Statistical Inference After Adaptive Sampling for Longitudinal Data
This work addresses the need for reliable statistical inference in digital intervention experiments using online reinforcement learning, which is crucial for post-experiment analyses but is incremental as it builds on existing Z-estimation methods.
The paper tackles the problem of statistical inference in longitudinal data collected by adaptive sampling algorithms, which pool data across users to optimize treatment decisions, and shows that standard variance estimators underestimate true variance; the authors develop a corrected sandwich estimator that provides consistent variance estimates.
Online reinforcement learning and other adaptive sampling algorithms are increasingly used in digital intervention experiments to optimize treatment delivery for users over time. In this work, we focus on longitudinal user data collected by a large class of adaptive sampling algorithms that are designed to optimize treatment decisions online using accruing data from multiple users. Combining or "pooling" data across users allows adaptive sampling algorithms to potentially learn faster. However, by pooling, these algorithms induce dependence between the sampled user data trajectories; we show that this can cause standard variance estimators for i.i.d. data to underestimate the true variance of common estimators on this data type. We develop novel methods to perform a variety of statistical analyses on such adaptively sampled data via Z-estimation. Specifically, we introduce the \textit{adaptive} sandwich variance estimator, a corrected sandwich estimator that leads to consistent variance estimates under adaptive sampling. Additionally, to prove our results we develop novel theoretical tools for empirical processes on non-i.i.d., adaptively sampled longitudinal data which may be of independent interest. This work is motivated by our efforts in designing experiments in which online reinforcement learning algorithms optimize treatment decisions, yet statistical inference is essential for conducting analyses after experiments conclude.