A model of Bradley-Terry type for paired comparisons is considered. In addition to the usual parameters, the model allows for individual parameters and corresponding parameters for the choice-alternatives. The model is applied to a set of data from a Danish investigation of the attitude of blue collar workers towards alternative social gains. The proposed model is finally compared to a model recently suggested by Schönemann and Wang.

Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the “triple product” [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater than every dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.

Following the work of Tucker, Cooper, and Meredith, image and anti-image covariance matrices from a correlation matrix that may be singular are derived.

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.

Extending the definitions of part and bipartial correlation to sets of variates, the notion of part and bipartial canonical correlation analysis are developed and illustrated.

The three most commonly used statistics, the arithmetic mean, variance, and the product-moment correlation, are most unfortunate choices when data are not strictly Gaussian. A new measure of correlation and a measure of scale are proposed which are substantially more robust than their least squares counterparts. An illustration shows how increased robustness can be obtained through the use of equal regression weights without severe loss in accuracy. The paper also shows how incorporating knowledge about the theoretical structure of the regression coefficients into their estimation can aid substantially in increasing their robustness.

Orthogonal rotation of canonical variates is shown to preserve the major properties of the canonical solution and may increase its interpretability.

Two forms of stationarity prior to criterion in absorbing Markov chains are examined. Both forms require that the probability of a particular response on a particular trial before absorption be independent of trial number. The stronger of these forms holds that this is true independent of starting state; the weaker, only for a specified set of starting probabilities. Simple, necessary and sufficient conditions for both forms are developed and applied to several examples.

A weaker generalized inverse (Rao's g-inverse; Graybill's c-inverse) can be used in place of the Moore-Penrose generalized inverse to obtain multiple and canonical correlations from singular covariance matrices.

A general question is raised concerning the possible consequences of employing the very popular INDSCAL multidimensional scaling model in cases where the assumptions of that model may be violated. Simulated data are generated which violate the INDSCAL assumption that all individuals perceive the dimensions of the common object space to be orthogonal. INDSCAL solutions for these various sets of data are found to exhibit extremely high goodness of fit, but systematically distorted object spaces and negative subject weights. The author advises use of Tucker’s three-mode model for multidimensional scaling, which can account for non-orthogonal perceptions of the object space dimensions. It is shown that the INDSCAL model is a special case of the three-mode model.

Interactive Scaling with Individual Subjects (ISIS) developed by Young & Cliff [1972], is a method involving interaction between subject and computer in real time to determine which judgments made by the subject are critical to the definition of a dimensional structure. The procedure is based on the mathematical fact that it is possible to define a space of R dimensions in terms of only the interpoint distances between all stimuli being scaled and a subset of (R +1) of these stimuli. For errorless judgments, any subset of (R + 1) stimuli is appropriate. However, fallible data require that the subset consist of stimuli that are maximally dissimilar, and the ISIS procedure is designed to obtain such an optimum subset (a “basis”).

This research evaluates a modified version of ISIS with respect to (a) a metric MDS analysis based on all possible pairs of the stimuli, and (b) a metric MDS analysis based on a subset of one-third of the possible pairs, or about the same number as that required by ISIS. Results show that the ISIS method achieves better fit than (b) at low error levels, and may also achieve better fit than (b) at higher error levels if the size of the basis is increased. The more stimuli in the basis the more indices of fit approach those of (a).

A new method of introducing error in MDS studies is used in the evaluation.

A probabilistic model for the validation of behavioral hierarchies is presented. Estimation is by means of iterative convergence to maximum likelihood estimates, and two approaches to assessing the fit of the model to sample data are discussed. The relation of this general probabilistic model to other more restricted models which have been presented previously is explored and three cases of the general model are applied to exemplary data.

A method is developed to investigate the additive structure of data that (a) may be measured at the nominal, ordinal or cardinal levels, (b) may be obtained from either a discrete or continuous source, (c) may have known degrees of imprecision, or (d) may be obtained in unbalanced designs. The method also permits experimental variables to be measured at the ordinal level. It is shown that the method is convergent, and includes several previously proposed methods as special cases. Both Monte Carlo and empirical evaluations indicate that the method is robust.

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.

This article is concerned with estimation of components of maximum generalizability in multifacet experimental designs involving multiple dependent measures. Within a Type II multivariate analysis of variance framework, components of maximum generalizability are defined as those composites of the dependent measures that maximize universe score variance for persons relative to observed score variance. The coefficient of maximum generalizability, expressed as a function of variance component matrices, is shown to equal the squared canonical correlation between true and observed scores. Emphasis is placed on estimation of variance component matrices, on the distinction between generalizability- and decision-studies, and on extension to multifacet designs involving crossed and nested facets. An example of a two-facet partially nested design is provided.

The kind of mastery score problem often encountered in educational and psychological testing may be roughly described as follows: given that a certain unit of instruction must be mastered by a subject, a test is administered at the end of this unit and, on the basis of the observed score or an estimate of the true score, a decision is made about the grant of the mastery status. If the subject is declared master of the unit, he will be allowed to proceed to the next unit of instruction. Otherwise, remedial work will be given. Given relevant consequences of misclassification, the problem is to determine an optimum decision rule.

A method is discussed which extends canonical regression analysis to the situation where the variables may be measured at a variety of levels (nominal, ordinal, or interval), and where they may be either continuous or discrete. There is no restriction on the mix of measurement characteristics (i.e., some variables may be discrete-ordinal, others continuous-nominal, and yet others discrete-interval). The method, which is purely descriptive, scales the observations on each variable, within the restriction imposed by the variable's measurement characteristics, so that the canonical correlation is maximal. The alternating least squares algorithm is discussed. Several examples are presented. It is concluded that the method is very robust. Inferential aspects of the method are not discussed.

Based on the test theory model for ordinal measurements proposed by Schulman and Haden, the present paper considers correlations between tests, attenuation, regressions involving true and observed scores, and prediction of test reliability.

The population correlation between tests is shown to be related to the expected sample correlation for samples of size n1 and n2. Errors of estimation, measurement and prediction are found to be similar to their counterparts in interval test theory, while attenuation is identical to its counterpart. The bias in estimating population reliability from sample data is compared for Kendall's tau and Spearman's rho.