In industrial acceptance sampling one frequently makes use of operating characteristic curves to describe the discriminating power of a particular sampling plan. Similarly, it is possible to demonstrate the selective efficiency of a test battery in terms of (a) the Applicant's Operating Characteristic (A.O.C.); (b) the Selector's Operating Characteristic (S.O.C.). The A.O.C. determines the chance of selection by means of a test for any given level of true ability. The S.O.C. connects functionally probability of success on the criterion with the predictor scores of a battery. For the case of a normal bivariate distribution the exact mathematical expressions of the OC curves are derived in terms of the correlation coefficient ρ, the cut-off points α and β, and the predictor and criterion scores X and Y (in standard measures). The Efficiency Index H is defined as the percentage of successful subjects gained by the use of a test battery, taking chance selection as a yardstick for comparison. Its optimum, for fixed ρ and α, is derived. The distribution law of the criterion scores of selectees is deduced and its first four moments are shown to depart little from normality for cases usually encountered in practice. A “Quality-Gainae” diagram graphically illustrates the improvements secured. Another simple device, the “Cost-Utility” diagram, explains to management the full implications of selecting personnel by means of a test battery. Neither of the diagrams requires an understanding of the correlation coefficient. The confidence belt of the OC curves, the standard error of the mean criterion score of selectees and the standard error of the predicted number of successful applicants are determined. Finally, the full theory is applied in detail to a real test battery.

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.

In certain investigations involving an experimental group and a control group, the effect of the experimental treatment may be expected to manifest itself by increasing the scores of some subjects but by decreasing the scores of other subjects. Examples arise in the area of defensive behavior. Customary tests of differences between means are inappropriate for assessing the existence of effect of the experimental treatment in such cases. A test based on the ranks of the observations is proposed; it will be sensitive to extreme scores (either large or small or both) for experimentals.

Thurstone's multiple group method of factor analysis has been widely used as a basis for rotation to simple structure. To make the most of the economy offered by this method, factor axes may be located directly by the correlation clusters; and methods of doing this are here discussed.

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.

Free-recall verbal learning is analyzed in terms of a probability model. The general theory assumes that the probability of recalling a word on any trial is completely determined by the number of times the word has been recalled on previous trials. Three particular cases of this general theory are examined. In these three cases, specific restrictions are placed upon the relation between probability of recall and number of previous recalls. The application of these special cases to typical experimental data is illustrated. An interpretation of the model in terms of set theory is suggested but is not essential to the argument.

A rapid method is described for machine computation of biserial correlations in item analysis with several criteria. This method has been found to yield biserial correlations from punched IBM cards at the rate of about 41 per hour.

The average rating (or normalized rank) of a person rated by a larger number of judges will in general be closer to the group average than will the average rating of a person rated by a smaller number of judges, as a result of rating unreliability and regression. This note presents a technique for correcting that bias.

The personnel-classification problems considered in this paper are related to those studied by Brogden (2), Lord (6), and Thorndike (8). Section 1 gives an approach to personnel classification. A basic problem and variations of it are treated in section 2; and the computation of a solution is illustrated in section 3. Two extensions of the basic problem are presented in section 4. Most of the methods indicated for computing solutions are applications of the “simplex” method used in linear programming (see 1, Chs. XXII, XXIII). The capabilities of a high speed computer in regard to the simplex method are discussed briefly (see section 1).

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.

A factor analysis was made of 39 experimental printed aptitude tests and seven reference tests selected from the Army Air Forces Aircrew Classification Battery. Thirteen factors were extracted and two independent orthogonal rotational solutions were completed. Twelve factors were interpreted. Of these, seven were clearly identifiable with previously known factors: numerical, perceptual-speed, spatial-relations, visualization, visual-memory, paired-associates-memory, and length-estimation factors. A planning factor was not as clearly identifiable. A reasoning factor was probably a composite of two or more factors that failed to separate. A new factor possibly has to do with orientation with respect to the points of the compass. Two factors were doublets, each apparently specific to one kind of test. Better conceptions were gained of the spatial-relations and visualization factors and of the kinds of tests that measure them best. Efforts to improve measures of unique factors were not uniformly successful. The attempt to duplicate a psychomotor test rather directly by analogy in printed form failed almost completely.

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.

An equation is derived for predicting the effect of chance success, relative to item difficulty, on item-test correlation. The values predicted by this equation and by equations derived by Guilford and Carroll for predicting the effect of chance success on item difficulty and test reliability are compared with empirical values in an experiment which used identical test items in multiple-choice and answer-only form.

Twenty-five group tests, assembled with certain hypotheses concerning the first and second closure factors in mind, were administered to 154 subjects, mostly graduate students. The intercorrelations were analyzed factorially, yielding eight factors that were rotated to an oblique simple structure. The factors were interpreted as: speed of closure, C1; flexibility of closure, C2; verbal closure, C3; word fluency, W; reasoning, R; perceptual speed, P; the first space factor, S1; and speed of handwriting, H. Four second-order factors were tentatively described as analytical ability, synthetic ability, speed of perception, and word fluency. Three of the reasoning tests had their highest loadings on C2 and one on C3, which seems to be evidence that flexibility of closure generalizes in the cognitive domain and is associated with analytical ability.

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.

A method is derived for finding the average Spearman rank correlation coefficient of N sets of ranks with a single dependent or criterion ranking of n items without computing any of the individual coefficients. Procedures for calculating the exact distribution of ρav for small values of N and n are described for the null case. The first four moments about zero of this distribution are derived, and it is concluded that for samples as small as N = 4 and n = 4 the normal distribution can be used safely in testing the hypothesis ρav = 0.

In order to facilitate garment and pattern construction, a research which involved taking a minimum of fifty-five measurements on several thousand women was conducted by the Bureau of Home Economics. Partial correlations with age held constant were computed for a representative group of 4,128 of the women. The correlations among twenty-nine of these variables served as the basis of the present study. By a combination of the multiple-group and the centroid method of factoring, five factors were extracted. After twenty-nine rotations, simple structure was evident, and the factors were interpreted as bone length, size of joints, circumference below the waist, circumference of extremities, and circumference above the waist. The intercorrelations of the primaries were computed, and two second-order factors were extracted. One of these seems to be primarily related to the growth of fatty tissue and the other to the development of the bones.

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.

This paper provides the formulas necessary for testing the significance of the differences between mean values of different multivariate normal populations by Hotelling's generalization of “Student'” ratio. It also indicates the methods, proposed by Mahalanobis, and by Rao, of classifying different multivariate populations and individuals. The methods are illustrated by means of personality test data obtained from students preparing for the General Elementary and the General Secondary Teaching Credentials.