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Fitting and Testing Carroll’s Weighted Unfolding Model for Preferences

Published online by Cambridge University Press:  01 January 2025

Mark L. Davison
Mark L. Davison*
Affiliation:
University of Minnesota
*
Requests for reprints should be addressed to Mark L. Davison, Department of Social, Psychological, and Philosophical Foundations of Education, University of Minnesota, Minneapolis, Minnesota 55455.
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Abstract

A quadratic programming algorithm is presented for fitting Carroll’s weighted unfolding model for preferences to known multidimensional scale values. The algorithm can be applied directly to pairwise preferences; it permits nonnegativity constraints on subject weights; and it provides a means of testing various preference model hypotheses. While basically metric, it can be combined with Kruskal’s monotone regression to fit ordinal data. Monte Carlo results show that (a) adequacy of “true” preference recovery depends on the number of data points and the amount of error, and (b) the proportion of data variance accounted for by the model sometimes only approximately reflects “true” recovery.

Keywords

multidimensional scalingutility theorymultiple regressionquadratic programming

Information

Type
Original Paper
Information
Psychometrika , Volume 41 , Issue 2 , June 1976 , pp. 233 - 247
Copyright
Copyright © 1976 The Psychometric Society

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Footnotes

This study is based on a doctoral dissertation submitted to the University of Illinois at Urbana-Champaign. The author wishes to thank the members of his dissertation committee—Lawrence E. Jones, Chairman, Charles Lewis, Stephen Golding, Ledyard Tucker, and Nancy Wiggins—for their helpful comments.

References

Reference Notes

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Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., & Nerlove, S. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences. Vol. I: Theory, 1972, New York: Seminar Press.Google Scholar
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Coombs, C. H. A note on the relation between the vector model and the unfolding model for preferences. Psychometrika, 1975, 40, 115–116.CrossRefGoogle Scholar
Kendall, M. G. and Stuart, A. The advanced theory of statistics (Vol. 2, 3rd ed., London: Griffin, 1973.Google Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. and Carroll, J. D. Geometric models and badness-of-fit functions. In Krishniah, P. R. (Eds.), Multivariate Analysis II. New York: Academic Press. 1969, 639–670.Google Scholar
Lingoes, J. C. An IBM 7090 program for Guttman-Lingoes smallest space analysis RI. Behavioral Science, 1966, 11, 332–332.Google Scholar
McNemar, Quinn Psychological Statistics, 4th ed., New York: John Wiley and Sons, 1969.Google Scholar
Ramsey, J. O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Rao, R. C. Linear statistical inference and its application, 1965, New York: Wiley.Google Scholar
Schönemann, P. On metric multidimensional unfolding. Psychometrika, 1970, 35, 349–366.CrossRefGoogle Scholar
Shocker, A. D. and Srinivasan, V. A consumer-based methodology for the identification of new product ideas. Management Science, 1973, 16, 147–167.Google Scholar
Srinivasan, V. and Shocker, A. D. Linear programming techniques for multidimensional analysis of preferences. Psychometrika, 1973, 38, 337–369.CrossRefGoogle Scholar
Srinivasan, V., Shocker, A. D., and Weinstein, A. G. Measurement of a composite criterion of managerial success. Organizational Behavior and Human Performance, 1973, 9, 147–167.CrossRefGoogle Scholar
Theil, H. and Van de Panne, T. Quadratic programming as an extension of conventional quadratic maximization. Management Science, 1960, 7, 1–20.CrossRefGoogle Scholar
Tucker, L. R. Intra-individual and inter-individual multidimensionality. In Gulliksen, H. and Messick, S. (Eds.), Psychological scaling: Theory and applications. New York: Wiley. 1960, 155–167.Google Scholar
Young, F. W. and Torgerson, W. S. TORSCA, a FORTRAN IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 1967, 12, 498–498.Google Scholar