P-values based on approximate conditioning and p*

Abstract

P-values based on higher order asymptotic formulas such as p * are now readily available for practitioners. However, it is not always clear what these P-values mean for discrete models. For a canonical parameter, p* should approximate a tail probability of the conditional distribution. Yet when this conditional distribution becomes degenerate, p* still gives a non-degenerate answer. So there is the need for a more general interpretation of p*. Pierce and Peters (1999) have argued that p* approximates an approximately conditional P-value and, implicitly, that this is an inferentially sensible quantity worth approximating. We investigate these twin claims for the simple case of 2 × 2 tables. W..