Permutation Tests: Multivariate

Abstract

This article deals with the multivariate extension 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 univariate transformation of q-dimensional individual 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 univariate situations. Problems in (iii) are more difficult and are solvable if the permutation testing principle is applicable and if the overall hypotheses are broken down in a finite set of subhypotheses, each provided with an unbiased partial test. The methodological tool that allows us to cope with these kind of problems is the NonParametric Combination (NPC) of dependent permutation tests. To show the potential of NPC, an example with multivariate mixed data and restricted alternatives is discussed. Among the many applications of NPC, we mention the following: the one-way Multivariate Analysis of Variance (MANOVA), when some of the variables are quantitative and others categorical; analysis of repeated measures with dependent random effects and dependent errors; analysis of multivariate restricted alternatives; analysis of problems when some data are missing and the underlying missing process is not ignorable; analysis of multivariate ordered categorical data; the multivariate Behrens–Fisher problem; some multivariate stochastic dominance problems; exact separate tests for main effects and interactions in balanced and unbalanced factorial designs; closed testing procedures; and multiaspect testing problems. One important feature of the NPC methodology, which is particularly useful for most applications, is that it frees the researcher from the need to define and model the dependence relations among responses and partial tests. Keywords: closed testing; combined tests; conditional inference; multiple tests; nonparametric test combination; permutation tests; randomization tests; restricted alternatives; stochastic dominance