Causal inference in spatial settings is met with unique challenges and opportunities. On one hand, a unit's outcome can be affected by the exposure at many locations, leading to interference. On the other hand, unmeasured spatial variables can confound the effect of interest. Our work has two overarching goals. First, using causal diagrams, we illustrate that spatial confounding and interference can manifest as each other, meaning that investigating the presence of one can lead to wrongful conclusions in the presence of the other, and that statistical dependencies in the exposure variable can render standard analyses invalid. This can have crucial implications for analyzing data with spatial or other dependencies, and for understanding the effect of interventions on dependent units. Secondly, we propose a parametric approach to mitigate bias from local and neighborhood unmeasured spatial confounding and account for interference simultaneously. This approach is based on simultaneous modeling of the exposure and the outcome while accounting for the presence of spatially-structured unmeasured predictors of both variables. We illustrate our approach with a simulation study and with an analysis of the local and interference effects of sulfur dioxide emissions from power plants on cardiovascular mortality.

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The analysis of environmental mixtures is of growing importance in environmental epidemiology, and one of the key goals in such analyses is to identify exposures and their interactions that are associated with adverse health outcomes. Typical approaches utilize flexible regression models combined with variable selection to identify important exposures and estimate a potentially nonlinear relationship with the outcome of interest. Despite this surge in interest, no approaches to date can identify exposures and interactions while controlling any form of error rates with respect to exposure selection. We propose two novel approaches to estimating the health effects of environmental mixtures that simultaneously 1) Estimate and provide valid inference for the overall mixture effect, and 2) identify important exposures and interactions while controlling the false discovery rate. We show that this can lead to substantial power gains to detect weak effects of environmental exposures. We apply our approaches to a study of persistent organic pollutants and find that our approach is able to identify more interactions than existing approaches.

Summary The analysis of environmental mixtures is of growing importance in environmental epidemiology, and one of the key goals in such analyses is to identify exposures and their interactions that are associated with adverse health outcomes. Typical approaches utilize flexible regression models combined with variable selection to identify important exposures and estimate a potentially nonlinear relationship with the outcome of interest. Despite this surge in interest, no approaches to date can identify exposures and interactions while controlling any form of error rates with respect to exposure selection. We propose two novel approaches to estimating the health effects of environmental mixtures that simultaneously (i) estimate and provide valid inference for the overall mixture effect and (ii) identify important exposures and interactions while controlling the false discovery rate (FDR). We show that this can lead to substantial power gains to detect weak effects of environmental exposures. We apply our approaches to a study of persistent organic pollutants and find that controlling the FDR leads to substantially different conclusions.

The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either restrict to specific sub-classes of sparsity patterns and/or are not scalable to settings with hundreds of responses and predictors. Bayesian approaches which only focus on estimating the posterior mode are scalable, but do not generate samples from the posterior distribution for uncertainty quantification. Using a bi-convex regression based generalized likelihood and spike-and-slab priors, we develop an algorithm called Joint Regression Network Selector (JRNS) for joint regression and covariance selection which (a) can accommodate general sparsity patterns, (b) provides posterior samples for uncertainty quantification, and (c) is scalable and orders of magnitude faster than the state-ofthe-art Bayesian approaches providing uncertainty quantification. We demonstrate the statistical and computational efficacy of the proposed approach on synthetic data and through the analysis of selected cancer data sets. We also establish highdimensional posterior consistency for one of the developed algorithms.