, Volume 42, Issue 2, pp 243–258 | Cite as

Three dimensions of the survival curve: horizontalization, verticalization, and longevity extension

  • Siu Lan Karen Cheung
  • Jean-Marie Robine
  • Edward Jow-Ching Tu
  • Graziella Caselli


Three dimensions of the survival curve have been developed: (1) “horizontalization,” which corresponds to how long a cohort and how many survivors can live before aging-related deaths significantly decrease the proportion of survivors; (2) “verticalization,” which corresponds to how concentrated aging-related (“normal”) deaths are around the modal age at death (M); and (3) “longevity extension,” which corresponds to how far the highest normal life durations can exceed M. Our study shows that the degree of horizontalization increased relatively less than the degree of verticalization in Hong Kong from 1976 to 2001. After age normalization, the highest normal life durations moved closer to M, implying that the increase in human longevity is meeting some resistance.


Survival Curve Premature Mortality Maximum Life Span Human Longevity Life Duration 
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  1. Barbi, E., G. Caselli, and J. Vallin. 2003. “Trajectories of Extreme Survival in Heterogeneous Populations.” Population 58:43-66.Google Scholar
  2. Bell, F.C., A.H. Wade, and S.C. Goss. 1992. “Life Tables for the United States: 1900-2080.” Actuarial Study, No. 107. SSA Publication No. 11-11536. Washington, DC: Office of the Chief Actuary, Social Security Administration.Google Scholar
  3. Benjamin, B. 1959. “Actuarial Aspects of Human Lifespans.” Pp. 2-20 in The Life Span of Animals, Vol. 5, CIBA Foundation Colloquia on Aging, edited by G.E.W. Wolstenholme and M. O’Connor. Boston: Little, Brown.Google Scholar
  4. Boldsen, J.L. and B. Jeune. 1990. “Distribution of Age at Menopause in Two Danish Samples.” Human Biology 62:291-300.Google Scholar
  5. Cheung, J. 2001. “The Long-Term Trend of Non-Maori Mortality and Its More Recent Compression Effect.” Paper presented at the biannual conference of the Population Association of New Zealand, Wellington, June 28-29.Google Scholar
  6. Cheung, S.L.K. 2003. Scalar Expansion and Normal Longevity in Hong Kong. Doctoral dissertation. Hong Kong University of Science and Technology.Google Scholar
  7. Cheung, S.L.K., J.-M. Robine, E.J.-C. Tu, and G. Caselli. 2005. “Change in the Distribution of Normal Life Durations Through the Demographic Transition: The Case of Hong Kong.” Unpublished manuscript. INSERM/Démographie et Santé, Montpellier, France.Google Scholar
  8. Coale, A.J. 1996. “Age Patterns and Time Sequence of Mortality in National Populations With the Highest Expectation of Life at Birth.” Population and Development Review 22:127-35.CrossRefGoogle Scholar
  9. Comfort, A. 1956. The Biology of Senescence. London: Routledge & Kegan Paul.Google Scholar
  10. Comfort, A. 1964. Aging: The Biology of Senescence. Holt, Rinehart and Winston.Google Scholar
  11. Deevey, E.S. 1947. “Life Tables for Natural Populations of Animals.” Quarterly Review of Biology 22:283-314.Google Scholar
  12. Eakin, T. and M. Witten. 1995. “How Square Is the Survival Curve of a Given Species?” Experimental Gerontology 30:33-64.CrossRefGoogle Scholar
  13. Faber, J.F. 1982. “Life Tables for the United States: 1900-2050.” Actuarial Study, No. 87. SSA Publication No. 11-11534. Washington, DC: Office of the Chief Actuary, Social Security Administration.Google Scholar
  14. Faber, J.F. and A.H. Wade. 1983. “Life Tables for the United States: 1900-2050.” Actuarial Study, No. 89. SSA Publication No. 11-11536. Washington, DC: Office of the Chief Actuary, Social Security Administration.Google Scholar
  15. Finch, C.E. and M.C. Pike. 1996. “Maximum Life Span Predictions From the Gompertz Mortality Model.” Journal of Gerontology: Biological Sciences 51A(3):B183-94.Google Scholar
  16. Fries, J.F. 1980. “Aging, Natural Death, and the Compression of Morbidity.” New England Journal of Medicine 303:130-35.CrossRefGoogle Scholar
  17. — 1984. “The Compression of Morbidity: Miscellaneous Comments About a Theme.” The Gerontologist 24:354-59.Google Scholar
  18. — 2002. “Reducing Disability in Older Age.” Journal of the American Medical Association 288:3164-66.CrossRefGoogle Scholar
  19. Frommer, D.J. 1964. “Changing Age of the Menopause.” British Medical Journal 2:349-51.CrossRefGoogle Scholar
  20. Gavrilov, L.A. and N.S. Gavrilova. 2001. “The Reliability Theory of Aging and Longevity.” Journal of Theoretical Biology 213:527-45.CrossRefGoogle Scholar
  21. Go, C.G., J.E. Brustrom, M.F. Lynch, and C.M. Aldwin. 1995. “Ethnic Trends in Survival Curves and Mortality.” The Gerontologist 35:318-26.Google Scholar
  22. Gordon, T.J. 1980. “Changing Mortality Patterns.” Pp. 1-23 in The Future of Life Expectancy: Economic, Social and Medical Implications of Living Longer. Chicago: Society of Actuaries.Google Scholar
  23. Greenwood, M. and J.O. Irwin. 1939. “The Biostatistics of Senility.” Human Biology 11(1):1-23.Google Scholar
  24. Hayflick, L. 1981. “Prospects for Human Life Extension by Genetic Manipulation.” Pp. 162-79 in Aging: A Challenge to Science and Society, Vol. 1, edited by D. Danon, N.W. Shock, and M. Marois. New York: Oxford University Press.Google Scholar
  25. Herskind, A.M., M. McGue, N.W. Holm, T.I. Sorense, B. Harvale, and J.W. Vaupel. 1996. “The Heritability of Human Longevity: A Population-Based Study of 2,872 Danish Twin Pairs Born 1870–1900.” Human Genetics 97:319-23.CrossRefGoogle Scholar
  26. Hill, G.B. 1993. “The Entropy of the Survival Curve: An Alternative Measure.” Canadian Studies in Population 20(1):43-57.Google Scholar
  27. Hong Kong Census and Statistics Department. 2002. Hong Kong Life Tables 1996–2031. Hong Kong: Hong Kong Government Printer.Google Scholar
  28. Horiuchi, S. 2003. “Interspecies Differences in the Life Span Distribution: Humans Versus Invertebrates.” Pp. 127–51 in Population and Development Review, edited by J.R. Carey and S. Tuljapurkar. 29(Suppl.).Google Scholar
  29. Hubert, H.B., D.A. Bloch, J.W. Oehlert, and J.F. Fries. 2002. “Lifestyle Habits and Compression of Morbidity.” Journal of Gerontology 57A:M347-M351.Google Scholar
  30. Kannisto, V. 1999. “Measuring the Compression of Mortality.” Paper distributed at the European Population Conference, The Hague, The Netherlands, August 30-September 3.Google Scholar
  31. Kannisto, V. 2000a. “Central and Dispersion Indicators of Individual Life Duration: New Methods.” Paper presented at the International Union for the Scientific Study of Population seminar on “Human Longevity, Individual Life Durations and Growth of the Oldest-Old,” Montpellier, October 23–26.Google Scholar
  32. Kannisto, V. 2000b. “Measuring the Compression of Mortality.” Demographic Research 3(Article 6). Available on-line at Scholar
  33. — 2001. “Mode and Dispersion of the Length of Life.” Population: An English Selection 13:159-71.Google Scholar
  34. Kannisto, V., J. Lauritsen, A.R. Thatcher, and J.W. Vaupel. 1994. “Reductions in Mortality at Advanced Ages: Several Decades of Evidence From 27 Countries.” Population and Development Review 20:793-810.CrossRefGoogle Scholar
  35. Laplace, P.S. 1812. Théorie Analytique des Probabilités [Analytical theory of probabilities]. Paris: Veuve Courcier.Google Scholar
  36. Levy, M.L. 1996. “La Rectangularisation de la Courbe des Survivants” [Rectangularization of the survival curve]. Pp. 576-79 in Morbidite, Mortalite: Problemes de Mesure, Facteurs d’evolution, Essai de Prospective, Aidelf n8.Google Scholar
  37. Lexis, W. 1878. “Sur la Durée Normale de la Vie Humaine et sur la Théorie de la Stabilité des Rapports Statistiques” [On the normal human lifespan and on the theory of the stability of the statistical ratios]. Annales de Démographie Internationale 2:447-60.Google Scholar
  38. MacMahon, B and J. Worcester. 1966. Age at Menopause: United States-1960-1962. Washington, DC: NCHS National Center for Health Statistics U.S. Department of Health, Education, and Welfare.Google Scholar
  39. Manton, K.G. and B. Singer. 1994. “What’s the Fuss About the Compression of Mortality?” Chance 7:21-30.Google Scholar
  40. Manton, K.G and E. Stallard. 1996. “Longevity in the United States: Age and Sex-Specific Evidence on Life Span Limits From Mortality Patterns 1960–1990.” Journal of Gerontology: Biological Sciences 51A:B362-75.Google Scholar
  41. Manton, K.G. and H.D. Tolley. 1991. “Rectangularization of the Survival Curve: Implications of an Ill-Posed Question.” Journal of Aging and Health 3:172-93.CrossRefGoogle Scholar
  42. Martel, S. and R. Bourbeau. 2003. “Compression de la Mortalité et Rectangularisation de la Courbe de Survie au Québec au Cours du XXe Siècle” [Mortality compression and rectangularization of the survival curve in Quebec during the twentieth century]. Cahiers Québécois de Démographie 32(1):43-75.Google Scholar
  43. McGue, M., J.W. Vaupel, N. Holm, and B. Harvald. 1993. “Longevity Is Moderately Heritable in a Sample of Danish Twins Born 1870–80.” Journal of Gerontology 48:B237-44.Google Scholar
  44. Mirer, T.W. 1988. Economic Statistics and Econometrics, 2nd ed. New York: Macmillan.Google Scholar
  45. Myers, G.C. and K.G. Manton. 1984a. “Compression of Mortality: Myth or Reality?” The Gerontologist 24:346-53.Google Scholar
  46. — 1984b. “Recent Changes in the U.S. Age at Death Distribution: Further Observations.” The Gerontologist 24:572-75.Google Scholar
  47. Nagnur, D. 1986. “Rectangularization of the Survival Curve and Entropy: The Canadian Experience 1921–1981.” Canadian Studies in Population 13:83-102.Google Scholar
  48. Nusselder, W.J. and J.P. Mackenbach. 1996. “Rectangularization of the Survival Curve in the Netherlands, 1950–1992.” The Gerontologist 36:773-82.Google Scholar
  49. —. 1997. “Rectangularization of the Survival Curve in the Netherlands: An Analysis of Underlying Causes of Death.” Journal of Gerontology: Social Sciences 52B:S145-S154.Google Scholar
  50. Oeppen, J. and J.W. Vaupel. 2002. “Enhanced: Broken Limits to Life Expectancy.” Science 296:1029-31.CrossRefGoogle Scholar
  51. Olshansky S.J., B.A. Carnes, A. Désesquelles. 2001. “Prospects for Human Longevity in an Aging World.” Science 291:1491-92.CrossRefGoogle Scholar
  52. Paccaud, F., C. Sidoti Pinto, A. Marazzi, J. Mili. 1998. “Age at Death and Rectangularization of Survival Curve: Trends in Switzerland, 1969–1994.” Journal of Epidemiology and Community Health 52:412-15.CrossRefGoogle Scholar
  53. Pearl, R. and C.R. Doering. 1923. “A Comparison of the Mortality of Certain Lower Organisms With That of Man.” Science 1468:209-12.Google Scholar
  54. Pearl, R. and J.R. Miner. 1935. “Experimental Studies on the Duration of Life. XIV. The Comparative Mortality of Certain Lower Organisms.” Quarterly Review of Biology 10:60-79.Google Scholar
  55. Pelletier, F., J. Legare, and R. Bourbeau. 1997. “Mortality in Quebec During the Nineteenth Century: From the State to the Cities.” Population Studies 51:93-103.CrossRefGoogle Scholar
  56. Quetelet, A. 1991[1835]. Sur l’homme et le developpement de ses facultés, essai d’une physique sociale [On man and the development of his abilities, essay on social physics]. Paris: Fayard.Google Scholar
  57. — 1848. Du systéme social et des lois qui le régissent [On the social system and its laws]. Paris, Guillaumin.Google Scholar
  58. — 1871. Anthropométrie ou mesure des différentes facultés de l’homme [Anthropometry or the assessment of the different abilities of man]. Brussels: Muquardt.Google Scholar
  59. Robine, J.-M. 2001. “Redefining the Stages of the Epidemiological Transition by a Study of the Dispersion of Life Spans: The Case of France.” Population: An English Selection 13: 173-94.Google Scholar
  60. Robine, J.-M. 2003. “Life Course, Environmental Change and Life Span.” Pp. 229–38 in Population and Development Review, edited by J.R. Carey and S. Tuljapurkar. 29(Suppl.).Google Scholar
  61. Robine, J.-M., G. Caselli, Y. Saito. 2003. “An Unprecedented Increase in the Number of Centenarians.” Paper presented at the IUSSP Seminar on Longevity and Health, Rockefeller University, New York, October 20–22.Google Scholar
  62. Robine, J.-M. and F. Paccaud. 2005. “Nonagenarians and Centenarians in Switzerland, 1860-2001: A Demographic Analysis.” Journal of Epidemiology and Community Health 59:31-37.CrossRefGoogle Scholar
  63. Robine, J.-M. and Y. Saito. 2003. “Survival Beyond Age 100: Acceleration of the Evolution in Japan.” Pp. 208–27 in Population and Development Review, edited by J.R. Carey and S. Tuljapurkar. 29(Suppl.).Google Scholar
  64. Robine, J.-M., Y. Saito, and C. Jagger. 2003. “The Emergence of Extremely Old People: The Case of Japan.” Experimental Gerontology 37:739-43.Google Scholar
  65. Robine, J.-M. and J.W. Vaupel. 2001. “Supercentenarians: Slower Ageing Individuals or Senile Elderly?” Experimental Gerontology 36:915-30.Google Scholar
  66. —. 2002. “Emergence of Supercentenarians in Low-Mortality Countries.” North American Actuarial Journal 6(3):54-63.Google Scholar
  67. Rothenberg, R., H.R. Lentzner, and R.A. Parker. 1991. “Population Aging Patterns: The Expansion of Mortality.” Journal of Gerontology: Social Science 46:S66-70.Google Scholar
  68. Schneider, E.L. and J.A. Brody. 1983. “Aging, Natural Death, and the Compression of Morbidity: Another View.” New England Journal of Medicine 304:854-55.Google Scholar
  69. Smith, D.W.E. 1994. “The Tails of Survival Curves.” Bio Essays 16:907-11.Google Scholar
  70. Strehler, B.L. 1975. “Implications of Aging Research for Society.” Federation Proceedings 34(1):5-8.Google Scholar
  71. Thatcher, A.R. 1999. “The Demography of Centenarians in England and Wales.” Population Trends 162, Part 1:5-43.Google Scholar
  72. — 2001. “The Demography of Centenarians in England and Wales.” Population 13:139-56.Google Scholar
  73. Vaupel, J.W. 2002. “Post-Darwinian Longevity.” MPIDR Working Paper WP 2002-043, September. Demographic Research. Available on-line at http://www.demogr.mpg.deGoogle Scholar
  74. Vaupel, J.W., J.R. Carey, and K. Christensen. 2003. “Aging. It’s Never Too Late.” Science 301:1679-81.CrossRefGoogle Scholar
  75. Vaupel, J.W. and B. Jeune. 1995. “The Emergence and Proliferation of Centenarians.” Pp. 109-16 in Exceptional Longevity: From Prehistory to the Present, edited by B. Jeune and J.W. Vaupel. Odense, Denmark: Odense University Press.Google Scholar
  76. Vaupel, J.W., K.G. Manton, and E. Stallard. 1979. “The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality.” Demography 16:439-54.CrossRefGoogle Scholar
  77. Véron, J. and J.-M. Rohrbasser. 2003. “Wilhelm Lexis: The Normal Length of Life as an Expression of the ’Nature of Things.’” Population: An English Selection 58:303-22.Google Scholar
  78. Vita, A.J., R.B. Terry, H.B. Hubert, and J.F. Fries. 1998. “Aging, Health Risks, and Cumulative Disability.” New England Journal of Medicine 338:1035-41.CrossRefGoogle Scholar
  79. Wilmoth, J.R. 1998. “The Future of Human Longevity: A Demographer’s Perspective.” Science 280(17):395-97.Google Scholar
  80. — 2000. “Demography of Longevity: Past, Present, and Future Trends.” Experimental Gerontology 35(9–10):1111-29.Google Scholar
  81. — 2002. “Human Longevity in Historical Perspective.” Pp. 11-24 in Physiological Basis of Aging and Geriatrics, 3rd ed., edited by P.S. Timiras. Boca Raton, FL: CRC Press.Google Scholar
  82. Wilmoth, J.R., L.J. Deegan, H. Lundström, and S. Horiuchi. 2000. “Increase of Maximum Lifespan in Sweden, 1861–1999.” Science 289(29):2366-68.Google Scholar
  83. Wilmoth, J.R. and S. Horiuchi. 1999. “Rectangularization Revisited: Variability of Age at Death Within Human Populations.” Demography 36:475-95.CrossRefGoogle Scholar
  84. Wilmoth, J.R. and H. Lundström. 1996. “Extreme Longevity in Five Countries: Presentation of Trends With Special Attention to Issues of Data Quality.” European Journal of Population 12:63-93.CrossRefGoogle Scholar
  85. Yashin, A.I., A.S. Begun, S.I. Boiko, S.V. Ukraintseva, and J. Oeppen. 2001. “The New Trends in Survival Improvement Require a Revision of Traditional Gerontological Concepts.” Experimental Gerontology 37:157-67.CrossRefGoogle Scholar

Copyright information

© Population Association of America 2005

Authors and Affiliations

  • Siu Lan Karen Cheung
    • 1
  • Jean-Marie Robine
    • 2
  • Edward Jow-Ching Tu
    • 3
  • Graziella Caselli
    • 4
  1. 1.INSERM/D’emographie et Santé, Val d’Aurelle, Parc EuromédecineMontpellier cedex 5France
  2. 2.Démographie et SantéINSERMFrance
  3. 3.Hong Kong University of Science and TechnologyChina
  4. 4.University of Rome “La Sapienza.”RomeItaly

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