Gaining power in multiple testing of interval hypotheses via conditionalization

In this article, we introduce a novel procedure for improving power of multiple testing procedures (MTPs) of interval hypotheses. When testing interval hypotheses the null hypothesis $P$-values tend to be stochastically larger than standard uniform if the true parameter is in the interior of the null hypothesis. The new procedure starts with a set of $P$-values and discards those with values above a certain pre-selected threshold, while the rest are corrected (scaled-up) by the value of the threshold. Subsequently, a chosen family-wise error rate (FWER) or false discovery rate MTP is applied to the set of corrected $P$-values only. We prove the general validity of this procedure under independence of $P$-values, and for the special case of the Bonferroni method, we formulate several sufficient conditions for the control of the FWER. It is demonstrated that this "filtering" of $P$-values can yield considerable gains of power.