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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
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

Roskam, E. I. Metric analysis of ordinal data in psychology: Models and numerical methods for metric analysis of conjoint ordinal data in psychology. Unpublished doctoral dissertation, Katholieke Universitat te Nijmegen, 1968.Google Scholar
Kruskal, J. B. How to use MDSCAL, a program to do multidimensional unfolding, 1968, Murray Hill, New Jersey: Bell Telephone Laboratories.Google Scholar
Gleason, T. C. Multidimensional scaling of sociometric data. Unpublished doctoral dissertation, University of Michigan, 1969.Google Scholar
Carroll, J. D. and Chang, J. J. Relating preference data to multidimensional scaling solutions via a generalization of Coombs’ unfolding model. Paper presented at the meeting of the Psychometric Society, Madison, Wisconsin, April 1967.Google Scholar
Davison, M. L. Fitting a set of points to a space defined by a second set: An extension of multidimensional scaling and unfolding models. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign, 1974.Google Scholar
Davison, M. L. and Jones, L. E. A similarity-attraction model for predicting sociometric choice from perceived group structure. Manuscript submitted for publication, 1975.Google Scholar
Nygren, T. E. Individual differences in perception of political candidates. Unpublished masters thesis, University of Illinois at Urbana-Champaign, 1973.Google Scholar
Bechtel, G. G., Tucker, L. R., and Chang, W. A. scaler product model for the multidimensional scaling of choice. Psychometrika, 1971, 36, 369–388.CrossRefGoogle Scholar
Bennett, J. R. and Hays, W. L. Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 1960, 25, 27–43.CrossRefGoogle Scholar
Boot, J. C. G. Quadratic programming, 1964, Chicago: Rand McNally.Google Scholar
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
Coombs, C. H. A theory of data, 1964, New York: John Wiley and Sons.Google Scholar
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
Bechtel, G. G., Tucker, L. R., and Chang, W. A. scaler product model for the multidimensional scaling of choice. Psychometrika, 1971, 36, 369–388.CrossRefGoogle Scholar
Bennett, J. R. and Hays, W. L. Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 1960, 25, 27–43.CrossRefGoogle Scholar
Boot, J. C. G. Quadratic programming, 1964, Chicago: Rand McNally.Google Scholar
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
Coombs, C. H. A theory of data, 1964, New York: John Wiley and Sons.Google Scholar
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