Silvan Tomldns [1962] once remarked that it seemed to him that human personality was “organized as a language is organized, with elements of varying degrees of complexity–from letters, words, phrases, and sentences to styles–and with a set of rules of combination which enable the generation of both endless novelty and the very high order of redundancy which we call style.” He went on to note that “if we had to be blind about one or the other of these types of components, we should sacrifice the elements for the rules,” although “factor analysis appears to have made the opposite decision. It would tell what letters, or words, or phrases, or even styles were invariant and characteristic of a personality or of a number of personalities,” but by itself it does not and cannot “generate the rules of combination which together with the elements constitute personality” [Tomkins, 1962, p. 287].

This paper is devoted to the study of a certain statistic, u, defined on samples from a bivariate population with variances σ11, σ22 and correlation ρ. Let the parameter corresponding to u be υ. Under binormality assumptions the following is demonstrated. (i) If σ11 = σ22, then the distribution of u can be obtained rapidly from the F distribution. Statistical inferences about ρ = υ may be based on F. (ii) In the general case, allowing for σ11 ≠ σ22, a certain quantity involving u, r (sample correlation between the variables) and υ follows a t distribution. Statistical inferences about υ may be based on t. (iii) In the general case a quantity t′ may be constructed which involves only the statistic u and only the parameter υ. If treated like a t distributed magnitude, t′ admits conservative statistical inferences. (iv) The F distributed quantity mentioned in (i) is equivalent to a certain t distributed quantity as follows from an appropriate transformation of the variable. (v) Three test statistics are given which can be utilized in making statistical inferences about ρ = υ in the case σ11 = σ22. A comparison of expected lengths of confidence intervals for ρ obtained from the three test statistics is made. (vi) The use of the formulas derived is illustrated by means of an application to coefficient alpha.

A combination is achieved of two lines of psychometric interest: a) multidimensional scaling and b) factor analysis. This is accomplished with the use of three-mode factor analysis of scalar product matrices, one for each subject. Two of the modes are the groups of objects scaled and the third mode is the sample of subjects. Results are an object space, a person space, and a system for changing weights given to dimensions and of angles between dimensions in the object space for individuals located at different places in the person space. The development is illustrated with data from an adjective similarity study.

A model of individual differences in pair comparisons is developed, assuming the affective values to have a quasi-simplex structure. Procedures for fitting the model to data are suggested. An empirical example is given for one of the procedures.

Aitken’s generalized least squares (GLS) principle, with the inverse of the observed variance-covariance matrix as a weight matrix, is applied to estimate the factor analysis model in the exploratory (unrestricted) case. It is shown that the GLS estimates are scale free and asymptotically efficient. The estimates are computed by a rapidly converging Newton-Raphson procedure. A new technique is used to deal with Heywood cases effectively.

A multivariate logistic latent trait model for items scored in two or more nominal categories is proposed. Statistical methods based on the model provide 1) estimation of two item parameters for each response alternative of each multiple choice item and 2) recovery of information from “wrong” responses when estimating latent ability. An application to a large sample of data for twenty vocabulary items shows excellent fit of the model according to a chi-square criterion. Item and test information curves are compared for estimation of ability assuming multiple category and dichotomous scoring of these items. Multiple scoring proves substantially more precise for subjects of less than median ability, and about equally precise for subjects above the median.

A metric multidimensional scaling (MDS) procedure based on computer-subject interaction is developed, and an experiment designed to validate the procedure is presented. The interactive MDS system allows generalization of current MDS systems in two directions: (a) very large numbers of stimuli may be scaled; and (b) the scaling is performed with individual subjects, facilitating the investigation of individual as well as group processes. The experiment provided positive support for the interactive MDS system. Specifically, (a) individual data are amenable to meaningful interpretation, and they provide a tentative basis for quantitative investigation; and (b) grouped data provide meaningful interpretive and quantitative results which are equivalent to results from standard paired-comparisons methods.

Experiments in short-term memory usually collect binary data and pool the results of several subjects. The observed pooled proportion of recall is then transformed by the angular transformation to stabilize the variance. In this paper the variance of the transformed proportion is derived and a simple test statistic is obtained to check the validity of the binomial model. Some numerical comparisons for selected poolings are made.

Considerable attention has been given in the psychological literature to techniques of data reduction that partition a set of objects into optimally homogeneous groups. This paper is an attempt to extend the hierarchical partitioning algorithms proposed by Johnson and to emphasize a general connection between these clustering procedures and the mathematical theory of lattices. A goodness-of-fit statistic is first proposed that is invariant under monotone increasing transformations of the basic similarity matrix. This statistic is then applied to three illustrative hierarchical clusterings: two obtained by the Johnson algorithms and one obtained by an algorithm that produces the same chain under hypermonotone increasing transformations of the similarity measures.

A local independence latent structure model, which assumes m latent classes, requires a minimum of 2m-1 items for the solution of the 2m2 latent parameters. If one adds 3 items to the test and if one assumes local “dependence” between pairs of items, thereby adding\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {\begin{array}{*{20}c} {2m + 2}\\ 2\\ \end{array} } \right)$$\end{document} additional latent parameters, ξij, representing the association between items i and j, then it is possible to obtain estimates for all of the latent parameters: latent class frequencies latent probabilities, and measures of association between pairs of items. The solution consists of (1) forming (m + 1) × (m + 1) matrices of manifest data, which are singular, (2) solving for the ξij in equations that result from the singularity of the data matrices, (3) “correcting” the manifest data by removing the “contamination” due to local dependence, and (4) estimating the remaining latent parameters from the corrected data, using methods outlined in earlier literature.

A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures.

In one well-known model for psychological distances, objects such as stimuli are placed in a hierarchy of clusters like a phylogenetic tree; in another common model, objects are represented as points in a multidimensional Euclidean space. These models are shown theoretically to be mutually exclusive and exhaustive in the following sense. The distances among a set of n objects will be strictly monotonically related either to the distances in a hierarchical clustering system, or else to the distances in a Euclidean space of less than n − 1 dimensions, but not to both. Consequently, a lower bound on the number of Euclidean dimensions necessary to represent a set of objects is one less than the size of the largest subset of objects whose distances satisfy the ultrametric inequality, which characterizes the hierarchical model.

Special cases of the factor analysis model are developed for four selection situations. Methods are suggested whereby parameters in each case can be estimated using a maximum likelihood procedure recently developed by Jöreskog. Also, a numerical illustration is presented for each case.

Some relations between maximum likelihood factor analysis and factor indeterminacy are discussed. Bounds are derived for the minimum average correlation between equivalent sets of correlated factors which depend on the latent roots of the factor intercorrelation matrix Ψ. Empirical examples are presented to illustrate some of the theory and indicate the extent to which it can be expected to be relevant in practice.

A model for the analysis of paired comparison data is presented which combines features of the BTL-model with features of the Unfolding model. The model is metric, mathematically tractable, and has an exact algebraic solution. Since it is multidimensional and allows for individual differences, it is thought to be more realistic for some choice situations than either the Thurstone model or the BTL-model. No claim is made that the present model will be appropriate for all conceivable choice situations. Rather, it is argued that the fact that it is explicitly falsifiable is a point in its favor.

The rank test Rn of Cronholm and Revusky [1965] in effect doubles the number of comparisons carried out in the Mann-Whitney test between “Treated” and “Control” subjects chosen from a single group of n and triples the asymptotic Pitman efficiency against shift alternatives by using a series of n – 1 subexperiments. However, the full series can be too costly or time-consuming. It is shown that j appropriately chosen subexperiments will permit one to make at least j/(j + 1) of the number of comparisons possible under Rn, with large sample efficiency of j/(j + 2) relative to Rn against shift alternatives and small sample efficiency greater than that, j = 2, 3, …, n − 2. The resulting test criterion is a sum of j independent Mann-Whitney test statistics. Its null distribution is tabulated for n ≤ 10 and small sample efficiency comparisons are carried out for n = 10.