Fitting Bradley Terry Models Using a Multiplicative Algorithm

Abstract

We consider the problem of estimating the parameters of a Bradley Terry Model by the method of maximum likelihood, given data from a paired comparisons experiment. The parameters of a basic model can be taken to be weights which are positive and sum to one. Hence they correspond to design weights and optimality theorems and numerical techniques developed in the optimal design arena can be transported to this estimation problem. Furthermore extensions of the basic model to allow for a factorial structure in the treatments leads to an optimisation problem with respect to several sets of weights or distributions. We can extend techniques to this case. In section 1 we introduce the notion of paired comparisons experiments and the Bradley Terry Model. In section 2 the parameter estimation problem is outlined with optimality results and a general class of multiplicative algorithms outlined in sections 3 and 4 respectively. A specific algorithm is applied to the Bradley Terry log-likelihood in section 5 and treatments with a factorial structure are considered in section 6. Finally in section 7 extensions to triple comparisons and to extended rankings are briefly outlined.