Advertisement

One or more rates of ageing? The extended gamma-Gompertz model (EGG)

  • Giambattista Salinari
  • Gustavo De SantisEmail author
Original Paper
  • 31 Downloads

Abstract

Hidden heterogeneity poses serious challenges to survival analysis because the observed (aggregate) and the unobservable (individual) hazard functions may differ markedly from each other. However, the recent discovery of the so-called “mortality plateau” (i.e., the approximately constant value when mortality levels off, at very old ages) has brought new insights and pushed researchers towards the use of the gamma-Gompertz mortality model. Among the assumptions of this model, two are particularly relevant here: the shape, not the level, of the individual hazard function is a constant and so is the rate of ageing, i.e., the relative increase in mortality risks as people get older. The latter, however, does not pass empirical tests: the rate of ageing seems to vary (albeit only slightly) by age, gender, birth cohort and country. In this paper, we propose a new model (EGG, or extended gamma-Gompertz) which overcomes this limitation by allowing the rate of ageing to increase gradually with age before converging to a constant value, as in Gompertz. While preserving all the fine theoretical and empirical properties of its simpler predecessor, the EGG model adapts better to empirical reality, i.e., in this paper, the mortality profile of the cohorts born between 1820 and 1899 in five countries with high-quality data. The advantages of this more refined mortality model are discussed.

Keywords

Mortality models Gompertz Gamma-Gompertz Frailty Selection bias 

Notes

Acknowledgements

We gratefully acknowledge financial support from the 2016 JPI-MYBL (Joint Programme Initiative - More Years Better Lives), CREW Project (“Care, retirement and wellbeing of older people across different welfare regimes”), Decree: n. 3266/2018. Two anonymous referees helped to improve this article. All remaining errors are our own.

References

  1. Abbring J, van den Berg GJ (2007) The unobserved heterogeneity distribution in duration analysis. Biometrika 94:87–99MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barbi E (2003) Assessing the rate of ageing of the human population. MPIDR, WP 2003-008Google Scholar
  3. Barbi E, Caselli G, Vallin J (2003) Trajectories of extreme survival in heterogeneous populations. Popul-E 58(1):43–65CrossRefGoogle Scholar
  4. Barbi E, Lagona F, Marsili M, Vaupel JW, Wachter KW (2018) The plateau of human mortality: demography of longevity pioneers. Science 360:1459–1461.  https://doi.org/10.1126/science.aat3119 MathSciNetCrossRefGoogle Scholar
  5. Beard RE (1959) Note on some mathematical mortality models. In: Wolstenholme GEW, O’Conner M (eds) The lifespan of animals. Ciba Foundation colloquium on ageing. Little, Brown, Boston, pp 302–311Google Scholar
  6. Beltrán-Sánchez H, Finch CE, Crimmins EM (2015) Twentieth century surge of excess adult male mortality. PNAS 112(29):8993–8998CrossRefGoogle Scholar
  7. Brillinger DR (1986) The natural variability of vital rates and associated statistics. Biometrics 42:693–734MathSciNetCrossRefzbMATHGoogle Scholar
  8. Burger O, Missov TI (2016) Evolutionary theory of ageing and the problem of correlated Gompertz parameters. J Theor Biol 408:34–41CrossRefzbMATHGoogle Scholar
  9. Carey J, Liedo P, Orozco D, Vaupel JW et al (1992) Slowing of mortality rates at older ages in large medfly cohorts. Science 258:457CrossRefGoogle Scholar
  10. Carey J, Liedo P, Vaupel JW (1995) Mortality dynamics of density in the Mediterranean fruit fly. Exp Gerontol 30(6):605–629CrossRefGoogle Scholar
  11. Carnes BA, Witten TM (2014) How long must humans live? J Gerontol 69:965–970CrossRefGoogle Scholar
  12. Colchero F et al (2016) The emergence of longevous populations. PNAS 113(48):E7681–E7690CrossRefGoogle Scholar
  13. Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B 34:187–220MathSciNetzbMATHGoogle Scholar
  14. Curtis D, Dijkman J, Lambrecht T, Vanhaute E (2017) Low countries. In: Gráda CÓ (ed) Alfani G. Famine in European history. Cambrige University Press, Cambridge, pp 119–140Google Scholar
  15. Dribe M, Olsson M, Svensson P (2017) Nordic Europe. In: Alfani G, Ó Gráda C (eds) Famine in European history. Cambrige University Press, Cambridge, pp 185–211CrossRefGoogle Scholar
  16. Engelman M, Seplaki CL, Varadhan R (2017) A quiescent phase in human mortality? Exploring the ages of least vulnerability. Demography 54(3):1097–1118.  https://doi.org/10.1007/s13524-017-0569-z CrossRefGoogle Scholar
  17. Finkelstein M, Esaulova V (2006) Asymptotic behavior of a general class of mixture failure rates. Adv Appl Probab 38:242–262MathSciNetCrossRefzbMATHGoogle Scholar
  18. Gampe J (2010) Human mortality beyond age 110. In: Maier H, Gampe J, Jeune B, Robine JM, Vaupel JW (eds) Supercentenarians. Demogr Res Monogr, No. 7. Springer, Heidelberg, pp 219–230CrossRefGoogle Scholar
  19. Goldstein JR (2011) A secular trend toward earlier male sexual maturity: evidence from shifting ages of male young adult mortality. PLoS ONE 6:1–5Google Scholar
  20. Greenwood M, Irwin JO (1939) The biostatistics of senility. Hum Biol 11(1):1–23Google Scholar
  21. Gurven M, Kaplan H (2007) Longevity among hunters-gatherers: a cross-cultural examination. Popul Dev Rev 33(2):321–365CrossRefGoogle Scholar
  22. Hanagal DD (2011) Modelling survival data using frailty models. Chapman & Hall, Boca RatonCrossRefGoogle Scholar
  23. Horiuchi S (2003) Interspecies differences in the life span distribution: humans versus invertebrates. Popul Dev Rev 29:127–151Google Scholar
  24. Horiuchi S, Wilmoth JR (1998) Deceleration in the age pattern of mortality at older ages. Demography 4:391–412CrossRefGoogle Scholar
  25. Hougaard P (1984) Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71(1):75–83MathSciNetCrossRefzbMATHGoogle Scholar
  26. Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). www.mortality.org. Accessed June 2017
  27. Jazwinski S, Kim S, Lai C, Benguria A (1998) Epigenetic stratification: the role of individual change in the biological aging process. Exp Gerontol 33(6):571–580CrossRefGoogle Scholar
  28. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  29. Loudon I (1988) Maternal mortality: 1880–1950. Some regional and international comparisons. Soc Hist Med 1:183–228CrossRefGoogle Scholar
  30. McGilchrist C, Aisbett C (1991) Regression with frailty in survival analysis. Biometrics 47(2):461–466CrossRefGoogle Scholar
  31. Missov TI, Finkelstein M (2011) Admissible mixing distributions for a general class of mixture survival models with known asymptotics. Theor Popul Biol 80:64–70CrossRefzbMATHGoogle Scholar
  32. Missov TI, Vaupel JW (2015) Mortality implications of mortality plateaus. SIAM Rev 57:61–70MathSciNetCrossRefzbMATHGoogle Scholar
  33. Olshansky SJ (1998) On the biodemography of aging: a review essay. Popul Dev Rev 24(2):381–393CrossRefGoogle Scholar
  34. Pampel F (2010) Divergent patterns of smoking across high-income nations. In: Crimmins EM, Preston SH, Cohen B (eds) International differences in mortality at older ages. The National Academies Press, Washington, pp 132–164Google Scholar
  35. Preston SH (1970) An international comparison of excessive adult mortality. Popul Stud 24:5–20CrossRefGoogle Scholar
  36. Salinari G, De Santis G (2014) Comparing the rate of individual senescence across time and space. Popul-E 69(2):165–190Google Scholar
  37. Salinari G, De Santis G (2015) On the beginning of mortality acceleration. Demography 52:39–60CrossRefGoogle Scholar
  38. Thatcher A, Kannisto V, Vaupel JW (1998) The force of mortality at ages 80 to 120, vol 5. Odense monographs on population ageing. Odense University Press, OdenseGoogle Scholar
  39. Valkonen T, Van Poppel F (1997) The contribution of smoking to sex differences in life expectancy. Four Nordic countries and The Netherlands 1970–1989. Eur J Public Health 7:302–310CrossRefGoogle Scholar
  40. Vaupel JW (1997) Trajectories of mortality at advanced ages. In: Wachter KW, Finch CE (eds) Between Zeus and the salmon: the biodemography of longevity. National Academy Press, Washington DCGoogle Scholar
  41. Vaupel JW (2010) Biodemography of human ageing. Nature 464:536–542CrossRefGoogle Scholar
  42. Vaupel JW, Missov T (2014) Unobserved population heterogeneity: a review of formal relationships. Demogr Res 31(22):659–686.  https://doi.org/10.4054/demres.2014.31.22 CrossRefGoogle Scholar
  43. Vaupel JW, Yashin A (1985) Heterogeneity’s ruses: some surprising effects of selection on population dynamics. Am Stat 39(3):176–185MathSciNetGoogle Scholar
  44. Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16(3):439–454CrossRefGoogle Scholar
  45. Vaupel JW, Johnson T, Lithgow G, Curtsinger J, Fukui H, Xiu L, Khazaeli A, Pletcher S, Wang J, Muller H et al (1994) Rates of mortality in populations of Caenorhabditis elegans. Science 266:826–828CrossRefGoogle Scholar
  46. Vaupel JW, Carey JR, Christensen K, Johnson TE, Yashin AI, Holm NV, Iachine IA, Kannisto V, Khazaeli AA, Liedo P, Longo VD, Zeng Y, Manton KG, Curtsinger JW (1998) Biodemographic trajectories of longevity. Science 280:855–860CrossRefGoogle Scholar
  47. Wienke A (2011) Frailty models in survival analysis. Chapman & Hall, Boca RatonGoogle Scholar
  48. Yashin A, Iachine IA (1997) How frailty models can be used for evaluating longevity limits: taking advantages of an interdisciplinary approach. Demography 34(1):31–48CrossRefGoogle Scholar
  49. Zarulli V (2013) The effect of mortality shocks on the age-pattern of adult mortality. Popul-E 68(2):265–292CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di scienze economiche e aziendaliUniversity of SassariSassariItaly
  2. 2.DiSIA - Dip. di Statistica, Informatica, ApplicazioniUniversity of FlorenceFlorenceItaly

Personalised recommendations