🤖 AI Summary
In spatial regression, spatial confounding arises when spatial covariates correlate with high-frequency components of the latent spatial random effect, biasing coefficient estimates. Conventional two-stage frequentist approaches (e.g., spatial+) fail to propagate uncertainty from the first stage and cannot guarantee orthogonality between covariate residuals and the spatial effect in the high-frequency domain. This paper proposes the first Bayesian joint modeling framework that end-to-end integrates spatial covariate adjustment with spatial effect estimation. We introduce a novel joint prior on smoothing parameters that explicitly decouples high-frequency collinearity, balancing deconfounding and predictive accuracy. Built upon a hierarchical Gaussian random field and inference via Integrated Nested Laplace Approximation (INLA), our method demonstrates, in simulations and real-data applications, substantially reduced coefficient bias, improved nominal coverage of posterior credible intervals, and more robust, interpretable statistical inference.
📝 Abstract
Spatial confounding is a common issue in spatial regression models, occurring when spatially indexed covariates that model the mean of the response are correlated with a spatial effect included in the model. This dependence, particularly at high spatial frequencies combined with smoothing, can introduce bias in the regression coefficient estimates. The spatial+ framework is a widely used two-stage frequentist approach to mitigate spatial confounding by explicitly modeling and removing the spatial structure in the confounding covariate, replacing it with residuals in the second-stage model for the response. However, frequentist spatial+ does not propagate uncertainty from the first-stage estimation to the second stage, and inference can be cumbersome in a frequentist setting. In contrast, a Bayesian joint modeling framework inherently propagates uncertainty between stages and allows for direct inference on the model parameters. Despite its advantages, the original spatial+ method does not ensure the residuals and spatial effects in the second-stage model are free of shared high spatial frequencies without additional assumptions. To address this, we propose a novel joint prior for the smoothness parameters of the spatial effects that mitigates this issue while preserving the predictive power of the model. We demonstrate the efficacy of our approach through simulation studies and real-world applications.