1. Mortality Measurement at Advanced Ages 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. Additional files http://health-studies.org/tutorial/gompertz.xls http://health-studies.org/tutorial/Sullivan.xls

3. The Concept of Life Table Life table is a classic demographic format of describing a population's mortality experience with age. Life Table is built of a number of standard numerical columns representing various indicators of mortality and survival. The concept of life table was first suggested in 1662 by John Graunt. Before the 17th century, death was believed to be a magical or sacred phenomenon that could not and should not be quantified. The invention of life table was a scientific breakthrough in mortality studies.

4. Life Table Cohort life table as a simple example Consider survival in the cohort of fruit flies born in the same time

5. Number of dying, d(x)

6. Number of survivors, l(x)

7. Number of survivors at the beginning of the next age interval: l(x+1) = l(x) – d(x) Probability of death in the age interval: q(x) = d(x)/l(x)

8. Probability of death, q(x)

9. Person-years lived in the interval, L(x) L(x) are needed to calculate life expectancy. Life expectancy, e(x), is defined as an average number of years lived after certain age. L(x) are also used in calculation of net reproduction rate (NRR)

10. Calculation of life expectancy, e(x) Life expectancy at birth is estimated as an area below the survival curve divided by the number of individuals at birth

11. Life expectancy, e(x) T(x) = L(x) + … + Lω where Lω is L(x) for the last age interval. Summation starts from the last age interval and goes back to the age at which life expectancy is calculated. e(x) = T(x)/l(x) where x = 0, 1, …,ω

12. Life Tables for Human Populations In the majority of cases life tables for humans are constructed for hypothetical birth cohort using cross- sectional data Such life tables are called period life tables Construction of period life tables starts from q(x) values rather than l(x) or d(x) as in the case of experimental animals

13. Formula for q(x) using age-specific mortality rates a(x) called the fraction of the last interval of life is usually equal to 0.5 for all ages except for the first age (from 0 to 1) Having q(x) calculated, data for all other life table columns are estimated using standard formulas.

14. Life table probabilities of death, q(x), for men in Russia and USA. 2005

15. Period life table for hypothetical population Number of survivors, l(x), at the beginning is equal to 100,000 This initial number of l(x) is called the radix of life table

16. Life table number of survivors, l(x), for men in Russia and USA. 2005.

17. Life table number of dying, d(x), for men in Russia and USA. 2005

18. Life expectancy, e(x), for men in Russia and USA. 2005

19. Trends in life expectancy for men in Russia, USA and Estonia

20. Trends in life expectancy for women in Russia, USA and Estonia

21. A growing number of persons living beyond age 80 emphasizes the need for accurate measurement and modeling of mortality at advanced ages.

22. What do we know about late-life mortality?

23. 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.

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

26. Mortality Leveling-Off in House Fly Musca domestica Based on life table of 4,650 male house flies published by Rockstein & Lieberman, 1959

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

28. Mortality Deceleration in Animal Species Invertebrates: Nematodes, shrimps, bdelloid rotifers, degenerate medusae (Economos, 1979) Drosophila melanogaster (Economos, 1979; Curtsinger et al., 1992) Housefly, blowfly (Gavrilov, 1980) Medfly (Carey et al., 1992) Bruchid beetle (Tatar et al., 1993) Fruit flies, parasitoid wasp (Vaupel et al., 1998) Mammals: Mice (Lindop, 1961; Sacher, 1966; Economos, 1979) Rats (Sacher, 1966) Horse, Sheep, Guinea pig (Economos, 1979; 1980) However no mortality deceleration is reported for Rodents (Austad, 2001) Baboons (Bronikowski et al., 2002)

29. Existing Explanations of Mortality Deceleration Population Heterogeneity (Beard, 1959; Sacher, 1966). “… sub-populations with the higher injury levels die out more rapidly, resulting in progressive selection for vigour in the surviving populations” (Sacher, 1966) Exhaustion of organism’s redundancy (reserves) at extremely old ages so that every random hit results in death (Gavrilov, Gavrilova, 1991; 2001) Lower risks of death for older people due to less risky behavior (Greenwood, Irwin, 1939) Evolutionary explanations (Mueller, Rose, 1996; Charlesworth, 2001)

30. Mortality at Advanced Ages – 20 years ago Source: Gavrilov L.A., Gavrilova N.S. The Biology of Life Span: A Quantitative Approach, NY: Harwood Academic Publisher, 1991

31. Mortality at Advanced Ages, Recent Study Source: Manton et al. (2008). Human Mortality at Extreme Ages: Data from the NLTCS and Linked Medicare Records. Math.Pop.Studies

32. Mortality force (hazard rate) is the best indicator to study mortality at advanced ages Does not depend on the length of age interval Has no upper boundary and theoretically can grow unlimitedly Famous Gompertz law was proposed for fitting age-specific mortality force function (Gompertz, 1825)

33. Problems in Hazard Rate Estimation At Extremely Old Ages 1. Mortality deceleration in humans may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect) 2. Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high 3. Ages of very old people may be highly exaggerated

34. Social Security Administration’s Death Master File (SSA’s DMF) Helps to Alleviate the First Two Problems Allows to study mortality in large, more homogeneous single-year or even single-month birth cohorts Allows to estimate mortality in one- month age intervals narrowing the interval of hazard rates estimation

35. What Is SSA’s DMF ? As a result of a court case under the Freedom of Information Act, SSA is required to release its death information to the public. SSA’s DMF contains the complete and official SSA database extract, as well as updates to the full file of persons reported to SSA as being deceased. SSA DMF is no longer a publicly available data resource (now is available from Ancestry.com for fee) We used DMF full file obtained from the National Technical Information Service (NTIS). Last deaths occurred in September 2011.

36. SSA’s DMF Advantage Some birth cohorts covered by DMF could be studied by the method of extinct generations Considered superior in data quality compared to vital statistics records by some researchers

37. Social Security Administration’s Death Master File (DMF) Was Used in This Study: To estimate hazard rates for relatively homogeneous single-year extinct birth cohorts (1881-1895) To obtain monthly rather than traditional annual estimates of hazard rates To identify the age interval and cohort with reasonably good data quality and compare mortality models

38. Monthly Estimates of Mortality are More Accurate Simulation assuming Gompertz law for hazard rate Stata package uses the Nelson- Aalen estimate of hazard rate: H(x) is a cumulative hazard function, d x is the number of deaths occurring at time x and n x is the number at risk at time x before the occurrence of the deaths. This method is equivalent to calculation of probabilities of death:

39. Hazard rate estimates at advanced ages based on DMF Nelson-Aalen monthly estimates of hazard rates using Stata 11

40. More recent birth cohort mortality Nelson-Aalen monthly estimates of hazard rates using Stata 11

41. Hypothesis Mortality deceleration at advanced ages among DMF cohorts may be caused by poor data quality (age exaggeration) at very advanced ages If this hypothesis is correct then mortality deceleration at advanced ages should be less expressed for data with better quality

42. Quality Control (1) Study of mortality in the states with different quality of age reporting: Records for persons applied to SSN in the Southern states were found to be of lower quality (Rosenwaike, Stone, 2003) We compared mortality of persons applied to SSN in Southern states, Hawaii, Puerto Rico, CA and NY with mortality of persons applied in the Northern states (the remainder)

43. Mortality for data with presumably different quality: Southern and Non-Southern states of SSN receipt The degree of deceleration was evaluated using quadratic model

44. Quality Control (2) Study of mortality for earlier and later single-year extinct birth cohorts: Records for later born persons are supposed to be of better quality due to improvement of age reporting over time.

45. Mortality for data with presumably different quality: Older and younger birth cohorts The degree of deceleration was evaluated using quadratic model

46. At what age interval data have reasonably good quality? A study of age-specific mortality by gender

47. Women have lower mortality at advanced ages Hence number of females to number of males ratio should grow with age

48. Observed female to male ratio at advanced ages for combined 1887-1892 birth cohort

49. Age of maximum female to male ratio by birth cohort

50. Modeling mortality at advanced ages Data with reasonably good quality were used: Northern states and 88-106 years age interval Gompertz and logistic (Kannisto) models were compared Nonlinear regression model for parameter estimates (Stata 11) Model goodness-of-fit was estimated using AIC and BIC

51. Fitting mortality with logistic and Gompertz models

52. Bayesian information criterion (BIC) to compare logistic and Gompertz models, men, by birth cohort (only Northern states) Birth cohort 1886188718881889189018911892189318941895 Cohort size at 88 years 35928363994080340653407874272345345457194666446698 Gompertz -139505.4-139687.1-170126.0-167244.6-189252.8-177282.6-188308.2-191347.1-192627.8-191304.8 logistic -134431.0-134059.9-168901.9-161276.4-189444.4-172409.6-183968.2-187429.7-185331.8-182567.1 Better fit (lower BIC) is highlighted in red Conclusion: In nine out of ten cases Gompertz model demonstrates better fit than logistic model for men in age interval 88-106 years

53. Bayesian information criterion (BIC) to compare logistic and Gompertz models, women, by birth cohort (only Northern states) Birth cohort 1886188718881889189018911892189318941895 Cohort size at 88 years 6834070499793708229885319905899606599474102697106291 Gompertz -340845.7-366590.7-421459.2-417066.3-416638.0-453218.2-482873.6-529324.9-584429-566049.0 logistic -339750.0-366399.1-420453.5-421731.7-408238.3-436972.3-470441.5-513539.1-562118.8-535017.6 Better fit (lower BIC) is highlighted in red Conclusion: In nine out of ten cases Gompertz model demonstrates better fit than logistic model for women in age interval 88-106 years

54. Comparison to mortality data from the Actuarial Study No.116 1900 birth cohort in Actuarial Study was used for comparison with DMF data – the earliest birth cohort in this study 1894 birth cohort from DMF was used for comparison because later birth cohorts are less likely to be extinct Historical studies suggest that adult life expectancy in the U.S. did not experience substantial changes during the period 1890- 1900 (Haines, 1998)

55. In Actuarial Study death rates at ages 95 and older were extrapolated We used conversion formula (Gehan, 1969) to calculate hazard rate from life table values of probability of death: µ x = -ln(1-q x )

56. Mortality at advanced ages, males: Actuarial 1900 cohort life table and DMF 1894 birth cohort Source for actuarial life table: Bell, F.C., Miller, M.L. Life Tables for the United States Social Security Area 1900- 2100 Actuarial Study No. 116 Hazard rates for 1900 cohort are estimated by Sacher formula

57. Mortality at advanced ages, females: Actuarial 1900 cohort life table and DMF 1894 birth cohort Source for actuarial life table: Bell, F.C., Miller, M.L. Life Tables for the United States Social Security Area 1900- 2100 Actuarial Study No. 116 Hazard rates for 1900 cohort are estimated by Sacher formula

58. Estimating Gompertz slope parameter Actuarial cohort life table and SSDI 1894 cohort 1900 cohort, age interval 40-104 alpha (95% CI): 0.0785 (0.0772,0.0797) 1894 cohort, age interval 88-106 alpha (95% CI): 0.0786 (0.0786,0.0787) Hypothesis about two- stage Gompertz model is not supported by real data

59. Which estimate of hazard rate is the most accurate? Simulation study comparing several existing estimates:  Nelson-Aalen estimate available in Stata  Sacher estimate (Sacher, 1956)  Gehan (pseudo-Sacher) estimate (Gehan, 1969)  Actuarial estimate (Kimball, 1960)

60. Simulation study to identify the most accurate mortality indicator Simulate yearly l x numbers assuming Gompertz function for hazard rate in the entire age interval and initial cohort size equal to 10 11 individuals Gompertz parameters are typical for the U.S. birth cohorts: slope coefficient (alpha) = 0.08 year -1 ; R 0 = 0.0001 year -1 Focus on ages beyond 90 years Accuracy of various hazard rate estimates (Sacher, Gehan, and actuarial estimates) and probability of death is compared at ages 100- 110

61. Simulation study of Gompertz mortality Compare Sacher hazard rate estimate and probability of death in a yearly age interval Sacher estimates practically coincide with theoretical mortality trajectory Probability of death values strongly undeestimate mortality after age 100

62. Simulation study of Gompertz mortality Compare Gehan and actuarial hazard rate estimates Gehan estimates slightly overestimate hazard rate because of its half-year shift to earlier ages Actuarial estimates undeestimate mortality after age 100

63. Deaths at extreme ages are not distributed uniformly over one-year interval 85-year olds102-year olds 1894 birth cohort from the Social Security Death Index

64. Accuracy of hazard rate estimates Relative difference between theoretical and observed values, % Estimate100 years110 years Probability of death11.6%, understate26.7%, understate Sacher estimate0.1%, overstate Gehan estimate4.1%, overstate Actuarial estimate1.0%, understate4.5%, understate

65. Mortality of 1894 birth cohort Monthly and Yearly Estimates of Hazard Rates using Nelson-Aalen formula (Stata)

66. Sacher formula for hazard rate estimation (Sacher, 1956; 1966) l x - survivor function at age x; ∆x – age interval Hazard rate Simplified version suggested by Gehan (1969): µ x = -ln(1-q x )

67. Mortality of 1894 birth cohort Sacher formula for yearly estimates of hazard rates

68. Conclusions Deceleration of mortality in later life is more expressed for data with lower quality. Quality of age reporting in DMF becomes poor beyond the age of 107 years Below age 107 years and for data of reasonably good quality the Gompertz model fits mortality better than the logistic model (no mortality deceleration) Sacher estimate of hazard rate turns out to be the most accurate and most useful estimate to study mortality at advanced ages

69. Another example of human data demonstrating no mortality deceleration 1681 centenarian siblings born before 1880 and lived 60 years and more

70. Mortality Deceleration in Other Species Invertebrates: Nematodes, shrimps, bdelloid rotifers, degenerate medusae (Economos, 1979) Drosophila melanogaster (Economos, 1979; Curtsinger et al., 1992) Medfly (Carey et al., 1992) Housefly, blowfly (Gavrilov, 1980) Fruit flies, parasitoid wasp (Vaupel et al., 1998) Bruchid beetle (Tatar et al., 1993) Mammals: Mice (Lindop, 1961; Sacher, 1966; Economos, 1979) Rats (Sacher, 1966) Horse, Sheep, Guinea pig (Economos, 1979; 1980) However no mortality deceleration is reported for Rodents (Austad, 2001) Baboons (Bronikowski et al., 2002)

71. Recent developments “none of the age- specific mortality relationships in our nonhuman primate analyses demonstrated the type of leveling off that has been shown in human and fly data sets” Bronikowski et al., Science, 2011 "

72. What about other mammals? Mortality data for mice: Data from the NIH Interventions Testing Program, courtesy of Richard Miller (U of Michigan) Argonne National Laboratory data, courtesy of Bruce Carnes (U of Oklahoma)

73. Mortality of mice (log scale) Miller data Actuarial estimate of hazard rate with 10-day age intervals malesfemales

74. Laboratory rats Data sources: Dunning, Curtis (1946); Weisner, Sheard (1935), Schlettwein-Gsell (1970)

75. Mortality of Wistar rats Actuarial estimate of hazard rate with 50-day age intervals Data source: Weisner, Sheard, 1935 malesfemales

76. Acknowledgments This study was made possible thanks to: generous support from the National Institute on Aging (R01 AG028620) Stimulating working environment at the Center on Aging, NORC/University of Chicago

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