Spatial+: a novel approach to spatial confounding

TL;DR

Spatial+ is a new method that reduces spatial confounding bias in spatial regression models by replacing covariates with residuals, improving inference and extending to non-Gaussian responses.

Contribution

We introduce spatial+, a novel approach that mitigates spatial confounding bias by residualizing covariates, applicable across various spatial models including non-Gaussian responses.

Findings

Spatial+ avoids bias in effect estimates.

Simulation confirms improved inference with spatial+.

Method is compatible with existing software.

Abstract

In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered modelling forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. The mechanism behind spatial confounding is poorly understood and methods for dealing with it are limited. We propose a novel approach, spatial+, in which collinearity is reduced by replacing the covariates in the spatial model by their residuals after spatial dependence has been regressed away. Using a thin plate spline model formulation, we recognise spatial confounding as a smoothing-induced bias identified by Rice (1986), and through asymptotic analysis of the effect estimates, we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straight-forward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non-Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.