A method is given for finding a linear combination of binary item scores that minimizes the expected frequency of misclassification, in discriminating between two groups. The item scores are not assumed to be stochastically independent. The method uses the theory of threshold functions, developed by electrical engineers. Since psychometricians may not be familiar with this theory an elementary introduction to the required material is also given.

Given the complete set R of rank orders obtained from any configuration of n stimulus points in r dimensions in accordance with the unfolding model, a configuration from which just these orders may be derived will be described as a solution for R. The space is assumed to be Euclidean. Necessary and sufficient conditions are derived for a configuration to be a solution for R. The geometrical constraints which are necessary and sufficient to determine the subset of pairs of orders and opposites contained in R are also identified and constitute the constraint system for the ordinal vector model. The relationship between the two models is discussed.

This paper offers a new methodology for analyzing individual differences in preference judgments with regard to a set of stimuli prespecified in a multidimensional attribute space. The individual is modelled as possessing an “ideal point” denoting his most preferred stimulus location in this space and a set of weights which reveal the relative saliences of the attributes. He prefers those stimuli which are “closer” to his ideal point (in terms of a weighted Euclidean distance measure). A linear programming model is proposed for “external analysis” i.e., estimation of the coordinates of his ideal point and the weights (involved in the Euclidean distance measure) by analyzing his paired comparison preference judgments on a set of stimuli, prespecified by their coordinate locations in the multidimensional space. A measure of “poorness of fit” is developed and the linear programming model minimizes this measure over all possible solutions. The approach is fully nonmetric, extremely flexible, and uses paired comparison judgments directly. The weights can either be constrained nonnegative or left unconstrained. Generalizations of the model to consider ordinal or interval preference data and to allow an orthogonal transformation of the attribute space are discussed. The methodology is extended to perform “internal analysis,”i.e., to determine the stimuli locations in addition to weights and ideal points by analyzing the preference judgments of all subjects simultaneously. Computational results show that the methodology for external analysis is “unbiased”—i.e., on an average it recovers the “true” ideal point and weights. These studies also indicate that the technique performs satisfactorily even when about 20 percent of the paired comparison judgments are incorrectly specified.

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman [1957] and by Harris [1963] provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates.

It is shown that an important distinction can be made between determinacy of common-factors and determinacy of unique-factors and that determinacy can be used as a criterion to establish a lower bound for the ratio of number of variables to number of factors necessary if specified levels of common- and unique-factor determinacy are to be attained.

A model containing linear and nonlinear parameters (e. g., a spatial multidimensional scaling model) is viewed as a linear model with free and constrained parameters. Since the rank deficiency of the design matrix for the linear model determines the number of side conditions needed to identify its parameters, the design matrix acts as a guide in identifying the parameters of the nonlinear model. Moreover, if the design matrix and the uniqueness conditions constitute an orthogonal linear model, then the associated error sum of squares may be expressed in a form which separates the free and constrained parameters. This immediately provides least squares estimates of the free parameters, while simplifying the least squares problem for those which are constrained. When the least squares estimates for a nonlinear model are obtained in this way, i.e. by conceptualizing it as a submodel, the final error sum of squares for the nonlinear model will be a restricted minimum whenever the side conditions of the model become real restrictions upon its submodel. In this case the design matrix for the embracing orthogonal model serves as a guide in introducing parameters into the nonlinear model as well as in identifying these parameters. The method of overwriting a nonlinear model with an orthogonal linear model is illustrated with two different spatial analyses of a three-way preference table.

A procedure is proposed whereby questionnaire data, which is usually ordinal in nature and often error-ridden, may be transformed to reduce the error variance in the data and to improve the metric properties of the individual variables. The technique is suggested by a result of Eckart and Young. The properties of the method are investigated by means of a Monte Carlo study. Various matrices were generated representing the usual concept of “true scores”. These matrices were distorted using two levels of random errors and two kinds of categorization. The distorted matrices were in turn transformed by the proposed methods and compared to the “true scores”. In all cases an overall measure of similarity reveals the transformed matrices are better approximations to the “true scores” than the untransformed data. Some properties of the transformation are discussed and some possible applications of the general technique are suggested.

A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser's D2 is also derived from the general formulation. A comparison of the Mahalonobis and D2 approaches indicates that score vectors using D2 distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.