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.

翻译：空间线性混合模型（SLMM）包含可能相互混杂的固定效应与空间随机效应。限制性空间回归（RSR）模型将空间随机效应限制在协变量的正交列空间中，从而实现了对SLMM的“去混杂”。近期研究显示，在某种对RSR的解释框架下，RSR的表现通常逊于SLMM。本文证明任何可加模型均可重新参数化为去混杂模型，形成我们称之为可加模型的线性重参数化（LRAM）框架。在此重参数化下，协变量的系数（称为去混杂回归效应）不同于SLMM中的（混杂）回归效应。研究表明，在LRAM解释框架下，现有的去混杂空间模型在贝叶斯语境中产生的去混杂回归效应估计、空间预测及空间预测方差与SLMM完全等价。此外，广义RSR（GRSR）与SLMM在混杂回归效应推断上具有一致性。虽然我们的结论与近期批评意见完全吻合，但基于LRAM解释的新发现提供了关键澄清，从而得出不同甚至相反的结论。最后，我们探讨了去混杂方法在推断与计算层面的优势，并通过模拟实验加以验证。