Abstract
Bradley–Terry models parsimoniously summarize preferences expressed in paired comparisons of objects through a set of latent “merit” or “preference” parameters linked to the individual objects rather than to the pairs. When comparisons of all pairs of k objects are observed, the resulting k(k - 1)/2 linearly independent proportions are reduced to k - 1 latent parameters, or potentially fewer if these are further modeled using characteristics of the objects. This entry introduces the basic model, discusses its relationship with the probit, logit, and loglinear models developed in other contexts, and describes some extensions that allow for data with ties, varying degrees of preference, or simultaneous expressions of preferences along multiple dimensions of assessment.
References
- 1 Abelson, R. M. & Bradley, R. A. (1954). A 2 (2 factorial with paired comparisons, Biometrics 10, 487–502.
- 2 Agresti, A. (1990). Categorical Data Analysis. Wiley, New York, pp. 261–305.
- 3 Agresti, A. (1992). Analysis of ordinal paired comparison data, Applied Statistics 41, 287–297.
- 4 Beaver, R. J. (1974). Locally asymptotically most stringent tests for paired comparison experiments, Journal of the American Statistical Association 69, 423–427.
- 5 Beaver, R. J. & Gokhale, D. V. (1975). A model to incorporate within-pair order effects in paired comparisons, Communications in Statistics 4, 923–939.
- 6
Bhandari, S. K.,
Hande, S. N. &
Ali, M. M.
(1993).
An optimal sequential procedure for ranking pairwise compared treatments,
Calcutta Statistical Association Bulletin
27,
191–197.
10.1177/0008068319930304 Google Scholar
- 7 Bock, R. D. & Jones, L. V. (1968). The Measurement and Prediction of Judgment and Choice. Holden-Day, San Francisco.
- 8 Bradley, R. A. (1954). The rank analysis of incomplete block designs. II. Additional tables for the method of paired comparisons, Biometrika 41, 502–537.
- 9 Bradley, R. A. (1976). Science, statistics, and paired comparisons (with discussion), Biometrics 32, 213–232.
- 10 Bradley, R. A. (1984). Paired comparisons: some basic procedures and examples, in Handbook of Statistics, Vol. 4, P. R. Krishnaiah & P. K. Sen, eds. North-Holland, Amsterdam, pp. 299–326.
- 11 Bradley, R. A. (1985). Paired comparisons, in Encylopedia of Statistical Sciences, Vol. 6, S. Kotz, N. L. Johnson & C. Read, eds. Wiley, New York, pp. 555–560.
- 12 Bradley, R. A. & Terry, M. B. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons, Biometrika 39, 324–345.
- 13 Breslow, N. & Clayton, D. G. (1993). Approximate inference in generalized linear models, Journal of the American Statistical Association 88, 9–25.
- 14
Daniels, H. E.
(1969).
Round-robin tournament scores,
Biometrika
56,
295–299.
10.1093/biomet/56.2.295 Google Scholar
- 15 David, H. A. (1988). The Method of Paired Comparisons, 2nd Ed. Wiley, New York.
- 16
David, H. A. &
Andrews, D. M.
(1987).
Closed adaptive sequential paired-comparison selection procedures,
Journal of Statistical Computation and Simulation
27,
127–141.
10.1080/00949658708810986 Google Scholar
- 17 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.
- 18 Davidson, R. R. & Beaver, R. J. (1977). On extending the Bradley-Terry model to incorporate within-pair order effects, Biometrics 33, 693–702.
- 19 Davidson, R. R. & Bradley, R. A. (1969). Multivariate paired comparisons: the extension of a univariate model and associated estimation and test procedures, Biometrika 56, 81–95.
- 20 Davidson, R. R. & Bradley, R. A. (1971). A regression relationship for multivariate paired comparisons, Biometrika 58, 555–560.
- 21 Davidson, R. R. & Farquhar, P. H. (1976). A bibliography on the method of paired comparisons, Biometrics 32, 233–240.
- 22 Diggle, P. J., Liang, K.-Y. & Zeger, S. L. (1994). Analysis of Longitudinal Data. Clarendon, Oxford, pp. 146–189.
- 23 Dillon, W. R., Kumar, A. & de Borrero, M. S. (1993). Capturing individual differences in paired comparisons.: an extended BTL model incorporating descriptor variables, Journal of Marketing Research 30, 42–51.
- 24 El-Helbawy, A. T. (1992). Optimal paired comparison designs, in Order Statistics and Nonparametrics: Theory and Applications, P. K. Sen & I. A. Salama, eds. North-Holland, Amsterdam, pp. 349–361.
- 25 Fienberg, S. E. & Larntz, K. (1976). Log linear representation for paired and multiple comparisons models, Biometrika 63, 245–254.
- 26 Finney, D. J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London, pp. 349–403.
- 27
Ford, L. R., Jr
(1957).
Solution of a ranking problem from binary comparisons,
American Mathematical Monthly
64,
28–33.
10.2307/2308513 Google Scholar
- 28 Glenn, W. A. & David, H. A. (1960). Ties in paired-comparison experiments using a modified Thurstone-Mosteller model, Biometrics 16, 86–109.
- 29 Hirji, K. F., Mehta, C. R. & Patel, N. R. (1987). Computing distributions for exact logistic regression, Journal of the American Statistical Association 82, 1110–1117.
- 30 Imrey, P. B., Johnson, W. D. & Koch, G. G. (1976). An incomplete contingency table approach to paired-comparison experiments, Journal of the American Statistical Association 71, 614–623.
- 31 Kousgaard, N. (1976). Models for paired comparisons with ties, Scandinavian Journal of Statistics 3, 1–14.
- 32 Leonard, T. (1977). An alternative Bayesian approach to the Bradley-Terry model for paired comparisons, Biometrics 33, 121–132.
- 33 Luce, R. D. (1959). Individual Choice Behavior. Wiley, New York, pp. 1–37.
- 34 McCullagh, P. (1980). Regression models for ordinal data (with discussion), Journal of the Royal Statistical Society, Series B 42, 109–142.
- 35 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.
- 36 Stigler, S. S. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press, Cambridge, Mass, pp. 239–254.
- 37 Thompson, W. A., Jr & Singh, J. (1967). The use of limit theorems in paired comparison model building, Psychometrika 32, 255–264.
- 38 Thurstone, L. L. (1927). Psychophysical analysis, American Journal of Psychology 38, 368–389.
- 39 Virginia Agricultural Experiment Station (1951). Statistical Methods for Sensory Difference Tests of Food Quality: Bi-Annual Report No. 2. Virginia Agricultural Experiment Station, Blacksburg.
- 40 Walker, S. H. & Duncan, D. B. (1967). Estimation of the probability of an event as a function of several independent variables, Biometrika 54, 167–179.
- 41
Zermelo, E.
(1929).
Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung,
Mathematische Zeitschrift
29,
436–460.
10.1007/BF01180541 Google Scholar