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Paired Comparison, Triple Comparison, and Ranking Experiments as Generalized Linear Models, and Their Implementation on GLIM

Published online by Cambridge University Press:  01 January 2025

Douglas E. Critchlow  and
Michael A. Fligner
Douglas E. Critchlow
Affiliation:
Department of Statistics, The Ohio State University
Michael A. Fligner*
Affiliation:
Department of Statistics, The Ohio State University
*
Requests for reprints should be sent to Michael A. Fligner, Department of Statistics, The Ohio State University, 1958 Neil Avenue, Columbus, Ohio 43210.
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Abstract

A wide variety of paired comparison, triple comparison, and ranking experiments may be viewed as generalized linear models. These include paired comparison models based on both the Bradley-Terry and Thurstone-Mosteller approaches, as well as extensions of these models that allow for ties, order of presentation effects, and the presence of covariates. Moreover, the triple comparison model of Pendergrass and Bradley, as well as models for complete rankings of more than three items, can also be represented as generalized linear models. All such models can be easily fit by maximum likelihood, using the widely available GLIM computer package. Additionally, GLIM enables the computation of likelihood ratio statistics for testing many hypotheses of interest. Examples are presented that cover a variety of cases, along with their implementation on GLIM.

Keywords

paired comparisonsranking modelsgeneralized linear modelsBradley-Terry modelsThurstone-Mosteller models
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 3 , September 1991 , pp. 517 - 533
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The work of both authors was supported by the National Science Foundation. The first author was also supported by an Ohio State University Seed Grant. The authors thank Edward Richter and Maria Vargo Thomas for supplying the salad dressing data in Example 3, and the Associate Editor and referees for suggestions which led to a much improved version of the manuscript.

References

Aitkin, M., Anderson, D., Francis, B., Hinde, J. (1989). Statistical modelling in GLIM, Oxford: Clarendon Press.Google Scholar
Baker, R. J. (1985). GLIM 3.77 Reference guide. GLIM 3.77 reference manual, Oxford: Numerical Algorithms Group, Royal Statistical Society.Google Scholar
Beaver, R. J. (1977). Weighted least-squares analysis of several univariate Bradley-Terry models. Journal of the American Statistical Association, 72, 629–634.CrossRefGoogle Scholar
Beaver, R. J. (1977). Weighted least squares response surface fitting in factorial paired comparisons. Communications in Statistics—Theory and Methods, A6(13), 1275–1287.CrossRefGoogle Scholar
Bishop, Y. M. M., Fienberg, S. E., Holland, P. W. (1975). Discrete multivariate analysis: Theory and practice, Cambridge: M.I.T. Press.Google Scholar
Bradley, R. A. (1976). Science, statistics and paired comparisons. Biometrics, 32, 213–232.CrossRefGoogle ScholarPubMed
Bradley, R. A. (1984). Paired comparisons: some basic procedures and examples. In Krishnaiah, P. R., Sen, P. K. (Eds.), Handbook of statistics, Volume 4 (pp. 299–326). Amsterdam: North-Holland.Google Scholar
Bradley, R. A., Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 39, 324–345.Google Scholar
Critchlow, D. E. (1985). Metric methods for analyzing partially ranked data (Lecture Notes in Statistics, Volume 34), New York: Springer-Verlag.CrossRefGoogle Scholar
David, H. A. (1988). The method of paired comparisons 2nd ed.,, London: Charles Griffin & Company.Google Scholar
Davidson, R. R. (1970). On extending the Bradley-Terry model to incorporate within-pair order effects. Biometrics, 33, 693–702.CrossRefGoogle Scholar
Diaconis, P. (1988). Group representations in probability and statistics, Hayward, CA: Institute of Mathematical Statistics.CrossRefGoogle Scholar
Fienberg, S. E. (1970). An interative procedure for estimation in contingency tables. Annals of Mathematical Statistics, 41, 901–917.CrossRefGoogle Scholar
Fienberg, S. E. (1979). Log linear representation for paired comparison models with ties and within-pair order effects. Biometrics, 35, 479–481.CrossRefGoogle Scholar
Fienberg, S. E., Larntz, K. (1976). Log linear representation for paired and multiple comparison models. Biometrika, 63, 245–254.CrossRefGoogle Scholar
Fligner, M. A., Verducci, J. S. (1986). Distance-based ranking models. Journal of the Royal Statitical Society, Series B, 83, 859–869.Google Scholar
Goodman, L. A. (1968). The analysis of cross-classified data: Independence, quasi-independence, and interactions in contingency tables with or without missing entries. Journal of the American Statistical Association, 63, 1091–1131.Google Scholar
Grizzle, J. E., Starmer, C. F., Koch, G. G. (1969). Analysis of categorical data by linear models. Biometrics, 25, 489–504.CrossRefGoogle ScholarPubMed
Harris, W. P. (1957). A revised law of comparative judgment. Psychometrika, 22, 189–198.CrossRefGoogle Scholar
Imrey, P. B., Johnson, W. D., Koch, G. G. (1976). Incomplete contingency table approach to paired-comparison experiments. Journal of the American Statistical Association, 71, 614–623.CrossRefGoogle Scholar
Lindsey, J. K. (1989). The analysis of categorical data using GLIM (Lecture Notes in Statistics, Volume 56), New York: Springer-Verlag.CrossRefGoogle Scholar
Mallows, C. L. (1957). Non-null ranking models. I.. Biometrika, 44, 114–130.CrossRefGoogle Scholar
McCullagh, P., Nelder, J. A. (1989). Generalized linear models 2nd ed.,, New York: Chapman and Hall.CrossRefGoogle Scholar
Mosteller, F. (1951). Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika, 16, 3–9.CrossRefGoogle Scholar
Nelder, J. A., Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series A, 135, 370–384.CrossRefGoogle Scholar
Pendergrass, R. N., Bradley, R. A. (1960). Ranking in triple comparisons. In Olkin, I., Ghurye, S. G., Hoeffding, W., Madow, W. G., Mann, H. B. (Eds.), Contributions to probability and statistics: Essays in honor of Harold Hotelling (pp. 331–351). Palo Alto, CA: Stanford University Press.Google Scholar
Rao, P. V., Kupper, L. L. (1967). Ties in paired-comparison experiments: A generalization of the Bradley-Terry model. Journal of the American Statistical Association, 62, 194–204.CrossRefGoogle Scholar
Sadasivan, G. (1983). Within-pair order effects in paired comparisons. Studia Scientiarum Mathematicarum Hungarica, 18, 229–238.Google Scholar
Sinclair, C. D. (1982). GLIM for preference. In Gilchrist, R. (Eds.), GLIM 82: Proceedings of the international conference on generalised linear models (pp. 164–178). New York: Springer-Verlag.CrossRefGoogle Scholar
Springall, A. (1973). Response surface fitting using a generalization of the Bradley-Terry paired comparison model. Applied Statistics, 22, 59–68.CrossRefGoogle Scholar
Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Vargo, M. D. (1989). Microbiological spoilage of a moderate acid food system using a dairy-based salad dressing model, Columbus, OH: Ohio State University, Department of Food Science and Nutrition.Google Scholar