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


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


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


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)


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)

advanced age mortality patterns and longevity predictors natalia s. gavrilova, ph.d. leonid a....

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