New Perspectives on Spatial Confounding

In the last two decades, considerable research has been devoted to a phenomenon known as spatial confounding. Spatial confounding is thought to occur when there is multicollinearity between a covariate and the random effect in a spatial regression model. This multicollinearity is considered highly problematic when the inferential goal is estimating regression coefficients and various methodologies have been proposed to attempt to alleviate it. In this paper, we offer a novel perspective of synthesizing the work in the field of spatial confounding. We propose that at least two distinct phenomena are currently conflated with the term spatial confounding. We refer to these as the "analysis model" and the "data generation" types of spatial confounding. We show that these two issues can lead to contradicting conclusions about whether spatial confounding exists and whether methods to alleviate it will improve inference. Our results also illustrate that in most cases, traditional spatial linear mixed models do help to improve inference on regression coefficients. Drawing on the insights gained, we offer a path forward for research in spatial confounding. 

The spatial linear mixed model (SLMM) consists of fixed and spatial random effects that may be confounded. Partially motivated as a means to address potential issues with confounding, Restricted spatial regression (RSR) models restrict the spatial random effects to be in the orthogonal column space of the covariates. Recent articles have shown that the misspecified RSR generally performs worse than the SLMM when the data is generated from the SLMM. However, we show that the misspecified RSR model's posterior distribution is equivalent up to a reparameterization to that of the SLMM's posterior distribution, under a certain prior assumption on the orthogonalized regression coefficients. This suggests that the RSR models are not sub-optimal as the subsequent Bayesian analysis can be interpreted as a type of SLMM Bayesian analysis. We also show that the RSR model's posterior distribution does not coincide with the original SLMM under a different prior specification for the orthogonalized regression coefficients. While our results are in complete agreement with results in the recent criticisms, our conclusions are contrary in the sense that we conclude that RSRs can be useful depending on your choice of prior distributions. Additionally, we develop this equivalence relationship further in the context of unmeasured confounders and nonlinearity, where we explore a semi-parametric property and develop new computational benefits. Several illustrations are presented. 

Adjusting for an unmeasured confounder is generally an intractable problem, but in the spatial setting it may be possible under certain conditions. In this paper, we derive necessary conditions on the coherence between the treatment variable of interest and the unmeasured confounder that ensure the causal effect of the treatment is estimable. We specify our model and assumptions in the spectral domain to allow for different degrees of confounding at different spatial resolutions. The key assumption that ensures identifiability is that confounding present at global scales dissipates at local scales. We show that this assumption in the spectral domain is equivalent to adjusting for global-scale confounding in the spatial domain by adding a spatially smoothed version of the treatment variable to the mean of the response variable. Within this general framework, we propose a sequence of confounder adjustment methods that range from parametric adjustments based on the Matern coherence function to more robust semi-parametric methods that use smoothing splines. These ideas are applied to areal and geostatistical data for both simulated and real datasets. 

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.