Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-08T04:13:46.639Z Has data issue: false hasContentIssue false
  • English
  • Français

The Repeated Insertion Model for Rankings: Missing Link between Two Subset Choice Models

Published online by Cambridge University Press:  01 January 2025

Jean-Paul Doignon ,
Aleksandar Pekeč  and
Michel Regenwetter
Jean-Paul Doignon
Affiliation:
Université Libre De Bruxelles
Aleksandar Pekeč
Affiliation:
Duke University
Michel Regenwetter*
Affiliation:
University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Michel Regenwetter, Department of Psychology, University of Illinois, 603 E. Daniel St., Champaign, IL 61820. E-mail: regenwet@uiuc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Several probabilistic models for subset choice have been proposed in the literature, for example, to explain approval voting data. We show that Marley et al.'s latent scale model is subsumed by Falmagne and Regenwetter's size-independent model, in the sense that every choice probability distribution generated by the former can also be explained by the latter. Our proof relies on the construction of a probabilistic ranking model which we label the “repeated insertion model.” This model is a special case of Marden's orthogonal contrast model class and, in turn, includes the classical Mallows φ-model as a special case. We explore its basic properties as well as its relationship to Fligner and Verducci's multistage ranking model.

Keywords

Approval votingprobabilistic choice modelsprobabilistic ranking modelssubset choice
Type
Theory And Methods
Information
Psychometrika , Volume 69 , Issue 1 , March 2004 , pp. 33 - 54
Copyright
Copyright © 2004 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors are grateful to the National Science Foundation for grants SES98-18756 to Regenwetter and Pekeč, and SBR97-30076 to Regenwetter. This collaborative research was carried out in the context of the conference Random Utility 2000 held at Duke University and sponsored by NSF, the Fuqua School of Business and the Center for International Business Education and Research. We thank the editor and four referees for helpful suggestions and we are grateful to Prof. J. I. Marden for providing useful information on contrast models. We thank Moon-Ho Ho for programming and running the data analyses.

References

Ahn, H. (1995). Nonparametric 2-stage estimation of conditional choice probabilities in a binary choice model under uncertainty. Journal of Econometrics, 67, 337–378CrossRefGoogle Scholar
Atkinson, D., Wampold, B., Lowe, S., Matthews, L., Ahn, H. (1998). Asian American preferences for counselor characteristics: Application of the Bradley-Terry-Luce model to paired comparison data. Counseling Psychologist, 26, 101–123CrossRefGoogle Scholar
Babington Smith, B. (1950). Discussion on Professor Ross's paper. Journal of the Royal Statistical Society, Series B, 12, 153–162Google Scholar
Baier, D., Gaul, W. (1999). Optimal product positioning based on paired comparison data. Journal of Econometrics, 89, 365–392CrossRefGoogle Scholar
Baltas, G., Doyle, P. (2001). Random utility models in marketing research: A survey. Journal of Business Research, 51, 115–125CrossRefGoogle Scholar
Barberá, S., Pattanaik, P.K. (1986). Falmagne and the rationalizability of stochastic choices in terms of random orderings. Econometrica, 54, 707–715CrossRefGoogle Scholar
Billot, A., Thisse, J. (1999). A discrete choice model when context matters. Journal of Mathematical Psychology, 43, 518–538CrossRefGoogle ScholarPubMed
Block, H.D., Marschak, J. (1960). Random orderings and stochastic theories of responses. In Olkin, I., Ghurye, S., Hoeffding, H., Madow, W., Mann, H. (Eds.), Contributions to Probability and Statistics (pp. 97–132). Palo Alto, CA: Stanford University PressGoogle Scholar
Böckenholt, U., Dillon, W. (1997). Modeling within-subject dependencies in ordinal paired comparison data. Psychometrika, 62, 411–434CrossRefGoogle Scholar
Chen, Z., Kuo, L. (2001). A note on the estimation of the multinomial logit model with random effects. American Statistician, 55, 89–95CrossRefGoogle Scholar
Courcoux, P., Semenou, M. (1997). Preference data analysis using a paired comparison model. Food Quality and Preference, 8, 353–358CrossRefGoogle Scholar
Critchlow, D.E., Fligner, M.A., Verducci, J.S. (1991). Probability models on rankings. Journal of Mathematical Psychology, 35, 294–318CrossRefGoogle Scholar
Critchlow, D.E., Fligner, M.A., Verducci, J.S. (1993). Probability Models and Statistical Analyses for Ranking Data. New York: SpringerGoogle Scholar
Doignon, J.-P., Fiorini, S. (2003). The approval-voting polytope: Combinatorial interpretation of the facets. Mathématiques, Informatique et Sciences Humaines, 161, 29–39Google Scholar
Doignon, J.-P., & Fiorini, S. (in press). The facets and the symmetries of the approval-voting polytope. Journal of Combinatorial Theory, Series B.Google Scholar
Doignon, J.-P., Regenwetter, M. (1997). An approval-voting polytope for linear orders. Journal of Mathematical Psychology, 41, 171–188CrossRefGoogle ScholarPubMed
Doignon, J.-P., Regenwetter, M. (2002). On the combinatorial structure of the approval voting polytope. Journal of Mathematical Psychology, 46, 554–563CrossRefGoogle Scholar
Falmagne, J.-C. (1978). A representation theorem for finite random scale systems. Journal of Mathematical Psychology, 18, 52–72CrossRefGoogle Scholar
Falmagne, J.-C., Regenwetter, M. (1996). Random utility models for approval voting. Journal of Mathematical Psychology, 40, 152–159CrossRefGoogle Scholar
Fishburn, P. (2001). Signed orders, choice probabilities, and linear polytopes. Journal of Mathematical Psychology, 45, 53–80CrossRefGoogle ScholarPubMed
Fligner, M.A., Verducci, J.S. (1988). Multi-stage ranking models. Journal of the American Statistical Association, 83, 892–901CrossRefGoogle Scholar
Huang, J., Nychka, D. (2000). A nonparametric multiple choice method within the random utility framework. Journal of Econometrics, 97, 207–225CrossRefGoogle Scholar
Ichimura, H., Thompson, T.S. (1998). Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution. Journal of Econometrics, 86, 269–295CrossRefGoogle Scholar
Knuth, D.E. (1998). The Art of Computer Programming, (vol. 3: Sorting and Searching, 2nd printing). Menlo Park, CA: Addison-WesleyGoogle Scholar
Koppen, M. (1995). Random utility representation of binary choice probabilities: Critical graphs yielding critical necessary conditions. Journal of Mathematical Psychology, 39, 21–39CrossRefGoogle Scholar
Lewbel, A. (2000). Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables. Journal of Econometrics, 97, 145–177CrossRefGoogle Scholar
Mallows, C.L. (1957). Non-null ranking models I. Biometrika, 44, 114–130CrossRefGoogle Scholar
Marden, J. (1992). Use of nested orthogonal contrasts in analyzing rank data. Journal of the American Statistical Association, 87, 307–318CrossRefGoogle Scholar
Marden, J.I. (1995). Analyzing and Modeling Rank Data. London: Chapman & HallGoogle Scholar
Marley, A.A.J. (1993). Aggregation theorems and the combination of probabilistic rank orders. In Critchlow, D. E., Fligner, M. A., Verducci, J.S. (Eds.), Probability Models and Statistical Analyses for Ranking Data (pp. 216–240). New York: SpringerCrossRefGoogle Scholar
McFadden, D., Train, K. (2000). Mixed MNL models for discrete response. Journal of Applied Econometrics, 15, 447–4703.0.CO;2-1>CrossRefGoogle Scholar
Norpoth, H. (1979). The parties come to order! Dimensions of preferential choice in the West German Electorate, 1961–1976. The American Political Science Review, 73, 724–736CrossRefGoogle Scholar
Peterson, G., Brown, T. (1998). Economic valuation by the method of paired comparison, with emphasis on evaluation of the transitivity axiom. Land Economics, 74, 240–261CrossRefGoogle Scholar
Regenwetter, M., Doignon, J.-P. (1998). The choice probabilities of the latent-scale model satisfy the size-independent model whenn is small. Journal of Mathematical Psychology, 42, 102–106CrossRefGoogle Scholar
Regenwetter, M., Marley, A., Joe, H. (1998). Random utility threshold models of subset choice. Australian Journal of Psychology: Special issue on mathematical psychology, 50, 175–185CrossRefGoogle Scholar
Tsai, R. (2000). Remarks on the identifiability of Thurstonian ranking models: Case V, case III, or neither?. Psychometrika, 65, 233–240CrossRefGoogle Scholar
Zeng, L. (2000). A heteroscedastic generalized extreme value discrete choice model. Sociological Methods & Research, 29, 118–144CrossRefGoogle Scholar