Probabilistic Foundation of Confirmatory Adaptive Designs

Adaptive designs allow the investigator of a confirmatory trial to react to unforeseen developments by changing the design. This broad flexibility comes at the price of a complex statistical model where important components, such as the adaptation rule, remain unspecified. It has thus been doubted whether Type I error control can be guaranteed in general adaptive designs. This criticism is fully justified as long as the probabilistic framework on which an adaptive design is based remains vague and implicit. Therefore, an indispensable step lies in the clarification of the probabilistic fundamentals of adaptive testing. We demonstrate that the two main principles of adaptive designs, namely the conditional Type I error rate and the conditional invariance principle, will provide Type I error rate control, if the conditional distribution of the second-stage data, given the first-stage data, can be described in terms of a regression model. A similar assumption is required for regression analysis where the distribution of the covariates is a nuisance parameter and the model needs to be identifiable independently from the covariate distribution. We further show that under the assumption of a regression model, the events of an arbitrary adaptive design can be embedded into a formal probability space without the need of posing any restrictions on the adaptation rule. As a consequence of our results, artificial constraints that had to be imposed on the investigator only for mathematical tractability of the model are no longer necessary.