1. Mortality Tables and Laws: Biodemographic Analysis and Reliability Theory Approach Dr. Natalia S. Gavrilova, Ph.D. Dr. Leonid A. Gavrilov, Ph.D. Center on Aging NORC and the University of Chicago Chicago, Illinois, USA

2. Questions of Actuarial Significance How far could mortality decline go? (absolute zero seems implausible) Are there any ‘biological’ limits to human mortality decline, determined by ‘reliability’ of human body? (lower limits of mortality dependent on age, sex, and population genetics) Were there any indications for ‘biological’ mortality limits in the past? Are there any indications for mortality limits now?

3. How can we improve the actuarial forecasts of mortality and longevity ? By taking into account the mortality laws summarizing prior experience in mortality changes over age and time: Gompertz-Makeham law of mortality Compensation law of mortality Late-life mortality deceleration

4. The Gompertz-Makeham Law μ(x) = A + R e αx A – Makeham term or background mortality R e αx – age-dependent mortality; x - age Death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age. risk of death

5. Gompertz Law of Mortality in Fruit Flies Based on the life table for 2400 females of Drosophila melanogaster published by Hall (1969). Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

6. Gompertz-Makeham Law of Mortality in Flour Beetles Based on the life table for 400 female flour beetles (Tribolium confusum Duval). published by Pearl and Miner (1941). Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

7. Gompertz-Makeham Law of Mortality in Italian Women Based on the official Italian period life table for 1964-1967. Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

8. How can the Gompertz- Makeham law be used? By studying the historical dynamics of the mortality components in this law: μ(x) = A + R e αx Makeham componentGompertz component

9. Historical Stability of the Gompertz Mortality Component Before the 1980s Historical Changes in Mortality for 40-year-old Swedish Males 1. Total mortality, μ 40 2. Background mortality (A) 3. Age-dependent mortality (Re α40 ) Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

10. Predicting Mortality Crossover Historical Changes in Mortality for 40-year-old Women in Norway and Denmark 1. Norway, total mortality 2. Denmark, total mortality 3. Norway, age- dependent mortality 4. Denmark, age- dependent mortality Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

11. Predicting Mortality Divergence Historical Changes in Mortality for 40-year-old Italian Women and Men 1. Women, total mortality 2. Men, total mortality 3. Women, age- dependent mortality 4. Men, age-dependent mortality Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

12. A Broader View on the Historical Changes in Mortality Swedish Females Data source: Human Mortality Database

13. Extension of the Gompertz-Makeham Model Through the Factor Analysis of Mortality Trends Mortality force (age, time) = = a 0 (age) + a 1 (age) x F 1 (time) + a 2 (age) x F 2 (time) Where: a i (age) – a set of numbers; each number is fixed for specific age group F j (time) – “factors,” a set of standardized numbers; each number is fixed for specific moment of time (mean = 0; st. dev. = 1)

14. Factor Analysis of Mortality Trends Swedish Females “Factor analysis of the time series of mortality confirms the preferential reduction in the mortality of old-aged and senile people [in recent years]…” Gavrilov, Gavrilova, The Biology of Life Span, 1991. Data source for the current slide: Human Mortality Database

15. Actuarial Implications Mortality trends before the 1950s are useless or even misleading for the current mortality forecasts because all the “rules of the game” has been changed dramatically

16. Preliminary Conclusions There was some evidence for ‘ biological’ mortality limits in the past, but these ‘limits’ proved to be responsive to the recent technological and medical progress. Thus, there is no convincing evidence for absolute ‘biological’ mortality limits now. Analogy for illustration and clarification: There was a limit to the speed of airplane flight in the past (‘sound’ barrier), but it was overcome by further technological progress. Similar observations seems to be applicable to current human mortality decline.

17. Compensation Law of Mortality (late-life mortality convergence) Relative differences in death rates are decreasing with age, because the lower initial death rates are compensated by higher slope of mortality growth with age (actuarial aging rate)

18. Compensation Law of Mortality Convergence of Mortality Rates with Age 1 – India, 1941-1950, males 2 – Turkey, 1950-1951, males 3 – Kenya, 1969, males 4 - Northern Ireland, 1950- 1952, males 5 - England and Wales, 1930- 1932, females 6 - Austria, 1959-1961, females 7 - Norway, 1956-1960, females Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

19. Compensation Law of Mortality in Laboratory Drosophila 1 – drosophila of the Old Falmouth, New Falmouth, Sepia and Eagle Point strains (1,000 virgin females) 2 – drosophila of the Canton-S strain (1,200 males) 3 – drosophila of the Canton-S strain (1,200 females) 4 - drosophila of the Canton-S strain (2,400 virgin females) Mortality force was calculated for 6-day age intervals. Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

20. Actuarial Implications Be prepared to a paradox that higher actuarial aging rates may be associated with higher life expectancy in compared populations (e.g., males vs females)

21. Mortality deceleration at advanced ages. After age 95, the observed risk of death [red line] deviates from the value predicted by an early model, the Gompertz law [black line]. Mortality of Swedish women for the period of 1990-2000 from the Kannisto-Thatcher Database on Old Age Mortality Source: Gavrilov, Gavrilova, “Why we fall apart. Engineering’s reliability theory explains human aging”. IEEE Spectrum. 2004.

23. M. Greenwood, J. O. Irwin. BIOSTATISTICS OF SENILITY

24. Mortality Leveling-Off in House Fly Musca domestica Our analysis of the life table for 4,650 male house flies published by Rockstein & Lieberman, 1959. Source: Gavrilov & Gavrilova. Handbook of the Biology of Aging, Academic Press, 2005, pp.1-40.

25. Non-Aging Mortality Kinetics in Later Life Source: Economos, A. (1979). A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 2: 74-76.

26. Classic Actuarial Publications on Late-Life Mortality Deceleration Perks, W. 1932. On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries 63:12. Beard, R.E. 1959. Note on some mathematical mortality models. In: The Lifespan of Animals. Little, Brown, Boston, 302-311 “It became clear in the early part of this century that it [Gompertz-Makeham law] was not universally applicable, particularly at the older ages where accumulating data suggested a slowing down of the rate of increase with age.” Beard, In: Biological Aspects of Demography, 1971.

27. Testing the “Limit-to-Lifespan” Hypothesis Source: Gavrilov L.A., Gavrilova N.S. 1991. The Biology of Life Span

28. Actuarial Implications There is no fixed “ω” – the last old age when mortality tables could be closed “for sure”

29. Latest Developments Was the mortality deceleration law overblown? A Study of the Real Extinct Birth Cohorts in the United States

30. Challenges in Death Rate Estimation at Extremely Old Ages Mortality deceleration may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect) Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high Ages of very old people may be highly exaggerated

31. U.S. Social Security Administration Death Master File Helps to Relax the First Two Problems Allows to study mortality in large, more homogeneous single-year or even single-month birth cohorts Allows to study mortality in one-month age intervals narrowing the interval of hazard rates estimation

32. What Is SSA DMF ? SSA DMF is a publicly available data resource (available at Rootsweb.com) Covers 93-96 percent deaths of persons 65+ occurred in the United States in the period 1937- 2004 Some birth cohorts covered by DMF could be studied by method of extinct generations Considered superior in data quality compared to vital statistics records by some researchers

34. Mortality at Advanced Ages by Sex

35. What are the explanations of mortality laws? Mortality and aging theories

36. Aging is a Very General Phenomenon!

37. Stages of Life in Machines and Humans The so-called bathtub curve for technical systems Bathtub curve for human mortality as seen in the U.S. population in 1999 has the same shape as the curve for failure rates of many machines.

38. Non-Aging Failure Kinetics of Industrial Materials in ‘Later Life’ (steel, relays, heat insulators) Source: Economos, A. (1979). A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 2: 74-76.

39. What Is Reliability Theory? Reliability theory is a general theory of systems failure.

40. Reliability Theory Reliability theory was historically developed to describe failure and aging of complex electronic (military) equipment, but the theory itself is a very general theory.

41. Redundancy Creates Both Damage Tolerance and Damage Accumulation (Aging) System with redundancy accumulates damage (aging) System without redundancy dies after the first random damage (no aging)

42. Reliability Model of a Simple Parallel System Failure rate of the system: Elements fail randomly and independently with a constant failure rate, k n – initial number of elements  nk n x n-1 early-life period approximation, when 1-e -kx  kx  k late-life period approximation, when 1-e -kx  1

43. Failure Rate as a Function of Age in Systems with Different Redundancy Levels Failure of elements is random

44. Standard Reliability Models Explain Mortality deceleration and leveling-off at advanced ages Compensation law of mortality

45. Standard Reliability Models Do Not Explain The Gompertz law of mortality observed in biological systems Instead they produce Weibull (power) law of mortality growth with age: μ(x) = a x b

46. An Insight Came To Us While Working With Dilapidated Mainframe Computer The complex unpredictable behavior of this computer could only be described by resorting to such 'human' concepts as character, personality, and change of mood.

47. Reliability structure of (a) technical devices and (b) biological systems Low redundancy Low damage load High redundancy High damage load X - defect

48. Model of organism with initial damage load Failure rate of a system with binomially distributed redundancy (approximation for initial period of life): x 0 = 0 - ideal system, Weibull law of mortality x 0 >> 0 - highly damaged system, Gompertz law of mortality where - the initial virtual age of the system The initial virtual age of a system defines the law of system’s mortality: Binomial law of mortality

49. People age more like machines built with lots of faulty parts than like ones built with pristine parts. As the number of bad components, the initial damage load, increases [bottom to top], machine failure rates begin to mimic human death rates.

50. Actuarial Implications If the initial damage load is really important, then we may expect significant effects of early-life conditions (like season-of-birth) on late-life mortality

51. Life Expectancy and Month of Birth In Real U.S. Birth Cohorts Source: L.A. Gavrilov, N.S. Gavrilova. (2005). "Mortality of Centenarians: A Study Based on the Social Security Administration Death Master File" Presented at the 2005 Annual Meeting of the Population Association of America.

53. Acknowledgments This study was made possible thanks to: generous support from the National Institute on Aging, and stimulating working environment at the Center on Aging, NORC/University of Chicago

54. For More Information and Updates Please Visit Our Scientific and Educational Website on Human Longevity: http://longevity-science.org

56. Gavrilov, L., Gavrilova, N. Reliability theory of aging and longevity. In: Handbook of the Biology of Aging. Academic Press, 6 th edition, 2005, pp.1-40.