Estimate an Exposure Response Function with Negative Controls: A Bayesian Nonparametric Approach

Unmeasured confounding bias threatens the validity of observational studies. Although sensitivity analyses and study designs have been proposed to address this problem, they often overlook the growing availability of auxiliary data. Using negative controls from these data is a promising new approach to reduce unmeasured confounding bias. In this article, we develop a Bayesian nonparametric method to estimate a causal exposureresponse function (CERF) using information from negative controls to adjust for unmeasured confounding for continuous exposures. We model the CERF as a mixture of linear models. This strategy captures the potential nonlinear shape of CERFs while maintaining computational efficiency, and it leverages closed-form results that hold under the linear model assumption. We assess the performance of our method through simulation studies. We find that the proposed method can recover the true shape of the CERF in the presence of unmeasured confounding under assumptions. To show the practical utility of our approach, we apply it to adjust for a possible unmeasured confounder when evaluating the relationship between long-term exposure to ambient PM2.5 and cardiovascular hospitalization rates among the elderly in the continental US. We implement our estimation procedure in open source software and have made the code publicly available to ensure reproducibility.