SciELO - Scientific Electronic Library Online

 
vol.35 special issue 2Integration Methods of Odds Ratio Based on Meta-AnalysisUsing Fixed and Random Effect Models Useful in Public Health author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.35 no.spe2 Bogotá June 2012

 

An Extension to the Scale Mixture of Normals for Bayesian Small-Area Estimation

Una extensión a la mezcla de escala de normales para la estimación Bayesiana en pequeñas áreas

FRANCISCO J. TORRES-AVILÉS1, GLORIA ICAZA2, REINALDO B. ARELLANO-VALLE3

1Universidad de Santiago de Chile, Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Santiago, Chile. Assistant professor. Email: francisco.torres@usach.cl
2Universidad de Talca, Instituto de Matemática y Física, Talca, Chile. Associate professor. Email: gicaza@utalca.cl
3Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Departamento de Estadística, Santiago, Chile. Professor. Email: reivalle@mat.puc.cl


Abstract

This work considers distributions obtained as scale mixture of normal densities for correlated random variables, in the context of the Markov random field theory, which is applied in Bayesian spatial intrinsically autoregressive random effect models. Conditions are established in order to guarantee the posterior distribution existence when the random field is assumed as scale mixture of normal densities. Lung, trachea and bronchi cancer relative risks and childhood diabetes incidence in Chilean municipal districts are estimated to illustrate the proposed methods. Results are presented using appropriate thematic maps. Inference over unknown parameters is discussed and some extensions are proposed.

Key words: Disease mapping, Markov random field, Hierarchical model, Incidence rate, Relative risk.


Resumen

Este trabajo aborda las distribuciones obtenidas como mezcla de escala de normales para variables aleatorias correlacionadas, en el contexto de la teoría de los campos markovianos, la cual es aplicada a modelos bayesianos espaciales con efectos aleatorios autoregresivos intrínsecos. Se establecen condiciones para garantizar la existencia de la distribución a posteriori cuando se asume una distribución mezcla de escala de normales para el campo markoviano propuesto. Para ilustrar los métodos propuestos, se estiman los riesgos relativos de cáncer de tráquea, bronquios y pulmón, y tasas de incidencia de diabetes tipo 1 en distritos municipales de Chile. Los resultados son presentados usando mapas temáticos apropiados. Se discute la inferencia sobre los parámetros desconocidos y se proponen algunas extensiones.

Palabras clave: campo aleatorio markoviano, mapeo de enfermedades, modelo jerárquico, riesgo relativo, tasa de incidencia.


Texto completo disponible en PDF


References

1. Andia, M., Hsing, A. W., Andreotti, G. & Ferreccio, C. (2008), 'Geographic variation of gallbladder cancer mortality and risk factors in Chile: A population-based ecologic study', International Journal of Cancer 123(6), 1411-1416.         [ Links ]

2. Andrews, D. F. & Mallows, C. L. (1974), 'Scale mixture of normal distributions', Journal of the Royal Statistical Society Series B 36(1), 99-102.         [ Links ]

3. Assunção, R. M., Potter, J. E. & Cavenaghi, S. M. (2002), 'A Bayesian space varying parameter model applied to estimating fertility schedules', Statistics in Medicine 21, 2057-2075.         [ Links ]

4. Banerjee, S., Carlin, B. & Gelfand, A. (2004a), Hierarchical Modeling and Analysis for Spatial Data, Monographs on Statistics and Applied Probability 101. Chapman and Hall, Boca Ratón, Florida.         [ Links ]

5. Banerjee, S., Carlin, B. & Gelfand, A. (2004b), Hierarchical Modeling and Analysis for Spatial Data, Monographs on Statistics and Applied Probability 101. Chapman and Hall, Boca Ratón, Florida.         [ Links ]

6. Besag, J. (1974), 'Spatial interaction and the statistical analysis of lattice systems', Journal of the Royal Statistical Society Series B 36(2), 192-236.         [ Links ]

7. Besag, J. (1986), 'On the statistical analysis of dirty pictures', Journal of the Royal Statistical Society Series B 48(3), 259-302.         [ Links ]

8. Besag, J., York, J. & Mollié, A. (1991), 'Bayesian image restoration, with two applications in spatial statistics', Annals of the Institute of Statistical Mathematics 43, 1-59.         [ Links ]

9. Best, N., Arnold, R., Thomas, A., Waller, L. & Collon, E. (1999), Bayesian models for spatially correlated disease and exposure data, 'Bayesian Statistics', Vol. 6, Oxford University Press, Oxford.         [ Links ]

10. Breslow, N. & Clayton, D. (1993), 'Approximate inference in generalized linear mixed models', Journal of the American Statistical Association 88, 9-25.         [ Links ]

11. Carrasco, E., Pérez-Bravo, F., Dorman, J., Mondragón, A. & Santos, J. L. (2006), 'Increasing incidence of type 1 diabetes in population from Santiago of Chile: Trends in a period of 18 years (1986-2003)', Diabetes/Metabolism Research and Reviews 22, 34-37.         [ Links ]

12. Clayton, D. & Kaldor, J. (1987), 'Empirical Bayes estimates of age-standardized relative risks for use in disease mapping', Biometrics 43, 671-681.         [ Links ]

13. Congdon, P. (2003), Applied Bayesian Modelling, Wiley & Sons, Chichester.         [ Links ]

14. Damien, P. & Walker, S. (2001), 'Sampling truncated normal, beta and gamma densities', Journal of Computational and Graphical Statistics 10(2), 206-215.         [ Links ]

15. Fang, K. T., Kotz, S. & Ng, K. W. (1990), Symmetric Multivariate and Related Distributions, Chapman and Hall, New York.         [ Links ]

16. Ferreccio, C., Rollán, A., Harris, P., Serrano, C., Gederlini, A., Margozzini, P., Gonzalez, C., Aguilera, X., Venegas, A. & Jara, A. (2007), 'Gastric cancer is related to early Helicobacter pylori infection in a high prevalence country', Cancer Epidemiology, Biomarkers & Prevention 16, 662-667.         [ Links ]

17. Gelfand, A. E. & Ghosh, S. K. (1998), 'Model choice: a minimum posterior predictive loss approach', Biometrika 85, 1-11.         [ Links ]

18. Geweke, J. (1993), 'Bayesian treatment of the independent Student-t linear model', Journal of Applied Econometrics 8, 519-540.         [ Links ]

19. Ghosh, M., Natarajan, K., Stroud, T. W. F. & Carlin, B. P. (1998), 'Generalized linear models for small-area estimation', Journal of the American Statistical Association 93(441), 273-282.         [ Links ]

20. Icaza, G., Núñez, L., Díaz, N. & Varela, D. (2006), Atlas de mortalidad por enfermedades cardiovasculares en Chile, 1997- 2003, Universidad de Talca y Ministerio de Salud, New York.         [ Links ]

21. Icaza, G., Núñez, L., Torres, F., Díaz, N. & Varela, D. (2007), 'Distribución geográfica de mortalidad por tumores malignos de tráquea, bronquios y pulmón, Chile 1997-2004', Revista Médica de Chile 135(11), 1397-1405.         [ Links ]

22. Kano, Y. (1994), 'Consistency property of elliptical probability density functions', Journal of Multivariate Analysis 51, 139-147.         [ Links ]

23. Lange, K. & Sinsheimer, J. S. (1993), 'Normal/independent distributions and their applications in robust regression', Journal of Computational and Graphical Statistics 2(2), 175-198.         [ Links ]

24. Lyu, S. & Simoncelli, E. P. (2007), Statistical modeling of images with fields of Gaussian scale mixtures, 'Advances in Neural Information Processing Systems', Vol. 19, MIT Press, Cambridge, p. 945-952.         [ Links ]

25. Mollié, A. (2000), Bayesian mapping of Hodgkin's disease in France, 'Spatial Epidemiology: Methods and Applications', Oxford University Press, New York, p. 267-285.         [ Links ]

26. Parent, O. & Lesage, J. P. (2008), 'Using the variance structure of the conditional autoregressive specification to model knowledge spillovers', Journal of Applied Economics 23, 235-256.         [ Links ]

27. Pascutto, C., Wakefield, J. C., Best, N. G., Richardson, S., Bernardinelli, L., Staines, A. & Elliott, P. (2000), 'Statistical issues in the analysis of disease mapping data', Statistics in Medicine 19(17-18), 2493-519.         [ Links ]

28. Roislien, J. & Omre, O. (2006), 'T-distributed random fields: a parametric model for heavy-tailed well-log data', Mathematical Geology 38(7), 821-849.         [ Links ]

29. Spiegelhalter, D. J., Best, N. G., Carlin, B. P. & Van der Linde, A. (2002), 'Bayesian measures of model complexity and fit', Journal of the Royal Statistical Society, Series B 64, 583-639.         [ Links ]

30. Torres-Avilés, F., Icaza, G., Carrasco, E. & Pérez-Bravo, F. (2010), 'Clustering of cases of type 1 diabetes in high socioeconomic communes in Santiago de Chile: Spatio-temporal and geographical analysis.', Acta Diabetologica 47(3), 251-257.         [ Links ]

31. Zellner, A. (1976), 'Bayesian and non-Bayesian analysis of the regression model with multivariate student-t error terms', Journal of the American Statistical Association 71(354), 400-405.         [ Links ]

[Recibido en septiembre de 2011. Aceptado en febrero de 2012]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv35n2a01,
AUTHOR = {Torres-Avilés, Francisco J. and Icaza, Gloria and Arellano-Valle, Reinaldo B.},
TITLE = {{An Extension to the Scale Mixture of Normals for Bayesian Small-Area Estimation}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2012},
volume = {35},
number = {2},
pages = {185-204}
}

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License