  A key goal of environmental health research is to assess the risk posed by
mixtures of pollutants. As epidemiologic studies of mixtures can be expensive
to conduct, it behooves researchers to incorporate prior knowledge about
mixtures into their analyses. This work extends the Bayesian multiple index
model (BMIM), which assumes the exposure-response function is a non-parametric
function of a set of linear combinations of pollutants formed with a set of
exposure-specific weights. The framework is attractive because it combines the
flexibility of response-surface methods with the interpretability of linear
index models. We propose three strategies to incorporate prior toxicological
knowledge into construction of indices in a BMIM: (a) constraining index
weights, (b) structuring index weights by exposure transformations, and (c)
placing informative priors on the index weights. We propose a novel prior
specification that combines spike-and-slab variable selection with informative
Dirichlet distribution based on relative potency factors often derived from
previous toxicological studies. In simulations we show that the proposed priors
improve inferences when prior information is correct and can protect against
misspecification suffered by naive toxicological models when prior information
is incorrect. Moreover, different strategies may be mixed-and-matched for
different indices to suit available information (or lack thereof). We
demonstrate the proposed methods on an analysis of data from the National
Health and Nutrition Examination Survey and incorporate prior information on
relative chemical potencies obtained from toxic equivalency factors available
in the literature.
