Demystifying Spatial Confounding — A Unified Analytical Framework for the Bias in Covariate Effect Estimates

In spatial regression models, unmeasured spatial variables, represented by spatial random effects, are typically not independent of observed covariates and can induce significant bias in covariate effect estimates. This fundamental problem, known as spatial confounding, has been extensively studied, but sometimes with puzzling and seemingly contradictory results. Here, we introduce a broad theoretical framework that brings mathematical clarity to spatial confounding. Our focus is the finite-sample bias affecting simulation results and practical applications. Our intention is not to challenge existing results in the literature, but rather to build intuition and provide a unifying perspective that helps explain and connect them. We derive a compact analytical expression where, in the metric induced by the precision structure of the chosen analysis model, the bias is expressed in terms of the empirical correlation between a covariate and its confounder and their relative sizes. Thus, our formulation directly identifies these relationships as the key driver of bias; moreover, using the eigendecomposition of the precision structure, we obtain detailed quantitative characterizations of its behavior. The eigendecomposition has a natural interpretation as spatial frequencies and nonspatial information, with the eigenvalues encoding the precise effect of spatial smoothing in the model. This allows the insights from our bias expression to be translated into interpretable scenarios and explain subtle and counter-intuitive behaviors. Finally, we propose a general strategy for addressing the bias in practice. When a covariate has nonspatial information, we show that a general form of the so-called spatial+ method can reduce the bias. If not, explicit assumptions for identifiability are needed, and we develop a procedure in which multiple capped versions of spatial+ are applied. We illustrate our approach with an application to air temperature in Germany.