Log Linear Representation for Paired Comparison Models with Ties and Within-Pair Order Effects

Davidson and Beaver (1977) have considered a generalization of the Bradley-Terry model for paired comparisons in the presence of ties and which examines the effect of the order of presentation of the objects within a pair. The present note shows how their model can be represented in loglinear form, and how the results for maximum likelihood (ML) estimation follow directly from known results for loglinear models fitted to categorical data (see Bishop, Fienberg, and Holland, 1975, Fienberg 1977, or Haberman 1974). The advantages of the loglinear model approach to paired comparisons problems, as have been noted by Fienberg and Larntz (1976), are three-fold (i) each variant on a standard model need not be considered de nova, since general results are available, (ii) standard methods of solving likelihood equations can be used without new proofs of convergence, and (iii) further extensions to a class of models are often much easier to visualize when the models are represented in loglinear form.