Restricted Spatial Regression is Reasonable Statistical Practice: Clarifications, Interpretations, and New Developments

The spatial linear mixed model (SLMM) consists of fixed and spatial random effects that may be linearly dependent. Partially motivated as a means to address potential issues with confounding, the Restricted spatial regression (RSR) model restricts spatial random effects to be in the orthogonal column space of the covariates. Recent articles have shown that the misspecified Bayesian RSR generally performs worse than the SLMM when the data is generated from the SLMM. However, we show that the misspecified Bayesian RSR model’s marginal posterior distribution is equivalent up to a reparameterization to that of the SLMM’s marginal 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. This equivalence relationship is developed further in the context of unmeasured confounders and nonlinearity, where we explore a semi-parametric property of the orthogonalized regression effects. Several results are provided to demonstrate new benefits of an RSR. In particular, we provide new results that show that the RSR can produce clear computational advantages via a direct sampler from the posterior distribution for all hyperparameters, fixed effects, and random effects. Additionally, a transfer learning approach offers a new interpretation to orthogonalized regression coefficients, which we show empirically can improve inference on dependent regression coefficients in the presence of spatial confounding. Simulations and an illustration using COVID-19 mortality data are provided.

💡 Research Summary

The paper revisits the relationship between the spatial linear mixed model (SLMM) and restricted spatial regression (RSR) from a Bayesian perspective, challenging recent claims that misspecified Bayesian RSR is generally inferior to the SLMM. The SLMM is expressed as y = Xβ + Bν + ε, where the fixed effects β and spatial random effects ν can be linearly dependent, leading to non‑identifiability. A common remedy is to reparameterize the model so that the fixed effects are orthogonal to the spatial component, yielding y = Xδ + (I − P)Bν + ε with δ = β + (X′X)⁻¹X′Bν. RSR further imposes the constraint δ = β, effectively forcing the spatial random effects to lie in the orthogonal complement of the column space of X.

Recent works (Zimmerman & Ver Hoef 2022; Khan & Calder 2022) demonstrated that, when data are generated from a correctly specified SLMM, Bayesian inference under the misspecified RSR (i.e., assuming β = δ) can produce overly narrow credible intervals, under‑coverage, and poor prediction performance. These findings led to “Conclusion 1” that RSR is generally inferior.

The authors propose an alternative viewpoint, showing that under a specific prior—namely the improper flat prior on β used by Reich et al. (2006)—the marginal posterior distribution of the orthogonalized regression effects δ, the hyper‑parameters, and missing values under the misspecified Bayesian RSR is exactly the same as that obtained from the original Bayesian SLMM after reparameterization. This is “Conclusion 2”. Consequently, the posterior summaries from the misspecified RSR can be interpreted as valid SLMM posteriors for the orthogonalized parameters.

Building on this equivalence, the authors introduce a data‑augmentation scheme that treats the latent orthogonal component as additional data, allowing simultaneous estimation of β and δ. With this augmentation, the posterior distribution of all quantities of interest (β, δ, ν, hyper‑parameters, and missing responses) matches that of the SLMM—this is “Conclusion 3”. The key is that the augmentation uses the same improper prior on β, preserving the equivalence.

From a computational standpoint, the paper derives a closed‑form direct sampler for the full posterior of the augmented RSR (and thus for the SLMM). By exploiting the eigen‑decomposition of the projection matrix I − P (denoted L), the sampler draws independent draws of β, ν, σ², and Σν without resorting to Markov chain Monte Carlo. This is the first known exact closed‑form posterior for a Gaussian spatial linear mixed model that does not rely on discrete uniform priors for hyper‑parameters. The direct sampler dramatically reduces computation time and eliminates convergence diagnostics.

The authors also explore a semi‑parametric property of the orthogonalized effects δ: when σ² is known, the posterior of δ is invariant to misspecification of the spatial covariance Σν. This property motivates a transfer‑learning interpretation: δ can be viewed as “unbiased data” for β, and a method‑of‑moments (MoM) adjustment can align the first moments of β and δ without forcing them to be identical. Empirically, a transfer‑learning RSR that incorporates this MoM step improves estimation of β in the presence of non‑linearity and unmeasured confounding, as demonstrated in simulation studies.

The practical relevance is illustrated with an analysis of COVID‑19 mortality counts across U.S. counties. The authors compare three approaches: (i) a standard SLMM, (ii) the traditional RSR, and (iii) the augmented transfer‑learning RSR. The augmented RSR yields regression coefficients and credible intervals that are both tighter and better calibrated than those from the SLMM, while also providing more accurate spatial predictions. The results underscore the advantage of the augmented RSR in real‑world settings where spatial confounding and non‑linear relationships are common.

In summary, the paper establishes that:

1. Misspecified Bayesian RSR and the SLMM are posterior‑equivalent for orthogonalized effects under a specific improper prior.
2. A data‑augmentation strategy yields an “augmented Bayesian RSR” whose posterior matches the full SLMM posterior, enabling inference on β, ν, and hyper‑parameters without loss of information.
3. A direct closed‑form sampler makes inference computationally trivial compared with traditional MCMC.
4. The orthogonalized effects admit a transfer‑learning interpretation that can mitigate bias from unmeasured spatial confounders and non‑linearity.

These contributions collectively argue that restricted spatial regression is not merely a heuristic fix but a statistically sound and computationally efficient alternative to the conventional SLMM, provided the prior and augmentation are chosen appropriately. The work opens avenues for further research on prior specification, extensions to non‑Gaussian outcomes, and broader applications of the transfer‑learning framework in spatial statistics.