Permutation Anderson-Darling type and moment-based test statistics for univariate ordered categorical data

In this article we consider permutation methods for univariate testing on ordered variables based on the nonparametric combination of permutation dependent tests (Pesarin, 2001). Several solutions have been proposed to cope with univariate testing problems on ordered categorical data, most of which are based on the restricted maximum likelihood ratio test. These solutions are generally criticized, because the degree of accuracy of their asymptotic null and alternative distributions is difficult to assess and to characterize. By working within the nonparametric combination of dependent permutation tests (NPC), it is possible to find exact solutions to that kind of problems. We examine some of these solutions and discuss one more new exact solution based on simultaneous analysis of a finite set of sampling moments of ranks, or general scores, assigned to ordered classes and processed by the NPC method. A simulation study analyzes the empirical power of the considered NPC solutions compared with the Wilcoxon test with ties correction. The new permutation test based on Fisher's combination of sampling moments shows a good behavior in power in all considered situations both for balanced and unbalanced sample sizes.