Spatial Deconfounding is Reasonable Statistical Practice: Interpretations, Clarifications, and New Benefits

The spatial linear mixed model (SLMM) consists of fixed and spatial random effects that can be confounded. Restricted spatial regression (RSR) models restrict the spatial random effects to be in the orthogonal column space of the covariates, which "deconfounds" the SLMM. Recent articles have shown that the RSR generally performs worse than the SLMM under a certain interpretation of the RSR. We show that every additive model can be reparameterized as a deconfounded model leading to what we call the linear reparameterization of additive models (LRAM). Under this reparameterization the coefficients of the covariates (referred to as deconfounded regression effects) are different from the (confounded) regression effects in the SLMM. It is shown that under the LRAM interpretation, existing deconfounded spatial models produce estimated deconfounded regression effects, spatial prediction, and spatial prediction variances equivalent to that of SLMM in Bayesian contexts. Furthermore, a general RSR (GRSR) and the SLMM produce identical inferences on confounded regression effects. While our results are in complete agreement with recent criticisms, our new results under the LRAM interpretation provide clarifications that lead to different and sometimes contrary conclusions. Additionally, we discuss the inferential and computational benefits to deconfounding, which we illustrate via a simulation.