A Unified Theory of Two-Stage Adaptive Designs

Adaptation of a clinical trial occurs when there is a change from the planned course of action in design, sample size, or method of analysis. There are many proposed adaptations in the literature, but no comprehensive mathematical theory to support them. The standard Neyman-Pearson theory is not suitable for the setting, and its use may result in invalid inference. This article provides a general formulation of adaptation, under which its validity can be broadly and rigorously established. The basic theory on distribution, stochastic independence, point estimation, confidence intervals, hypothesis testing, and asymptotics is developed. Various examples are given to illustrate applications of the theory. The goal is to provide a solid theoretical foundation to aid the development of adaptive methods for clinical investigations.