class selector capacity of the machine than does the method of wiring the plugboard when the numbers all have the same sign.