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 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.

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.

It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.

Performance measures, conditionalized on the last error and other events, have been of central concern in the development of absorbing Markov-chain models for learning and problem solving. Expressions for response probabilities and expected latencies conditional on the occurrence of the last error and other events are derived using matrix methods.

As a consequence of the complexity of the iterative optimum seeking procedures used by practical nonmetric multidimensional scaling algorithms, many of their computational properties have not been well understood. In particular, the following questions are of interest: (a) from the user’s point of view, are there significant differences between alternative available programs, (b) are suboptimal solutions frequently encountered, and how does this depend on the characteristics of the algorithm? Using Monte Carlo techniques to generate dissimilarity matrices with known underlying configurations, Kruskal’s M-D-SCAL, Guttman-Lingoes’ SSA-I, and Young-Torgerson’s TORSCA-9 programs were compared. It was found: (a) that differences between the solutions obtained by the algorithms were typically so small as to be of little practical importance, (b) deviant solutions were occasionally produced by each of the algorithms, but most often by M-D-SCAL, and furthermore, most frequently in one dimension. The likelihood of being trapped in a nonoptimal position is reduced by a good choice of initial configuration, and also possibly by constructing the iterative process to favor the possibility of stepping over small local depressions when far from the location of the optimum. The conclusions of this study are based on a total of 2160 scaling solutions.