There is abundant interest in assessing the joint effects of multiple exposures on human health. This is often referred to as the mixtures problem in environmental epidemiology and toxicology. Classically, studies have examined the adverse health effects of different chemicals one at a time, but there is concern that certain chemicals may act together to amplify each other's effects. Such amplification is referred to as synergistic interaction, while chemicals that inhibit each other's effects have antagonistic interactions. Current approaches for assessing the health effects of chemical mixtures do not explicitly consider synergy or antagonism in the modeling, instead focusing on either parametric or unconstrained nonparametric dose response surface modeling. The parametric case can be too inflexible, while nonparametric methods face a curse of dimensionality that leads to overly wiggly and uninterpretable surface estimates. We propose a Bayesian approach that decomposes the response surface into additive main effects and pairwise interaction effects, and then detects synergistic and antagonistic interactions. Variable selection decisions for each interaction component are also provided. This Synergistic Antagonistic Interaction Detection (SAID) framework is evaluated relative to existing approaches using simulation experiments and an application to data from NHANES.
There is a rich literature on Bayesian nonparametric methods for unknown densities. The most popular approach relies on Dirichlet process mixture models. These models characterize the unknown density as a kernel convolution with an unknown almost surely discrete mixing measure, which is given a Dirichlet process prior. Such models are very flexible and have good performance in many settings, but posterior computation relies on Markov chain Monte Carlo algorithms that can be complex and inefficient. As a simple and general alternative, we propose a class of nearest neighbor-Dirichlet processes. The approach starts by grouping the data into neighborhoods based on standard algorithms. Within each neighborhood, the density is characterized via a Bayesian parametric model, such as a Gaussian with unknown parameters. Assigning a Dirichlet prior to the weights on these local kernels, we obtain a simple pseudo-posterior for the weights and kernel parameters. A simple and embarrassingly parallel Monte Carlo algorithm is proposed to sample from the resulting pseudo-posterior for the unknown density. Desirable asymptotic properties are shown, and the methods are evaluated in simulation studies and applied to a motivating dataset in the context of classification.
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