Finite-sample consistency of combination-based permutation tests with application to repeated measures designs

In several application fields, e.g. genetics, image and functional analysis, several biomedical and social experimental and observational studies, etc. it may happen that the number of observed variables is much larger than that of subjects. It can be proved that, for a given and fixed number of subjects, when the number of variables increases and the noncentrality parameter of the underlying population distribution increases with respect to each added variable, then power of multivariate permutation tests based on Pesarin's combining functions [Pesarin, F. (2001), Multivariate Permutation Tests with Applications in Biostatistics, New York: Wiley, Chichester] is monotonically increasing. These results confirm and extend those presented by [Blair, Higgins, Karniski and Kromrey (1994), 'A Study of Multivariate Permutation Tests which May Replace Hotelling's T2 Test in Prescribed Circumstances', Multivariate Behavioral Research 29, 141-163]. Moreover, they allow us to introduce the property of finite-sample consistency for those kinds of combination-based permutation tests. Sufficient conditions are given in order that the rejection rate converges to one, for fixed sample sizes at any attainable -values, when the number of variables diverges. A simulation study and a real case study are presented