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A Simple Multivariate Probabilistic Model for Preferential and Triadic Choices

Published online by Cambridge University Press:  01 January 2025

Kenneth Mullen and
Daniel M. Ennis
Kenneth Mullen
Affiliation:
Department of Mathematics and Statistics, University of Guelph
Daniel M. Ennis*
Affiliation:
Philip Morris Research Center
*
Requests for reprints should be sent to Daniel Ennis, Philip Morris Research Center, PO Box 26583, Richmond, Virginia, 23261.
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Abstract

Multidimensional probabilistic models of behavior following similarity and choice judgements have proven to be useful in representing multidimensional percepts in Euclidean and non-Euclidean spaces. With few exceptions, these models are generally computationally intense because they often require numerical work with multiple integrals. This paper focuses attention on a particularly general triad and preferential choice model previously requiring the numerical evaluation of a 2n-fold integral, where n is the number of elements in the vectors representing the psychological magnitudes. Transforming this model to an indefinite quadratic form leads to a single integral. The significance of this form to multidimensional scaling and computational efficiency is discussed.

Keywords

multivariate triadsduo-triocanonical quadratic formschoice modelsmultidimensional scalingpreferenceunfolding

Information

Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 1 , March 1991, pp. 69 - 75
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The authors would like to thank Jean-Claude Falmagne and Norman Johnson for suggestions and advice concerning quadratic forms.

References

Ashby, F. G., & Perrin, N. (1988). Toward a unified theory of similarity and recognition. Psychological Review, 95, 124–150.CrossRefGoogle Scholar
Coombs, C. H. (1964). A theory of data, New York: Wiley.Google Scholar
David, H. A., & Trivedi, M. C. (1962). Pair, triangle and duo-trio tests, Blacksburg, VA: Virginia Polytechnic Institute, Department of Statistics.Google Scholar
Dennis, J. E. Jr., Schnabel, R. B. (1983). Numerical methods for unconstrained optimization and nonlinear equations, New Jersey: Prentice Hall.Google Scholar
De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28–41.CrossRefGoogle Scholar
Ennis, D. M. (1988b). Toward a universal law of generalization. Science, 242, 944–944.CrossRefGoogle Scholar
Ennis, D. M. (1988b). Confusable and discriminable stimuli: Comment on Nosofsky (1986) and Shepard (1986). Journal of Experimental Psychology, 4, 408–411.CrossRefGoogle Scholar
Ennis, D. M., & Mullen, K. (1986). A multivariate model for discrimination methods. Journal of Mathematical Psychology, 30, 206–219.CrossRefGoogle Scholar
Ennis, D. M., & Mullen, K. (1986). Theoretical aspects of sensory discrimination. Chemical Senses, 11, 513–522.CrossRefGoogle Scholar
Ennis, D. M., & Mullen, K., Frijters, J. E. R. (1988). Variants of the method of triads: Unidimensional Thurstonian models. British Journal of Mathematical and Statistical Psychology, 41, 25–36.CrossRefGoogle Scholar
Ennis, D. M., Palen, J. J., & Mullen, K. (1988). A multidimensional stochastic theory of similarity. Journal of Mathematical Psychology, 32, 449–465.CrossRefGoogle Scholar
Ennis, D. M., Mullen, K., de Doncker, E., & Kapenga, J. (1989). A general mathematical model for triadic and preference choices, Richmond, Virginia: Philip Morris Research Center.Google Scholar
Frijters, J. E. R. (1979). Variations of the triangular method and the relationship of its unidimensional probabilistic models to three-alternative forced choice signal detection theory models. British Journal of Mathematical and Statistical Psychology, 32, 229–241.CrossRefGoogle Scholar
Genz, A. C., & Malik, A. A. (1980). Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region. Journal of computing and Applied Mathematics, 6, 295–302.CrossRefGoogle Scholar
Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48, 419–426.CrossRefGoogle Scholar
Johnson, N. L., & Kotz, S. (1970). Continuous univariate distributions — 2, New York: Wiley.Google Scholar
MacKay, D. B. (1989). Probabilistic multidimensional scaling: An anisotropic model for distance judgments. Journal of Mathematical Psychology, 33, 187–205.CrossRefGoogle Scholar
Mullen, K., & Ennis, D. M. (1987). Mathematical formulation of multivariate Euclidean models for discrimination methods. Psychometrika, 52, 235–249.CrossRefGoogle Scholar
Mullen, K., Ennis, D. M., de Doncker, E., & Kapenga, J. A. (1988). Models for the duo-trio and triangular methods. Biometrics, 44, 1169–1175.CrossRefGoogle ScholarPubMed
Nosofsky, R. M. (1986). Attention, similarity and the identification-categorization relationship. Journal of Experimental Psychology: General, 115, 39–57.CrossRefGoogle ScholarPubMed
Nosofsky, R. M. (1988). On exemplar-based representations: Comment on Ennis (1988). Journal of Experimental Psychology: General, 117, 412–414.CrossRefGoogle Scholar
Ruben, H. (1960). Probability content of regions under spherical normal distributions, I.. Annals of Mathematical Statistics, 31, 598–619.CrossRefGoogle Scholar
Ruben, H. (1962). Probability content of regions under spherical normal distributions, IV. Annals of Mathematical Statistics, 33, 542–570.CrossRefGoogle Scholar
Ruben, H. (1963). A new result on the distribution of quadratic forms. Annals of Mathematical Statistics, 34, 1582–1584.CrossRefGoogle Scholar
Shepard, R. N. (1986). Discrimination and generalization in identification and classification: Comment on Nosofsky. Journal of Experimental Psychology: General, 115, 58–61.CrossRefGoogle Scholar
Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237, 1317–1323.CrossRefGoogle Scholar
Shepard, R. N. (1988). Time and distance in generalization and discrimination: Comment on Ennis (1988). Journal of Experimental Psychology, General, 117, 415–416.CrossRefGoogle Scholar
Shepard, R. N. (1988). Toward a universal law of generalization: Response. Science, 242, 944–944.CrossRefGoogle Scholar
Thurstone, L. L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Torgerson, W. S. (1958). Theory and methods of scaling, New York: Wiley.Google Scholar
Zinnes, J. L., & MacKay, D. B. (1987). Probabilistic multidimensional analysis of preference ratio judgements. Communications and Cognition, 20, 17–44.Google Scholar