Conditional overdispersed models: Application to count area data

Abstract

This paper proposes alternative models for the analysis of count data featuring a given spatial structure, which corresponds to geographical areas. We assume that the overdispersion data structure partially results from the existing and well justified spatial correlation between geographical adjacent regions, so an extension of existing overdispersion models that include spatial neighborhood structures within a Bayesian framework is proposed. These models allow practitioners to quantify the association explained by the considered neighborhood structures and the one modelled by additional factors. Finally, using the information provided by the Colombian National Demographic and Health Survey, the usefulness of the proposed models is illustrated by fitting them to infant mortality rates and to data including the proportion of mothers who, after giving birth to their last child, underwent a postnatal screening period in Colombia.

Keywords

Count area data infant mortality overdispersion postnatal screening period rates spatial statistics

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References

Lindsey

Jones

. Choosing among generalized linear models applied to medical data. Stat Med 1998; 17: 59–68.

Lee

Nelder

. Analysis of ulcer data using hierarchical gerealized linear models. Stat Med 2002; 21: 191–202.

Zhang

Lin

. Spatial scan statistics with overdispersion. Stat Med 2012; 31: 762–774.

McCullagh

Nelder

. Generalized linear models, London: Chapman and Hall, 1989.

Hinde

Demétrio

CGB

. Overdispersion: models and estimation. Computat Stat Data Anal 1998; 27: 151–170.

Williams

. The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity. Biometrics 1975; 31: 949–952.

Williams

. Extra-binomial variation in logistic linear models. Appl Stat 1982; 31: 144–188.

Margolin

Kaplan

Zeiger

. Statistical analysis of the Ames Salmonella/mocrosome test. Proc Natl Acad Sci USA 1981; 78: 3779–3783.

Breslow

. Extra-Poisson variation in log-linear models. Appl Stat 1984; 33: 38–44.

10.

Lawless

. Negative binomial and mixed Poisson regression. Can J Stat 1987; 15: 209–225.

11.

Poortema

. On modelling overdispersion of counts. Statistica Neerlandica 1999; 53: 5–20.

12.

Wakefield

. Disease mapping and spatial regression with count data. Biostatistics 2007; 8: 158–183.

13.

Mohebbi

Wolfe

Forbes

. Disease mapping and regression with count data in the presence of overdispersion and spatial autocorrelation. A Bayesian model averaging approach. Int J Environ Res Public Health 2014; 11: 883–902.

14.

Quintero-Sarmiento

Cepeda-Cuervo

Núñez-Antón

. Estimating infant mortality in Colombia: some overdispersion modelling approaches. J Appl Stat 2012; 39: 1011–1036.

15.

Thompson

Sharp

. Explaining heterogeneity in meta-analysis: a comparison of methods. Stat Med 1999; 18: 2693–2708.

16.

Thompson

Stephen

. Explaining heterogeneity in meta-analysis: a comparison of methods. Stat Med 1999; 18: 2693–2708.

17.

Cepeda-Cuervo

Urdinola

Rodríguez

. Double generalized spatial econometric models. Commun Stat – Simul Computat 2012; 41: 671–685.

18.

Krebs

. Ecological methodology, Menlo Park, CA: Benjamin/Cummings, 1999.

19.

Cepeda-Cuervo

Núñez-Antón

. Spatial double generalized beta regression models: extensions and application to study quality of education in Colombia. J Educ Behavior Stat 2013; 38: 604–628.

20.

Moran

PAP

. The interpretation of statistical maps. J Royal Stat Soc Ser B 1948; 10: 243–251.

21.

Geary

. The contiguity ratio and statistical mapping. Incorporate Statis 1954; 5: 115–146.

22.

Anselin

. Spatial econometrics: methods and models, Boston, MA: Kluwer Academic Publishers, 1988.

23.

Whittle

. On stationary process in the plane. Biometrika 1954; 41: 434–449.

24.

Besag

. Spatial interaction and the statistical analysis of lattice systems. J Royal Statis Soc Ser B 1974; 36: 192–236. with discussion.

25.

Best

Richardson

Thomson

. A comparison of Bayesian spatial models for disease mapping. Statis Meth Med Res 2005; 14: 35–59.

26.

Wakefield

Best

Waller

Bayesian approaches to disease mapping. In: Elliott

Wakefield

Best

Briggs

(eds). Spatial epidemiology: methods and appplications, Oxford, UK: Oxford University Press, 2000, pp. 104–127.

27.

Besag

York

Mollié

. A Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 1991; 43: 1–59. with discussion.

28.

Anselin

. Under the hood: Issues in the specification and interpretation of spatial regression models. Agri Econ 2002; 27: 247–267.

29.

Gelman

Carlin

Stern

et al.

Bayesian data analysis, Boca Raton, FL: Chapman & Hall/CRC, 2003.

30.

Fleming

Techniques for estimating spatially dependent discrete choice models. In: Anselin

Florax

LJGM

Rey

(eds). Advances in spatial econometrics. Methodology, tools and applications, New York, NY: Springer, 2004, pp. 145–168.

31.

Dormann

McPherson

Araújo

et al.

Methods to account for spatial autocorrelation in the analysis of species distributional data: a review. Ecography 2007; 30: 609–628.

32.

Dey

Gelfand

Peng

. Overdispersed generalized linear models. J Stat Plan Infer 1997; 64: 93–107.

33.

Dean

. Testing for overdispersion in Poisson and binomial regression models. J Am Stat Assoc 1992; 87: 451–457.

34.

Crowder

. Beta-Binomial anova for proportions. Appl Stat 1978; 27: 34–37.

35.

Crowder

. Gaussian estimation for correlated binomial data. J Royal Stat Soc Ser B 1985; 47: 229–237.

36.

Carlin

Louis

. Bayes and empirical Bayes methods for data analysis, 2nd ed. London, UK: Chapman and Hall/CRC, 2000.

37.

Spiegelhalter DJ, Thomas A, Best N, et al. WinBUGS user manual. MRC Biostatistics Unit: Cambridge, UK, 2003, www.mrc-bsu.cam.ac.uk/bugs.

38.

Spiegelhalter

Best

Carlin

et al.

Bayesian measures of model complexity and fit (with discussion). J Royal Stat Soc Ser B 2002; 64: 583–616.

39.

Watanabe

. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J Mach Learn Res 2010; 11: 3571–3594.

40.

Piironen

Vehtari

. Comparison of Bayesian predictive methods for model selection. Stat Comput 2017; 27: 711–735.

41.

Gelman

Hwang

Vehtari

. Understanding predictive information criteria for Bayesian models. Stat Comput 2013; 24: 997–1016.

42.

Preston

. The changing relation between mortality and level of economic development. Popul Studies 1975; 29: 231–248.

43.

Cortés

. Editorial Desarrollo humano, pobreza y salud. Colombia Médica 2006; 37: 5–5.

44.

Sepúlveda García de la Torre

. La Violencia de género como causa de maltrato infantil. Cuadernos de Medicina Forense 2006; 12: 149–164.

45.

Dammert AC. Access to health services and infant mortality in Peru. Investigaciones Breves, No. 18. Lima, Peru: Economic Research Consortium, 2001.

46.

Silva Ayçaguer

Duran Macho

. Mortalidad infantil y condiciones higiénico-sociales en las Américas. Un estudio de correlación. Revista de Saúde Pública 1990; 24: 473–480.

47.

Xiaping

Banerjee

Carlin

. Order-free co-regionalized areal data models with application to multiple-disease mapping. J Royal Stat Soc Ser B 2007; 69: 817–838.

48.

Lambert

Brown

Florax

RJGM

. A two-step estimator for a spatial lag model of counts: theory, small sample performance and an application. Region Sci Urban Econ 2010; 40: 241–252.

49.

Wall

. A close look at the spatial structure implied by the CAR and SAR models. J Stat Plan Infer 2004; 121: 311–324.

50.

Zimmerman

Núñez-Antón

. Antedependence models for longitudinal data, London, UK: Chapman and Hall, 2010.