  An important goal of environmental health research is to assess the risk
posed by mixtures of environmental exposures. Two popular classes of models for
mixtures analyses are response-surface methods and exposure-index methods.
Response-surface methods estimate high-dimensional surfaces and are thus highly
flexible but difficult to interpret. In contrast, exposure-index methods
decompose coefficients from a linear model into an overall mixture effect and
individual index weights; these models yield easily interpretable effect
estimates and efficient inferences when model assumptions hold, but, like most
parsimonious models, incur bias when these assumptions do not hold. In this
paper we propose a Bayesian multiple index model framework that combines the
strengths of each, allowing for non-linear and non-additive relationships
between exposure indices and a health outcome, while reducing the
dimensionality of the exposure vector and estimating index weights with
variable selection. This framework contains response-surface and exposure-index
models as special cases, thereby unifying the two analysis strategies. This
unification increases the range of models possible for analyzing environmental
mixtures and health, allowing one to select an appropriate analysis from a
spectrum of models varying in flexibility and interpretability. In an analysis
of the association between telomere length and 18 organic pollutants in the
National Health and Nutrition Examination Survey (NHANES), the proposed
approach fits the data as well as more complex response-surface methods and
yields more interpretable results.
