Demystifying Spatial Confounding

Abstract: Spatial confounding is a fundamental issue in spatial regression models which arises because spatial random effects, included to approximate unmeasured spatial variation, are typically not independent of covariates in the model. This can lead to significant bias in covariate effect estimates. The problem is complex and has been the topic of extensive research with sometimes puzzling and seemingly contradictory results. Here, we develop a broad theoretical framework that brings mathematical clarity to the mechanisms of spatial confounding, providing explicit analytical expressions for the resulting bias. We see that the problem is directly linked to spatial smoothing and identify exactly how the size and occurrence of bias relate to the features of the spatial model as well as the underlying confounding scenario. Using our results, we can explain subtle and counter-intuitive behaviours. Finally, we propose a general approach for dealing with spatial confounding bias in practice, applicable for any spatial model specification. When a covariate has non-spatial information, we show that a general form of the so-called spatial+ method can be used to eliminate bias. When no such information is present, the situation is more challenging but, under the assumption of unconfounded high frequencies, we develop a procedure in which multiple capped versions of spatial+ are applied to assess the bias in this case. We illustrate our approach with an application to air temperature in Germany.