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Avoiding Degeneracy in Multidimensional Unfolding by Penalizing on the Coefficient of Variation

Published online by Cambridge University Press:  01 January 2025

Frank M. T. A. Busing ,
Patrick J. K. Groenen  and
Willem J. Heiser
Frank M. T. A. Busing*
Affiliation:
Leiden University
Patrick J. K. Groenen
Affiliation:
Erasmus University Rotterdam
Willem J. Heiser
Affiliation:
Leiden University
*
Requests for reprints should be sent to Frank M.T.A. Busing, Department of Psychology, Leiden University, P.O. Box 9555, 2300 RB Leiden. E-mail: busing@fsw.LeidenUniv.nl
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Abstract

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

Keywords

unfoldingdegeneracypenaltyStressiterative majorizationPREFSCAL
Type
Article
Information
Psychometrika , Volume 70 , Issue 1 , March 2005 , pp. 71 - 98
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

The authors would like to thank the editor, an associate editor, and three reviewers for their valuable comments and suggestions to improve the quality of this work.

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