Skip to main content
SpringerLink
Log in

Geographically Weighted Negative Binomial Regression—incorporating overdispersion

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Global regression assumes that a single model adequately describes all parts of a study region. However, the heterogeneity in the data may be sufficiently strong that relationships between variables can not be spatially constant. In addition, the factors involved are often sufficiently complex that it is difficult to identify them in the form of explanatory variables. As a result Geographically Weighted Regression (GWR) was introduced as a tool for the modeling of non-stationary spatial data. Using kernel functions, the GWR methodology allows the model parameters to vary spatially and produces non-parametric surfaces of their estimates. To model count data with overdispersion, it is more appropriate to use a negative binomial distribution instead of a Poisson distribution. Therefore, we propose the Geographically Weighted Negative Binomial Regression (GWNBR) method for the modeling of data with overdispersion. The results obtained using simulated and real data show the superiority of this method for the modeling of non-stationary count data with overdispersion compared with competing models, such as global regressions, e.g., Poisson and negative binomial and Geographically Weighted Poisson Regression (GWPR). Moreover, we illustrate that these competing models are special cases of the more robust model GWNBR.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  • Casella, G., Berger, R.L.: Statistical Inference, 2nd edn. Duxbury, N. Scituate (2001)

    Google Scholar 

  • Cleveland, W.S., Devlin, S.J.: Locally-weighted regression: and approach to regression analysis by local fitting. J. Am. Stat. Assoc. 83, 596–610 (1988)

    Article  MATH  Google Scholar 

  • Farber, S., Paez, A.: A systematic investigation of cross-validation in gwr model estimation: empirical analysis and Monte Carlo simulations. J. Geogr. Syst. 9(4), 371–396 (2007)

    Article  Google Scholar 

  • Farber, S., Paez, A., Volz, E.: Topology and dependency tests in spatial and network autoregressive models. Geogr. Anal. 41(2), 158–180 (2009)

    Article  Google Scholar 

  • Fotheringham, A.S., Brunsdon, C., Charlton, M.: Geographically Weighted Regression. Wiley, New York (2002)

    Google Scholar 

  • Geniaux, G., Ay, J.S., Napoléone, C.: A spatial hedonic approach on land use change anticipations. J. Reg. Sci. 51(5), 967–986 (2011)

    Article  Google Scholar 

  • Hadayeghi, A., Shalaby, A.S., Persaud, B.N.: Development of planning level transportation safety tools using geographically weighted Poisson regression. Accid. Anal. Prev. 42, 678–688 (2010)

    Article  Google Scholar 

  • Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. Chapman & Hall/CRC, London (1990)

    MATH  Google Scholar 

  • Hilbe, J.M.: Negative Binomial Regression, 2nd edn. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  • Hope, A.C.A.: A simplified Monte Carlo significance test procedure. J. R. Stat. Soc. 30, 582–598 (1968)

    MATH  Google Scholar 

  • Kobayashi, T., Lane, B.: Spatial heterogeneity and transit use. In: 11th World Conference on Transportation Research (2007)

    Google Scholar 

  • Leung, Y., Mei, C.L., Zhang, W.X.: Statistical tests for spacial nonstationarity based on the geographically weighted regression model. Environ. Plan. A 32, 9 (2000a)

    Article  Google Scholar 

  • Leung, Y., Mei, C.L., Zhang, W.X.: Testing for spatial autocorrelation among the residuals of the geographically weighted regression. Environ. Plan. A 32, 871–890 (2000b)

    Article  Google Scholar 

  • Nakaya, T., Fotheringham, A.S., Brunsdon, C., Charlton, M.: Geographically weighted Poisson regression for disease association mapping. Stat. Med. 24, 2695–2717 (2005)

    Article  MathSciNet  Google Scholar 

  • Nelder, J.A., Wedderburn, R.W.M.: Generalized linear models. J. R. Stat. Soc. 135, 370–384 (1972)

    Google Scholar 

  • Paez, A., Farber, S., Wheeler, D.: A simulation-based study of geographically weighted regression as a method for investigating spatially varying relationships. Environ. Plan. A 43(12), 2992–3010 (2011)

    Article  Google Scholar 

  • Staniswalis, J.G.: The kernel estimate of a regression function in likelihood-based models. J. Am. Stat. Assoc. 84, 276–283 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • Wheeler, D., Tiefelsdorf, M.: Multicollinearity and correlation among local regression coefficients in geographically weighted regression. J. Geogr. Syst. 7(2), 161–187 (2005)

    Article  Google Scholar 

  • Wood, S.: Generalized Additive Models: An Introduction with R, 2nd edn. Chapman & Hall/CRC, London (2006)

    Google Scholar 

Download references

Acknowledgements

We wish to thank two anonymous referees and the AE for their valuable comments, which improved the quality of this manuscript.

Author information

Authors and Affiliations

  1. Departamento de Estatistica, Universidade de Brasilia, Brasilia, Brazil

    Alan Ricardo da Silva & Thais Carvalho Valadares Rodrigues

Authors
  1. Alan Ricardo da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thais Carvalho Valadares Rodrigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alan Ricardo da Silva.

Rights and permissions

Reprints and permissions

About this article

Cite this article

da Silva, A.R., Rodrigues, T.C.V. Geographically Weighted Negative Binomial Regression—incorporating overdispersion. Stat Comput 24, 769–783 (2014). https://doi.org/10.1007/s11222-013-9401-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-013-9401-9

Keywords

Search

Navigation