Bayesian Analysis

Alleviating Spatial Confounding for Areal Data Problems by Displacing the Geographical Centroids

Marcos Oliveira Prates, Renato Martins Assunção, and Erica Castilho Rodrigues

Full-text: Open access


Spatial confounding between the spatial random effects and fixed effects covariates has been recently discovered and showed that it may bring misleading interpretation to the model results. Techniques to alleviate this problem are based on decomposing the spatial random effect and fitting a restricted spatial regression. In this paper, we propose a different approach: a transformation of the geographic space to ensure that the unobserved spatial random effect added to the regression is orthogonal to the fixed effects covariates. Our approach, named SPOCK, has the additional benefit of providing a fast and simple computational method to estimate the parameters. Also, it does not constrain the distribution class assumed for the spatial error term. A simulation study and real data analyses are presented to better understand the advantages of the new method in comparison with the existing ones.

Article information

Bayesian Anal., Volume 14, Number 2 (2019), 623-647.

First available in Project Euclid: 18 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Areal Data Bayesian Statistics Spatial Confounding Spatial Regression

Creative Commons Attribution 4.0 International License.


Prates, Marcos Oliveira; Assunção, Renato Martins; Rodrigues, Erica Castilho. Alleviating Spatial Confounding for Areal Data Problems by Displacing the Geographical Centroids. Bayesian Anal. 14 (2019), no. 2, 623--647. doi:10.1214/18-BA1123.

Export citation


  • Besag, J. (1974). Spatial interaction and the statistical analysis of lattice data systems (with discussion). Journal of the Royal Statistical Society, Series B 36, 192–225.
  • Besag, J., J. York, and A. Mollie (1991). Bayesian image restoration with two application in spatial statistics (with discussion). Annals of the Institute Statistical Mathematics 43, 1–59.
  • Breslow, N. E. and D. G. Clayton (1993). Approximate inference in generalized linear mixed models. Journal of the American statistical Association 88(421), 9–25.
  • Clayton, D., L. Bernardinelli, and C. Montomoli (1993). Spatial correlation in ecological analysis. International Journal of Epidemiology 6, 1193–1202.
  • Cressie, N. (1991). Statistics for spatial data. John Wiley & Sons.
  • Gelman, A. and F. Tuerlinckx (2000). Type s error rates for classical and bayesian single and multiple comparison procedures. Computational Statistics 15(3), 373–390.
  • Hanks, E. M., E. M. Schliep, M. B. Hooten, and J. A. Hoeting (2015). Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification. Environmetrics 26(4), 243–254.
  • Hefley, T. J., M. B. Hooten, E. M. Hanks, R. E. Russell, and D. P. Walsh (2017). The bayesian group lasso for confounded spatial data. Journal of Agricultural, Biological and Environmental Statistics 22(1), 42–59.
  • Hodges, J. S. and B. J. Reich (2011, January). Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love. The American Statistician 64(4), 325–334.
  • Hughes, J. and X. Cui (2017). ngspatial: Fitting the Centered Autologistic and Sparse Spatial Generalized Linear Mixed Models for Areal Data. Denver, CO. R package version 1.2.
  • Hughes, J. and M. Haran (2013). Dimension reduction and alleviation of confounding for spatial generalized linear mixed models. Journal of the Royal Statistical Society, Series B 75, 139–159.
  • Lee, D. (2013). CARBayes: An R package for Bayesian spatial modeling with conditional autoregressive priors. Journal of Statistical Software 55(13), 1–24.
  • Leroux, B. G., X. Lei, and N. Breslow (1999). Estimation of disease rates in small areas: A new mixed model for spatial dependence. In M. E. Halloran and D. Berry (Eds.), In Statistical Models in Epidemiology; the Environment and Clinical Trials, pp. 179–192. New York: Springer–Verlag.
  • Lunn, D. J., A. Thomas, N. Best, and D. Spiegelhalter (2000). WinBUGS – a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10, 325–337.
  • Menzel, U. (2012). CCP: Significance Tests for Canonical Correlation Analysis (CCA). R package version 1.1.
  • Murakami, D. and D. A. Griffith (2015). Random effects specifications in eigenvector spatial filtering: a simulation study. Journal of Geographical Systems 17(4), 311–331.
  • Paciorek, C. J. (2010). The importance of scale for spatial-confounding bias and precision of spatial regression estimators. Statistical Science 25, 107–125.
  • Prates, M. O., Assunção, R. M., and Rodrigues, E. C. (2018). Alleviating spatial confounding for areal data problems by displacing the geographical centroids: Supplementary Material. Bayesian Analysis.
  • R Development Core Team (2011). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.
  • Reich, B. J., J. S. Hodges, and V. Zadnik (2006). Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models. Biometrics 62, 1197–1206.
  • Rodrigues, E. C. and R. Assunção (2012). Bayesian spatial models with a mixture neighborhood structure. Journal of Multivariate Analysis 109(0), 88–102.
  • Rue, H. and L. Held (2005). Gaussian Markov random fields: Theory and applications. Chapman & Hall.
  • Rue, H., S. Martino, and N. Chopin (2009). Approximate bayesian inference for latent gaussian models using integrated nested laplace approximations (with discussion). Journal of the Royal Statistical Society, Series B 71, 319–392.
  • Sampson, P. D. and P. Guttorp (1992). Nonparametric estimation of nonstationary spatial covariance structure. Journal of the American Statistical Association 87(417), 108–119.
  • Wilks, S. (1935). On the independence of k sets of normally distributed statistical variables. Econometrica, Journal of the Econometric Society, 309–326.
  • Zadnik, V. and B. J. Reich (2006). Analysis of the relationship between socioeconomic factors and stomach cancer incidence in Slovenia. Neoplasma 53, 103–110.

Supplemental materials

  • Alleviating spatial confounding for areal data problems by displacing the geographical centroids: Supplementary Material. The supplementary material present some results for the simulation studies of the paper.