A probabilistic multidimensional scaling with unique axes1

Abstract: A probabilistic multidimensional scaling model is proposed. The model assumes that the coordinates of each stimulus are normally distributed with variance Σ_i = diag(σ²₁, … σ²_Ri). The advantage of this model is that axes are determined uniquely. The distribution of the distance between two stimuli is obtained by polar coordinates transformation. The method of maximum likelihood estimation for means and variances using the EM algorithm is discussed. Further, simulated annealing is suggested as a means of obtaining initial values in order to avoid local maxima. A simulation study shows that the estimates are accurate, and a numerical example concerning the location of Japanese cities shows that natural axes can be obtained without introducing individual parameters.     

A new computational method to fit the weighted euclidean distance model

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an analytic solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other analytic procedures. The procedure, which we call SUMSCAL (subjectivemetricscaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.Comments by J. Douglas Carroll and J. B. Kruskal have been very helpful in preparing this paper.     

Multivariate stochastic processes compatible with “aspect” models of similarity and choice

Various recent works have developed feature or aspect models of similarity and preference. These models are more concerned with the fine detail of the judgment process than were prior models, but nevertheless they have not in general developed an underlying stochastic process compatible with the assumed structure. In this paper, we show that a particular class of multivariate stochastic processes, namely those associated with the Marshall-Olkin multivariate exponential distribution, generates several of these models. In particular, such stochastic processes (appropriately interpreted) yield Tversky's elimination by aspects model, Edgell and Geisler's (normal) additive random aspects model, and Shepard and Arabie's additive cluster model.This work was supported by Natural Science and Engineering Research Council of Canada Grant A8124 to A.A.J. Marley.     

Tests for stability in linear structural equation systems

Interpretations regarding the effects of exogenous and endogenous variables on endogenous variables in linear structural equation systems depend upon the convergence of a matrix power series. Convergence depends upon the eigenvalues of the structural coefficient matrix. The test for convergence developed by Jöreskog and Sörbom is shown to be only sufficient, not necessary and sufficient.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.     

Spatial and conjoint models based on pairwise comparisons of dissimilarities and combined effects: Complete and incomplete designs

In pairwise multidimensional scaling, a spatial representation for a set of objects is determined from comparisons of the dissimilarity of any two objects drawn from the set to the dissimilarity of other pairs of objects drawn from that set. In pairwise conjoint scaling, comparisons among the joint effects produced by pairs of objects, where the objects in a pair are drawn from separate sets, are used to determine numerical representations for the objects in each set. Monte Carlo simulations of both pairwise dissimilarities and pairwise conjoint effects show that Johnson's algorithm can provide good metric recovery in the presence of high levels of error even when only a small percentage of the complete set of pairwise comparisons are tested.     

Restricted multidimensional scaling models for asymmetric proximities

Restricted multidimensional scaling models [Bentler & Weeks, 1978] allowing constraints on parameters, are extended to the case of asymmetric data. Separate functions are used to model the symmetric and antisymmetric parts of the data. The approach is also extended to the case in which data are presumed to be linearly related to squared distances. Examples of several models are provided, using journal citation data. Possible extensions of the models are considered. This research was supported in part by USPHS Grant 0A01070, P. M. Bentler, principal investigator, and NIMH Grant MH-24819, E. J. Anthony and J. Worland, principal investigators. The authors wish to thank E. W. Holman and several anonymous reviewers for their valuable suggestions concerning this research.     

A note on the investigation of local and global identifiability

In general, nonlinear models such as those commonly employed for the analysis of covariance structures, are not globally identifiable. Any investigation of local identifiability must either yield a mapping of identifiability onto the entire parameter space, which will rarely be feasible in any applications of interest, or confine itself to the neighbourhood of such points of special interest as the maximum likelihood point.The author would like to thank J. Jack McArdle and Colin Fraser for their comments on this paper.     

The use of latent class models for assessing prerequisite relations and transference among traits

This paper presents a strategy for pairwise assessment which may be used to evaluate the nature of both prerequisite and transference relations existing among a set of traits. This strategy is appropriate for use both within a confirmatory context, in which an attempt is made to establish the validity of some specified set of relations among traits, as well as within an exploratory context, in which a search is made for unconjectured prerequisite and transference relations existing between pairs of traits. Both uses of this strategy are based on a variety of latent class models which are representative of various possible relational states existing between pairs of traits. Thus, the nature of trait relations may be investigated through the use of statistical assessments of both absolute and relative fit attained by these models. An application is presented to exemplify how this strategy may be used within the exploratory context.The author is obliged to Clifford Clogg, C. Mitchell, Dayton, and William Schafer for helpful comments made regarding a previous draft of this paper as well as to Mary Papageorgiou who provided access to the data which was considered in this study.