Some Equivalence Classes in Paired Comparisons

In a paired comparison experiment n judges give a preference in some or all of the (t/2) pairs of t items. Frequently the purpose of the experiment is to test the null hypothesis that every preference is equally likely against a vaguely defined alternative of consistency. Our purpose is to study several of the tests used, from the point of view of a natural equivalence relation which arises in graph theory. In the first section we introduce graph theory notation, the equivalence relation, and some results on partial and strict orderings on the equivalence classes. The succeeding section applies these notions to Kendall and Babington Smith's statistic in detail (hereafter simply referred to as Kendall's statistic), and mentions applications in the Bradley-Terry model, and the strong-stochastic ordering model.