Coherent Tests for Interval Null Hypotheses

In a celebrated 1996 article, Schervish showed that, for testing interval null hypotheses, tests typically viewed as optimal can be logically incoherent. Specifically, one may fail to reject a specific interval null, but nevertheless—testing at the same level with the same data—reject a larger null, in which the original one is nested. This result has been used to argue against the widespread practice of viewing p-values as measures of evidence. In the current work we approach tests of interval nulls using simple Bayesian decision theory, and establish straightforward conditions that ensure coherence in Schervish’s sense. From these, we go on to establish novel frequentist criteria—different to Type I error rate—that, when controlled at fixed levels, give tests that are coherent in Schervish’s sense. The results suggest that exploring frequentist properties beyond the familiar Neyman–Pearson framework may ameliorate some of statistical testing’s well-known problems.