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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

Information

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.

References

Bennett, J.F., & Hayes, W.L. (1960). Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 25, 36–48.CrossRefGoogle Scholar
Borg, I., & Bergermaier, R. (1982). Degenerationsprobleme im Unfolding und Ihre ösung. Zeitschrift ür Sozialpsychologie, 13, 287–299.Google Scholar
Borg, I., & Groenen, P.J.F. (1997). Modern Multidimensional Scaling: Theory and Applications. New York: Springer.CrossRefGoogle Scholar
Borg, I., & Lingoes, J. (1987). Multidimensional Similarity Structure Analysis. Berlin: Springer.CrossRefGoogle Scholar
Busing, F.M.T.A., & Heiser, W.J. (2003). PREFSCAL Progress Report: two-way models (Tech. Rep.) Leiden. The Netherlands: Leiden University. (working paper)Google Scholar
Carroll, J.D. (1972). In Shepard, R.N., Romney, A.K., & Nerlove, S.B. (Eds), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Vol. 1, pp 105–155) New York: Seminar Press.Google Scholar
Coombs, C.H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 148–158.CrossRefGoogle ScholarPubMed
Coombs, C.H. (1964). A Theory of Data. New York: Wiley.Google Scholar
Coombs, C.H., & Kao, R.C. (1960). On a connection between factor analysis and multidimensional unfolding. Psychometrika, 25, 219–231.CrossRefGoogle Scholar
Dagpunar, A. (1988). Principles of Random Variate Generation. Oxford: Clarendon Press.Google Scholar
De Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In Barra, J.R., Brodeau, F., Romier, G., & van Cutsem, B. (Eds.), Recent Developments in Statistics (pp. 133–145). The Netherlands: North-Holland: Amsterdam.Google Scholar
De Leeuw, J. (1983). On Degenerate Nonmetric Unfolding Solutions (Tech. Rep.). Department of Data Theory, FSW/RUL.Google Scholar
De Leeuw, J., & Heiser, W.J. (1980). Multidimensional scaling with restrictions on the configuration. In Krishnaiah, P.R. (Eds.), Multivariate Analysis (Vol 5) (pp. 501–522). The Netherlands: North-Holland: Amsterdam.Google Scholar
Dennis, J.E., & Schnabel, R.B. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, New Jersey: Prentice-Hall. (Republished by SIAM, Philadelphia, in 1996 as Vol. 16 of Classics in Applied Mathematics)Google Scholar
De Sarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via alternating weighted least squares. Psychometrika), 50(3), 275–300.CrossRefGoogle Scholar
De Sarbo, W.S., & Rao, V.R. (1984). GENFOLD2: A set of models and algorithms for the GENeral unFOLDing analysis of preference/dominance data. Journal of Classification, 1, 147–186.CrossRefGoogle Scholar
DeSarbo, W.S., Young, M.R., & Rangaswamy, A. (1997). A parametric Multidimensional Unfolding Procedure for Incomplete Nonmetric Preference/Choice Set Data in Marketing Research (Tech. Rep.). The Pennsylvania State University.(Working paper)Google Scholar
Dinkelbach, W. (1967). On nonlinear fractional programming. Management Science, 13, 492–498.CrossRefGoogle Scholar
Gower, J.C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53, 325–338.CrossRefGoogle Scholar
Green, P.E., & Rao, V. (1972). Applied multidimensional scaling. Hinsdale, IL: Dryden Press.Google Scholar
Groenen, P.J.F. (1993). The majorization Approach to Multidimensional Scaling: Some Problems and Extensions. DSWO Press: Leiden, The Netherlands.Google Scholar
Groenen, P.J.F., & Heiser, W.J. (1996). The tunneling method for global optimization in multidimensional scaling. Psychometrika, 61, 529–550.CrossRefGoogle Scholar
Hardy, G.H., Littlewood, J.E., & Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press.Google Scholar
Hayes, W.L., & Bennett, J.F. (1961). Multidimensional unfolding: Determining configuration from complete rank order preference data. Psychometrika, 26, 221–238.CrossRefGoogle Scholar
Heiser, W.J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, Leiden University.Google Scholar
Heiser, W.J. (1987). Joint ordination of species and sites: The unfolding technique. In Legendre, P., & Legendre, L. (Eds.), Developments in Numerical Ecology (pp. 189–221). Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
Heiser, W.J. (1989). Order invariant unfolding analysis under smoothness restrictions. In De Soete, G., Feger, H., & Klauer, K.C. (Eds.), New Developments in Psychological Choice Modeling (pp. 3–31). North-Holland: Amsterdam.CrossRefGoogle Scholar
Heiser, W.J. (1995). Convergent computation by iterative majorization: Theory and applications in multidimensional data analysis. In Krzanowski, W.J. (Eds.), Recent Advances in Descriptive Multivariate Analysis (pp. 157–189). Oxford: Oxford University Press.CrossRefGoogle Scholar
Heiser, W.J., & De Leeuw, J. (1979). How to Use SMACOF-III: A Program for Metric Multidimensional Unfolding (Tech. Rep.). Leiden University, Department of Data Theory.Google Scholar
Katsnelson, J., & Kotz, S. (1957). On the upper limits of some measures of variability. Archivfür Meterologie, Geophysik and Bioklimatologie (B), 8, 103.CrossRefGoogle Scholar
Kim, C., Rangaswamy, A., & DeSarbo, W.S. (1999). A quasi-metric approach to multidimensional unfolding for reducing the occurence of degenerate solutions. Multivariate Behavioral Research, 34(2), 143–180.CrossRefGoogle Scholar
Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J.B. (1977). Multidimensional scaling and other methods for discovering structure. In Enslein, K., Ralston, A., & Wilf, H.S. (Eds.), Mathematical Methods for Digital Computers (Vol. 2 (pp. 296–339). New York: Wiley.Google Scholar
Kruskal, J.B., & Carroll, J.D. (1969). Geometrical models and badness-of-fit functions. In Krishnaiah, P.R. (Eds.), Multivariate Analysis(vol. 2) (pp. 639–671). New York: Academic Press.Google Scholar
Kruskal, J.B., Young, F.W., & Seery, J.B. (1978). How to Use KYST, A Very Flexible Program to do Multidimensional Scaling and Unfolding (Tech. Rep.). Murray Hill, NJ: Bell Laboratories.Google Scholar
L’Ecuyer, P. (1999). Tables of maximally equidistributed combined LFSR generators. Mathematics of Computing, 68, 261–269.CrossRefGoogle Scholar
Marden, J.I. (1995). Analyzing and Modeling Rank Data. London: Chapman and Hall.Google Scholar
McClelland, G.H., & Coombs, C.H. (1975). ORDMET: A general algorithm for constructing all numerical solutions to ordered metric solutions. Psychometrika, 40, 269–290.CrossRefGoogle Scholar
Pearson, K. (1896). Regression, heridity, and panmixia. Philosophical Transactions of the Royal Society of London, Series A, 187, 253–318.Google Scholar
Roskam, E.E.Ch.I. (1968). Metric Analysis of Ordinal Data. Voorschoten: VAM. Shepard, R.N. (1974) Representation of structure in similarity data: Problems and prospects. Psychometrika, 39(4), 373–421.Google Scholar
Shepard, R.N. (1974). Representation of structure in similarity data: Problems and prospects. Psychometrika,, 39(4), 373–421.CrossRefGoogle Scholar
Takane, Y., Young, F.W., & Leeuw, J. (1977). Nonmetric individual differences MDS: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7–67.CrossRefGoogle Scholar
Torgerson, W.S. (1958). Theory and Methods of Scaling. New York: Wiley.Google Scholar
Trosset, M.W. (1998). A new formulation of the nonmetric STRAIN problem in multidimensional scaling. Journal of Classification, 15, 15–35.CrossRefGoogle Scholar
Van Blokland-Vogelesang, A.W. (1989). Unfolding and consensus ranking: A prestige ladder for technical occupations. In De Soete, G., Feger, H., & Klauer, K.C. (Eds.), New Developments in Psychological Choice Modeling (pp. 237–258). The Netherlands: North-Holland: Amsterdam.CrossRefGoogle Scholar
Van Blokland-Vogelesang, A.W. (1993). A nonparametric distance model for unidimensional unfolding. In Fligner, M.A., & Verducci, J.S. (Eds.), Probability Models and Statistical Analyses for Ranking Data (pp. 241–276). New York: Springer-Verlag.CrossRefGoogle Scholar
Wagenaar, W.A., & Padmos, P. (1971). Quantitative interpretation of Stress in Kruskal’s method multidimensional scaling technique. British Journal of Mathematical and Statistical Psychology, 24, 101–110.CrossRefGoogle Scholar
Winsberg, S., & Carroll, J.D. (1989). A quasi-nonmetric method for multidimensional scaling via an extended Euclidean model. Psychometrika, 54, 217–229.CrossRefGoogle Scholar
Young, F.W. (1972). A model for polynomial conjoint analysis algorithms. In Shepard, R.N., Romney, A.K., & Nerlove, S.B. (Eds.), Multidimensional Scaling, Theory (Vol. I (pp. 69–104). New York: Springer.Google Scholar
Young, F.W., & Torgerson, W.S. (1967). T0RSCA, a Fortran IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 12, 498.Google Scholar