A Bayesian Gaussian Process for Estimating a Causal Exposure Response Curve in Environmental Epidemiology

Motivated by environmental policy questions, we address the challenges of estimation, change point detection, and uncertainty quantification of a causal exposure-response function (CERF). Under a potential outcome framework, the CERF describes the relationship between a continuously varying exposure (or treatment) and its causal effect on an outcome. We propose a new Bayesian approach that relies on a Gaussian process (GP) model to estimate the CERF nonparametrically. To achieve the desired separation of design and analysis phases, we parametrize the covariance (kernel) function of the GP to mimic matching via a Generalized Propensity Score (GPS). The hyper-parameters as well as the form of the kernel function of the GP are chosen to optimize covariate balance. Our approach achieves automatic uncertainty evaluation of the CERF with high computational efficiency, and enables change point detection through inference on derivatives of the CERF. We provide theoretical results showing the correspondence between our Bayesian GP framework and traditional approaches in causal inference for estimating causal effects of a continuous exposure. We apply the methods to 520,711 ZIP-code-level observations to estimate the causal effect of long-term exposures to PM2.5, ozone, and NO2 on all-cause mortality among Medicare enrollees in the US. A computationally efficient implementation of the proposed GP models is provided in the GPCERF R package, which is available on CRAN.