Sensitivity Analysis under the f-Sensitivity Models: Definition, Estimation and Inference
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The digitization of the economy has witnessed an explosive growth of available observational data across a variety of domains, which has provided an exciting opportunity for causal inference that aims to discern the effectiveness of different treatment options. However, unlike in experimental data – where the designer has the full control to randomize treatment choices based on observable covariates – causal inference using observational data faces the key challenge of confoundedness: there might exist unobservable covariates that affect both the treatment choice and the outcome. In this paper, we aim to add to the emerging line of tools for sensitivity analysis that assesses the robustness of causal conclusions against potential unmeasured confounding. We propose a new sensitivity model where, in contrast to uniform bounds on the selection bias in the literature, we assume the selection bias is bounded "on average". It generalizes the widely used Γ-marginal selection model of Tan (2006) and is particularly suitable for situations where the selection bias grows unbounded locally but, due to the impact of such region being small overall, is controlled at the population level. We study the partial identification of treatment effects under the new sensitivity model. From a distributional-shift perspective, we represent the bounds on counterfactual means via certain distributionally robust optimization programs. We then design procedures to estimate these bounds, and show that our procedure is doubly robust to the estimation error of nuisance components and remains valid even when the optimization step is off. In addition, we establish the Wald-type inference guarantee of our procedure that is again robust to the optimization step. Finally, we demonstrate our method and verify its validity with numerical experiments.
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