Identifying main effects and interactions among exposures using Gaussian processes

Abstract

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.

Powering Research through Innovative Methods for Mixtures in Epidemiology (PRIME) Program: Novel and Expanded Statistical Methods

“…Another major advancement from PRIME is refined methods for estimating and testing interactions within an exposure-response framework. Ferrari et al developed Mixselect, which uses a Gaussian process to parameterize the multivariate exposure-response surface and partitions this surface into main effects and higher order interactions [16]. In closely related work, Antonelli et al employed Bayesian sparsity priors with a semiparametric regression framework to produce variable importance scores for each exposure in the mixture as well as for each pairwise interaction (NLinteraction) [29].…”

“…Ferrari et al developed Bayesian factor analysis models (FIN) that do well modeling pairwise and higher-order interactions among many variables (see Section 3.5), and because it parameterizes the full exposure-response relationship to be one implied by the association between the outcome and a smaller number of factors, a full exposure-response surface can be estimated by the method [12]. Ferrari et al followed this up with MixSelect, which decomposes a Gaussian process regression, a form of kernel regression with a Gaussian kernel, into main effects and interaction components [16]. These estimates could also be used to obtain an overall exposure response relationship.…”

Spike‐and‐slab least absolute shrinkage and selection operator generalized additive models and scalable algorithms for high‐dimensional data analysis

“…Such integration can be achieved by setting up a structural hyperprior on the inclusion indicator of the smooth function null space $$$ {\boldsymbol{\gamma}}^0 $$$. A similar strategy has been used in Ferrari and Dunson 44 …”

Spike-and-Slab LASSO Generalized Additive Models and Scalable Algorithms for High-Dimensional Data Analysis

“…Such integration can be achieved by setting up a structural hyperprior on the inclusion indicator of the smoothing function null space 0 . The similar strategy has been used in Ferrari and Dunson 50 .…”