Tractable Ridge Regression for Paired Comparisons

ABSTRACT 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.