The latent structure model considered here postulates that a population of individuals can be divided into m classes such that each class is “homogeneous” in the sense that for the individuals in the class the responses to N dichotomous items or questions are statistically independent. A method is given for deducing the proportions of the population in each latent class and the probabilities of positive responses to each item for individuals in each class from knowledge of the probabilities of positive responses for individuals from the population as a whole. For estimation of the latent parameters on the basis of a sample, it is proposed that the same method of analysis be applied to the observed data. The method has the advantages of avoiding implicitly defined and unobservable quantities, and of using relatively simple computational procedures of conventional matrix algebra, but it has the disadvantages of using only a part of the available information and of using that part asymmetrically.

One of the most difficult parts of multiple-factor analysis for the student to learn is the rotation of the coordinate axes to simple structure. The complete graphical methods have been the most dependable. The work of interpreting these diagrams is not only tedious but it requires the development of a skill that many otherwise competent students are unable to master. The writer has tried many times to develop an analytical method which would eliminate the graphical methods of successive approximation. The method to be described here seems at last to be successful and practically feasible. It is equally applicable to unimodal and bimodal hyperplanes so that no restrictive assumptions need to be made about the positive or negative signs of the factorial components.

The personnel classification problem is identified mathematically with other problems in the social and biological sciences. This mathematical problem is shown to be a special case of the general mathematical problem of linear programming. It is proposed here that the personnel classification problem may be solved directly by methods particularly appropriate to it as well as by the simplex method, which is a standard method for solving the general linear programming problem. The method of optimal regions is derived and illustrated in this paper.

A multivariate analysis based on transmitted information is presented. It is shown that sample transmitted information provides a simple method for measuring and testing association in multi-dimensional contingency tables. Relations with analysis of variance are pointed out, and statistical tests are described.

Latent distance analysis provides a probability model for the non-perfect Guttman scale; the restricted latent distance structure is simpler to compute than the general structure. Since no sampling theory for latent structure analysis is available, the advantages of the general structure cannot be expressed formally. The two structures are compared in terms of their fit to fifteen sets of empirical data. The computation schemes used are summarized.

A generalized method for collecting data on interstimulus similarity is presented and its special cases evaluated by use of information theory. A method of analyzing the data by the Unfolding Technique is presented which permits the study of the latent attribute structure underlying the similarity of stimuli for a single individual.

The concept of parsimony in factor analysis is discussed. Arguments are advanced to show that this concept bears an analogic relationship to entropy in statistical mechanics and information in communication theory. A formal explication of the term parsimony is proposed which suggests approaches to the final resolution of the rotational problem. This paper provides the rationale underlying Carroll's (2) analytical solution for approximating simple structure, and the solutions of Saunders (7) and Neuhaus and Wrigley (5).

In descriptive factor analysis, the factor matrix is merely an alternate mode of expression of the meaning expressed by the correlation matrix. In inferential factor analysis, the factor matrix expresses material inferred from the correlation matrix but not contained in the correlation matrix. A major problem has been whether or not the latter type is possible. Granit's color cone study is interpreted as a factor study in the hope of showing that inferential factor analysis actually exists. It is argued that proof of these and other points in factor theory can be accomplished only by logical analysis.

It is first demonstrated that Aitken's selection formulas are equivalent to a linear transformation in the factor space. On this basis the Thomson-Ledermann theorem concerning the invariance of the number of common factors under selection, and a theorem concerning the invariance of factor loadings under selection are derived. A mathematical proof of the results of Thurstone, which are concerned with the invariance of simple structure under selection, is given. The paper provides a conclusive answer to the question, considered by Thurstone and Thomson, whether a multivariate selection is always reducible to successive univariate selections.

A least squares solution has been presented for the Law of Categorical Judgment and the Method of Successive Intervals, which is formally equivalent to Horst's solution for the matrix of incomplete data. A simplified approximation which is recommended for use with any matrix of complete data is also given. A procedure, analogous to that originally devised by Thurstone and presented by Saffir, is outlined for the typical experimental matrix of incomplete data. It is essentially a “point and slope” method of fitting a straight line which comes rather directly from the least squares solution.

A simple algebraic proof of a theorem defining the optimal solution to the personnel classification problem is given. If a set of constants, one for each job, are known, the theorem indicates that each individual should be classified by adding the constants to the estimates of the individual's productivity in the several jobs and selecting the job for which the resulting sum is highest.

The square root method for the solution of a set of simultaneous linear equations or the reduction of a matrix has been known for some time under a variety of names. Because of its usefulness in statistical work, especially in factor analysis, the square root method is presented in general terms and an example given. Several independently developed “multiple group methods” for factor analysis are compared and synthesized. Their fundamental concepts are set forth and an appropriate system of notation developed. Detailed computational procedures are outlined, and the square root method is emphasized as a computing aid in multiple group analysis.

A rationale and a procedure for constructing questionnaire keys so as to minimize the effect of distortion or faking by the respondents is developed. This rationale is based on the supposition that suppressor items can be identified to reduce distortion in, and thus add to, the validity of questionnaire keys. The procedure is designed primarily for application to the construction of forced choice items.

The results of an empirical test of the efficicacy of the forced choice pairing procedure developed as a consequence of this rationale are presented. A key based on forced choice pairs developed by this procedure gave a validity of .33; a second key based on forced choice pairs developed by more conventional procedures gave a validity of .23.

A formula is derived which gives the maximum expected correlation between two multiple-choice tests as a function of the distributions of proportions correct for the items in the two tests and the probability of chance success. The formula is similar to one derived by Carroll based on “true” item difficulties. A numerical example is provided.

Fifty-three tests designed to measure aspects of creative thinking were administered to 410 air cadets and student officers. The scores were intercorrelated and 16 factors were extracted. Orthogonal rotations resulted in 14 identifiable factors, a doublet, and a residual. Nine previously identified factors were: verbal comprehension, numerical facility, perceptual speed, visualization, general reasoning, word fluency, associational fluency, ideational fluency, and a factor combining Thurstone's closure I and II. Five new factors were identified as originality, redefinition, adaptive flexibility, spontaneous flexibility, and sensitivity to problems.

A solution for Case III of the Law of Comparative Judgment, modeled after Thurstone's solution for Case IV but eliminating the restrictive assumption of relatively equal discriminal dispersions, is developed.

This paper reports on the third of a series of four experiments using similar concepts and methods for objective personality measurement, and overlapping test batteries. One hundred students were measured with 115 tests. The scores were correlated and factored by a re-iterated multiple group centroid method. The 17 factors thus obtained were rotated toward a clear simple structure. The relation of the rotated factors to earlier ones is indicated, but no extensive interpretation is attempted.

Let R be any correlation matrix of order n, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrix U2 such that G = R − U2 is Gramian and of minimum rank r. Let s1 be the number of latent roots of R which are greater than or equal to unity. Then it is proved here that r ≥ s1. Two further lower bounds to r are also established that are better than s1. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices in R is small, but the minimum rank r for a Gramian G is nevertheless very large compared with n. Heuristic criteria are given for testing the hypothesis that a finite r exists for the infinite universe of content from which the sample of n observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.