Bayesian spatial+: A joint model perspective

Spatial confounding is a common issue in spatial regression models, occurring when spatially varying covariates correlate with the spatial effect included in the model. This dependence, particularly at high spatial frequencies, can introduce bias in regression coefficient estimates when combined with smoothing penalties. The spatial+ framework is a widely used two-stage frequentist approach that mitigates spatial confounding by explicitly modeling and removing the spatial structure in the confounding covariate, then using the corresponding residuals in the second-stage model for the response. However, it does not propagate first-stage uncertainty, does not discuss a general inferential framework, and, crucially, cannot guarantee that covariate residuals and spatial effects in the response model are free of shared high-frequency structure, so confounding may persist. We propose Bayesian spatial+, a joint modeling approach that simultaneously addresses these limitations. Our framework naturally propagates uncertainty and enables straightforward posterior inference, while ensuring separation of spatial frequencies through specialized joint priors on smoothness parameters. We further introduce a cut-feedback strategy that prevents feedback between model components from reintroducing confounding. Simulation studies and real-world applications show substantial gains in bias reduction and interval coverage relative to existing approaches. Notably, in our comparisons, Bayesian spatial+ is the only method for which credible interval coverage remains stable as the sample size increases.