Spatial Confounding in Multivariate Areal Data Analysis

TL;DR

This work examines spatial confounding in Bayesian multivariate areal regression, contrasting analysis- and data-generation perspectives to understand variance inflation and estimator efficiency. It extends the BYM2 framework to a multivariate, coregionalized setting with both spatial and non-spatial latent factors, and derives exact expressions for how fixed-effect estimates behave under confounding. Through simulations and a US-county application involving obesity, diabetes, and cancer mortality, the authors show that a properly specified multivariate spatial model can improve estimation accuracy and maintain valid uncertainty quantification, even when the spatial structure is misspecified. They argue that spatial models remain advantageous in disease mapping contexts and highlight the importance of considering multivariate dependencies and model misspecification when addressing spatial confounding.

Abstract

We investigate spatial confounding in the presence of multivariate disease dependence. In the "analysis model perspective" of spatial confounding, adding a spatially dependent random effect can lead to significant variance inflation of the posterior distribution of the fixed effects. The "data generation perspective" views covariates as stochastic and correlated with an unobserved spatial confounder, leading to inferior statistical inference over multiple realizations. Although multiple methods have been proposed for adjusting statistical models to mitigate spatial confounding in estimating regression coefficients, the results on interactions between spatial confounding and multivariate dependence are very limited. We contribute to this domain by investigating spatial confounding from the analysis and data generation perspectives in a Bayesian coregionalized areal regression model. We derive novel results that distinguish variance inflation due to spatial confounding from inflation based on multicollinearity between predictors and provide insights into the estimation efficiency of a spatial estimator under a spatially confounded data generation model. We demonstrate favorable performance of spatial analysis compared to a non-spatial model in our simulation experiments even in the presence of spatial confounding and a misspecified spatial structure. In this regard, we align with several other authors in the defense of traditional hierarchical spatial models (Gilbert et al., 2025; Khan and Berrett, 2023; Zimmerman and Ver Hoef, 2022) and extend this defense to multivariate areal models. We analyze county-level data from the US on obesity / diabetes prevalence and diabetes-related cancer mortality, comparing the results with and without spatial random effects.

Figures (2)

• Figure 1: Density plot of posterior expectations of $D_{KL}(p(Y_0) || q(Y_0))$ for the non-spatial model, conditioned CAR and SAR spatial models, and unconditioned CAR and SAR spatial models over 300 datasets simulated from a CAR data generation model.
• Figure 2: Estimates of US county-level obesity prevalence in 2015, diabetes prevalence in 2015, diabetes-related cancer mortality rates from 2010 to 2020, and percentage of residents that were Hispanic in 2010. Counties with a missing covariate or response are shown in gray and excluded from the analysis.