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Modeling Within-Subject Dependencies in Ordinal Paired Comparison Data

Published online by Cambridge University Press:  01 January 2025

Ulf Böckenholt  and
William R. Dillon
Ulf Böckenholt*
Affiliation:
University of Illinois
William R. Dillon*
Affiliation:
Southern Methodist University
*
Requests for reprints should be sent to Ulf Böckenholt, Department of Psychology, University of Illinois, Champaign, IL 61820, or to William R. Dillon, School of Business, Southern Methodist University, Dallas, TX 75275.
Requests for reprints should be sent to Ulf Böckenholt, Department of Psychology, University of Illinois, Champaign, IL 61820, or to William R. Dillon, School of Business, Southern Methodist University, Dallas, TX 75275.
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Abstract

This paper presents two probabilistic models based on the logistic and the normal distribution for the analysis of dependencies in individual paired comparison judgments. It is argued that a core assumption of latent class choice models, independence of individual decisions, may not be well-suited for the analysis of paired comparison data. Instead, the analysis and interpretation of paired comparison data may be much simplified by allowing for within-person dependencies that result from repeated evaluations of the same options in different pairs. Moreover, by relating dependencies among the individual-level responses to (in)consistencies in the judgmental process, we show that the proposed graded paired comparison models reduce to ranking models under certain conditions. Three applications are presented to illustrate the approach.

Keywords

repeated measurementsrankingsintransitivitiesThurstonian models
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 3 , September 1997 , pp. 411 - 434
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

This research was partially supported by NSF grant SBR-9409531. The authors are grateful to the reviewers, Alan Agresti and Herbert Hoijtink for their helpful comments on this research.

References

Agresti, A. (1992). Analysis of ordinal paired comparison data. Applied Statistics, 41, 287–297.CrossRefGoogle Scholar
Bishop, Y. M. M., Fienberg, S. E., Holland, P. W. (1975). Discrete multivariate analysis, Cambridge: MIT Press.Google Scholar
Babington-Smith, B. (1950). Discussion of Professor Ross' paper. Journal of the Royal Statistical Society, Series B, 12, 153–162.Google Scholar
Bock, R. D., Jones, L. V. (1968). The measurement and prediction of judgment and choice, San Francisco: Holden-Day.Google Scholar
Böckenholt, U. (1992). Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, 45, 31–49.CrossRefGoogle Scholar
Böckenholt, I., Gaul, W. (1986). Analysis of choice behavior via probabilistic ideal point and vector models. Applied Stochastic Models and Data Analysis, 2, 209–226.CrossRefGoogle Scholar
Bradley, R. A., Terry, M. E. (1952). Rank analysis of incomplete block designs: The method of paired comparisons. Biometrika, 39, 324–345.Google Scholar
Carroll, J. D., DeSoete, G. (1991). Toward a new paradigm for the study of multiattribute choice behavior. American Psychologist, 46, 342–351.CrossRefGoogle Scholar
Cook, W. D., Kress, M. (1985). Ordinal ranking with intensity of preference. Management Science, 31, 26–32.CrossRefGoogle Scholar
David, H. A. (1988). The method of paired comparisons, London: Griffin.Google Scholar
Davidson, R. R. (1970). On extending the Bradley-Terry model to accommodate ties in paired comparison experiments. Journal of the American Statistical Association, 65, 317–328.CrossRefGoogle Scholar
DeSoete, G., Winsberg, S. (1993). A Thurstonian pairwise choice model with univariate and multivariate spline transformations. Psychometrika, 58, 233–256.CrossRefGoogle Scholar
Dillon, W. R., Kumar, A., de Borrero, M. (1993). Capturing individual differences in paired comparisons: An extended BTL model incorporating descriptor variables. Journal of Marketing Research, 30, 42–51.CrossRefGoogle Scholar
Edwards, A. L., Thurstone, L. L. (1952). An interval consistency check for the method of successive intervals and the method of graded dichotomies. Psychometrika, 17, 169–180.CrossRefGoogle Scholar
Fienberg, S. E., Larntz, K. (1976). Log-linear representation for paired and multiple comparisons models. Biometrika, 63, 245–254.CrossRefGoogle Scholar
Formann, A. K. (1989). Constrained latent class models: Some further applications. The British Journal of Mathematical and Statistical Psychology, 42, 37–54.CrossRefGoogle Scholar
Formann, A. K. (1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 87, 476–486.CrossRefGoogle Scholar
Glenn, W. A., David, H. A. (1960). Ties in paired comparison experiments using a modified Thurstone-Mosteller model. Biometrics, 16, 86–109.CrossRefGoogle Scholar
Goodman, L. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, 537–552.CrossRefGoogle Scholar
Haberman, S. (1988). A stabilized Newton-Raphson algorithm for log-linear models for frequency tables derived by indirect observations. In Clogg, C. C. (Eds.), Sociological methodology 1988 (pp. 193–212). Oxford: Blackwell.Google Scholar
Halff, H. M. (1976). Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 14, 244–246.CrossRefGoogle Scholar
Joe, H. (1988). Majorization, entropy and paired comparisons. Annals of Statistics, 16, 915–925.CrossRefGoogle Scholar
Kendall, M. G. (1962). Ranks and measures. Biometrika, 49, 133–137.CrossRefGoogle Scholar
Kendall, M. G. (1975). Rank correlation methods, London: Griffin.Google Scholar
Kroeger, K. (1992). Unpublished data set. University of Illinois, Urbana-Champaign.Google Scholar
Luce, R. D. (1959). Individual choice behavior, New York: Wiley.Google Scholar
Mellers, B. A., Biagini, K. (1994). Similarity and choice. Psychological Review, 101, 505–518.CrossRefGoogle Scholar
Morrison, H. W. (1963). Testable conditions for triads of paired comparison choices. Psychometrika, 28, 369–390.CrossRefGoogle Scholar
Park, C. W. (1978). A seven-point scale and a decision-maker's simplifying choice strategy: An operationalized satisficing-plus model. Organizational Behavior and Human Performance, 21, 252–271.CrossRefGoogle Scholar
Pendergrass, P. N., Bradley, R. A. (1960). Ranking in triple comparison. In Olkin, I. (Eds.), Contributions to probability and statistics (pp. 331–351). Palo Alto: Stanford University Press.Google Scholar
Rumelhart, D. L., Greeno, J. G. (1971). Similarity between stimuli: An experimental test of the Luce and Restle choice models. Journal of Mathematical Psychology, 8, 370–381.CrossRefGoogle Scholar
Sjöberg, L. (1967). Successive categories scaling of paired comparisons. Psychometrika, 32, 297–308.CrossRefGoogle Scholar
Stevens, S. S. (1986). Psychophysics: Introduction to its perceptual, neural, and social prospects, New Brunswick: Transaction Books.Google Scholar
Takane, Y. (1980). Maximum likelihood estimation in the generalized cases of Thurstone's law of comparative judgment. Japanese Psychological Research, 22, 188–196.CrossRefGoogle Scholar
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary choice data. Cognition and Communication, 20, 45–62.Google Scholar
Takane, Y., deLeeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408.CrossRefGoogle Scholar
Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Torgerson, W. S. (1958). Theory and method of scaling, New York: John Wiley & Sons.Google Scholar
Tutz, G. (1986). Bradley-Terry-Luce models with an ordered response. Journal of Mathematical Psychology, 30, 306–316.CrossRefGoogle Scholar
Tversky, A. (1969). Intransitivity of preference. Psychological Review, 76, 31–48.CrossRefGoogle Scholar
Vermunt, J. K. (1993). Lem: Log-linear and event history analysis with missing data using the EM algorithm. Unpublished manuscript, Tilburg University.Google Scholar
van Acker, P. (1990). Transitivity revisited. Annals of Operations Research, 23, 1–35.CrossRefGoogle Scholar
Yakowitz, S. J., Spragins, (1968). On the identifiability of finite mixtures. Annals of Mathematical Statistics, 39, 209–214.CrossRefGoogle Scholar