Abstract

Spatial confounding is a common challenge in spatial regression models, arising when spatially varying covariates correlate with the spatial random effects included to capture unmeasured spatial variation. This dependence — particularly at high spatial frequencies — can introduce substantial bias in covariate effect estimates when combined with smoothing of the spatial effect. Despite extensive research, the mechanisms underlying spatial confounding have remained incompletely understood, with results that are sometimes puzzling or seemingly contradictory. This is in part because the complexity of spatial models means that conclusions are often based on simulation studies, potentially sensitive to the specific simulation setup. Moreover, a common approach for identifying confounding bias has been to focus on the difference between estimates from spatial and non-spatial analyses — yet, as our results confirm, this difference does not in itself determine whether either estimate is biased.

We develop a broad theoretical framework for spatial confounding in spatial generalized additive models (GAMs) (Dupont, Marques, Kneib, 2026+). Central to our findings is an explicit analytical expression for the bias, which shows that — in the metric defined by the precision matrix of the chosen spatial model — the bias is determined by the correlation between the covariate and the confounder, scaled by the size of the confounder relative to the size of the covariate. We can precisely characterize how bias depends on the spatial frequency content of both the covariate and the confounder. In particular, we show that bias is directly linked to spatial smoothing and can become arbitrarily large when confounding occurs at high spatial frequencies. Conversely, confounding at low frequencies tends to have negligible impact on the spatial model estimate. These results provide a unified explanation for findings that have previously appeared contradictory. It also helps explain why and when certain methods work.

Building on this theoretical foundation, we propose practical tools for bias detection and mitigation applicable under any spatial model specification. We introduce capped spatial+, a diagnostic procedure that leverages our bias expression to assess the presence of spatial confounding by fitting multiple capped versions of the spatial+ estimator across a range of frequency thresholds. Stable estimates below a given cap provide evidence of an unbiased spatial model, while instability signals the presence of confounding. We illustrate our approach with an application to monthly mean air temperature in Germany.

I further discuss Bayesian spatial+ (Marques & Wiemann, 2026), a joint modeling approach that addresses key limitations of the classical frequentist spatial+ method, including uncertainty propagation and frequency separation through joint priors on smoothness parameters. Simulation studies confirm the theoretical predictions across a range of confounding scenarios and demonstrate substantial improvements in bias reduction relative to existing approaches, namely as sample size increases.