Mixed Bradley-Terry Models

In a basic Bradley-Terry model for paired comparisons, a separate ’ability’ is estimated for each ’player’ and the odds of player i beating player j is given by the ratio of these abilities. The model may be represented as either a logistic regression model or a log-linear model and fitted using standard glm software or more specialised packages, such as BradleyTerry or prefmod in R. Often however, the substantive interest lies not in estimating the abilities of a particular set of players, but modelling player abilities by player covariates. Usually this is implemented by replacing the individual player abilities by linear predictors in the glm model. An important drawback of this approach is that it does not allow for variability between players with the same covariate values. Clearly this can be overcome by incorporating random effects for the players. Whilst mixed Bradley-Terry models could be handled in principle by any glmm software, specialised software is desirable to enable natural representation of the models and provide useful summary functions. Here we present recent developments of the BradleyTerry package, which extend its capabilities to fit mixed Bradley-Terry models and also to incorporate contest-specific effects.