Journal of Geographical Systems

, Volume 9, Issue 2, pp 145–166 | Cite as

An assessment of coefficient accuracy in linear regression models with spatially varying coefficients

  • David C. WheelerEmail author
  • Catherine A. Calder
Original Article


The realization in the statistical and geographical sciences that a relationship between an explanatory variable and a response variable in a linear regression model is not always constant across a study area has led to the development of regression models that allow for spatially varying coefficients. Two competing models of this type are geographically weighted regression (GWR) and Bayesian regression models with spatially varying coefficient processes (SVCP). In the application of these spatially varying coefficient models, marginal inference on the regression coefficient spatial processes is typically of primary interest. In light of this fact, there is a need to assess the validity of such marginal inferences, since these inferences may be misleading in the presence of explanatory variable collinearity. In this paper, we present the results of a simulation study designed to evaluate the sensitivity of the spatially varying coefficients in the competing models to various levels of collinearity. The simulation study results show that the Bayesian regression model produces more accurate inferences on the regression coefficients than does GWR. In addition, the Bayesian regression model is overall fairly robust in terms of marginal coefficient inference to moderate levels of collinearity, and degrades less substantially than GWR with strong collinearity.


Bayesian regression Geographically weighted regression MCMC Simulation study Spatial statistics Collinearity 


  1. Agarwal DK, Gelfand AE (2005) Slice sampling for simulation based fitting of spatial data models. Stat Comput 15:61–69CrossRefGoogle Scholar
  2. Banerjee S, Johnson GA (2005) Coregionalized single- and multi-resolution spatially-varying growth curve modelling with application to weed growth. UMN Biostat Tech ReportGoogle Scholar
  3. Banerjee S, Carlin BP, Gelfand AE (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  4. Casella G, George EI (1992) Explaining the Gibbs sampler. Am Stat 46(3):167–174CrossRefGoogle Scholar
  5. Chib S, Greenberg E (1995) Understanding the Metropolis–Hastings algorithm. Am Stat 49(4):327–335CrossRefGoogle Scholar
  6. Fotheringham AS, Brunsdon C, Charlton M (2002) Geographically weighted regression: the analysis of spatially varying relationships. Wiley, West SussexGoogle Scholar
  7. Gelfand AE, Kim H, Sirmans CF, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98:387–396CrossRefGoogle Scholar
  8. Gelfand AE, Schmit AM, Banerjee S, Sirmans CF (2004) Nonstationary multivariate process modeling through spatially varying coregionalization. Test 13(2):263–312CrossRefGoogle Scholar
  9. Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis. Chapman & Hall, LondonGoogle Scholar
  10. Goldstein M (1976) Bayesian analysis of regression problems. Biometrika 63(1):51–58CrossRefGoogle Scholar
  11. Huang Y, Leung Y (2002) Analysing regional industrialisation in Jiangsu province using geographically weighted regression. J Geogr Syst 4:233–249CrossRefGoogle Scholar
  12. Loader C (1999) Local regression and likelihood. Springer, New YorkGoogle Scholar
  13. Longley PA, Tobón C (2004) Spatial dependence and heterogeneity in patterns of hardship: an intra-urban analysis. Ann Assoc Am Geogr 94:503–519CrossRefGoogle Scholar
  14. MacEachern SN, Berliner LM (1994) Subsampling the Gibbs sampler. Am Stat 48(3):190–193CrossRefGoogle Scholar
  15. Nakaya T (2001) Local spatial interaction modelling based on the geographically weighted regression approach. GeoJournal 53:347–358CrossRefGoogle Scholar
  16. Neal RM (2003) Slice sampling. Ann Stat 31(3):705–767CrossRefGoogle Scholar
  17. Neter J, Kutner MH, Nachtsheim CJ, Wasserman W (1996) Applied linear regression models. Irwin, ChicagoGoogle Scholar
  18. Wheeler D (2006) Diagnostic tools and a remedial method for collinearity in geographically weighted regression. Environ Plann A (in press)Google Scholar
  19. Wheeler D, Tiefelsdorf M (2005) Multicollinearity and correlation among local regression coefficients in geographically weighted regression. J Geogr Syst 7:161–187CrossRefGoogle Scholar
  20. Wheeler D, Calder C (2006) Bayesian spatially varying coefficient models in the presence of collinearity. In: 2006 Proceedings of the American Statistical Association, Spatial Modeling Section [CD-ROM], Alexandria, VA: American Statistical AssociationGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of BiostatisticsEmory UniversityAtlantaUSA
  2. 2.Department of StatisticsThe Ohio State UniversityColumbusUSA

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