On the Limiting Behavior of the “Probability of Claiming Superiority” in a Bayesian Context

Abstract

In the context of Bayesian sample size determination in clinical trials, a quantity of interest is the marginal probability that the posterior probability of an alternative hypothesis of interest exceeds a specified threshold. This marginal probability is the same as “average power”; that is, the average of the power function with respect to the prior distribution when using a test based on a Bayesian rejection region. We give conditions under which this marginal probability (or average power) converges to the prior probability of the alternative hypothesis as the sample size increases. This same large sample behavior also holds for the average power of a (frequentist) consistent test. We also examine the limiting behavior of “conditional average power”; that is, power averaged with respect to the prior distribution conditional on the alternative hypothesis being true.