Efficient Statistical Estimation for Sequential Adaptive Experiments with Implications for Adaptive Designs

Adaptive experimental designs have gained popularity in clinical trials and online experiments. Unlike traditional, fixed experimental designs, adaptive designs can dynamically adjust treatment randomization probabilities and other design features in response to data accumulated sequentially during the experiment. These adaptations are useful to achieve diverse objectives, including reducing uncertainty in the estimation of causal estimands or increasing participants' chances of receiving better treatments during the experiment. At the end of the experiment, it is often desirable to answer causal questions from the observed data. However, the adaptive nature of such experiments and the resulting dependence among observations pose significant challenges to providing valid statistical inference and efficient estimation of causal estimands. Building upon the Targeted Maximum Likelihood Estimator (TMLE) framework tailored for adaptive designs (van der Laan, 2008), we introduce a new Adaptive-Design-Likelihood-based TMLE (ADL-TMLE) to estimate a wide class of causal estimands from adaptive experiment data, including the average treatment effect as our primary example. We establish asymptotic normality and semiparametric efficiency of ADL-TMLE under relaxed positivity and design stabilization assumptions for adaptive experiments. Motivated by these results, we further propose a novel adaptive design aimed at minimizing the variance of the estimator based on data generated under that design. Simulations show that ADL-TMLE demonstrates superior variance-reduction performance across different types of adaptive experiments, and that the proposed adaptive design attains lower variance than the standard efficiency-oriented adaptive design. Finally, we generalize our framework to broader settings, including those with longitudinal structures.