Doubly robust proximal causal inference under confounded outcome-dependent sampling

Outcome-dependent sampling is widely used in epidemiological and socioeconomic research to efficiently and economically study the treatment-outcome relationship. However, skepticism about unmeasured confounding bias and selection bias poses challenges for causal inference. In particular, a hidden common cause of the treatment, outcome, and sample selection process will generally induce both (a) unmeasured confounding, and (b) selection bias, thus rendering standard causal parameters unidentifiable without additional assumptions. In this article, we review and extend a recently proposed proximal causal inference approach for test-negative design studies to general outcome-dependent sampling designs, which leverages a pair of proxies of hidden factors at the source of unmeasured confounding and selection bias to jointly correct for both biases. Our contributions include: (i) a new set of conditions for the proximal identification of a causal odds ratio parameter when (a) and (b) co-exist; and (ii) a characterization of new proximal doubly robust estimators for the odds ratio effect under (a) and (b). We illustrate the performance of our estimators through extensive simulations and data from an application to University of Michigan Health System.