Tractable Ridge Regression for Paired Comparisons

Paired-comparison models, such as Bradley–Terry and Thurstone–Mosteller, are commonly used to estimate relative strengths of pairwise compared items in tournament-style data. We discuss estimation of paired-comparison models with a ridge penalty. A new approach is derived which combines empirical Bayes and composite likelihoods without any need to refit the model, as a convenient alternative to cross-validation of the ridge tuning parameter. Simulation studies demonstrate much better predictive accuracy of the new approach relative to ordinary maximum likelihood. A widely used alternative, the application of a standard bias-reducing penalty, is also found to improve appreciably the performance of maximum likelihood; but the ridge penalty, with tuning as developed here, yields greater accuracy still. The methodology is illustrated through application to 28 seasons of English Premier League football.