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On the Effects of Spatial Confounding in Hierarchical Models

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  • Widemberg S. Nobre
  • Alexandra M. Schmidt
  • João B. M. Pereira

Abstract

Usually, in spatial generalised linear models, covariates that are spatially smooth are collinear with spatial random effects. This affects the bias and precision of the regression coefficients. This is known in the spatial statistics literature as spatial confounding. We discuss the problem of confounding in the case of multilevel spatial models wherein there are multiple observations within clusters. We show that even under the standard multilevel model, which allows for independent (i.e. not spatially correlated) cluster effects, the cluster‐level fixed effects might be biased depending on the structure of the ‘true’ generating mechanism of the processes. We provide simulation studies in order to investigate the effects of confounding in the estimation of fixed effects present in random intercept models under different scenarios of confounding. One remedy to spatial confounding is restricted spatial regression wherein the spatial random effects are constrained to be orthogonal to the fixed effects of the model. We propose one way to fit a restricted spatial regression model for multilevel data and illustrate it with artificial data analyses. We also briefly describe the issue of confounding in random intercept and slope models.

Suggested Citation

  • Widemberg S. Nobre & Alexandra M. Schmidt & João B. M. Pereira, 2021. "On the Effects of Spatial Confounding in Hierarchical Models," International Statistical Review, International Statistical Institute, vol. 89(2), pages 302-322, August.
    • Handle: RePEc:bla:istatr:v:89:y:2021:i:2:p:302-322
    DOI: 10.1111/insr.12407
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    References listed on IDEAS

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    1. Garritt L. Page & Yajun Liu & Zhuoqiong He & Donchu Sun, 2017. "Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 780-797, September.
    2. Alan Gelfand & Alexandra Schmidt & Sudipto Banerjee & C. Sirmans, 2004. "Nonstationary multivariate process modeling through spatially varying coregionalization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 263-312, December.
    3. John Hughes & Murali Haran, 2013. "Dimension reduction and alleviation of confounding for spatial generalized linear mixed models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 139-159, January.
    4. Ephraim M. Hanks & Erin M. Schliep & Mevin B. Hooten & Jennifer A. Hoeting, 2015. "Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification," Environmetrics, John Wiley & Sons, Ltd., vol. 26(4), pages 243-254, June.
    5. Hauke Thaden & Thomas Kneib, 2018. "Structural Equation Models for Dealing With Spatial Confounding," The American Statistician, Taylor & Francis Journals, vol. 72(3), pages 239-252, July.
    6. Brian J. Reich & James S. Hodges & Vesna Zadnik, 2006. "Effects of Residual Smoothing on the Posterior of the Fixed Effects in Disease-Mapping Models," Biometrics, The International Biometric Society, vol. 62(4), pages 1197-1206, December.
    7. Hodges, James S. & Reich, Brian J., 2010. "Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love," The American Statistician, American Statistical Association, vol. 64(4), pages 325-334.
    8. Gelfand, Alan E. & Banerjee, Sudipto & Sirmans, C.F. & Tu, Yong & Eng Ong, Seow, 2007. "Multilevel modeling using spatial processes: Application to the Singapore housing market," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3567-3579, April.
    9. João B. M. Pereira & Widemberg S. Nobre & Igor F. L. Silva & Alexandra M. Schmidt, 2020. "Spatial confounding in hurdle multilevel beta models: the case of the Brazilian Mathematical Olympics for Public Schools," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(3), pages 1051-1073, June.
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    Cited by:

    1. Alexandra M. Schmidt, 2022. "Discussion on “Spatial+: a novel approach to spatial confounding” by Emiko Dupont, Simon N. Wood, and Nicole H. Augustin," Biometrics, The International Biometric Society, vol. 78(4), pages 1300-1304, December.
    2. Emiko Dupont & Nicole H. Augustin, 2024. "Spatial Confounding and Spatial+ for Nonlinear Covariate Effects," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 29(3), pages 455-470, September.

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