Bayesian kernel machine regression for heteroscedastic health outcome data

The field of environmental epidemiology has placed an increasing emphasis on understanding the health effects of mixtures of metals, chemicals, and pollutants in recent years. Bayesian Kernel Machine Regression (BKMR) is a statistical method that has gained significant traction in environmental mixture studies due to its ability to account for complex non-linear relationships between the exposures and health outcome and its ability to identify interaction effects between the exposures. However, BKMR makes the crucial assumption that the error terms have a constant variance, and this assumption is not typically checked in practice. In this paper, we create a diagnostic function for checking this constant variance assumption in practice and develop Heteroscedastic BKMR (HBKMR) for environmental mixture analyses where this assumption is not met. By specifying a Bayesian hierarchical variance model for the error term variance parameters, HBKMR produces updated estimates of the environmental mixture's health effects and their corresponding 95% credible intervals. We apply HBKMR in two real-world case studies that motivated this work: 1) Examining the effects of prenatal metal exposures on behavioral problems in toddlers living in Suriname and 2) Assessing the impacts of metal exposures on simple reaction time in children living near coal-fired power plants in Kentucky. In both case studies, HBKMR provides a substantial improvement in model fit compared to BKMR, with differences in some of the mixture effect estimates and typically narrower 95% credible intervals after accounting for the heteroscedasticity.