Sensitivity Analysis Under the f-Sensitivity Model: A Distributional Robustness Perspective

The growing availability of observational data has ushered in an exciting new era of personalized data-driven decision making. A key impediment to this vision is that observational data are generally prone to unobserved confounding, so inferential schemes that assume it away—as is often done in the literature—are inadequate. In this paper, we introduce the f-sensitivity model, which characterizes the violation of unconfoundedness by assuming that the selection bias because of unmeasured confounding is only bounded “on average” in contrast to the existing stringent uniform bound sensitivity models. As such, the f-sensitivity model allows for large unmeasured confounding by incorporating the likelihood of such a large magnitude. Within the f-sensitivity model, we propose a framework for sensitivity analysis where the optimal solutions to a new class of distributionally robust optimization (DRO) problems provide bounds to the counterfactual means and hence, the treatment effects. We then construct point estimators for these bounds by adopting a novel debiasing technique to estimate the solution of its dual problem. Our estimators are shown to (1) converge to the exact bounds when the dual problem’s optimal solution can be consistently estimated and even if not (as we operate in general nonparametric domains), (2) converge to valid bounds on counterfactual means provided that any nuisance component can be estimated consistently. We further establish asymptotic normality and Wald-type inference results for these estimators under slower-than-root-n convergence rates of the estimated nuisance components. Finally, the efficacy of our algorithms is demonstrated with several numerical experiments with synthetic data (and hence, known ground truths).

Funding: This research was supported by the Office of Naval Research [Grant 13983263], the Directorate for Engineering [Grants CCF-2312205, DMS-2413135, and ECCS-2419564], the New York University Center for Global Economy and Business [Grant 2026], the National Institutes of Health [Grants R56HG010812, R01MH113078, and R01MH123157], Army Research Office [Grant 2003514594], and Wharton Analytics.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.0001.