, Volume 55, Issue 4, pp 1363–1388 | Cite as

Bayesian Estimation of Age-Specific Mortality and Life Expectancy for Small Areas With Defective Vital Records

  • Carl P. SchmertmannEmail author
  • Marcos R. Gonzaga


High sampling variability complicates estimation of demographic rates in small areas. In addition, many countries have imperfect vital registration systems, with coverage quality that varies significantly between regions. We develop a Bayesian regression model for small-area mortality schedules that simultaneously addresses the problems of small local samples and underreporting of deaths. We combine a relational model for mortality schedules with probabilistic prior information on death registration coverage derived from demographic estimation techniques, such as Death Distribution Methods, and from field audits by public health experts. We test the model on small-area data from Brazil. Incorporating external estimates of vital registration coverage though priors improves small-area mortality estimates by accounting for underregistration and automatically producing measures of uncertainty. Bayesian estimates show that when mortality levels in small areas are compared, noise often dominates signal. Differences in local point estimates of life expectancy are often small relative to uncertainty, even for relatively large areas in a populous country like Brazil.


Mortality Small areas Bayesian models Data quality 



This research was supported by the Capes Foundation of Brazil’s Ministry of Education. Marcos R. Gonzaga gratefully acknowledges support from Research Projects 470866/2014-4 (Estimativas de mortalidade e construção de tabelas de vida para pequenas áreas no Brasil, 1980 a 2010 MCTI/CNPQ/MEC/Capes/Ciências Sociais Aplicadas) and 454223/2014-5 (Estimativas de mortalidade e construção de tabelas de vida para pequenas áreas no Brasil, 1980 a 2010/MCTI/CNPQ/Universal 14/2014).


  1. Agostinho, C. (2009). Estudo sobre a mortalidade adulta, para Brasil entre 1980 e 2000 e Unidades da Federação em 2000: Uma aplicação dos métodos de distribuição de mortes [A study of adult mortality in Brazil between 1980 and 2000, and in Brazilian states in 2000] (Unpublished doctoral dissertation). Faculdade de Ciências Econômicas, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil.Google Scholar
  2. Alexander, M., Zagheni, E., & Barbieri, M. (2017). A flexible Bayesian model for estimating subnational mortality. Demography, 54, 2025–2041.CrossRefGoogle Scholar
  3. Alkema, L., Kantorová, V., Menozzi, C., & Biddlecom, A. (2013). National, regional, and global rates and trends in contraceptive prevalence and unmet need for family planning between 1990 and 2015: A systematic and comprehensive analysis. Lancet, 381, 1642–1652.CrossRefGoogle Scholar
  4. Alkema, L., Raftery, A. E., Gerland, P., Clark, S. J., Pelletier, F., Buettner, T., & Heilig, G. K. (2011). Probabilistic projections of the total fertility rate for all countries. Demography, 48, 815–839.CrossRefGoogle Scholar
  5. Bennett, N. G., & Horiuchi, S. (1981). Estimating the completeness of death registration in a closed population. Population Index, 47, 207–221.CrossRefGoogle Scholar
  6. Bennett, N. G., & Horiuchi, S. (1984). Mortality estimation from registered deaths in less developed countries. Demography, 21, 217–233.CrossRefGoogle Scholar
  7. Bernardinelli, L., & Montomoli, C. (1992). Empirical Bayes versus fully Bayesian analysis of geographical variation in disease risk. Statistics in Medicine, 11, 983–1007.CrossRefGoogle Scholar
  8. Bhat, P. N. M. (2002). Completeness of India’s sample registration system: An assessment using the general growth balance method. Population Studies, 56, 119–134.Google Scholar
  9. Bignami-Van Assche, S. (2005, March–April). Province-specific mortality in China 1990–2000. Paper presented at the annual meeting of the Population Association of America, Philadelphia, PA.Google Scholar
  10. Borges, D., Miranda, D., Duarte, T., Novaes, F., Ettel, K., Guimarães, T., & Ferreira, T. (2012). Mortes violentas no Brasil: Uma análise do fluxo de informações [Violent deaths in Brazil: An analysis of the flow of information.]. Rio de Janeiro, Brazil: LAV/UERJ.Google Scholar
  11. Brass, W. (1971). Mortality models and their uses in demography. Transactions of the Faculty of Actuaries, 33, 123–142.CrossRefGoogle Scholar
  12. Brass, W. (1975). Methods for estimating fertility and mortality from limited and defective data, based on seminars held 16–24 September 1971 at Centro Latinamerico de Demografia (CELADE) San Jose, Costa Rica (Report). Chapel Hill, NC: International Program of Laboratories for Population Statistics.Google Scholar
  13. Campos, N. O. B., & Rodrigues, R. N. (2004). Ritmo de declínio nas taxas de mortalidade dos idosos nos estados do Sudeste, 1980–2000 [The pace of decline in mortality rates of the elderly in states of the Southeast, 1980–2000]. Revista Brasileira de Estudos de População, 21, 323–342.Google Scholar
  14. Carpenter, B., Gelman, A., Hoffman, M. D., Lee, G., Goodrich, B., Betancourt, M., . . . Riddell, A. (2017). Stan: A probabilistic programming language. Journal of Statistical Software, 76, 1–32.
  15. Congdon, P. (2009). Life expectancies for small areas: A Bayesian random effects methodology. International Statistical Review, 77, 222–240.CrossRefGoogle Scholar
  16. de Beer, J. (2012). Smoothing and projecting age-specific probabilities of death by TOPALS. Demographic Research, 27(article 20), 543–592. CrossRefGoogle Scholar
  17. de Boor, C. (2001). Applied mathematical sciences: Vol. 27. A practical guide to splines (Revised ed.). New York, NY: Springer-Verlag.Google Scholar
  18. de Mello Jorge, M. H. P., Gawryszewski, V. P., & Latorre, M. D. R. D. D. O. (1997). Análise dos dados de mortalidade [Analysis of mortality data.]. Revista de Saúde Pública, 31, 5–25. CrossRefGoogle Scholar
  19. de Mello Jorge, M. H. P., Laurenti, R., & Davidson Gotlieb, S. L. (2007). Análise da qualidade das estatísticas vitais brasileiras: A experiência de implantação do SIM e do SINASC [Quality analysis of Brazilian vital statistics: The experience of implementing the SIM and SINASC systems]. Ciência e Saúde Coletiva, 12, 643–654.CrossRefGoogle Scholar
  20. de Oliveira, G. L., Loschi, R. H., & Assunção, R. M. (2017). A random-censoring Poisson model for underreported data. Statistics in Medicine. Advance online publication. doi:
  21. Freire, F. H., Lima, E. C., Queiroz, B. L., Gonzaga, M. R., & Souza, F. H. (2015, May). Mortality estimates and construction of life tables for small areas in Brazil, 2010. Paper presented at the annual meeting of the Population Association of America, San Diego, CA.Google Scholar
  22. Frias, P. G., Szwarcwald, C. L., de Souza, P. R. B., Jr., Almeida, W. D. S., & Lira, P. I. C. (2013). Correcting vital information: Estimating infant mortality, Brazil, 2000–2009. Revista de Saúde Pública, 47, 1048–1058.CrossRefGoogle Scholar
  23. Gerland, P., Raftery, A. E., Ševčíková, H., Li, N., Gu, D., Spoorenberg, T., . . . Wilmoth, J. (2014). World population stabilization unlikely this century. Science, 346, 234–237.Google Scholar
  24. Glen, A. G., & Leemis, L. M. (Eds.). (2017). International series in operations research & management science. Computational probability applications. Cham, Switzerland: Springer.Google Scholar
  25. Gonzaga, M. R., & Schmertmann, C. P. (2016). Estimating age-and sex-specific mortality rates for small areas with TOPALS regression: An application to Brazil in 2010. Revista Brasileira de Estudos de População, 33, 629–652.CrossRefGoogle Scholar
  26. Greene, W. H. (1997). Econometric analysis (3rd ed.). Upper Saddle River, NJ: Prentice Hall.Google Scholar
  27. Hill, K. (2007). Methods for measuring adult mortality in developing countries: A comparative review (Global Burden of Disease 2000 in Aging Populations Research Paper No. 13). Cambridge, MA: Harvard Burden of Disease Unit.Google Scholar
  28. Hill, K., & Queiroz, B. (2010). Adjusting the general growth balance method for migration. Revista Brasileira de Estudos de População, 27, 7–20.CrossRefGoogle Scholar
  29. Hill, K., You, D., & Choi, Y. (2009). Death distribution methods for estimating adult mortality: Sensitivity analysis with simulated data errors. Demographic Research, 21(article 9), 235–254. CrossRefGoogle Scholar
  30. Hill, K. H. (1987). Estimating census and death registration completeness. Asian and Pacific Population Forum, 1(3), 8–13, 23.Google Scholar
  31. Instituto Brasileiro de Geografia e Estatística (Ed.). (2013). Tábuas abreviadas de mortalidade por sexo e idade: Brasil, grandes regiões e unidades da federação, 2010. Estudos e pesquisas. Informação demográfica e socioeconomic [Sex- and age-specific abbreviated life tables for Brazilian states and major regions in 2010: Studies and research. Demographic and socioeconomic information]. Rio de Janeiro, Brazil: Instituto Brasileiro de Geografia e Estatística (IBGE).Google Scholar
  32. Jonker, M. F., Van Lenthe, F. J., Congdon, P. D., Donkers, B., Burdorf, A., & Mackenbach, J. P. (2012). Comparison of Bayesian random-effects and traditional life expectancy estimations in small-area applications. American Journal of Epidemiology, 176, 929–937.CrossRefGoogle Scholar
  33. Lynch, S. M. (2007). Introduction to applied Bayesian statistics and estimation for social scientists. New York, NY: Springer.CrossRefGoogle Scholar
  34. Målqvist, M., Eriksson, L., Nga, N. T., Fagerland, L. I., Hoa, D. P., Wallin, L., . . . Persson, L.-Å. (2008). Unreported births and deaths, a severe obstacle for improved neonatal survival in low-income countries: A population based study. BMC International Health and Human Rights, 8, 4.
  35. Mathers, C. D., Ma Fat, D., Inoue, M., Rao, C., & Lopez, A. D. (2005). Counting the dead and what they died from: An assessment of the global status of cause of death data. Bulletin of the World Health Organization, 83, 171–177.Google Scholar
  36. Matos, K., de Godoy, M., & Baccarat, C. (2013). Mortalidade por causas externas em crianças, adolescentes e jovens: Uma revisão bibliográfica [Mortality from external causes in children, teenagers, and young adults: A bibliographic review]. Espaço para a Saúde-Revista de Saúde Pública do Paraná, 14(1/2), 82–93.Google Scholar
  37. Moreno, E., & Girón, J. (1998). Estimating with incomplete count data: A Bayesian approach. Journal of Statistical Planning and Inference, 66, 147–159.CrossRefGoogle Scholar
  38. Murray, C. J. L., Rajaratnam, J. K., Marcus, J., Laakso, T., & Lopez, A. D. (2010). What can we conclude from death registration? Improved methods for evaluating completeness. PLoS Medicine, 7(4), 1000262. CrossRefGoogle Scholar
  39. Ocaña-Riola, R., & Mayoral-Cortés, J.-M. (2010). Spatio-temporal trends of mortality in small areas of Southern Spain. BMC Public Health, 10 , 1.
  40. Paes, N. A. (2005). Avaliação da cobertura dos registros de óbitos dos estados brasileiros em 2000 [Assessment of completeness of death reporting in Brazilian states in 2000]. Revista de Saúde Pública, 39, 882–890.CrossRefGoogle Scholar
  41. Paes, N. A., & Albuquerque, M. E. E. (1999). Avaliação da qualidade dos dados populacionais e cobertura dos registros de óbitos para as regiões Brasileiras [Evaluation of population data quality and death registration coverage for Brazilian regions]. Revista de Saúde Pública, 33, 33–43.CrossRefGoogle Scholar
  42. Pletcher, S. D. (1999). Model fitting and hypothesis testing for age-specific mortality data. Journal of Evolutionary Biology, 12, 430–439.CrossRefGoogle Scholar
  43. Preston, S., Coale, A. J., Trussell, J., & Weinstein, M. (1980). Estimating the completeness of reporting of adult deaths in populations that are approximately stable. Population Index, 46, 179–202.CrossRefGoogle Scholar
  44. Preston, S., & Hill, K. (1980). Estimating the completeness of death registration. Population Studies, 34, 349–366.CrossRefGoogle Scholar
  45. Queiroz, B. L. (2012, November). Estimativas do grau de cobertura e da esperança de vida para as unidades da federação no Brasil entre 2000 e 2010 [Estimates of the degree of coverage and life expectancy for Brazilian states between 2000 and 2010]. Paper presented at the XVIII Encontro de Estudos de População da ABEP, Aguas de Lindóia.Google Scholar
  46. Queiroz, B. L., Freire, F. H. M. A., Gonzaga, M. R., & Lima, E. E. C. (2017). Completeness of death-count coverage and adult mortality (45q15) for Brazilian states from 1980 to 2010. Revista Brasileira de Epidemiologia, 20, 21–33.CrossRefGoogle Scholar
  47. Queiroz, B. L., Lima, E. C., Freire, F. H., & Gonzaga, M. R. (2013). Adult mortality estimates for small areas in Brazil, 1980–2010: A methodological approach. Lancet, 381, S120.CrossRefGoogle Scholar
  48. Raftery, A. E. (1988). Inference for the binomial N parameter: A hierarchical Bayes approach. Biometrika, 75, 223–228.Google Scholar
  49. Raftery, A. E., Chunn, J. L., Gerland, P., & Ševčíková, H. (2013). Bayesian probabilistic projections of life expectancy for all countries. Demography, 50, 777–801.Google Scholar
  50. Raftery, A. E., Lalic, N., Gerland, P., Li, N., & Heilig, G. (2014). Joint probabilistic projection of female and male life expectancy. Demographic Research, 30(article 27), 795–822.
  51. Riggan, W. B., Manton, K. G., Creason, J. P., Woodbury, M. A., & Stallard, E. (1991). Assessment of spatial variation of risks in small populations. Environmental Health Perspectives, 96, 223–238.Google Scholar
  52. Ševčíková, H., Li, N., Kantorová, V., Gerland, P., & Raftery, A. E. (2016). Age-specific mortality and fertility rates for probabilistic population projections. In R. Schoen (Ed.), Dynamic demographic analysis (pp. 285–310). Cham, Switzerland: Springer.Google Scholar
  53. Soares Filho, A. M., Souza, M. F. M., Gazal-Carvalho, C., Malta, D. C., Alencar, A. P., Silva, M. M. A., & Morais Neto, O. L. (2007). Análise da mortalidade por homicídios no Brasil [Analysis of homicide mortality in Brazil]. Epidemiologia e Serviços de Saúde, 16, 7–18.Google Scholar
  54. Stephens, A. S., Purdie, S., Yang, B., & Moore, H. (2013). Life expectancy estimation in small administrative areas with non-uniform population sizes: Application to Australian New South Wales local government areas. BMJ Open, 3(12), e003710.
  55. Szwarcwald, C. L., Morais Neto, O. L., Frias, P. G., de Souza, P. R. B., Jr., Escalante, J. J. C., de Lima, R. B., & Viola, R. C. (2011). Busca ativa de óbitos e nascimentos no Nordeste e na Amazônia Legal: Estimação das coberturas do SIM e do SINASC nos municípios Brasileiros [Active search for deaths and births in the Northeast and in the Legal Amazon: Estimation of coverage of SIM and SINASC in Brazilian municipalities]. In Ministry of Health (Ed.), Saúde Brasil 2010: Uma análise da situação e de evidências selecionadas de impacto de ações de vigilância em saúde [Saúde Brasil 2010: An analysis of the health situation and selected evidence of the impact of health surveillance actions] (pp. 79–98). Brasília: Ministério da Saúde.Google Scholar
  56. Tsimbos, C., Kalogirou, S., & Verropoulou, G. (2014). Estimating spatial differentials in life expectancy in Greece at local authority level. Population, Space and Place, 20, 646–663.CrossRefGoogle Scholar
  57. Wilmoth, J., Zureick, S., Canudas-Romo, V., Inoue, M., & Sawyer, C. (2012). A flexible two-dimensional mortality model for use in indirect estimation. Population Studies, 66, 1–28.CrossRefGoogle Scholar
  58. You, D., Hug, L., Ejdemyr, S., Idele, P., Hogan, D., Mathers, C., . . . Alkema, L. (2015). Global, regional, and national levels and trends in under-5 mortality between 1990 and 2015, with scenario-based projections to 2030: A systematic analysis by the UN Inter-agency Group for Child Mortality Estimation. Lancet, 386, 2275–2286.Google Scholar

Copyright information

© Population Association of America 2018

Authors and Affiliations

  1. 1.Center for Demography and Population HealthFlorida State UniversityTallahasseeUSA
  2. 2.Universidade Federal do Rio Grande do NorteNatalBrazil

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