臺灣人口平均餘命之趨緩成長

Why is the Increase in Life Expectancy Slowing Down? An Analysis Based on Gompertz Function

10.6191/JPS.2015.50.02

董宜禎(Yi-Jhen Dong)；陳寬政(Kuan-Jeng Chen)；王德睦(Te-Mu Wang)；吳郁婷(Yu-Ting Wu)

出生時平均餘命 ； 區間餘命 ； 人口熵 ； 存活函數 ； 死亡年齡分布變異 ； life expectancy at birth ； temporary life expectancies ； population entropy ； survival function ； variability of age at death

人口學刊

50期（2015 / 06 / 01）

29 - 60

繁體中文

臺灣死亡率自日治時期以來快速下跌，平均餘命乃大幅上漲；死亡率下降過程中出現年齡分配變動的現象，下降的動力由早期嬰幼兒人口逐漸移入中高齡人口，晚近則高齡死亡率在年齡軸上趨平發展，使得平均餘命趨向極值與否成為討論的焦點。本文先使用人口粗死亡率的定義解析指出，年齡標準化的死亡水準雖然持續下降卻有趨緩的動向，對應著人口出生時平均餘命之趨緩成長，建立本文所要分析的事實基礎。接著引入文獻說明平均餘命之趨緩成長可能關聯著人類細胞分裂次數的限制，以及物競天擇的進化原理，使得「自然」死亡的年齡分配在40歲與已知最高歲數之間，而若最高齡的限制紋風不動，則死亡年齡分布的變異會因平均餘命增長而產生壓縮的現象。進一步使用區間餘命、死亡年齡分布變異數、人口熵、與Gompertz存活函數來檢討平均餘命趨向遲緩成長的原因，本文使用簡易生命表說明幼年及青壯人口的死亡率迫近自然下限，區間餘命迫近自然上限，使得平均餘命的成長動力轉為老年人口的死亡率改善；固然老年死亡率距離自然下限較遠，有較大幅度改善的空間，卻因長期累積致命風險與慢性疾病不容易治癒而較難改善，使得死亡水準下降與平均餘命的成長趨緩。接著使用完全生命表套入Gompertz存活函數以延伸年齡到100歲以上的單齡歲數，突破人口統計設定的最高齡組限制，並關聯死亡年齡分布的變異數與人口熵。分析結果發現平均餘命趨向遲緩成長的同時，死亡年齡分布的變異數與人口熵皆伴隨下降，而Gompertz存活函數的兩個主要參數也正好可以連結存活曲線的矩形化，指出臺灣人口的死亡年齡分布在高齡端117歲前後似乎形成一個牆角，因而產生死亡壓縮的現象。

Mortality in Taiwan has declined since 1920, and life expectancy has dramatically increased. This paper attempts to explain how the mortality decline and the increase in life expectancy have been slowing down. Adopting the notion of temporary life expectancy, it is shown that while the temporary life expectancies of those aged under 60 have approached the age interval as a natural upper limit, the main force of continued mortality decline shifted to the older age groups, and the momentum of mortality decline began to dwindle away. Applying Gompertz's survival function simply A inverted, measuring the level of survival, the q parameter measures the steepness of the decline of survivors along with age. It is found that q is consistently related to Keyfitz's entropy measure evaluated at age 0 and the variance of death numbers d(x) derived from the survival function l(x). All three measures moved south as the increase in life expectancy at birth kept slowing down. A mortality compression is obviously in place, hindering the continuing increase in life expectancy. It is suggested that the b and q parameters of the Gompertz survival function, as simple and natural as they are, can be applied to the examination of the rectangularization of the survival curve in place of the angle and other intriguing measurements.

1. 王德睦、李大正(2009)。臺灣的存活曲線矩型化與壽命延長。人口學刊，38，1-31。
   連結：
2. 陳寬政、林子瑜、邱毅潔、紀筱涵(2009)。人口老化、疾病擴張、與健保醫療費用。人口學刊，39，59-83。
   連結：
3. Carrel, A. 1912. “On the Permanent Life of Tissues Outside the Organism.”Journal of Experimental Medicine 15(5): 516-528
4. 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
5. Greville, T. N. E. 1943. “Short Methods of Constructing Abridged Life Tables.” Record of the American Institute of Actuaries 32(65): 29-42
6. Ahlburg, D. A.,Vaupel, J. W.(1990).Alternative Projections of the US Population.Demography,27(4),639-652.
7. Arriaga, E. E.(1984).Measuring and Explaining the Change in Life Expectancies.Demography,21(1),83-96.
8. Baudisch, A.(2008).Inevitable Aging? Contributions to Evolutionary-Demographic Theory.Heidelberg, Germany:Springer.
9. Beard, R. E.(1963).A Theory of Mortality Based on Actuarial, Biological, and Medical Considerations.Proceedings of International Population Conference, New York, 1961,London:
10. Bengtsson, T.(ed.)(2006).Perspectives on Mortality Forecasting - III. The Linear Rise in Life Expectancy: History and Prospects.Swedish:Swedish Social Insurance Agency.
11. Bongaarts, J.(2006).How Long Will We Live?.Population and Development Review,32(4),605-628.
12. Carnes, B. A.,Olshansky, S. J.(1993).Evolutionary Perspectives on Human Senescence.Population and Development Review,19(4),793-806.
13. Carnes, B. A.,Olshansky, S. J.(2007).A Realist View of Aging, Mortality and Future Longevity.Population and Development Review,33(2),367-381.
14. Cheung, S. L. K.,Robine, J. M.(2007).Increase in Common Longevity and the Compression of Mortality: The Case of Japan.Population Studies,61(1),85-97.
15. Cheung, S. L. K.,Robine, J. M.,Tu, E. J.,Caselli, G.(2005).Three Dimensions of the Survival Curve: Horizontalization, Verticalization, and Longevity Extension.Demography,42(2),243-258.
16. Chiang, C. L.(1972).On Constructing Current Life Tables.Journal of the American Statistical Association,67(339),538-541.
17. Chiang, C. L.,Norris, F. D.,Olsen, F. E.,Shipley, P. W.,Supplee, H. E.(1961).Determination of the Fraction of Last Year of Life and Its Variation by Age, Race, and Sex.Annual Meeting of the American Public Health Association,Detroit:
18. 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(1),127-135.
19. Fries, J. F.(1980).Aging, Natural Death, and the Compression of Morbidity.New England Journal of Medicine,303(3),130-135.
20. Fries, J. F.(2000).Compression of Morbidity in the Elderly.Vaccine,18(16),1584-1589.
21. Fries, J. F.,Bruce, B.,Chakravarty, E.(2011).Compression of Morbidity 1980-2011: A Focused Review of Paradigms and Progress.Journal of Aging Research,2011,1-10.
22. 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(3),432-447.
23. Hayflick, L.(1997).Mortality and Immortality at the Cellular Level: A Review.Biochemistry(Moscow),62(11),1180-1190.
24. Hayflick, L.(1965).The Limited in Vitro Lifetime of Human Diploid Cell Strains.Experimental Cell Research,37,614-636.
25. Hill, G.(1993).The Entropy of the Survival Curve: An Alternative Measure.Canadian Studies in Population,20(1),43-57.
26. Horiuchi, S.,Coale, A. J.(1990).Age Patterns of Mortality for Older Women: An Analysis Using the Age Specific Rate of Mortality Change with Age.Mathematical Population Studies,2(4),245-267.
27. Kannisto, V.(2000).Measuring the Compression of Mortality.Demographic Research,3(6),1-24.
28. Kannisto, V.(2001).Mode and Dispersion of the Length of Life.Population: An English Selection,13(1),159-171.
29. Keyfitz, N.(1977).Applied Mathematical Demography.New York:John Wiley and Sons.
30. 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(5),456-483.
31. Manton, K. G.,Stallard, E.,Tolley, H. D.(1991).Limits to Human Life Expectancy: Evidence, Prospects, and Implications.Population and Development Review,17(4),603-637.
32. Mirzaee, M.(1979).Graduate Center for Demography, University of Pennsylvania.
33. Nagnur, D.(1986).Rectangularization of the Survival Curve and Entropy: The Canadian Experience: 1921-1981.Canadian Studies in Population,13(1),83-102.
34. Olshansky, S. J.(2011).Aging, Health, and Longevity in the 21st Century.International Symposium on "Health, Longevity, and Society",Taipei:
35. Olshansky, S. J.,Carnes, B. A.(1994).Demographic Perspectives on Human Senescence.Population and Development Review,20(1),57-80.
36. Olshansky, S. J.,Carnes, B. A.,Cassel, C. K.(1990).In Search of Methuselah: Estimating the Upper Limits to Human Longevity.Science,250,634-640.
37. Olshansky, S. J.,Carnes, B. A.,Cassel, C. K.(1993).The Aging of the Human Species.Scientific American,268(4),46-52.
38. Shay, J. W.,Wright, W. E.(2000).Hayflick, His Limit, and Cellular Ageing.Nature Reviews: Molecular Cell Biology,1,72-76.
39. Strehler, B. L.(1975).Implications of Aging Research for Society.Federation Proceedings,34(1),5-8.
40. Taylor, W.(1951).Changing Mortality from 1841 to 1947 Measured by the Life Table.British Journal of Social Medicine,5,162-176.
41. Tissenbaum, H. A.,Johnson, T. E.(2008).Aging Processes in Caenorhabditis Elegans.Molecular Biology of Aging,New York:
42. Tuljapurkar, S. D.(1982).Why Use Population Entropy? It Determines the Rate of Convergence.Journal of Mathematical Biology,13,325-337.
43. Ueda, K.(2008).IMF Working PaperIMF Working Paper,未出版
44. Uhlenberg, P.(ed.)(2009).International Handbook of Population Aging.New York:Springer.
45. Vaupel, J. W.(2010).Biodemography of Human Ageing.Nature,464,536-542.
46. Vaupel, J. W.,Carey, J. R.,Christensen, K.,Johnson, T. E.,Yashin, I. A.,Iachine, N. V.(1998).Biodemographic Trajectories of Longevity.Science,280(5365),855-860.
47. Wilmoth, J.,Horiuchi, S.(1999).Rectangularization Revisited: Variability of Age at Death within Human Populations.Demography,36(4),475-495.
48. 林佳瑩、陳信木(2012)。終壽成數的估計與應用。2012年臺灣健康與社會學社年會暨社會變遷與健康發展研討會，臺北:
49. 陳紹馨(1979)。臺灣的人口變遷與社會變遷。臺北:聯經。
50. 陳寬政、董宜禎(2014)。高齡死亡率的趨平發展。健康與社會，2，23-45。
51. 陳寬政、劉正、涂肇慶(1999)。出生時平均餘命的長期趨勢。臺灣社會學研究，3，87-114。

1. 劉議謙,宮可倫,李永琮(2019)。死亡率模型之比較：以臺灣資料為例。人口學刊，58，1-37。