The distribution of the power function induced by a mixture design prior

In the hybrid Bayesian-frequentist approach to testing, the power function, i.e. the probability of rejecting the null hypothesis, is a random variable whose distribution is induced by a design prior that embodies the goal of the trial. Inspection of the probability distribu- tion of the power is useful to establish whether its expected value, the so-called Probability of Success (PoS), is adequate or not for sample size determination (SSD). In this article we consider the use of a mix- ture prior for those trials where multiple scenarios have to be taken into consideration at the design stage. We provide closed-form results for the distribution of the power and PoS in the case of a normal model. The crucial role of mixture components’ weights on the evaluation of success and on SSD are illustrated in an application to the design of a superiority trial.