1. Mortality Patterns at Advanced Ages
Dr. Natalia S. Gavrilova, Ph.D.
Dr. Leonid A. Gavrilov, Ph.D.
Center on Aging
NORC at The University of Chicago
Chicago, Illinois, USA

2. Extremely Important Topic
Because more and more people survive to advanced ages, we need to have more accurate and reliable estimates of mortality and mortality trends at extreme old ages

3. Brief History and Background

4. Three scenarios for mortality at advanced ages
Source: Gavrilov L.A., Gavrilova N.S. The Biology of Life Span:
A Quantitative Approach, NY: Harwood Academic Publisher, 1991

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

6. Earlier studies suggested that the exponential growth of mortality with age (Gompertz law) is followed by a period of deceleration, with slower rates of mortality increase.

7. Mortality deceleration at advanced ages
After age 95, the observed risk of death [red line] deviates from the values predicted by the Gompertz law [black line].
Mortality of Swedish women for the period of 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

8. Mortality Leveling-Off in House Fly Musca domestica
Based on life table of 4,650 male house flies published by Rockstein & Lieberman, 1959
Source: Gavrilov, Gavrilova, Handbook of the Biology of Aging, 2006

9. 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)

10. 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 10

11. Study of the Social Security Administration Death Master File
North American Actuarial Journal, 2011, 15(3):

12. Birth cohort mortality, DMF data
Nelson-Aalen monthly estimates of hazard rates using Stata 11

13. Gompertz model of old-age mortality
Study of 20 single-year extinct U.S. birth cohorts based on the Social Security Administration Death Master File (DMF) found no mortality deceleration up to ages years (Gavrilova, Gavrilov, 2011).
However, data quality problems did not allow us to study mortality trajectories after age 107 or 110 years using this source of data.

14. The second studied dataset: U. S
The second studied dataset: U.S. cohort death rates taken from the Human Mortality Database

15. Nature (2016)
Based on data from the International Database on Longevity (IDL)
Note: After 2000 number of supercentenarians exposed to death in IDL rapidly declines

16. Why do we see more indications for apparent limit to human lifespan?

17. Two Observations

18. No mortality improvement for centenarians
Observation #1
No mortality improvement for centenarians
Found for Swedish and UK centenarians
Drefahl, S., H. Lundstrom, K. Modig, and A. Ahlbom "The Era of Centenarians: Mortality of the Oldest Old in Sweden." Journal of Internal Medicine 272(1):
Continuous Mortality Investigation "Initial Report on the Features of High Age Mortality. Working Paper 85.". London: Continuous Mortality Investigation Ltimited, UK

19. Recent scientific publications suggest that human longevity records stopped increasing.
Our finding that the mortality of U.S. centenarians has not decreased noticeably in recent decades is consistent with this suggestion.

20. Mortality of U.S. centenarians does not decline over time
Men
Women

21. Mortality of 1898-1902 birth cohorts both sexes

22. Improvement of age reporting
Observation #2
Improvement of age reporting

23. Is Mortality Deceleration Caused by Age Misreporting?
It was demonstrated that age misstatement biases mortality estimates downwards at the oldest ages, which contributes to mortality deceleration (Preston et al., 1999).
If this hypothesis is correct then mortality deceleration should be more prevalent among historically older birth cohorts

24. 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

25. Quality Control
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.

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

27. Historical Evolution of Mortality Trajectories 1880-1899 U. S
Historical Evolution of Mortality Trajectories U.S. birth cohorts. Men BIC values for fitting Gompertz and Kannisto models
Fitting age-specific cohort death rates taken from the Human Mortality Database

28. 1880-1899 U.S. birth cohorts. Women BIC values for fitting Gompertz and Kannisto models
Fitting age-specific cohort death rates taken from the Human Mortality Database

29. Conclusion
Mortality deceleration is more prevalent in historically older birth cohorts when age reporting was less accurate

30. At what ages data have reasonably good quality?
A study of age-specific mortality by gender using indirect measure of data quality

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

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

33. Male to female ratio for survivors to specific age for 1898-1902 birth cohorts in DMF

34. The validity of our method of gender assignment in DMF: No difference between gender-specific mortality estimates in DMF and vital statistics with known gender

35. What is the quality of age reporting in DMF across ages and birth cohorts?
A study of data quality for five
single-year birth cohorts
Supported by the Society of Actuaries

36. Age validation procedure
Age validation was conducted by linkage of DMF records to early historical sources (U.S. censuses, birth and marriage records, draft registration cards).
DMF records were scored according to reliability of age reporting. The scoring system included the following scores:
1 – several early historical sources agree about birth date
2 – one early historical sources agrees about birth date
3 – later sources agree about birth date
4 – early sources disagree
5 – foreign-born individual arrived in the U.S. later in life
6 – not found in any sources

37. An Example of Age Validation using Ancestry service
Finding person first in Ancestry database

38. Confirmation of birth date in early marriage record

39. Percent of records with questionable quality as a function of age
Percent of records with questionable quality as a function of age. 1898, 1900 and 1902 birth cohorts
Results of age validation study for samples of 100 records, by age group.
For ages 109 and 110+ years sample sizes were slightly higher than 100.

40. Percent of records with questionable quality at extreme old ages
Percent of records with questionable quality at extreme old ages birth cohorts

41. Regression model for percentage of poor quality records
Percentage of poor records is modeled as a linear function of binary (dummy) variables representing birth cohorts and ages.
where percent is percentage of poor quality records, AGE and COHORT represent sets of dummy variables (103, 105, 109 for AGE at death with 100 years used as a reference level and 1899, 1900, 1901, 1902 for COHORT birth year with 1898 used as a reference level), β1 and β2 are regression coefficients

42. Regression model for percentage of poor quality data
Variable
Regression coefficients
P-value
95% confidence intervals
1898 cohort
reference
1899 cohort
2.00
0.588
1900 cohort
-1.75
0.419
1901 cohort
-1.56e-15
1.000
1902 cohort
4.75
0.057
Age 100
Age 103
4.67
0.092
Age 105
4.33
0.112
Age 109
16.67
<0.001
Intercept
14.33

43. Force of mortality by the data quality score 1900 birth cohort, both sexes

44. Force of mortality after conducting data cleaning 1898-1902 birth cohort, both sexes

45. Is this assumption critical?
Are monthly estimates of the force of mortality less subjected to mortality deceleration?
We used actuarial estimate of hazard rate (calculated as central death rate).
This estimate assumes uniform distribution of deaths in the age interval.
Is this assumption critical?

46. 1894 birth cohort from the Social Security Death Index
Deaths at extreme ages are not distributed uniformly over one-year interval
85-year olds
102-year olds
1894 birth cohort from the Social Security Death Index

47. Simulation study of Gompertz mortality Compare Gehan and actuarial hazard rate estimates
Simplified Sacher estimates slightly overestimate hazard rate because of its half-year shift to earlier ages
Actuarial estimates (death rates) undeestimate mortality after age 100

48. Force of mortality by monthly and yearly estimates 1898 birth cohort, both sexes

49. Force of mortality by monthly and yearly estimates after data cleaning 1898 birth cohort, both sexes

50. Regional mortality at advanced ages
Study of mortality according to region of last residence and region of SSN application

51. Mortality, by region of last residence 1898-1902 birth cohort, women

52. Mortality, by region of SSN application 1898-1902 birth cohort, women

53. Conclusions
Gompertzialization of old-age mortality trajectories over time.
Mortality stagnation over time for ages 100 years and over.
If continued, these trends may lead to accelerating pattern of mortality increase with age
As a result, the number of centenarians in the future may be significantly lower than expected

54. Acknowledgments This study was made possible thanks to:
generous support from the
Society of Actuaries
National Institute on Aging (R21 AG052670)
Stimulating working environment at the Center on Aging, NORC/University of Chicago

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