On Deconfounding Spatial Confounding in Linear Models

Spatial confounding, that is, collinearity between fixed effects and random effects in a spatial generalized linear mixed model, can adversely affect estimates of the fixed effects. Restricted spatial regression methods have been proposed as a remedy for spatial confounding. Such methods replace inference for the fixed effects of the original model with inference for those effects under a model in which the random effects are restricted to a subspace orthogonal to the column space of the fixed effects model matrix; thus, they "deconfound" the two types of effects. We prove, however, that frequentist inference for the fixed effects of a deconfounded linear model is generally inferior to that for the fixed effects of the original spatial linear model; in fact, it is even inferior to inference for the corresponding nonspatial model. We show further that deconfounding also leads to inferior predictive inferences, though its impact on prediction appears to be relatively small in practice. Based on these results, we argue that deconfounding a spatial linear model is bad statistical practice and should be avoided.