[Paper Review] A Bayesian Gaussian Process for Estimating a Causal Exposure Response Curve in Environmental Epidemiology

Boyu Ren, Xiao Wu+3 more|arXiv (Cornell University)|May 7, 2021

Air Quality and Health Impacts35 references4 citations

TL;DR

This paper proposes a Bayesian Gaussian process model that nonparametrically estimates a causal exposure-response function (CERF) for continuous exposures like air pollutants, using a kernel function designed to mimic covariate balance via the Generalized Propensity Score (GPS). The method enables automatic uncertainty quantification, change point detection through derivative inference, and achieves robust performance even under GPS model misspecification, with application to PM₂.₅, ozone, and NO₂ on Medicare mortality data.

ABSTRACT

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.

Motivation & Objective

• To develop a robust, nonparametric method for estimating causal exposure-response functions (CERF) in environmental epidemiology with continuous exposures.
• To address challenges in uncertainty quantification, change point detection, and covariate balance in causal inference for continuous treatments.
• To create a scalable, computationally efficient Bayesian framework that integrates design and analysis phases through kernel parametrization inspired by GPS matching.
• To enable principled inference on the derivatives of the CERF for detecting thresholds or phase transitions in health effects.
• To provide a flexible, uncertainty-aware tool for policy-relevant environmental health research, especially for setting air quality standards.

Proposed method

• The method employs a Gaussian process (GP) with a kernel function specifically parametrized to induce covariate balance, emulating GPS-based matching in the design phase.
• Hyperparameters of the GP kernel are optimized to minimize imbalance in observed covariates, ensuring robustness to GPS model misspecification.
• The model leverages the property that derivatives of a GP are also GP processes, enabling full Bayesian inference on the derivative to detect change points.
• A computationally efficient implementation uses a nearest-neighbor GP (nnGP) approximation to scale to large datasets, such as 520,711 ZIP-code-level observations.
• The framework is embedded in a Bayesian hierarchical model that allows for uncertainty propagation from the GPS estimation to the final CERF.
• The method is implemented in the GPCERF R package on CRAN, supporting reproducibility and wide application.

Experimental results

Research questions

• RQ1 Can a Bayesian Gaussian process model nonparametrically estimate a causal exposure-response function while ensuring covariate balance?
• RQ2 How can uncertainty in the CERF be quantified automatically and efficiently in a Bayesian framework?
• RQ3 Can the derivatives of the CERF be used to detect statistically significant change points in the causal effect of continuous exposures?
• RQ4 How does the method perform under GPS model misspecification compared to existing approaches?
• RQ5 What are the policy-relevant implications of detecting non-linear thresholds in the causal effects of air pollutants on mortality?

Key findings

• The proposed Bayesian GP model achieves comparable performance to the GPS matching method of Wu et al. (2022) in estimating the CERF, with a slight increase in mean squared error but improved robustness to GPS model misspecification.
• The change point detection algorithm based on derivative inference demonstrated excellent performance in simulations, accurately identifying phase transitions in the exposure-response relationship.
• Application to US Medicare data revealed potential change points in the causal effect of PM₂.₅ on all-cause mortality, particularly in the low-to-mid exposure range, suggesting a non-linear threshold effect.
• The model successfully scaled to large datasets (520,711 observations) using the nnGP approximation, maintaining computational efficiency and accurate uncertainty quantification.
• Results were consistent with prior literature, reinforcing the need for stricter PM₂.₅ standards, especially at low exposure levels where mortality risk increases sharply.
• Posterior credible intervals were found to be overly optimistic when outcome uncertainty was ignored; future work should incorporate proper likelihood models for count data.