Advertisement

Demography

, Volume 55, Issue 6, pp 2025–2044 | Cite as

Separating the Signal From the Noise: Evidence for Deceleration in Old-Age Death Rates

  • Dennis M. FeehanEmail author
Article
First Online:

Abstract

Widespread population aging has made it critical to understand death rates at old ages. However, studying mortality at old ages is challenging because the data are sparse: numbers of survivors and deaths get smaller and smaller with age. I show how to address this challenge by using principled model selection techniques to empirically evaluate theoretical mortality models. I test nine models of old-age death rates by fitting them to 360 high-quality data sets on cohort mortality after age 80. Models that allow for the possibility of decelerating death rates tend to fit better than models that assume exponentially increasing death rates. No single model is capable of universally explaining observed old-age mortality patterns, but the log-quadratic model most consistently predicts well. Patterns of model fit differ by country and sex. I discuss possible mechanisms, including sample size, period effects, and regional or cultural factors that may be important keys to understanding patterns of old-age mortality. I introduce mortfit, a freely available R package that enables researchers to extend the analysis to other models, age ranges, and data sources.

Keywords

Mortality Aging Longevity Statistics Model selection 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The author thanks Vladimir Canudas-Romo, Scott Lynch, Matthew J. Salganik, and three anonymous reviews for helpful comments on early drafts of the manuscript.

Supplementary material

13524_2018_728_MOESM1_ESM.pdf (395 kb)
ESM 1 (PDF 394 kb)

References

  1. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.CrossRefGoogle Scholar
  2. Barbi, E., Lagona, F., Marsili, M., Vaupel, J. W., Wachter, K. W. (2018). The plateau of human mortality: Demography of longevity pioneers. Science, 360, 1459–1461.Google Scholar
  3. Beard, R. E. (1959). Appendix: Note on some mathematical mortality models. In G. E. W. Wolstenholme & M. O’Conner (Eds.), The lifespan of animals: Ciba Foundation Colloquium on Ageing (Vol. 5, pp. 302–311). Boston, MA: Brown Little.Google Scholar
  4. Beard, R. E. (1971). Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. In W. Brass (Ed.), Biological aspects of demography: Symposia of the Society for the Study of Human Biology (Vol. 10, pp. 57–68). Oxford, UK: Taylor & Francis.Google Scholar
  5. Bebbington, M., Green, R., Lai, C.-D., & Zitikis, R. (2014). Beyond the Gompertz law: Exploring the late-life mortality deceleration phenomenon. Scandinavian Actuarial Journal, 2014, 189–207.CrossRefGoogle Scholar
  6. Berkman, L. F., & Syme, S. L. (1979). Social networks, host resistance, and mortality: A nine-year follow-up study of Alameda County residents. American Journal of Epidemiology, 109, 186–204.CrossRefGoogle Scholar
  7. Black, D. A., Hsu, Y.-C., Sanders, S. G., Schofield, L. S., & Taylor, L. J. (2017). The Methuselah effect: The pernicious impact of unreported deaths on old age mortality estimates (NBER Working Paper No. 23574). Cambridge, MA: National Bureau of Economic Research.Google Scholar
  8. Bongaarts, J. (2005). Long-range trends in adult mortality: Models and projection methods. Demography, 42, 23–49.CrossRefGoogle Scholar
  9. Burnham, K. P., & Anderson, D. (2003). Model selection and multimodel inference: A practical information-theoretic approach (2nd ed.). New York, NY: Springer-Verlag.Google Scholar
  10. Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods & Research, 33, 261–304.CrossRefGoogle Scholar
  11. Byberg, L., Melhus, H., Gedeborg, R., Sundström, J., Ahlbom, A., Zethelius, B.,... Michaëlsson, K. (2009). Total mortality after changes in leisure time physical activity in 50 year old men: 35 year follow-up of population based cohort. BMJ, 338, b688.  https://doi.org/10.1136/bmj.b688 CrossRefGoogle Scholar
  12. Chiang, C. L. (1960). A stochastic study of the life table and its applications: I. Probability distributions of the biometric functions. Biometrics, 16, 618–635.CrossRefGoogle Scholar
  13. Claeskens, G., & Hjort, N. L. (2008). Model selection and model averaging (Vol. 330). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  14. Coale, A. J., & Kisker, E. E. (1990). Defects in data on old-age mortality in the United States: New procedures for calculating mortality schedules and life tables at the highest ages. Asian & Pacific Population Forum, 4(1), 1–31. Retrieved from http://hdl.handle.net/10125/3598
  15. Daróczi, G., & Tsegelskyi, R. (2015). Pander: An R pandoc writer [R package version 0.6.0]. Retrieved from https://cran.mtu.edu/web/packages/pander/index.html
  16. Dong, X., Milholland, B., & Vijg, J. (2016). Evidence for a limit to human lifespan. Nature, 538, 257–259.CrossRefGoogle Scholar
  17. Gavrilov, L. A., & Gavrilova, N. S. (1991). The biology of life span: A quantitative approach. Chur, Switzerland: Harwood Academic Publishers.Google Scholar
  18. Gavrilov, L. A., & Gavrilova, N. S. (2011). Mortality measurement at advanced ages: A study of the Social Security Administration Death Master File. North American Actuarial Journal, 15, 432–447.CrossRefGoogle Scholar
  19. Gavrilova, N. S., & Gavrilov, L. A. (2015). Biodemography of old-age mortality in humans and rodents. Journals of Gerontology, Series A: Biological Sciences & Medical Sciences, 70, 1–9.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. Gjonca, A., Maier, H., & Brockmann, H. (2000). Old-age mortality in Germany prior to and after reunification. Demographic Research, 3(article 1).  https://doi.org/10.4054/DemRes.2000.3.1
  22. Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513–583.CrossRefGoogle Scholar
  23. Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (Springer Series in Statistics, 2nd ed.). New York, NY: Springer.CrossRefGoogle Scholar
  24. Haveman-Nies, A., de Groot, L. C., & van Staveren, A. W. (2003). Dietary quality, lifestyle factors and healthy ageing in Europe: The SENECA study. Age and Ageing, 32, 427–434.CrossRefGoogle Scholar
  25. Himes, C., Preston, S., & Condran, G. (1994). A relational model of mortality at older ages in low mortality countries. Population Studies, 48, 269–291.CrossRefGoogle Scholar
  26. Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14, 382–401.CrossRefGoogle Scholar
  27. Horiuchi, S., & Wilmoth, J. R. (1998). Deceleration in the age pattern of mortality at older ages. Demography, 35, 391–412.CrossRefGoogle Scholar
  28. Jdanov, D. A., Jasilionis, D., Soroko, E. L., Rau, R., & Vaupel, J. W. (2008). Beyond the Kannisto-Thatcher database on old-age mortality: An assessment of data quality at advanced ages (MPIDR Working Paper No. WP-2008-013). Rostock, Germany: Max Planck Institute for Demographic Research.Google Scholar
  29. Kannisto, V., Lauritsen, J., Thatcher, A. R., & Vaupel, J. W. (1994). Reductions in mortality at advanced ages: Several decades of evidence from 27 countries. Population and Development Review, 20, 793–810.CrossRefGoogle Scholar
  30. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
  31. Kinsella, K., & Phillips, D. R. (2005). Global aging: The challenge of success (Population Bulletin, Vol. 60 No. 1). Washington, DC: Population Reference Bureau. Retrieved from https://assets.prb.org//pdf05/60.1GlobalAging.pdf
  32. Lenart, A., & Vaupel, J. W. (2017). Questionable evidence for a limit to human lifespan. Nature, 546, E13–E14.  https://doi.org/10.1038/nature22790 CrossRefGoogle Scholar
  33. Li, J. S. H., Ng, A. C. Y., & Chan, W. S. (2011). Modeling old-age mortality risk for the populations of Australia and New Zealand: An extreme value approach. Mathematics and Computers in Simulation, 81, 1325–1333.CrossRefGoogle Scholar
  34. Luy, M., Wegner, C., & Lutz, W. (2011). Adult mortality in Europe. In R. G. Rogers & E. M. Crimmins (Eds.), International handbook of adult mortality (pp. 49–81). New York, NY: Springer.CrossRefGoogle Scholar
  35. Lynch, S. M., & Brown, J. S. (2001). Reconsidering mortality compression and deceleration: An alternative model of mortality rates. Demography, 38, 79–95.CrossRefGoogle Scholar
  36. Lynch, S. M., Brown, J. S., & Harmsen, K. G. (2003). Black-white differences in mortality compression and deceleration and the mortality crossover reconsidered. Research on Aging, 25, 456–483.CrossRefGoogle Scholar
  37. Makeham, W. M. (1860). On the law of mortality and the construction of annuity tables. Assurance Magazine, and Journal of the Institute of Actuaries, 8, 301–310.CrossRefGoogle Scholar
  38. Manton, K. G., Stallard, E., & Vaupel, J. W. (1981). Methods for comparing the mortality experience of heterogeneous populations. Demography, 18, 389–410.CrossRefGoogle Scholar
  39. Martin, L., & Preston, S. (Eds.). (1994). Demography of aging. Washington, DC: National Academies Press.Google Scholar
  40. Max Planck Institute for Demographic Research (MPIDR). (2014). Kannisto-Thatcher database on old-age mortality [Database]. Rostock, Germany: MPIDR. http://www.demogr.mpg.de/databases/ktdb/introduction.htm
  41. Meslé, F., & Vallin, J. (2006). Diverging trends in female old-age mortality: The United States and the Netherlands versus France and Japan. Population and Development Review, 32, 123–145.CrossRefGoogle Scholar
  42. Olshansky, S. J., & Carnes, B. A. (1997). Ever since Gompertz. Demography, 34 , 1–15.CrossRefGoogle Scholar
  43. Ouellette, N. (2016). La forme de la courbe de mortalité des centenaires canadiens-français [The shape of the mortality curve of Canadian-French centenarians]. Gérontologie et Société, 38(3), 41–53.CrossRefGoogle Scholar
  44. Perks, W. (1932). On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries, 63, 12–57.CrossRefGoogle Scholar
  45. Pletcher, S. D. (1999). Model fitting and hypothesis testing for age-specific mortality data. Journal of Evolutionary Biology, 12, 430–439.CrossRefGoogle Scholar
  46. Promislow, D. E. L., Tatar, M., Pletcher, S., & Carey, J. R. (1999). Below-threshold mortality: Implications for studies in evolution, ecology and demography. Journal of Evolutionary Biology, 12, 314–328.CrossRefGoogle Scholar
  47. R Core Team. (2014). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from http://www.R-project.org/
  48. Rau, R., Soroko, E., Jasilionis, D., & Vaupel, J. W. (2008). Continued reductions in mortality at advanced ages. Population and Development Review, 34, 747–768.CrossRefGoogle Scholar
  49. Schloerke, B., Crowley, J., Cook, D., Hofmann, H., Wickham, H., Briatte, F.,...Thoen, E. (2014). GGally: Extension to ggplot2 [R package version 0.5. 0]. Retrieved from https://libraries.io/cran/GGally/0.5.0
  50. Sheshinski, E. (2007). The economic theory of annuities. Princeton, NJ: Princeton University Press.Google Scholar
  51. Slowikowski, K., & Irisson, J.-O. (2016). Ggrepel: Repulsive text and label geoms for “ggplot2.” [R package version 0.6. 5]. Retrieved from https://github.com/slowkow/ggrepel
  52. Staetsky, L. (2009). Diverging trends in female old-age mortality: A reappraisal. Demographic Research, 21(article 30), 885–914.  https://doi.org/10.4054/DemRes.2009.21.30 CrossRefGoogle Scholar
  53. Steinsaltz, D. (2005). Re-evaluating a test of the heterogeneity explanation for mortality plateaus. Experimental Gerontology, 40, 101–113.CrossRefGoogle Scholar
  54. Steinsaltz, D., & Wachter, K. (2006). Understanding mortality rate deceleration and heterogeneity. Mathematical Population Studies, 13, 19–37.CrossRefGoogle Scholar
  55. Strehler, B. L., & Mildvan, A. S. (1960). General theory of mortality and aging. Science, 132, 14–21.CrossRefGoogle Scholar
  56. Tabeau, E., van den Berg Jeths, A., & Heathcote, C. (2001). Forecasting mortality in developed countries: Insights from a statistical, demographic and epidemiological perspective. Dordrecht, the Netherlands: Springer.Google Scholar
  57. Thatcher, A. R., Kannisto, V., & Vaupel, J. W. (1998). The force of mortality at ages 80 to 120 (Odense Monographs on Population Aging No. 5). Odense, Denmark: Odense University Press.Google Scholar
  58. United Nations Population Division. (2015). World population prospects: The 2015 revision, methodology of the United Nations population estimates and projections (Working Paper No. ESA/P/WP.242). New York, NY: United Nations Department of Economic and Social Affairs.Google Scholar
  59. Vaupel, J. W., Carey, J. R., & Christensen, K. (2003). It’s never too late. Science, 301, 1679–1681.CrossRefGoogle Scholar
  60. Vaupel, J. W., Carey, J. R., Christensen, K., Johnson, T. E., Yashin, A. I., Holm, N. V.,... Curtsinger, J. W. (1998). Biodemographic trajectories of longevity. Science, 280, 855–860.CrossRefGoogle Scholar
  61. Vaupel, J. W., Manton, K. G., & Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16, 439–454.CrossRefGoogle Scholar
  62. Wang, J.-L., Muller, H.-G., & Capra, W. B. (1998). Analysis of oldest-old mortality: Lifetables revisited. Annals of Statistics, 26, 126–163.CrossRefGoogle Scholar
  63. Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293–297.Google Scholar
  64. Wickham, H. (2009). Ggplot2: Elegant graphics for data analysis. New York, NY: Springer.CrossRefGoogle Scholar
  65. Wickham, H., François, R., Henry, L., & Müller, K. (2016). Dplyr: A grammar of data manipulation [R package version 0.5. 0 database tool]. Vienna, Austria: R Core Development Team. Retrieved from https://cran.r-project.org/package=dplyr
  66. Wilmoth, J. R. (1995). Are mortality rates falling at extremely high ages? An investigation based on a model proposed by Coale and Kisker. Population Studies, 49, 281–295.CrossRefGoogle Scholar
  67. Yashin, A. I., Vaupel, J. W., & Iachine, I. A. (1994). A duality in aging: The equivalence of mortality models based on radically different concepts. Mechanisms of Ageing and Development, 74, 1–14.CrossRefGoogle Scholar
  68. Yi, Z., & Vaupel, J. W. (2003). Oldest old mortality in China. Demographic Research, 8(article 7), 215–244.  https://doi.org/10.4054/DemRes.2003.8.7 CrossRefGoogle Scholar

Copyright information

© Population Association of America 2018

Authors and Affiliations

  1. 1.Department of DemographyUniversity of CaliforniaBerkeleyUSA

Personalised recommendations

Cite article

Buy options