An omnibus test for the global null hypothesis

Abstract: Global hypothesis tests are a useful tool in the context of, e.g, clinical trials, genetic studies or meta analyses, when researchers are not interested in testing individual hypotheses, but in testing whether none of the hypotheses is false. There are several possibilities how to test the global null hypothesis when the individual null hypotheses are independent. If it is assumed that many of the individual null hypotheses are false, combinations tests have been recommended to maximise power. If, however, it is assumed that only one or a few null hypotheses are false, global tests based on individual test statistics are more powerful (e.g., Bonferroni or Simes test). However, usually there is no a-priori knowledge on the number of false individual null hypotheses. We therefore propose an omnibus test based on the combination of p-values. We show that this test yields an impressive overall performance. The proposed method is implemented in the R-package omnibus.