Nonparametric relevance-shifted multiple testing procedures for the analysis of high-dimensional multivariate data with small sample sizes

This article addresses the use of relevance-shifted tests on ratios for a multivariate parallel two-sample group design. Two empirical procedures are proposed which embed the relevance-shifted test on ratios. As both procedures test a hypothesis for each variable, the resulting multiple testing problem has to be considered. Hence, the procedures include a multiplicity correction. Both procedures are extensions of available procedures for point null hypotheses achieving exact control of the familywise error rate. Whereas the shift of the null hypothesis alone would give straight-forward solutions, the problems that are the reason for the empirical considerations discussed here arise by the fact that the shift is considered in both directions and the whole parameter space in between these two limits has to be accepted as null hypothesis.The first algorithm to be discussed uses a permutation algorithm, and is appropriate for designs with a moderately large number of observations. However, many experiments have limited sample sizes. Then the second procedure might be more appropriate, where multiplicity is corrected according to a concept of data-driven order of hypotheses.Nowadays, high-dimensional multivariate data are used in agriculture, biology and medicine. A recent example are microarray data, where two groups, for example normal and diseased tissue, are compared using tens of thousands of genes. The aim is to identify those genes with relevant over- or under-expression. Therefore, only two-sided tests are considered here. Nevertheless, directional one-sided relevance-shifted versions are also available [1]. Distinguishing between formal statistical significance and biological relevance is a frequently discussed issue [2]. One reason is that the commonly used point-zero null-hypothesis H0 : μ2 - μ1 = 0 is often biologically inappropriate, because depending on sample size and variance, biologically irrelevant small differences can be marked as statistically differ