Improving power of multivariate combination-based permutation tests

Developing powerful hypothesis testing procedures devoted at comparing multivariate populations is quite a common and relevant topic either from the methodological and the practical point of view and in this connection the NonParametric Combination (NPC) permutation methodology provides a more flexible and effective background for many multivariate testing problems (Pesarin and Salmaso in Permutation tests for complex data: theory, applications and software, 2010a). The goal of this paper is to propose some specific procedures aimed at possibly improving power of NPC Tests in the context of the additive linear model. It will be shown by an extensive simulation study, the improved-in-power NPC Tests are certainly good alternatives with respect to the traditional multivariate tests such as Hotelling T (2) and multivariate rank-based tests, especially in cases of heavy-tailed distributions. Moreover, the NPC methodology offers several advantages since it provides robust solutions with respect to the true underlying random error distribution and it is not affected by the problem of the loss of degrees of freedom when keeping fixed the number of observations. Indeed, unlike traditional methods, when the number of informative variables increases its power monotonically increases as well (leading to the so-called finite-sample consistency property of NPC Test, Pesarin and Salmaso in J. Nonparametr. Stat. 22(5):669-684, 2010b).