On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the proposed model yields a reasonable representation of actual experimentation.