  High-dimensional tests are applied to find relevant sets of variables and
relevant models. If variables are selected by analyzing the sums of products
matrices and a corresponding mean-value test is performed, there is the danger
that the nominal error of first kind is exceeded. In the paper, well-known
multivariate tests receive a new mathematical interpretation such that the
error of first kind of the combined testing and selecting procedure can more
easily be kept. The null hypotheses on mean values are replaced by hypotheses
on distributional sphericity of the individual score responses. Thus, model
choice is possible without too strong restrictions. The method is presented for
all linear multivariate designs. It is illustrated by an example from
bioinformatics: The selection of gene sets for the comparison of groups of
patients suffering from B-cell lymphomas.
