Use of Nonnull Models for Rank Statistics in Bivariate, Two-Sample, and Analysis of Variance Problems

A nonnull model for rank statistics based on Kendall's tau has received quite a bit of attention in the ranking literature lately. In this article, we analyze this model with an eye towards applying it to situations in which nonparametric statistics such as Kendall's tau, the Mann-Whitney/Wilcoxon U, the Jonckheere-Terpstra statistic, the Kruskal-Wallis statistic, and Friedman's statistic for randomized blocks designs are often used. The data for the basic model are a pair of vectors of length n, where both are permutations of the first n integers or one is a fixed and arbitrary design vector and the other a permutation. The model is Mallow's φ model, a one-parameter exponential family with natural statistic the number of discordant pairs between the two vectors. This model originated with Mann and has appeared in many articles since. We extend the model to cases in which there may be ties in the observations and to analysis of variance problems (in which there are several parameters). The first four sections describe the models and verify the asymptotic normality of the statistics and the maximum likelihood estimators of the parameters, as well as the asymptotic chi-squared nature of the likelihood ratio statistics under the null hypotheses. In Section 5 we compare the model to some other typical models. Section 6 contains examples, including analysis of the 1970 Draft Lottery data, and a 2 × 2 × 2 analysis of variance design.