Forced classification: A simple application of a quantification method

This study formulates a property of a quantification method, the principle of equivalent partitioning (PEP). When the PEP is used together with Guttman's principle of internal consistency (PIC) in a simple way, the combination offers an interesting way of analyzing categorical data in terms of the variate(s) chosen by the investigator, a type of canonical analysis. The study discusses applications of the technique to multiple-choice, rank-order, and paired comparison data.This study was supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A7942). Comments on the earlier drafts from anonymous reviewers and the editor were much appreciated.     

Redundancy analysis for qualitative variables

Redundancy analysis (also called principal components analysis of instrumental variables) is a technique for two sets of variables, one set being dependent of the other. Its aim is maximization of the explained variance of the dependent variables by a linear combination of the explanatory variables. The technique is generalized to qualitative variables; it then gives implicitly a simultaneous optimal scaling of the dependent, qualitative variables. Examples are taken from the Dutch Life Situation Survey 1977, using Satisfaction with Life and Happiness as dependent variables. The analysis leads to one well-being scale, defined by the explanatory variables Marital status, Schooling, Income and Activity.The views expressed in this paper are those of the author and do not necessarily reflect the policies of the Netherlands Central Bureau of Statistics.     

French validation of the Twenty-Item Value Inventory

IntroductionPersonal values are important for understanding psychological phenomena by serving as predictors of individuals’ attitudes and behavior. The Twenty-Item Value Inventory (TwIVI, Sandy et al., 2017) was developed to assess values with items from the Portrait Values Questionnaire (PVQ, Schwartz et al., 2001).ObjectiveThe goal of this study was to assess the reliability and validity of the French version of the TwIVI.MethodTwo French-Canadian samples of adults (N1 = 825; N2 = 429) were recruited in order to test the reliability and validity of the TwIVI.ResultsResults revealed that the French version of the TwIVI showed satisfying temporal stability, convergence validity with the PVQ, and concurrent validity with personality traits. Some issues were observed for internal consistency and structural validity, assessed with both multidimensional scaling and exploratory structural equation modeling.ConclusionOverall, these findings partially support the psychometric qualities of the French version of the TwIVI. Implications for theory and research on values are discussed.     

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.     

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.     

Fitting longitudinal reduced-rank regression models by alternating least squares

An alternating least squares method for iteratively fitting the longitudinal reduced-rank regression model is proposed. The method uses ordinary least squares and majorization substeps to estimate the unknown parameters in the system and measurement equations of the model. In an example with cross-sectional data, it is shown how the results conform closely to results from eigenanalysis. Optimal scaling of nominal and ordinal variables is added in a third substep, and illustrated with two examples involving cross-sectional and longitudinal data.Financial support by the Institute for Traffic Safety Research (SWOV) in Leidschendam, The Netherlands is gratefully acknowledged.     

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.     

Generalized eigenvalue problems with positive semi-definite matrices

In a recent paper in this journal McDonald, Torii, and Nishisato show that generalized eigenvalue problems in which both matrices are singular can sometimes be solved by reducing them to similar problems of smaller order. In this paper a more extensive analysis of such problems is used to sharpen and clarify the results of McDonald, Torii, and Nishisato. Possible extensions are also indicated. The relevant mathematical literature is reviewed briefly.