  There is wide interest in studying how the distribution of a continuous
response changes with a predictor. We are motivated by environmental
applications in which the predictor is the dose of an exposure and the response
is a health outcome. A main focus in these studies is inference on dose levels
associated with a given increase in risk relative to a baseline. Popular
methods either dichotomize the continuous response or focus on modeling changes
with the dose in the expectation of the outcome. Such choices may lead to
information loss and provide inaccurate inference on dose-response
relationships. We instead propose a Bayesian convex mixture regression model
that allows the entire distribution of the health outcome to be unknown and
changing with the dose. To balance flexibility and parsimony, we rely on a
mixture model for the density at the extreme doses, and express the conditional
density at each intermediate dose via a convex combination of these extremal
densities. This representation generalizes classical dose-response models for
quantitative outcomes, and provides a more parsimonious, but still powerful,
formulation compared to nonparametric methods, thereby improving
interpretability and efficiency in inference on risk functions. A Markov chain
Monte Carlo algorithm for posterior inference is developed, and the benefits of
our methods are outlined in simulations, along with a study on the impact of
DDT exposure on gestational age.
