Moderated Paired Comparisons: A Generalized Bradley–Terry Model for Continuous Data Using a Discontinuous Penalized Likelihood Function

Abstract

Summary Standard paired comparison models are widely used in circumstances where a measure of preference is available on a sequence or collection of pairwise comparisons between a group of objects or treatments. Typically, the measure of preference observations takes the form of simple binary outcomes, indicating which of the two objects or treatments in a particular comparison is preferred. Common examples of such situations include models of choice behaviour in politics or marketing, comparisons of medical treatments or in the realm of sports rankings. We investigate situations where the observed measure of preference for the paired comparisons is instead a continuous outcome indicating not simply the direction of preference but the degree of preference as well; in particular, we present, as a motivating example, a ranking analysis of the top 12 international limited overs cricket teams, with an appropriately defined margin of victory in individual matches playing the role of the measure of preference. We propose a new method, which is termed moderated paired comparisons, that is based on fitting a penalized likelihood model to the observed margins of victory. Importantly, the structure of the penalty function chosen allows for the model to assign differential importance to the information that is contained solely in the dichotomous win–loss outcome of a match as against that contained in the actual margin of the victory.