Permutation Tests: Multivariate

Abstract

Abstract This article deals with the multivariate methodology of permutation (or randomization) tests. Three different situations are met: (i) the first assumes that a single overall statistic is available; (ii) the second takes into consideration a unidimensional transformation of q ‐dimensional responses; (iii) in the third a single overall statistic is not available, or is too difficult to justify, or a single derived variable is not sufficient to capture all aspects of interest for the analysis. Testing problems in (i) and (ii) are equivalent to unidimensional ones. Problems in (iii) require recurs to the permutation testing principle provided that the global hypotheses are broken down into a finite set of subhypotheses, each provided with an unbiased partial test, and solution is found by acting in accordance with Roy's union‐intersection (UI) principle. The methodological tool to cope with these problems is the nonparametric combination (NPC) of dependent permutation tests (PTs). To illustrate the potential of UI‐NPC, two examples with multivariate data and restricted alternatives are briefly discussed. One important feature of the UI‐NPC methodology is that it frees researchers from the necessity to model and to estimate the dependence coefficients on responses and on partial tests.