James C. Lingoes

Some boundary conditions for a monotone analysis of symmetric matrices

This paper gives a rigorous and greatly simplified proof of Guttmans theorem for the least upper-bound dimensionality of arbitrary real symmetric matricesS, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements ofS. A comparable and more easily proven theorem for the vector model is also introduced. At mostn-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.

A direct approach to individual differences scaling using increasingly complex transformations

A family of models for the representation and assessment of individual differences for multivariate data is embodied in a hierarchically organized and sequentially applied procedure called PINDIS. The two principal models used for directly fitting individual configurations to some common or hypothesized space are the dimensional salience and perspective models. By systematically increasing the complexity of transformations one can better determine the validities of the various models and assess the patterns and commonalities of individual differences. PINDIS sheds some new light on the interpretability and general applicability of the dimension weighting approach implemented by the commonly used INDSCAL procedure.

Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm

In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,es, which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.

A MODEL AND ALGORITHM FOR MULTIDIMENSIONAL SCALING WITH EXTERNAL CONSTRAINTS ON THE DISTANCES

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onXs distances are expressed by the relations amongRs numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,Ri,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.

Multiple Scalogram Analysis: A Set-Theoretic Model for Analyzing Dichotomous Items

AMONG the various criticisms that have been directed against the scaling technique of Guttman (1944), a major one has been in reference to his concept of a “universe of content,’ (see, for example, Festinger [ 19471 and Loevinger [ 19481 ) . This particular concept lies a t the basis for item construction and selection in Guttman’s scalogram method. Quite generally what is meant by this phrase is the set of all statements which may be made in reference to a single variable or trait, as for example, Yove of country,”

A Solution to the Weighted Procrustes Problem in Which the Transformation is in Agreement with the Loss Function.

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.

WHAT WEIGHT SHOULD WEIGHTS HAVE IN INDIVIDUAL DIFFERENCES SCALING

Recently, individual difference scaling has become one of the most active fields of research in psychometrics. Numerous models and algorithms have been proposed but relatively little evidence as to the validity of the produced representations is available so far. To shed some new light on this issue we will reanalyze some data collected by Green and Rao (1972) via PINDIS (Procrustrean INdividual Difference Scaling). PINDIS is intimately related to the model underlying all presently available individual difference scaling algorithms but differs in important aspects which will allow a deeper insight into validity and interpretability of individual weights generated by them. To make our points clear, we will compare our results with those produced by INDSCAL which is (a) presently the most popular procedure, and (b) also the method of analysis chosen originally by Green and Rao (1972).

On evaluating the equivalency of alternative MDS representations

A typical question in MDS is whether two alternative configurations that are both acceptable in terms of data fit may be considered “practically the same”. To answer such questions on the equivalency of MDS solutions. Lingoes & Borg (1983) have recently proposed a quasistatistical decision strategy that allows one to take various features of the situation into account. This paper adds another important piece of information to this approach: for the Lingoes-Borg decision criterion R, we compute what proportion of R-values is greater/less than the observed coefficient if one were to consider all possible alternative distance sets within certain bounds defined by the observed fit coefficients for two alternative MDS solutions, what are the limits of acceptability for such fit coefficients, and how are the observed MDS configurations interrelated.

How similar are the different results

An increasing number of studies in the social sciences utilize Smallest-Space Analysis-I (SSA-I) (e.g., Laumann and Guttman, 1966; Schlesinger and Guttman, 1969; Levy and Guttman, 1975a, b; Elizur and Guttman, 1976; Levy, 1976; Guttman and Guttman, 1976; Ben-Sira, 1977; Maimon, 1978). This paper deals with some aspects of comparisons between SSA-I results, and analyzes existing methods for comparing two or more SSA-I solutions obtained from different populations or from different samples of the same population. Alternative measures of the degree of similarity between the various solutions are also discussed. We begin by very briefly discussing the SSA-I technique. Then, we move to the questions involved in comparing SSA-I solutions. A particular technique for comparison, PINDIS, is then explained in some detail. Finally, the results of an empirical study utilizing four SSA-I solutions are analyzed by PINDIS and by other approaches.

A Fortran IV Program Generalizing the Schonemann-Carroll Matrix Fitting Algorithm to Monotone and Linear Fitting of Configurations.

SCHONEMANN and Carroll (1970) proposed a generalization of the orthogonal Procrustes problem for obtaining a least-squares fit of a given matrix X to a target matrix Y under a choice of an orthogonal rotation, a translation, and a central dilation. Such linear displacements leave invariant the relative magnitudes of interpoint distances and the monotone measures of goodness of fit. Lingoes and Schonemann (1973) suggested alternative measures of fit which would permit comparisons among any number of such fitted pairs of matrices. Since a number of recent techniques for data analysis impose only ordinal restrictions for obtaining configurations, it would be desirable to extend the Schonemann-Carroll algorithm to monotone displacements, so that one could more clearly &dquo;see&dquo; how similar