Demystifying spatial confounding

Music City Center 

Spatial regression models, i.e. regression models for data collected at different geographical locations, use spatial random effects to approximate unmeasured spatial variation in the response variable. However, as spatial random effects are typically not independent of the covariates in the model, this can lead to significant bias in covariate effect estimates of interest, making the estimation potentially unreliable. This fundamental problem is referred to as spatial confounding. There has been much interest in spatial confounding, particularly in recent years, not least because the most established methods based on orthogonalisation of the spatial random effects for dealing with the problem were proven to be ineffective. However, research into the topic has sometimes led to puzzling and seemingly contradictory results. Here, we develop a broad theoretical framework that brings mathematical clarity to the mechanisms of spatial confounding, providing explicit and interpretable analytical expressions for the resulting bias. From these, we see that it is a problem directly linked to spatial smoothing, and we can identify exactly how the features of the model and the data generation process affect the size and occurrence of bias. 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 specification of the spatial random effects. 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.

Confounding bias

Spatial regression

Spatial random effects

Smoothing

Spatial+