A procedure is derived for obtaining an orthogonal transformation which most nearly transforms one given matrix into another given matrix, according to some least-squares criterion of fit. From this procedure, three analytic methods are derived for obtaining an orthogonal factor matrix which closely approximates a given oblique factor matrix. The case is considered of approximating a specified subset of oblique vectors by orthogonal vectors.

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 solution is developed in implicit form for the problem of assigning N men to n jobs, the proportion of men to be assigned to each job being specified in advance.

For multiple-choice tests where no a priori key exists, the initial selection of a key for maximum validity may be made on the basis of the number of persons choosing each alternative and their mean criterion score. The keying formula is derived. Once the initial keying has been done, further precision in keying and item selection may use, in addition, the mean total test score for persons choosing each alternative. Item-selection formulas suggested by Horst and by Gulliksen for maximizing test validity are both in the form of a ratio, an “item-validity index” divided by an “item-reliability index.” The formula derived here is shown to be equivalent to the numerators of these formulas. The expression in the denominators uses the total test score. Although a radical appears in the denominator of Horst's formula and not in the denominator of Gulliksen's formula, both of them select the same items in practice.

The rolling totals method of computing sums, sums of squares, and sums of cross-products appears to have several advantages over usual methods, in that it saves time and requires less equipment. Only an IBM sorter and a tabulator equipped with Card Cycle Total Transfer Device are used. The method provides an immediate independent visual check on the accuracy of the sums of cross-products of each successive variable as it is run. Since controlling is done by sorting, the necessity for re-wiring after each run on the tabulator is eliminated. The wiring, illustrated by a diagram, is simple and straightforward.

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 theory is advanced that personality factors obtained in the first order may often represent combinations of temperament traits that occur in the experimental population. Under these circumstances an investigation of the second order represents a purification process and yields factors which represent the more basic or pervasive characteristics of the original behavior items included in the factorial study. These second-order factors can be obtained directly in the first order by a careful selection of the variables which enter into the analysis. A second-order analysis was undertaken of the nine factors inherent in three of J. P. Guilford's inventories, and four clearly interpretable second-order factors were obtained. Three of these factors were obtained directly in the first order in a new factorial study of twenty-two behavior items. Attention is drawn to the similarities between these factors and traits of temperament postulated by an independent investigator.

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.

A table of values of Chi-square for two degrees of freedom corresponding to values of P from .001 to .999 is presented, together with a description and an example of its use in combining probabilities from two or more independent samples to obtain an aggregate probability.

A quick approximation of the best-fitting orthogonal simple structure from the known oblique simple structure is here developed.

The purpose of this article is to present a set of nomographs for testing the significance of the difference between uncorrelated percentages, and for determining necessary sample sizes when planning studies which will involve the comparison of percentages. Examples of these two uses are given as well as a description of the formula upon which the nomographs are based.

Test battery selection is justified on the grounds that it is more efficient than the conventional methods. More precise definitions of efficiency have shown that the additional cost of test administration can often outweigh the gains of selecting a more competent group of persons. Final evaluation of the profit and loss account can be made only when no further economies can be effected in the selection procedures themselves. A psychometric pre-screening technique which will reduce this cost considerably is developed here. Its psychometric structure is mathematically defined, and its advantages illustrated in terms of testing economy (E) and losses (L) of good material for different selection ratios (S.R.). It is shown that, under many circumstances, the testing economy can be considerable, and that in all practical cases “Losses” will be ≤ 1.5% for all values of ρ ≥ .8 (between the pre-screening test and the final battery). The technique in practice is illustrated from a set of data.

Scores on five differently speeded parts of a nonverbal reasoning test were analyzed to determine the relationship between the abilities measured by the speeded and the unspeeded parts. Two orthogonal factors were found: One is described as the ability to answer the problems correctly; the other is described as the tendency to answer the problems quickly. The first factor was found to be somewhat more valid than the second for predicting grades at the U. S. Naval Academy. The suggestion is made that descriptions of tests are incomplete unless they include some specification describing the speededness of the test.