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.
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References
Aitkin, M., Anderson, D., Francis, B., & Hinde, J. (1989).Statistical modelling in GLIM. Oxford: Clarendon Press.
Baker, R. J. (1985). GLIM 3.77 Reference guide. InGLIM 3.77 reference manual. Oxford: Numerical Algorithms Group, Royal Statistical Society.
Beaver, R. J. (1977a). Weighted least-squares analysis of several univariate Bradley-Terry models.Journal of the American Statistical Association, 72, 629–634.
Beaver, R. J. (1977b). Weighted least squares response surface fitting in factorial paired comparisons.Communications in Statistics—Theory and Methods, A6(13), 1275–1287.
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge: M.I.T. Press.
Bradley, R. A. (1976). Science, statistics and paired comparisons.Biometrics, 32, 213–232.
Bradley, R. A. (1984). Paired comparisons: some basic procedures and examples. In P. R. Krishnaiah & P. K. Sen (Eds.),Handbook of statistics, Volume 4, (pp. 299–326). Amsterdam: North-Holland.
Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons.Biometrika, 39, 324–345.
Critchlow, D. E. (1985).Metric methods for analyzing partially ranked data (Lecture Notes in Statistics, Volume 34). New York: Springer-Verlag.
David, H. A. (1988),The method of paired comparisons (2nd ed.). London: Charles Griffin & Company.
Davidson, R. R. (1970). On extending the Bradley-Terry model to incorporate within-pair order effects,Biometrics, 33, 693–702.
Diaconis, P. (1988).Group representations in probability and statistics. Hayward, CA: Institute of Mathematical Statistics.
Fienberg, S. E. (1970). An interative procedure for estimation in contingency tables.Annals of Mathematical Statistics, 41, 901–917.
Fienberg, S. E. (1979). Log linear representation for paired comparison models with ties and within-pair order effects.Biometrics, 35, 479–481.
Fienberg, S. E., & Larntz, K. (1976). Log linear representation for paired and multiple comparison models.Biometrika, 63, 245–254.
Fligner, M. A., & Verducci, J. S. (1986). Distance-based ranking models.Journal of the Royal Statitical Society, Series B, 83, 859–869.
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.
Grizzle, J. E., Starmer, C. F., & Koch, G. G. (1969). Analysis of categorical data by linear models.Biometrics, 25, 489–504.
Harris, W. P. (1957). A revised law of comparative judgment.Psychometrika, 22, 189–198.
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.
Lindsey, J. K. (1989).The analysis of categorical data using GLIM (Lecture Notes in Statistics, Volume 56). New York: Springer-Verlag.
Mallows, C. L. (1957). Non-null ranking models. I.Biometrika, 44, 114–130.
McCullagh, P., & Nelder, J. A. (1989).Generalized linear models (2nd ed.). New York: Chapman and Hall.
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.
Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models.Journal of the Royal Statistical Society, Series A, 135, 370–384.
Pendergrass, R. N., & Bradley, R. A. (1960). Ranking in triple comparisons. In I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow, & H. B. Mann (Eds.),Contributions to probability and statistics: Essays in honor of Harold Hotelling, (pp. 331–351). Palo Alto, CA: Stanford University Press.
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.
Sadasivan, G. (1983). Within-pair order effects in paired comparisons.Studia Scientiarum Mathematicarum Hungarica, 18, 229–238.
Sinclair, C. D. (1982). GLIM for preference. In R. Gilchrist (Ed.),GLIM 82: Proceedings of the international conference on generalised linear models (pp. 164–178, Lecture Notes in Statistics, Volume 14). New York: Springer-Verlag.
Springall, A. (1973). Response surface fitting using a generalization of the Bradley-Terry paired comparison model.Applied Statistics, 22, 59–68.
Thurstone, L. L. (1927). A law of comparative judgment.Psychological Review, 34, 273–286.
Vargo, M. D. (1989).Microbiological spoilage of a moderate acid food system using a dairy-based salad dressing model. Unpublished masters thesis, Ohio State University, Department of Food Science and Nutrition, Columbus, OH.
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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.
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Critchlow, D.E., Fligner, M.A. Paired comparison, triple comparison, and ranking experiments as generalized linear models, and their implementation on GLIM. Psychometrika 56, 517–533 (1991). https://doi.org/10.1007/BF02294488
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DOI: https://doi.org/10.1007/BF02294488