Comparison of exact parametric tests for high-dimensional data

Several proposals for exact or at least conservative parametric multivariate tests in the general linear model are considered that are applicable also for high-dimensional data, where the dimension of the observations may exceed the sample size. The common feature is the inclusion of principal component transformations into the test. Whereas a test proposal by Srivastava and von Rosen [Srivastava, M.S., von Rosen, D., 2004. MANOVA with singular variance matrix. Acta et Commentationes Universitatis Tartuensis de Mathematica 8, 253-269] originally assumes that the multivariate data have a known reduced rank which is used in the construction of the test, several versions of so-called PC tests by Läuter and colleagues accept a reduction of variance and utilize it for a "stabilization" of the test in terms of power. The different tests are compared with respect to their philosophy as well as in their performance in two real data examples and in simulation studies. It is shown that the test of Srivastava and von Rosen is a conservative test, even when a rank is assumed that is smaller than the true one. It is, however, less conservative than the conservative version of the PC test derived for the construction of a convenient confidence region for the investigated effects. The exact PC tests turns out to have the largest power.