A general consonance principle for closure tests based on p-values

The closure principle is a powerful approach to constructing efficient testing procedures controlling the familywise error rate in the strong sense. For small numbers of hypotheses and the setting of independent elementary $p$-values we consider closed tests where each intersection hypothesis is tested with a $p$-value combination test. Examples of such combination tests are the Fisher combination test, the Stouffer test, the Omnibus test, the truncated test, or the Wilson test. Some of these tests, such as the Fisher combination, the Stouffer, or the Omnibus test, are not consonant and rejection of the global null hypothesis does not always lead to rejection of at least one elementary null hypothesis. We develop a general principle to uniformly improve closed tests based on $p$-value combination tests by modifying the rejection regions such that the new procedure becomes consonant. For the Fisher combination test and the Stouffer test, we show by simulations that this improvement can lead to a substantial increase in power.