A major shortcoming of the Bradley-Terry model is that the maximum likelihood estimates are infinite-valued in the presence of separation and may be unreliable when data are nearly separated. A well-known solution consists of the addition of Firth' s penalty term to the log-likelihood function, and solve this penalized likelihood through logistic regression.The maximum likelihood estimates with and without Firth's penalty are compared in a large and heterogeneous population of table-tennis players. We additionally show that exact penalized maximum likelihood estimates can be reasonably approximated using a well-chosen Minorization-Maximization (MM) algorithm.
"Evaluation of the Addition of Firth’s Penalty Term to the Bradley-Terry Likelihood,"
Journal of Modern Applied Statistical Methods: Vol. 15
, Article 17.
Available at: http://digitalcommons.wayne.edu/jmasm/vol15/iss2/17