An approach to multivariate rank tests in multivariate analysis of variance

A class of multivariate rank-like quantities is defined and used to develop multivariate tests to mimic popular one-dimensional rank tests such as the Mann-Whitney/Wilcoxon two-sample test, the Jonckheere-Terpstra test for trend, and the Kruskal-Wallis one-way analysis of variance test. Tests in one-way analysis of variance are developed based on qualitative orthogonal contrasts, allowing decomposition of an overall statistic into asymptotically independent components based on the contrasts. The class of tests includes the usual normal-theory tests and the componentwise rank tests, but the main focus is on the tests based on a particular definition of multivariate rank. A study of the Pitman efficiency of the latter tests to those based on multivariate medians shows them to be superior at the normal, slightly heavy-tailed, and light-tailed distributions, whereas the median-based tests are superior for heavy tails. These results are analogous to the univariate case.