  Humans are routinely exposed to mixtures of chemical and other environmental
factors, making the quantification of health effects associated with
environmental mixtures a critical goal for establishing environmental policy
sufficiently protective of human health. The quantification of the effects of
exposure to an environmental mixture poses several statistical challenges. It
is often the case that exposure to multiple pollutants interact with each other
to affect an outcome. Further, the exposure-response relationship between an
outcome and some exposures, such as some metals, can exhibit complex, nonlinear
forms, since some exposures can be beneficial and detrimental at different
ranges of exposure. To estimate the health effects of complex mixtures we
propose a flexible Bayesian approach that allows exposures to interact with
each other and have nonlinear relationships with the outcome. We induce
sparsity using multivariate spike and slab priors to determine which exposures
are associated with the outcome, and which exposures interact with each other.
The proposed approach is interpretable, as we can use the posterior
probabilities of inclusion into the model to identify pollutants that interact
with each other. We illustrate our approach's ability to estimate complex
functions using simulated data, and apply our method to two studies to
determine which environmental pollutants adversely affect health.
