Leonid A. Gavrilov Natalia S. Gavrilova Center on Aging, NORC/University of Chicago,
Advanced Age Mortality Patterns and Longevity Predictors Natalia S. Gavrilova, Ph.D. Leonid A....
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Transcript of Advanced Age Mortality Patterns and Longevity Predictors Natalia S. Gavrilova, Ph.D. Leonid A....
Advanced Age Mortality Patterns and Longevity
Predictors
Natalia S. Gavrilova, Ph.D.Leonid A. Gavrilov, Ph.D.
Center on Aging
NORC and The University of Chicago Chicago, Illinois, USA
The growing number of persons living beyond age 80 underscores the need
for accurate measurement of mortality
at advanced ages.
Recent projections of the U.S. Census Bureau
significantly overestimated the actual number of
centenarians
Views about the number of centenarians in the United
States 2009
New estimates based on the 2010 census are two times
lower than the U.S. Bureau of Census forecast
The same story recently happened in the Great Britain
Financial Times
Mortality at advanced ages is the key variablefor understanding population trends among the
oldest-old
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.
Mortality at Advanced Ages – more than 20 years ago
Source: Gavrilov L.A., Gavrilova N.S. The Biology of Life Span: A Quantitative Approach, NY: Harwood Academic Publisher, 1991
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
Problems with 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
Conclusions from our earlier study of Social Security
Administration Death Master File
Mortality deceleration at advanced ages among DMF cohorts is more expressed for data of lower quality
Mortality data beyond ages 106-107 years have unacceptably poor quality (as shown using female-to-male ratio test). The study by other authors also showed that beyond age 110 years the age of individuals in DMF cohorts can be validated for less than 30% cases (Young et al., 2010)
Source: Gavrilov, Gavrilova, North American Actuarial Journal, 2011, 15(3):432-447
Nelson-Aalen monthly estimates of hazard rates using Stata 11
Data Source: DMF full file obtained from the National Technical Information Service (NTIS). Last deaths occurred in September 2011.
Observed female to male ratio at advanced ages for combined 1887-1892
birth cohort
Selection of competing mortality models using DMF
data Data with reasonably good quality were
used: non-Southern states and 85-106 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
Fitting mortality with Kannisto and Gompertz models
Kannisto model
Gompertz model
Akaike information criterion (AIC) to compare Kannisto and Gompertz
models, men, by birth cohort (non-Southern states)
Conclusion: In all ten cases Gompertz model demonstrates better fit than Kannisto model for men in age interval 85-106 years
U.S. Males
-370000
-350000
-330000
-310000
-290000
-270000
-250000
1890 1891 1892 1893 1894 1895 1896 1897 1898 1899
Birth Cohort
Akai
ke c
riter
ion
Gompertz Kannisto
Akaike information criterion (AIC) to compare Kannisto and Gompertz models, women, by
birth cohort (non-Southern states)
Conclusion: In all ten cases Gompertz model demonstrates better fit than Kannisto model for women in age interval 85-106 years
U.S. Females
-900000
-850000
-800000
-750000
-700000
-650000
-600000
1890 1891 1892 1893 1894 1895 1896 1897 1898 1899
Birth Cohort
Akai
ke C
riter
ion
Gompertz Kannisto
The second studied dataset:U.S. cohort death rates taken
from the Human Mortality Database
Selection of competing mortality models using HMD
data Data with reasonably good quality were used:
80-106 years age interval Gompertz and logistic (Kannisto) models were
compared Nonlinear weighted regression model for
parameter estimates (Stata 11) Age-specific exposure values were used as
weights (Muller at al., Biometrika, 1997) Model goodness-of-fit was estimated using
AIC and BIC
Fitting mortality with Kannisto and Gompertz models, HMD U.S. data
Fitting mortality with Kannisto and Gompertz models, HMD U.S. data
Akaike information criterion (AIC) to compare Kannisto and Gompertz
models, men, by birth cohort (HMD U.S. data)
Conclusion: In all ten cases Gompertz model demonstrates better fit than Kannisto model for men in age interval 80-106 years
U.S.Males
-250
-230
-210
-190
-170
-150
1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900
Birth Cohort
Akai
ke C
riter
ion
Gompertz Kannisto
Akaike information criterion (AIC) to compare Kannisto and Gompertz
models, women, by birth cohort (HMD U.S. data)
Conclusion: In all ten cases Gompertz model demonstrates better fit than Kannisto model for women in age interval 80-106 years
U.S. Females
-250-240-230-220-210-200-190-180-170-160-150
1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900
Birth Cohort
Akai
ke C
riter
ion
Gompertz Kannisto
Compare DMF and HMD data Females, 1898 birth cohort
Hypothesis about two-stage Gompertz model is not supported by real dataAge, years
60 70 80 90 100 110
log
Haz
ard
rate
0.01
0.1
1
DMFHMD
Alternative way to study mortality trajectories at advanced ages:
Age-specific rate of mortality change
Suggested by Horiuchi and Coale (1990), Coale and Kisker (1990), Horiuchi and Wilmoth (1998) and later called ‘life table aging rate (LAR)’
k(x) = d ln µ(x)/dx
Constant k(x) suggests that mortality follows the Gompertz model. Earlier studies found that k(x) declines in the age interval 80-100 years suggesting mortality deceleration.
Typical result from Horiuchi and Wilmoth paper (Demography,
1998)
Age-specific rate of mortality change
Swedish males, 1896 birth cohort
Flat k(x) suggests that mortality follows the Gompertz law
Age, years
60 65 70 75 80 85 90 95 100
k x v
alue
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Study of age-specific rate of mortality change using cohort
data
Age-specific cohort death rates taken from the Human Mortality Database
Studied countries: Canada, France, Sweden, United States
Studied birth cohorts: 1894, 1896, 1898 k(x) calculated in the age interval 80-100 years k(x) calculated using one-year mortality rates
Slope coefficients (with p-values) for linear regression models of k(x)
on age
All regressions were run in the age interval 80-100 years.
Country Sex Birth cohort
1894 1896 1898
slope p-value slope p-value slope p-value
Canada F -0.00023 0.914 0.00004 0.984 0.00066 0.583
M 0.00112 0.778 0.00235 0.499 0.00109 0.678
France F -0.00070 0.681 -0.00179 0.169 -0.00165 0.181
M 0.00035 0.907 -0.00048 0.808 0.00207 0.369
Sweden F 0.00060 0.879 -0.00357 0.240 -0.00044 0.857
M 0.00191 0.742 -0.00253 0.635 0.00165 0.792
USA F 0.00016 0.884 0.00009 0.918 0.000006 0.994
M 0.00006 0.965 0.00007 0.946 0.00048 0.610
In previous studies mortality rates were calculated for five-year age
intervals
Five-year age interval is very wide for mortality estimation at advanced ages. Assumption about uniform distribution of deaths in the age interval does not work for 5-year interval Mortality rates at advanced ages are biased downward
k x =ln( )m x ln( )m x 5
5
Simulation study of mortality following the Gompertz law
Simulate yearly lx numbers assuming Gompertz function for hazard rate in the entire age interval and initial cohort size equal to 1011 individuals
Gompertz parameters are typical for the U.S. birth cohorts: slope coefficient (alpha) = 0.08 year-1; R0= 0.0001 year-1
Numbers of survivors were calculated using formula (Gavrilov et al., 1983):
where Nx/N0 is the probability of survival to age x, i.e. the number of hypothetical cohort at age x divided by its initial number N0. a and b (slope) are parameters of Gompertz equation
N x
N 0=N x0
N 0exp
ab
( )eb x eb x0
Age-specific rate of mortality change with age, kx, by age interval for mortality calculation
Simulation study of Gompertz mortality
Taking into account that underlying mortality follows the Gompertz law, the dependence of k(x) on age should be flat
Age, years70 75 80 85 90 95 100 105
k x v
alue
0.04
0.05
0.06
0.07
0.08
0.09
5-year age intervalone-year age interval
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
Age
90 100 110 120
haza
rd ra
te, l
og s
cale
0.1
1
theoretical trajectorySacher estimateqx q x =
d xl x
x =
12x
lnl x x
l x x +
Conclusions Below age 107 years and for data of
reasonably good quality the Gompertz model fits cohort mortality better than the Kannisto model (no mortality deceleration) for 20 studied single-year U.S. birth cohorts
Age-specific rate of mortality change remains flat in the age interval 80-100 years for 24 studied single-year birth cohorts of Canada, France, Sweden and the United States suggesting that mortality follows the Gompertz law
Longevity Predictors
Leonid A. GavrilovNatalia S. Gavrilova
Center on Aging
NORC and The University of Chicago Chicago, USA
General Approach
To study “success stories” in long-term avoidance of fatal diseases (survival to 100 years) and factors correlated with this remarkable survival success
Winnie ain’t quitting now.
Smith G D Int. J. Epidemiol. 2011;40:537-562
Published by Oxford University Press on behalf of the International Epidemiological Association © The Author 2011; all rights reserved.
An example of incredible resilience
Exceptional longevity in a family of Iowa farmers
Father: Mike Ackerman, Farmer, 1865-1939 lived 74 years Mother: Mary Hassebroek 1870-1961 lived 91 years
1. Engelke "Edward" M. Ackerman b: 28 APR 1892 in Iowa 101
2. Fred Ackerman b: 19 JUL 1893 in Iowa 1033. Harmina "Minnie" Ackerman b: 18 SEP 1895 in Iowa 1004. Lena Ackerman b: 21 APR 1897 in Iowa 1055. Peter M. Ackerman b: 26 MAY 1899 in Iowa 866. Martha Ackerman b: 27 APR 1901 in IA 957. Grace Ackerman b: 2 OCT 1904 in IA 1048. Anna Ackerman b: 29 JAN 1907 in IA 1019. Mitchell Johannes Ackerman b: 25 FEB 1909 in IA 85
Studies of centenarians require careful design and
serious work on age validation
The main problem is to find an appropriate control group
Approach
Compare centenarians and shorter-lived controls, which are randomly sampled from the same data universe: computerized genealogies
Approach used in this study Compare centenarians with their
peers born in the same year but died at age 65 years
It is assumed that the majority of deaths at age 65 occur due to chronic diseases related to aging rather than injuries or infectious diseases (confirmed by analysis of available death certificates)
Case-control study of longevity
Cases - 765 centenarians survived to age 100 and born in USA in 1890-91Controls – 783 their shorter-lived peers born in USA in 1890-91 and died at age 65 yearsMethod: Multivariate logistic regressionGenealogical records were linked to 1900 and 1930 US censuses providing a rich set of variables
Age validation is a key moment in human longevity studies
Death dates of centenarians were validated using the U.S. Social Security Death Index
Birth dates were validated through linkage of centenarian records to early U.S. censuses (when centenarians were children)
Genealogies and 1900 and 1930 censuses provide three
types of variables Characteristics of early-life
conditions
Characteristics of midlife conditions
Family characteristics
Early-life characteristics• Type of parental household (farm or non-farm, own or rented),
• Parental literacy,
• Parental immigration status
• Paternal (or head of household) occupation
• Number of children born/survived by mother
• Size of parental household in 1900
• Region of birth
Midlife Characteristics from 1930 census
• Type of person’s household
• Availability of radio in household
• Person’s age at first marriage
• Person’s occupation (husband’s occupation in the case of women)
• Industry of occupation
• Number of children in household
• Veteran status, Marital status
Family Characteristicsfrom genealogy
• Information on paternal and maternal lifespan
• Paternal and maternal age at person’s birth,
• Number of spouses and siblings
• Birth order
• Season of birth
Example of images from 1930 census (controls)
Parental longevity, early-life and midlife conditions and survival to age 100.
MenMultivariate logistic regression, N=723
Variable Odds ratio 95% CI P-value
Father lived 80+ 1.84 1.35-2.51 <0.001
Mother lived 80+ 1.70 1.25-2.32 0.001
Farmer in 1930 1.67 1.21-2.31 0.002
Born in North-East 2.08 1.27-3.40 0.004
Born in the second half of year 1.36 1.00-
1.84 0.050
Radio in household, 1930 0.87 0.63-1.19 0.374
Parental longevity, early-life and midlife conditions and survival to age 100
WomenMultivariate logistic regression, N=815
VariableOdd
s ratio
95% CI P-value
Father lived 80+ 2.19 1.61-2.98 <0.001
Mother lived 80+ 2.23 1.66-2.99 <0.001
Husband farmer in 1930 1.15 0.84-1.56 0.383Radio in household, 1930 1.61 1.18-2.20 0.003
Born in the second half of year 1.18 0.89-1.58 0.256Born in the North-East region 1.04 0.62-1.67 0.857
Variables found to be non-significant in multivariate
analyses Parental literacy and immigration
status, farm childhood, size of household in 1900, percentage of survived children (for mother) – a proxy for child mortality, sibship size, father-farmer in 1900
Marital status, veteran status, childlessness, age at first marriage
Paternal and maternal age at birth, loss of parent before 1910
Season of birth and survival to 100
Birth in the first half and the second half of the year among centenarians and controls died at age 65
Significant difference P=0.0080
10
20
30
40
50
60
1st half 2nd half
CentenariansControls
Significant difference:
p=0.008
Within-Family Study of Season of Birth and Exceptional Longevity
Month of birth is a useful proxy characteristic for environmental effects acting during in-utero and early infancy development
Siblings Born in September-November Have Higher Chances to
Live to 100Within-family study of 9,724 centenarians born in 1880-1895 and their siblings survived to
age 50
Possible explanationsThese are several explanations of season-of birth effects on longevity pointing to the effects of early-life events and conditions: seasonal exposure to infections,nutritional deficiencies, environmental temperature and sun exposure. All these factors were shown to play role in later-life health and longevity.
Conclusions Both midlife and early-life
conditions affect survival to age 100 Parental longevity turned out to be
the strongest predictor of survival to age 100
Information about such an important predictor as parental longevity should be collected in contemporary longitudinal studies
AcknowledgmentsThis 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
For More Information and Updates Please Visit Our Scientific and Educational
Website on Human Longevity:
http://longevity-science.org
And Please Post Your Comments at our Scientific Discussion Blog:
http://longevity-science.blogspot.com/
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)
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
x = ln( )1 q x
x
x2
+ =
2xl x l x x +
l x l x x + +
Age
100 105 110 115 120 125
haza
rd ra
te, l
og s
cale
1
theoretical trajectoryGehan estimateActuarial estimate
Simulation study of the Gompertz mortality
Kernel smoothing of hazard rates .2
.4.6
.8H
azar
d, lo
g sc
ale
80 90 100 110 120age
Smoothed hazard estimate
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, dx is the number of deaths occurring at time x and nx is the number at risk at time x before the occurrence of the deaths. This method is equivalent to calculation of probabilities of death:
q x =d xl x
x = H( )x H( )x 1 =
d xn x
Sacher formula for hazard rate estimation
(Sacher, 1956; 1966)x =
1x
( )ln lx
x2
ln lx
x2
+ =
12x
lnl x x
l x x +
lx - survivor function at age x; ∆x – age interval
Hazardrate
Simplified version suggested by Gehan (1969):
µx = -ln(1-qx)