The Annals of Statistics

MM algorithms for generalized Bradley-Terry models

David R. Hunter

Full-text: Open access


The Bradley-Terry model for paired comparisons is a simple and much-studied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization-maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms or a wide class of generalizations of the Bradley-Terry model. While algorithms for problems of this type have tended to be custom-built in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new.

Article information

Ann. Statist. Volume 32, Number 1 (2004), 384-406.

First available in Project Euclid: 12 March 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F07: Ranking and selection 65D15: Algorithms for functional approximation

Bradley-Terry model Luce's choice axiom maximum likelihood estimation MM algorithm Newton-Raphson Plackett-Luce model


Hunter, David R. MM algorithms for generalized Bradley-Terry models. Ann. Statist. 32 (2004), no. 1, 384--406. doi:10.1214/aos/1079120141.

Export citation


  • Agresti, A. (1990). Categorical Data Analysis. Wiley, New York.
  • Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39 324--345.
  • David, H. A. (1988). The Method of Paired Comparisons, 2nd ed. Oxford Univ. Press.
  • Davidson, R. R. (1970). On extending the Bradley--Terry model to accommodate ties in paired comparison experiments. J. Amer. Statist. Assoc. 65 317--328.
  • Davidson, R. R. and Farquhar, P. H. (1976). A bibliography on the method of paired comparisons. Biometrics 32 241--252.
  • Dykstra, O., Jr. (1956). A note on the rank of incomplete block designs---applications beyond the scope of existing tables. Biometrics 12 301--306.
  • Ford, L. R., Jr. (1957). Solution of a ranking problem from binary comparisons. Amer. Math. Monthly 64 28--33.
  • Hastie, T. and Tibshirani, R. (1998). Classification by pairwise coupling. Ann. Statist. 26 451--471.
  • Heiser, W. J. (1995). Convergent computation by iterative majorization. In Recent Advances in Descriptive Multivariate Analysis (W. J. Krzanowski, ed.) 157--189. Oxford Univ. Press.
  • Hunter, D. R. and Lange, K. (2000). Rejoinder to discussion of ``Optimization transfer algorithms using surrogate objective functions.'' J. Comput. Graph. Statist. 9 52--59.
  • Lange, K. (1995). A gradient algorithm locally equivalent to the EM algorithm. J. Roy. Statist. Soc. Ser. B 57 425--437.
  • Lange, K., Hunter, D. R. and Yang, I. (2000). Optimization transfer using surrogate objective functions (with discussion). J. Comput. Graph. Statist. 9 1--59.
  • Luce, R. D. (1959). Individual Choice Behavior. Wiley, New York.
  • Magnus, J. R. and Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, New York.
  • Marden, J. I. (1995). Analyzing and Modeling Rank Data. Chapman and Hall, London.
  • McLachlan, G. J. and Krishnan, T. (1997). The EM Algorithm and Extensions. Wiley, New York.
  • Meng, X.-L. and Rubin, D. B. (1991). Using EM to obtain asymptotic variance--covariance matrices: The SEM algorithm. J. Amer. Statist. Assoc. 86 899--909.
  • Meng, X.-L. and Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80 267--278.
  • Pendergrass, R. N. and Bradley, R. A. (1960). Ranking in triple comparisons. In Contributions to Probability and Statistics (O. Olkin et al., eds.) 331--351. Stanford Univ. Press.
  • Plackett, R. L. (1975). The analysis of permutations. Appl. Statist. 24 193--202.
  • Rao, P. V. and Kupper, L. L. (1967). Ties in paired-comparison experiments: A generalization of the Bradley--Terry model. J. Amer. Statist. Assoc. 62 194--204. [Corrigendum J. Amer. Statist. Assoc. 63 1550--1551.]
  • Sham, P. C. and Curtis, D. (1995). An extended transmission/disequilibrium test (TDT) for multi-allele marker loci. Ann. Human Genetics 59 323--336.
  • Simons, G. and Yao, Y.-C. (1999). Asymptotics when the number of parameters tends to infinity in the Bradley--Terry model for paired comparisons. Ann. Statist. 27 1041--1060.
  • Stigler, S. M. (1994). Citation patterns in the journals of statistics and probability. Statist. Sci. 9 94--108.
  • Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung. Math. Z. 29 436--460.