Evaluation of the Addition of Firth's Penalty Term to the Bradley-Terry Likelihood

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 solving 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, showing that exact penalized maximum likelihood estimates can be reasonably approximated using a well-chosen Minorization-Maximization (MM) algorithm.