Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model

Abstract The Bradley-Terry model for a paired-comparison experiment with t treatments postulates a set of t ‘true’ treatment ratings π1, π2, · · ·, π t such that π i ≥ 0, ∑ π i = 1 and the probability for preferring treatment i to treatment j is π i (π i + π j )−1. Thus, according to this model, every comparison of two treatments results in a definite preference for one of the two. This is an unrealistic restriction since when there is no difference between the responses due to two treatments, any method of expressing preference for one over the other is somewhat arbitrary. This paper considers a modification of the Bradley-Terry model by introducing an additional parameter, called threshold parameter, into the model. This permits ‘ties’ in the model. The problem of estimation and tests of hypotheses for the parameters of the modified model is also dealt with in the paper.