Restricted Spatial Regression Methods: Implications for Inference

The issue of spatial confounding between the spatial random effect and the fixed effects in regression analyses has been identified as a concern in the statistical literature. Multiple authors have offered perspectives and potential solutions. In this paper, for the areal spatial data setting, we show that many of the methods designed to alleviate spatial confounding can be viewed as special cases of a general class of models. We refer to this class as Restricted Spatial Regression (RSR) models, extending terminology currently in use. We offer a mathematically based exploration of the impact that RSR methods have on inference for regression coefficients for the linear model. We then explore whether these results hold in the generalized linear model setting for count data using simulations. We show that the use of these methods have counterintuitive consequences which defy the general expectations in the literature. In particular, our results and the accompanying simulations suggest that RSR methods will typically perform worse than non-spatial methods. These results have important implications for dimension reduction strategies in spatial regression modeling. Specifically, we demonstrate that the problems with RSR models cannot be fixed with a selection of "better" spatial basis vectors or dimension reduction techniques.

[1]  Noel A Cressie,et al.  Statistics for Spatio-Temporal Data , 2011 .

[2]  Michael Tiefelsdorf,et al.  Global and local spatial autocorrelation in bounded regular tessellations , 2000, J. Geogr. Syst..

[3]  Kaare Brandt Petersen,et al.  The Matrix Cookbook , 2006 .

[4]  Michael L. Stein,et al.  Limitations on low rank approximations for covariance matrices of spatial data , 2014 .

[5]  M. Wall.  A close look at the spatial structure implied by the CAR and SAR models , 2004 .

[6]  E. Lehmann.  Ordered Families of Distributions , 1955 .

[7]  Guangqing Chi,et al.  Applied Spatial Data Analysis with R , 2015 .

[8]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[9]  Cedric E. Ginestet.  ggplot2: Elegant Graphics for Data Analysis , 2011 .

[10]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[11]  V. Zadnik,et al.  Effects of Residual Smoothing on the Posterior of the Fixed Effects in Disease‐Mapping Models , 2006, Biometrics.

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

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

[14]  M. Fuentes.  Approximate Likelihood for Large Irregularly Spaced Spatial Data , 2007, Journal of the American Statistical Association.

[15]  Mevin B. Hooten,et al.  The Bayesian Group Lasso for Confounded Spatial Data , 2017 .

[16]  A. Gelfand,et al.  Gaussian predictive process models for large spatial data sets , 2008, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[17]  Renato M. Assunção,et al.  Alleviating Spatial Confounding for Areal Data Problems by Displacing the Geographical Centroids , 2019, Bayesian Analysis.

[18]  Murali Haran,et al.  Dimension reduction and alleviation of confounding for spatial generalized linear mixed models , 2010, 1011.6649.

[19]  Christopher K. Wikle,et al.  Low-Rank Representations for Spatial Processes , 2010 .

[20]  Jonathan R. Bradley,et al.  Multivariate spatio-temporal models for high-dimensional areal data with application to Longitudinal Employer-Household Dynamics , 2015, 1503.00982.

[21]  Noel A Cressie,et al.  The SAR Model for Very Large Datasets: A Reduced Rank Approach , 2015 .

[22]  Murali Haran,et al.  A Computationally Efficient Projection-Based Approach for Spatial Generalized Linear Mixed Models , 2016, Journal of Computational and Graphical Statistics.

[23]  Christopher Paciorek.  Technical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models , 2009 .

[24]  Murali Haran,et al.  Markov chain Monte Carlo: Can we trust the third significant figure? , 2007, math/0703746.

[25]  P. Diggle.  Applied Spatial Statistics for Public Health Data , 2005 .

[26]  Daniel A. Griffith,et al.  Random effects specifications in eigenvector spatial filtering: a simulation study , 2015, J. Geogr. Syst..

[27]  P. Moran.  Notes on continuous stochastic phenomena. , 1950, Biometrika.

[28]  L. Waller,et al.  Applied Spatial Statistics for Public Health Data: Waller/Applied Spatial Statistics , 2004 .

[29]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[30]  James S. Hodges,et al.  Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects , 2013 .

[31]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[32]  Noel A. C. Cressie,et al.  Statistics for Spatial Data: Cressie/Statistics , 1993 .

[33]  C Montomoli,et al.  Spatial correlation in ecological analysis. , 1993, International journal of epidemiology.

[34]  Garritt L. Page,et al.  Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models , 2017 .

[35]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[36]  S. Zelditch,et al.  Eigenfunctions of the Laplacian on a Riemannian manifold , 2017 .

[37]  B. Carlin,et al.  Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota. , 2003, Biostatistics.

[38]  P. Diggle,et al.  Model‐based geostatistics , 2007 .

[39]  Christopher J Paciorek,et al.  The importance of scale for spatial-confounding bias and precision of spatial regression estimators. , 2010, Statistical science : a review journal of the Institute of Mathematical Statistics.

[40]  Mevin B. Hooten,et al.  Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification , 2015 .