Design and Analysis Considerations for Causal Inference under Two-Phase Sampling in Observational Studies

TL;DR

This paper derives the efficiency bounds for causal effect estimators under two-phase sampling and proposes methods to improve their efficiency by utilizing phase-1 information, supported by extensive simulations.

Contribution

It provides the first semiparametric efficiency bounds for weighted average treatment effects under two-phase sampling and introduces more efficient estimators leveraging phase-1 data.

Findings

Efficiency gains are substantial when incorporating phase-1 information.

The proposed estimators outperform naive methods in simulations.

Outcome-dependent sampling benefits from phase-1 data integration.

Abstract

Two-phase sampling is a simple and cost-effective estimation strategy in survey sampling and is widely used in practice. Because the phase-2 sampling probability typically depends on low-cost variables collected at phase 1, naive estimation based solely on the phase-2 sample generally results in biased inference. This issue arises even when estimating causal parameters such as the average treatment effect (ATE), and there has been growing interest in recent years in the proper estimation of such parameters under complex sampling designs (e.g., Nattino et al., 2025). In this paper, we derive the semiparametric efficiency bound for a broad class of weighted average treatment effects (WATE), which includes the ATE, the average treatment effect on the treated (ATT), and the average treatment effect on the overlapped population (ATO), under two-phase sampling. In addition to straightforward weighting estimators based on the sampling probabilities, we also propose estimators that can attain strictly higher efficiency under suitable conditions. In particular, under outcome-dependent sampling, we show that substantial efficiency gains can be achieved by appropriately incorporating phase-1 information. We further conduct extensive simulation studies, varying the choice of phase-1 variables and sampling schemes, to characterize when and to what extent leveraging phase-1 information leads to efficiency gains.