Multi-Source Causal Inference Using Control Variates Under Outcome Selection Bias

Abstract

While many areas of machine learning have benefited from the increasing availability of large and varied datasets, the benefit to causal inference has been limited given the strong assumptions needed to ensure the identifiability of causal effects -- which are often not satisfied in real-world datasets. For example, many large observational datasets (e.g., case-control studies in epidemiology, click-through data in recommender systems) suffer from selection bias on the outcome, which makes the average treatment effect (ATE) non-identifiable. We propose an algorithm to estimate causal effects from multiple data sources, where the ATE may be identifiable only in some datasets but not others. The idea is to construct control variates across the datasets in which the ATE may not be identifiable, which provably reduces the variance of the ATE estimate. We focus on a setting where the observational datasets suffer from outcome selection bias, assuming access to an auxiliary small dataset from which we can obtain a consistent estimate of the ATE. We propose a construction of control variate by taking the difference of the conditional odds ratio estimates from the two datasets. Across simulations and two case studies with real data, we show that the control variate-based ATE estimator has consistently and significantly reduced variance against different baselines.