The Random Power Function for Tests Based on Pivotal Quantities

In clinical trials planning, evaluation of the probability of success of an experiment is of central interest, for instance, in sample size determination. This assessment typically involves analyses of the power function of a test on a parameter of interest, such as a relevant treatment effect. In this article, we adopt a hybrid frequentist-Bayesian approach that is lately becoming more and more popular in the literature. Specifically, we focus on superiority trials, and we study the distribution of the power function induced by a design prior assigned to the parameter. Under mild assumptions, we derive general expressions for the cumulative and density functions of the random power in terms of its inverse. We then specialise this result to tests based on pivotal quantities, and we consider some classes of problems, both exact and asymptotic, conventionally employed in clinical trials. Ideas are exposed by resorting to four biomedical settings adapted from real data applications.