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Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love

Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love Many statisticians have had the experience of fitting a linear model with uncorrelated errors, then adding a spatially-correlated error term (random effect) and finding that the estimates of the fixed-effect coefficients have changed substantially. We show that adding a spatially-correlated error term to a linear model is equivalent to adding a saturated collection of canonical regressors, the coefficients of which are shrunk toward zero, where the spatial map determines both the canonical regressors and the relative extent of the coefficients’ shrinkage. Adding a spatially-correlated error term can also be seen as inflating the error variances associated with specific contrasts of the data, where the spatial map determines the contrasts and the extent of error-variance inflation. We show how to avoid this spatial confounding by restricting the spatial random effect to the orthogonal complement (residual space) of the fixed effects, which we call restricted spatial regression. We consider five proposed interpretations of spatial confounding and draw implications about what, if anything, one should do about it. In doing so, we debunk the common belief that adding a spatially-correlated random effect adjusts fixed-effect estimates for spatially-structured missing covariates. This article has supplementary material online. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The American Statistician Taylor & Francis

Adding Spatially-Correlated Errors Can Mess Up the Fixed Effect You Love

The American Statistician , Volume 64 (4): 10 – Nov 1, 2010
10 pages

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References (15)

Publisher
Taylor & Francis
Copyright
© 2010 American Statistical Association
ISSN
1537-2731
eISSN
0003-1305
DOI
10.1198/tast.2010.10052
Publisher site
See Article on Publisher Site

Abstract

Many statisticians have had the experience of fitting a linear model with uncorrelated errors, then adding a spatially-correlated error term (random effect) and finding that the estimates of the fixed-effect coefficients have changed substantially. We show that adding a spatially-correlated error term to a linear model is equivalent to adding a saturated collection of canonical regressors, the coefficients of which are shrunk toward zero, where the spatial map determines both the canonical regressors and the relative extent of the coefficients’ shrinkage. Adding a spatially-correlated error term can also be seen as inflating the error variances associated with specific contrasts of the data, where the spatial map determines the contrasts and the extent of error-variance inflation. We show how to avoid this spatial confounding by restricting the spatial random effect to the orthogonal complement (residual space) of the fixed effects, which we call restricted spatial regression. We consider five proposed interpretations of spatial confounding and draw implications about what, if anything, one should do about it. In doing so, we debunk the common belief that adding a spatially-correlated random effect adjusts fixed-effect estimates for spatially-structured missing covariates. This article has supplementary material online.

Journal

The American StatisticianTaylor & Francis

Published: Nov 1, 2010

Keywords: Confounding; Missing covariate; Random effect; Spatial correlation; Spatial regression

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