Minimax Rates and Adaptivity in Combining Experimental and Observational Data

TL;DR

This paper explores how integrating observational data with randomized controlled trials can improve effect estimation efficiency, revealing that adaptation is possible for estimation but impossible for inference due to confounding bias.

Contribution

It provides the first theoretical minimax analysis of data integration from RCTs and observational studies, proposing an adaptive estimator and highlighting fundamental limits for inference.

Findings

Adaptive estimator attains optimal convergence rate for estimation.

Efficiency gain from data integration is achievable without knowing confounding bias.

Adaptation for confidence interval length is generally impossible due to confounding.

Abstract

Randomized controlled trials (RCTs) are the gold standard for evaluating the causal effect of a treatment; however, they often have limited sample sizes and sometimes poor generalizability. On the other hand, non-randomized, observational data derived from large administrative databases have massive sample sizes and better generalizability, but they are prone to unmeasured confounding bias. It is thus of considerable interest to reconcile effect estimates obtained from randomized controlled trials and observational studies investigating the same intervention, potentially harvesting the best from both realms. In this paper, we theoretically characterize the potential efficiency gain of integrating observational data into the RCT-based analysis from a minimax point of view. For estimation, we derive the minimax rate of convergence for the mean squared error, and propose a fully adaptive anchored thresholding estimator that attains the optimal rate up to poly-log factors. For inference, we characterize the minimax rate for the length of confidence intervals and show that adaptation (to unknown confounding bias) is in general impossible. A curious phenomenon thus emerges: for estimation, the efficiency gain from data integration can be achieved without prior knowledge on the magnitude of the confounding bias; for inference, the same task becomes information-theoretically impossible in general. We corroborate our theoretical findings using simulations and a real data example from the RCT DUPLICATE initiative [Franklin et al., 2021b].