Random power and friends: hybrid Bayesian-frequentist approaches in clinical trials design

Expected values may fail in representing the distributions they summarize. This thesis investigates how this issue affects the evaluation of the design of an experiment, especially in clinical trials, when a hybrid Bayesian-frequentist approach is employed. We focus on the probability of success (PoS) of a test, which is originally (but not uniquely) defined as the expected value of the random power, ie the traditional power function of a test with respect to a design prior assigned to the parameter under scrutiny. We review and compare alternative definitions of PoS, we investigate the distributions they summarize (the random power and friends), and we provide a decision-theoretic look at the problem which leads to a unifying, uncontroversial quantification of success. We then go beyond clinical trials and hypothesis testing, and we study the Bayes risk as a synthesis of the random risk function in the point and set estimation classes of problems. Results, discussions and comparisons are supported by theoretical results and accompanied by biomedical examples and applications.