This article is motivated by the problem of inference on interactions among chemical exposures impacting human health outcomes. Chemicals often co-occur in the environment or in synthetic mixtures and as a result exposure levels can be highly correlated. We propose a latent factor joint model, which includes shared factors in both the predictor and response components while assuming conditional independence. By including a quadratic regression in the latent variables in the response component, we induce flexible dimension reduction in characterizing main effects and interactions. We propose a Bayesian approach to inference under this Factor analysis for INteractions (FIN) framework. Through appropriate modifications of the factor modeling structure, FIN can accommodate higher order interactions.

This article is motivated by the problem of studying the joint effect of different chemical exposures on human health outcomes. This is essentially a nonparametric regression problem, with interest being focused not on a black box for prediction but instead on selection of main effects and interactions. For interpretability we decompose the expected health outcome into a linear main effect, pairwise interactions and a nonlinear deviation. Our interest is in model selection for these different components, accounting for uncertainty and addressing nonidentifiability between the linear and nonparametric components of the semiparametric model. We propose a Bayesian approach to inference, placing variable selection priors on the different components, and developing a Markov chain Monte Carlo (MCMC) algorithm. A key component of our approach is the incorporation of a heredity constraint to only include interactions in the presence of main effects, effectively reducing dimensionality of the model search. We adapt a projection approach developed in the spatial statistics literature to enforce identifiability in modeling the nonparametric component using a Gaussian process. We also employ a dimension reduction strategy to sample the nonlinear random effects that aids the mixing of the MCMC algorithm. The proposed MixSelect framework is evaluated using a simulation study, and is illustrated using data from the National Health and Nutrition Examination Survey (NHANES). Code is available on GitHub.

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