Tractable Ridge Regression for Paired Comparisons

TL;DR

This paper introduces a new ridge regression approach for paired comparison models that improves predictive accuracy and is computationally efficient, demonstrated through simulations and application to English Premier League data.

Contribution

A novel estimation method combining empirical Bayes and composite likelihoods for paired comparison models, avoiding re-fitting and cross-validation for tuning.

Findings

Ridge penalty improves predictive accuracy over maximum likelihood.

The new approach outperforms standard bias-reducing penalties.

Application to football data demonstrates practical utility.

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 re-fit 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.