Advances in the design of Bayesian clinical trials: error-based sample size determination and fast approximation of operating characteristics

Designing clinical trials requires selecting key features such as the sample size, interim decision rules, and adaptation strategies. These choices are evaluated through operating characteristics (OCs), including Type I and Type II error probabilities, power, and expected sample size. In Bayesian settings, formal error control and efficient OC evaluation remain central methodological challenges, particularly for complex designs. This thesis develops methodological tools to support the design of Bayesian clinical trials along two complementary lines of research. The first line of research concerns sample size determination based on the joint control of predictive error probabilities. We derive Bayesian analogues of frequentist Type I and Type II errors by introducing two design prior distributions, one under each hypothesis, and computing predictive error probabilities. We propose criteria based on weighted or joint control of these errors to determine optimal sample sizes in general testing problems, and we illustrate the approach in randomized controlled trials. The same framework is extended to construct a two-stage design for binary outcomes, in which the optimal design is selected by jointly controlling predictive error probabilities and the posterior probability of correctly proceeding at each stage. The second line of research addresses the computational cost of evaluating OCs. In most settings, OC evaluation relies on Monte Carlo simulation, which can become prohibitively expensive as design complexity increases. We introduce the Q-approximation of OCs, a general strategy based on a quadratic approximation of the log-likelihood combined with asymptotic arguments. The method applies to designs that satisfy the likelihood principle, including multi-stage designs with early stopping, adaptively randomized designs, and designs incorporating external data. In numerical experiments, standard Monte Carlo estimation required between 150 and 1900 times more computing time than the Q-approximation to achieve comparable accuracy.