MM algorithms for generalized Bradley-Terry models

The Bradley-Terry model for paired comparisons is a simple and muchstudied 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 for 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.

[1]  E. Zermelo.  Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung , 1929 .

[2]  P. Moran.  On the method of paired comparisons. , 1947, Biometrika.

[3]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS , 1952 .

[4]  R. A. Bradley,et al.  RANK ANALYSIS OF INCOMPLETE BLOCK DESIGNS THE METHOD OF PAIRED COMPARISONS , 1952 .

[5]  R. A. Bradley,et al.  Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons , 1952 .

[6]  O. Dykstra.  A Note on the Rank Analysis of Incomplete Block Designs -- Applications beyond the Scope of Existing Tables , 1956 .

[7]  L. R. Ford.  Solution of a Ranking Problem from Binary Comparisons , 1957 .

[8]  L. R. Ford.  Solution of a Ranking Problem from Binary Comparisons , 1957 .

[9]  R. Duncan Luce,et al.  Individual Choice Behavior , 1959 .

[10]  Ralph A. Bradley,et al.  RANKING IN TRIPLE COMPARISONS , 1959 .

[11]  H. A. David,et al.  The method of paired comparisons , 1966 .

[12]  P. V. Rao,et al.  Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model , 1967 .

[13]  R. Davidson.  On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments , 1970 .

[14]  Roger R. Davidson,et al.  A Bibliography on the Method of Paired Comparisons , 1973 .

[15]  R. Plackett.  The Analysis of Permutations , 1975 .

[16]  J. Magnus,et al.  Matrix Differential Calculus with Applications , 1988 .

[17]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics (Revised Edition) , 1999 .

[18]  H. A. David,et al.  The Method of Paired Comparisons (2nd ed.). , 1989 .

[19]  Ina Ruck,et al.  USA , 1969, The Lancet.

[20]  Xiao-Li Meng,et al.  Using EM to Obtain Asymptotic Variance-Covariance Matrices: The SEM Algorithm , 1991 .

[21]  A. Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[22]  Alan Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[23]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[24]  Xiao-Li Meng,et al.  Maximum likelihood estimation via the ECM algorithm: A general framework , 1993 .

[25]  S. Stigler.  Citation Patterns in the Journals of Statistics and Probability , 1994 .

[26]  D. Curtis,et al.  An extended transmission/disequilibrium test (TDT) for multi‐allele marker loci , 1995, Annals of human genetics.

[27]  K. Lange.  A gradient algorithm locally equivalent to the EM algorithm , 1995 .

[28]  D. A. Wolf.  Recent advances in descriptive multivariate analysis , 1996 .

[29]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[30]  J. Marden.  Analyzing and Modeling Rank Data , 1996 .

[31]  Robert Tibshirani,et al.  Classification by Pairwise Coupling , 1997, NIPS.

[32]  Yi-Ching Yao,et al.  Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons , 1999 .

[33]  Xiao-Li Meng,et al.  [Optimization Transfer Using Surrogate Objective Functions]: Discussion , 2000 .

[34]  Kenneth Lange,et al.  [Optimization Transfer Using Surrogate Objective Functions]: Rejoinder , 2000 .