Statistical Papers

, 49:531 | Cite as

Modelling count data with overdispersion and spatial effects

Regular Article


In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson (GP) distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding correlated spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. in Stat Comput 12(4):353–367, (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. in J R Stat Soc B64(4):583–640, (2002) and using proper scoring rules, see for example Gneiting and Raftery in Technical Report no. 463, University of Washington, (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, spatial Poisson models with spatially correlated or uncorrelated random effects are to be preferred over all other models according to the considered criteria.


Bayesian inference Count data Overdispersion Spatial regression models Zero inflated models 


  1. Agarwal DK, Gelfand AE, Citron-Pousty S (2002) Zero-inflated models with application to spatial count data. Environ Ecol Stat 9:341–355CrossRefMathSciNetGoogle Scholar
  2. Angers JF, Biswas A (2003) A Bayesian analysis of zero-inflated generalized Poisson model. Comput Stat Data Anal 42:37–46CrossRefMathSciNetMATHGoogle Scholar
  3. Banerjee S, Carlin B, Gelfand A (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC, New YorkMATHGoogle Scholar
  4. Besag J, Kooperberg C (1995) On conditional and intrinsic autoregressions. Biometrika 82:733–746MATHMathSciNetGoogle Scholar
  5. Brier G (1950) Verification of forecasts expressed in terms of probability. Mon Weather Rev 78 (1):1–3CrossRefGoogle Scholar
  6. Consul P (1989) Generalized Poisson distributions. Properties and Applications. Marcel Dekker, New YorkMATHGoogle Scholar
  7. Consul P, Jain G (1973) A generalization of the Poisson distribution. Technometrics 15:791–799MATHCrossRefMathSciNetGoogle Scholar
  8. Czado C, Prokopenko S (2004) Modeling transport mode decisions using hierarchical binary spatial regression models with cluster effects. Discussion paper 406, SFB 386 Statistische Analyse diskreter Strukturen Http:// Scholar
  9. Famoye F, Singh K (2003a) On inflated generalized Poisson regression models. Adv Appl Stat 3(2):145–158MATHMathSciNetGoogle Scholar
  10. Famoye F, Singh K (2003b) Zero inflated generalized Poisson regression model (submitted)Google Scholar
  11. Gelman A, Carlin J, Stern H, Rubin D (2004) Bayesian data analysis, 2nd edn. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar
  12. Gilks W, Wild P (1992) Adaptive rejection sampling for Gibbs sampling. Appl Stat 41(2):337–348MATHCrossRefGoogle Scholar
  13. Gilks W, Richardson S, Spiegelhalter D (1996) Markov Chain Monte Carlo in Practice. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar
  14. Gneiting T, Raftery AE (2004) Strictly proper scoring rules, prediction and estimation. Technical Report no. 463, Department of Statistics, University of WashingtonGoogle Scholar
  15. Gschlößl S (2006) Hierarchical Bayesian spatial regression models with applications to non-life insurance. PhD thesis, Munich University of TechnologyGoogle Scholar
  16. Han C, Carlin B (2001) Markov chain Monte Carlo methods for computing Bayes factors: a comparative review. J Am Stat Assoc 96:1122–1132CrossRefGoogle Scholar
  17. Hoeting J, Madigan D, Raftery A, Volinsky C (1999) Bayesian model averaging: a tutorial. Stat Sci 14(4):382–417MATHCrossRefMathSciNetGoogle Scholar
  18. Jin X, Carlin B, Banerjee S (2005) Generalized hierarchical multivariate CAR models for areal data. Biometrics 61:950–961MATHCrossRefMathSciNetGoogle Scholar
  19. Joe H, Zhu R (2005) Generalized Poisson distribution: the property of mixture of Poisson and comparison with Negative Binomial distribution. Biometric J 47:219–229CrossRefMathSciNetGoogle Scholar
  20. Kass R, Raftery A (1995) Bayes factors and model uncertainty. J Am Stat Assoc 90:773–795MATHCrossRefGoogle Scholar
  21. Lambert D (1992) Zero-inflated Poisson regression with and application to defects in manufacturing. Technometrics 34(1):1–14MATHCrossRefGoogle Scholar
  22. van der Linde A (2005) DIC in variable selection. Statistica Neerlandica 59(1):45–56MATHCrossRefMathSciNetGoogle Scholar
  23. Pettitt A, Weir I, Hart A (2002) A conditional autoregressive Gaussian process for irregularly spaced multivariate data with application to modelling large sets of binary data. Stat Comput 12(4):353–367CrossRefMathSciNetGoogle Scholar
  24. Rodrigues J (2003) Bayesian analysis of zero-inflated distributions. Commun Stat 32(2):281–289MATHCrossRefGoogle Scholar
  25. Spiegelhalter D, Best N, Carlin B, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B 64(4):583–640MATHCrossRefGoogle Scholar
  26. Sun D, Tsutakawa RK, Kim H, He Z (2000) Bayesian analysis of mortality rates with disease maps. Stat Med 19:2015–2035CrossRefGoogle Scholar
  27. Winkelmann R (2003) Econometric analysis of count data, 4th edn. Springer, Berlin Heidelberg, GermanyMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Center of Mathematical SciencesMunich University of TechnologyGarchingGermany

Personalised recommendations