Design Stability in Adaptive Experiments: Implications for Treatment Effect Estimation

Abstract We study the problem of estimating the average treatment effect under sequentially adaptive treatment assignment mechanisms. In contrast to classical completely randomized designs, the setting we consider is one in which the probability of assigning treatment to each experimental unit may depend on prior assignments and observed outcomes. Within the potential outcomes framework (Neyman, 1923), we propose and analyse two natural estimators for the average treatment effect: the inverse propensity weighted estimator and an augmented inverse propensity weighted estimator. The cornerstone of our analysis is the concept of design stability, which requires that as the number of units grows, either the assignment probabilities converge, or sample averages of the inverse propensity scores and of the inverse complement propensity scores converge in probability to fixed, nonrandom limits. Our main results establish central limit theorems for both estimators under design stability and provide explicit expressions for their asymptotic variances. We further propose estimators for these variances, enabling the construction of asymptotically valid confidence intervals. Finally, we illustrate our theoretical results in the context of Wei’s adaptive coin design (Wei, 1978) and Efron’s biased coin design (Efron, 1971), highlighting the applicability of our results to sequential experimental designs with adaptive randomization.