Design Stability in Adaptive Experiments: Implications for Treatment Effect Estimation

TL;DR

This paper investigates how adaptive treatment assignment affects the estimation of average treatment effects, introducing the concept of design stability and providing methods for valid inference.

Contribution

It introduces the concept of design stability for adaptive experiments and derives asymptotic properties for IPW and AIPW estimators under this framework.

Findings

Central limit theorems established for IPW and AIPW estimators under design stability.

Explicit formulas for asymptotic variances of the estimators.

Proposed variance estimators enable valid confidence intervals in adaptive experiments.

Abstract

We study the problem of estimating the average treatment effect (ATE) under sequentially adaptive treatment assignment mechanisms. In contrast to classical completely randomized designs, we consider a setting in which the probability of assigning treatment to each experimental unit may depend on prior assignments and observed outcomes. Within the potential outcomes framework, we propose and analyze two natural estimators for the ATE: the inverse propensity weighted (IPW) estimator and an augmented IPW (AIPW) 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, non-random limits. Our main results establish central limit theorems for both the IPW and AIPW 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 and Efron's biased coin design, highlighting the applicability of the proposed methods to sequential experimentation with adaptive randomization.