A robust Bayesian stopping rule for sequential trials

Dealing with sequential clinical trials in a Bayesian context, we denote by θ the parameter of interest representing treatment effect and we start formalizing pre-experimental information through a prior probability distribution on θ. By iteratively applying Bayes theorem, the posterior distribution is derived after each observation is collected. Hence we monitor the posterior probability that θ exceeds a minimally clinical relevant threshold as the sample size sequentially increases. Then the trial is terminated with success when it is larger than a given cutoff; otherwise, the treatment is declared ineffective. In this setting the sample size is a random variable associated to the chosen stopping rule. We show by simulation that its expectation is smaller than the non sequential optimal sample size. Moreover, we consider the issue of robustness with respect to the prior specification. We define a robust Bayesian stopping criterion that is, in general, more conservative than the non robust one. However, we show that, working sequentially, we can save observations even though a robust stopping rule is considered. More precisely, we evaluate the critical level of robustness we can afford to have a sample size that reaches the non sequential non robust optimal one.