Deconfounding a Spatial Linear Model is Reasonable Statistical Practice

Music City Center 

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.

change-of-variables

restricted spatial regression

spatial linear mixed model