Quantile Regression for Area Disease Counts: Bayesian Estimation using Generalized Poisson Regression





Hierarchical quantile regression. Relative risk. Risk intervals. Elevated risk. Self-harm.


Generalized linear models based on Poisson regression are commonly applied to count data for area morbidity outcomes, focused on modelling the conditional mean of the response as a function of a set of risk factors. Mean regression models may be sensitive to outliers and provide no information on other distributional features of the response. We consider instead a Poisson lognormal hierarchical approach to quantile regression of spatially configured count data, allowing for observed risk factors and spatially correlated unobserved risk factors. This technique has the advantage that a profile of the relative outcome risk across quantiles can be obtained, including estimates of uncertainty (e.g. the uncertainty attaching to 2.5% or 5% relative risk quantiles). An application involves counts of emergency hospitalisations for self-harm for 6791 small areas in England. Known risk factors are area deprivation, a measure of social fragmentation and a measure of rural status. It is shown that impact of these predictors varies between quantiles, and that hierarchical quantile regression generally produces narrower risk intervals, except for outlier areas, and leads to a higher number of areas being classed as high risk.


McDaid D, Bonin E, Park A, Hegerl U, Arensman E, Kopp M, Gusmao R. Making the case for investing in suicide prevention interventions: estimating the economic impact of suicide and non-fatal self harm events. Injury Prevention 2010; 16(Suppl 1): A257-8. DOI: https://doi.org/10.1136/ip.2010.029215.916

Wakefield J. Disease mapping and spatial regression with count data. Biostatistics 2006; 8(2): 158-83. https://doi.org/10.1093/biostatistics/kxl008 DOI: https://doi.org/10.1093/biostatistics/kxl008

Besag J, York J, Mollié A. Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics 1991; 43(1): 1-20. https://doi.org/10.1007/BF00116466 DOI: https://doi.org/10.1007/BF00116466

Best N, Richardson S, Thomson A. A comparison of Bayesian spatial models for disease mapping. Statistical Methods in Medical Research 2005; 14(1): 35-59. https://doi.org/10.1191/0962280205sm388oa DOI: https://doi.org/10.1191/0962280205sm388oa

Richardson S, Thomson A, Best N, Elliott P. Interpreting posterior relative risk estimates in disease-mapping studies. Environmental Health Perspectives 2004; 112(9): 1016. https://doi.org/10.1289/ehp.6740 DOI: https://doi.org/10.1289/ehp.6740

Koenker R, Hallock K. Quantile regression: An introduction. Journal of Economic Perspectives 2001; 15(4): 43-56. https://doi.org/10.1257/jep.15.4.143 DOI: https://doi.org/10.1257/jep.15.4.143

Yu K, Moyeed RA. Bayesian quantile regression. Statistics & Probability Letters 2001; 54(4): 437-47. https://doi.org/10.1016/S0167-7152(01)00124-9 DOI: https://doi.org/10.1016/S0167-7152(01)00124-9

Tsionas EG. Bayesian quantile inference. Journal of Statistical Computation and Simulation 2003; 73(9): 659-74. https://doi.org/10.1080/0094965031000064463 DOI: https://doi.org/10.1080/0094965031000064463

Machado JA, Silva JS. Quantiles for counts. Journal of the American Statistical Association 2005; 100(472): 1226-37. https://doi.org/10.1198/016214505000000330 DOI: https://doi.org/10.1198/016214505000000330

Lee D, Neocleous T. Bayesian quantile regression for count data with application to environmental epidemiology. Journal of the Royal Statistical Society: Series C (Applied Statistics). 2010; 59(5): 905-20. https://doi.org/10.1111/j.1467-9876.2010.00725.x DOI: https://doi.org/10.1111/j.1467-9876.2010.00725.x

Yue Y, Rue H. Bayesian inference for additive mixed quantile regression models. Computational Statistics & Data Analysis 2011; 55(1): 84-96. https://doi.org/10.1016/j.csda.2010.05.006 DOI: https://doi.org/10.1016/j.csda.2010.05.006

Neelon B, Li F, Burgette LF, Benjamin Neelon SE. A spatiotemporal quantile regression model for emergency department expenditures. Statistics in Medicine 2015; 34(17): 2559-75. https://doi.org/10.1002/sim.6480 DOI: https://doi.org/10.1002/sim.6480

Reich B, Fuentes M, Dunson DB. Bayesian spatial quantile regression. Journal of the American Statistical Association. 2011; 106(493): 6-20. https://doi.org/10.1198/jasa.2010.ap09237 DOI: https://doi.org/10.1198/jasa.2010.ap09237

Dreassi E, Ranalli MG, Salvati N. Semiparametric M-quantile regression for count data. Statistical Methods in Medical Research 2014; 23(6): 591-610. https://doi.org/10.1177/0962280214536636 DOI: https://doi.org/10.1177/0962280214536636

Requia WJ, Koutrakis P, Roig HL, Adams MD, Santos CM. Association between vehicular emissions and cardiorespiratory disease risk in Brazil and its variation by spatial clustering of socio-economic factors. Environmental Research 2016; 150: 452-60. https://doi.org/10.1016/j.envres.2016.06.027 DOI: https://doi.org/10.1016/j.envres.2016.06.027

Chiu C, Wen TH, Chien LC, Yu HL. A probabilistic spatial dengue fever risk assessment by a threshold-based-quantile regression method. PLOS One 2014; 9(10): e106334. DOI: https://doi.org/10.1371/journal.pone.0106334

O’Hara RB, Kotze DJ. Do not log‐transform count data. Methods in Ecology and Evolution 2010; 1(2): 118-22. https://doi.org/10.1111/j.2041-210X.2010.00021.x DOI: https://doi.org/10.1111/j.2041-210X.2010.00021.x

Trinh G, Rungie C, Wright M, Driesener C, Dawes J. Predicting future purchases with the Poisson log-normal model. Marketing Letters 2014; 25(2): 219-34. https://doi.org/10.1007/s11002-013-9254-1 DOI: https://doi.org/10.1007/s11002-013-9254-1

Mahaki B, Mehrabi Y, Kavousi A, Mohammadian Y, Kargar M. Applying and comparing empirical and full Bayesian models in study of evaluating relative risk of suicide among counties of Ilam province. Journal of Education and Health Promotion 2015; 4: 50. https://doi.org/10.4103/2277-9531.162331 DOI: https://doi.org/10.4103/2277-9531.162331

Connolly SR, Dornelas M, Bellwood DR, Hughes TP. Testing species abundance models: a new bootstrap approach applied to Indo‐Pacific coral reefs. Ecology 2009; 90(11): 3138-49. https://doi.org/10.1890/08-1832.1 DOI: https://doi.org/10.1890/08-1832.1

Al-Hamzawi R, Yu K, Pan J. Prior elicitation in Bayesian quantile regression for longitudinal data. J Biomet Biostat 2011; 2: 115. DOI: https://doi.org/10.4172/2155-6180.1000115

Ancelet S, Abellan JJ, Del Rio Vilas VJ, Birch C, Richardson S. Bayesian shared spatial‐component models to combine and borrow strength across sparse disease surveillance sources. Biometrical Journal 2012; 54(3): 385-404. https://doi.org/10.1002/bimj.201000106 DOI: https://doi.org/10.1002/bimj.201000106

Zhou J, Chang HH, Fuentes M. Estimating the health impact of climate change with calibrated climate model output. Journal of Agricultural, Biological, and Environmental Statistics 2012; 17(3): 377-394. https://doi.org/10.1007/s13253-012-0105-y DOI: https://doi.org/10.1007/s13253-012-0105-y

Department of Communities and Local Government (DCLG) The English Indices of Deprivation. Office of National Statistics and DCLG, London 2015.

O’Farrell IB, Corcoran P, Perry IJ. The area level association between suicide, deprivation, social fragmentation and population density in the Republic of Ireland: a national study. Social Psychiatry and Psychiatric Epidemiology 2016; 51(6): 839-47. https://doi.org/10.1007/s00127-016-1205-8 DOI: https://doi.org/10.1007/s00127-016-1205-8

Lunn DJ, Thomas A, Best N, Spiegelhalter D. WinBUGS-a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing 2000; 10(4): 325-37. https://doi.org/10.1023/A:1008929526011 DOI: https://doi.org/10.1023/A:1008929526011

Brooks SP, Gelman A. General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics 1998; 7(4): 434-55. https://doi.org/10.1080/10618600.1998.10474787 DOI: https://doi.org/10.1080/10618600.1998.10474787

Watanabe S. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research 2010; 11: 3571-94.

Reed, C., Yu, K. A partially collapsed Gibbs sampler for Bayesian quantile regression. Brunel University, Dept of Mathematics Research Papers 2009.

Berkhof J, Van Mechelen I, Hoijtink H. Posterior predictive checks: Principles and discussion. Computational Statistics 2000; 15(3): 337-54. https://doi.org/10.1007/s001800000038 DOI: https://doi.org/10.1007/s001800000038

Lunn D, Jackson C, Best N, Thomas A, Spiegelhalter D. The BUGS Book: A Practical Introduction to Bayesian Analysis. CRC Press 2012. DOI: https://doi.org/10.1201/b13613




How to Cite

Congdon, P. (2017). Quantile Regression for Area Disease Counts: Bayesian Estimation using Generalized Poisson Regression. International Journal of Statistics in Medical Research, 6(3), 92–103. https://doi.org/10.6000/1929-6029.2017.06.03.1



General Articles