Discussion of the manuscript: Spatial+ a novel approach to spatial confounding

I congratulate Dupont, Wood and Augustin (DWA hereon) for providing an easy-to-implement method for estimation in the presence of spatial confounding, and for addressing some of the complicated aspects on the topic. The method regresses the covariate of interest on spatial basis functions and uses the residuals of this model in an outcome regression. The authors show that, if the covariate is not completely spatial, this approach leads to consistent estimation of the conditional association between the exposure and the outcome. Below I discuss conceptual and operational issues that are fundamental to inference in spatial settings: (i) the target quantity and its interpretability, (ii) the non-spatial aspect of covariates and their relative spatial scales, and (iii) the impact of spatial smoothing. While DWA provide some insights on these issues, I believe that the audience might benefit from a deeper discussion. In what follows, I focus on the setting where a researcher is interested in interpreting the relationship between a given covariate and an outcome. I refer to the covariate of interest as the exposure to differentiate it from the rest.

[1]  Victor Chernozhukov,et al.  Inference on Treatment Effects after Selection Amongst High-Dimensional Controls , 2011 .

[2]  Giovanni Parmigiani,et al.  Bayesian Effect Estimation Accounting for Adjustment Uncertainty , 2012, Biometrics.

[3]  Ashkan Ertefaie,et al.  Outcome‐adaptive lasso: Variable selection for causal inference , 2017, Biometrics.

[4]  J. Hodges,et al.  Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love , 2010 .

[5]  J. Robins.  A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect , 1986 .

[6]  Jonas Peters,et al.  Towards Causal Inference for Spatio-Temporal Data: Conflict and Forest Loss in Colombia , 2021, Journal of the American Statistical Association.

[7]  Corwin M Zigler,et al.  Adjusting for unmeasured spatial confounding with distance adjusted propensity score matching , 2016, Biostatistics.

[8]  J. Schwartz,et al.  Spatial Multiresolution Analysis of the Effect of PM2.5 on Birth Weights. , 2016, The annals of applied statistics.

[9]  Joshua P. Keller,et al.  Selecting a scale for spatial confounding adjustment , 2019, Journal of the Royal Statistical Society. Series A,.

[10]  Thomas Kneib,et al.  Structural Equation Models for Dealing With Spatial Confounding , 2018 .

[11]  Elizabeth L. Ogburn,et al.  Network Dependence Can Lead to Spurious Associations and Invalid Inference , 2019, Journal of the American Statistical Association.

[12]  Shu Yang,et al.  A Review of Spatial Causal Inference Methods for Environmental and Epidemiological Applications , 2020, International statistical review = Revue internationale de statistique.

[13]  G. Parmigiani,et al.  High-Dimensional Confounding Adjustment Using Continuous Spike and Slab Priors. , 2017, Bayesian analysis.

[14]  Sherri Rose,et al.  Implementation of G-computation on a simulated data set: demonstration of a causal inference technique. , 2011, American journal of epidemiology.