A 2 X 2 Factorial with Paired Comparisons

Bradley and Terry [2] have presented a method of paired comparisons depending on ranking within the incomplete blocks of size two. The method may be regarded as a generalization of the sign test. The model has recently been reviewed [3, 9] and its appropriateness examined with reference to certain experimental results. It is the purpose of this paper to consider treatments that form a factorial set in paired comparisons. A brief review of the basic model for paired comparisons will assist in understanding the modifications to be introduced. t parameters are defined for t treatments, T1 , *.. , T,, in a paired comparisons experiment with n repetitions of the (2) comparisons. The treatment parameters are 7r1, * r, -rt with 7ri > 0, i = 1, *. , t, and 7ri-= 1. It is postulated that the parameters are such that, if Xi is an observation on Tj , then