Bayesian kernel machine regression for estimating the health effects of multi-pollutant mixtures

Abstract

Because humans are invariably exposed to complex chemical mixtures, estimating the health effects of multi-pollutant exposures is of critical concern in environmental epidemiology, and to regulatory agencies such as the U.S. Environmental Protection Agency. However, most health effects studies focus on single agents or consider simple two-way interaction models, in part because we lack the statistical methodology to more realistically capture the complexity of mixed exposures. We introduce Bayesian kernel machine regression (BKMR) as a new approach to study mixtures, in which the health outcome is regressed on a flexible function of the mixture (e.g. air pollution or toxic waste) components that is specified using a kernel function. In high-dimensional settings, a novel hierarchical variable selection approach is incorporated to identify important mixture components and account for the correlated structure of the mixture. Simulation studies demonstrate the success of BKMR in estimating the exposure-response function and in identifying the individual components of the mixture responsible for health effects. We demonstrate the features of the method through epidemiology and toxicology applications.

Keywords: Air pollution; Bayesian variable selection; Environmental health; Gaussian process regression; Metal mixtures.

Fig. 1.

Median (25%, 75%) of the PIPs from BKMR with component-wise variable selection, across 300 simulated datasets for each of three true functions. The vector of exposure data was generated either based on the Bangladesh data with mixture components, where the truly associated components were Pb for , and Pb and Mn for and ; or based on the Boston air pollution data with mixture components, where the truly associated components were Al for , and Al and Cu for and . The proportion of simulation iterations for which each mixture component had -value <0.05 under the garrote test for KMR is printed below the -axis. ; ; ; ; ; ; ; ; ; ; ; carbon; ; ; .

Fig. 2.

Median (25%, 75%) of the PIPs from BKMR with hierarchical variable selection, across 300 simulated datasets for each of three true functions. Exposure data were generated based on the Boston air pollution data with mixture components categorized within eight groups. The truly associated components were Al (one of four pollutants in group 1) for , and Al and Cu (sole pollutant in group 5) for and . Plots on left show the PIPs for each group, and plots on the right show the conditional PIPs for the components in group 1 given that group 1 was included in the model.

Fig. 3.

Relationship of manganese (Mn) and arsenic (As) with the MCS, for lead (Pb) fixed at its median. (A) Posterior mean of the bivariate exposure-response function for Mn and As. Horizontal lines correspond to the 10th, 50th, and 90th percentiles of As. (B) Posterior SD of . Points correspond to the observed data points. (C) Relationship of Mn with MCS at three levels of As together with pointwise 95% credible intervals.

Fig. 4.

PIPs for the toxicology application estimated from BKMR with component-wise variable selection (A) and hierarchical variable selection (B). Left panel shows group-specific PIPs and right panel shows conditional inclusion probabilities for the components in group 1.