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
Additional files

http://health-studies.org/tutorial/gompertz.xls

http://health-studies.org/tutorial/Sullivan.xls
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.
Life Table


Cohort life table as a simple
example
Consider survival in the cohort of
fruit flies born in the same time
Number of dying, d(x)
Number of survivors, l(x)
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)
Probability of death, q(x)
Person-years lived in the interval, L(x)
Lx = x
lx + lx +
x
2
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)
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
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
Life Tables for Human Populations



In the majority of cases life tables for
humans are constructed for
hypothetical birth cohort using crosssectional 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
Formula for q(x) using
age-specific mortality rates
qx =
Mx
1 + (1
ax ) Mx
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.
Life table probabilities of death, q(x), for
men in Russia and USA. 2005
1
0
10
20
30
40
50
60
log(q(x))
0.1
0.01
0.001
0.0001
Age
Russia
USA
70
80
90
100
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
Life table number of survivors, l(x), for men
in Russia and USA. 2005.
120000
100000
80000
Russia
60000
USA
40000
20000
0
0
10
20
30
40
50
60
70
80
90
100
Life table number of dying, d(x), for men in
Russia and USA. 2005
Russia
USA
3500
3000
d(x)
2500
2000
1500
1000
500
0
0
10
20
30
40
50
Age
60
70
80
90
100
e(x)
Life expectancy, e(x), for men in Russia and
USA. 2005
80
70
60
50
40
30
20
10
0
Russia
USA
0
10
20
30
40
50
Age
60
70
80
90 100
Trends in life expectancy for men
in Russia, USA and Estonia
Trends in life expectancy for
women in Russia, USA and Estonia
A growing number of persons
living beyond age 80 emphasizes
the need for accurate
measurement and modeling of
mortality at advanced ages.
What do we know about late-life
mortality?
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.
M. Greenwood, J. O. Irwin. BIOSTATISTICS OF SENILITY
Mortality Leveling-Off in House Fly
Musca domestica
Based on life
table of 4,650
male house flies
published by
Rockstein &
Lieberman, 1959
hazard rate, log scale
0.1
0.01
0.001
0
10
20
Age, days
30
40
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.
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)
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)
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
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
Mortality force (hazard rate) is the best
indicator to study mortality at advanced ages
x



=
dN x
N x dx
=
d ln(N x )
ln(N x )
dx
x
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)
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
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 onemonth age intervals narrowing the
interval of hazard rates estimation
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.
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
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
Monthly Estimates of Mortality are More Accurate
Simulation assuming Gompertz law for hazard rate
Stata package uses the NelsonAalen estimate of hazard rate:
x
= H(x )
H(x
1) =
dx
nx
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:
qx =
dx
lx
Hazard rate estimates at advanced ages based on DMF
Nelson-Aalen monthly estimates of hazard rates using Stata 11
More recent birth cohort mortality
Nelson-Aalen monthly estimates of hazard rates using Stata 11
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
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)
Mortality for data with presumably different quality:
Southern and Non-Southern states of SSN receipt
The degree of deceleration was evaluated using quadratic model
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.
Mortality for data with presumably different quality:
Older and younger birth cohorts
The degree of deceleration was evaluated using quadratic model
At what age interval data have
reasonably good quality?
A study of age-specific mortality by gender
Women have lower mortality at
advanced ages
Hence number of females to number of males ratio should grow with age
Observed female to male ratio at advanced
ages for combined 1887-1892 birth cohort
Age of maximum female to male ratio
by birth cohort
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
Fitting mortality with logistic and
Gompertz models
Bayesian information criterion (BIC) to
compare logistic and Gompertz models,
men, by birth cohort (only Northern states)
Birth
cohort
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
Cohort
size at 88
years
35928
36399
40803
40653
40787
42723
45345
45719
46664
46698
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
Bayesian information criterion (BIC) to compare
logistic and Gompertz models, women, by birth
cohort (only Northern states)
Birth
cohort
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
Cohort
size at 88
years
68340
70499
79370
82298
85319
90589
96065
99474
102697
106291
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
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 18901900 (Haines, 1998)
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:
µx = -ln(1-qx)
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 19002100
Actuarial Study No.
116
Hazard rates for
1900 cohort are
estimated by Sacher
formula
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 19002100
Actuarial Study No.
116
Hazard rates for
1900 cohort are
estimated by Sacher
formula
Estimating Gompertz slope parameter
Actuarial cohort life table and SSDI 1894 cohort
0
1894 birth cohort, SSDI
1900 cohort, U.S. actuarial life table
log (hazard rate)
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 twostage Gompertz model
is not supported by real
data
-1
80
90
100
Age
110
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 to identify the most
accurate mortality indicator

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

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 100110
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
1
hazard rate, log scale
x
=
1
2 x
ln
lx
x
lx +
x
Probability of death values
strongly undeestimate mortality
after age 100
theoretical trajectory
Sacher estimate
qx
0.1
90
100
110
Age
120
qx =
dx
lx
Simulation study of Gompertz mortality
hazard rate, log scale
Compare Gehan and actuarial hazard rate estimates
Gehan estimates slightly
overestimate hazard rate
because of its half-year shift
to earlier ages
1
x
x +
105
110
115
Age
120
qx )
Actuarial estimates
undeestimate mortality after
age 100
theoretical trajectory
Gehan estimate
Actuarial estimate
100
= ln(1
125
x
2
2 lx lx +
=
x lx + lx +
x
x
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
Accuracy of hazard rate estimates
Relative difference between theoretical and observed values, %
Estimate
100 years
110 years
Probability of death
11.6%, understate
26.7%, understate
Sacher estimate
0.1%, overstate
0.1%, overstate
Gehan estimate
4.1%, overstate
4.1%, overstate
Actuarial estimate
1.0%, understate
4.5%, understate
Mortality of 1894 birth cohort
Monthly and Yearly Estimates of Hazard Rates
using Nelson-Aalen formula (Stata)
Sacher formula for hazard rate estimation
(Sacher, 1956; 1966)
1
( ln l
x =
x
x
Hazard
rate
x
2
ln l
x +
x
2
) =
1
2 x
ln
lx - survivor function at age x; ∆x – age interval
Simplified version suggested by Gehan (1969):
µx = -ln(1-qx)
lx
x
lx +
x
Mortality of 1894 birth cohort
Sacher formula for yearly estimates of hazard rates
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
Another example of human data
demonstrating no mortality deceleration
1681 centenarian siblings born before 1880
and lived 60 years and more
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)
Recent developments
“none of the agespecific 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
"
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)
Mortality of mice (log scale)
Miller data
males

females
Actuarial estimate of hazard rate with 10-day age intervals
Laboratory rats

Data sources: Dunning, Curtis (1946);
Weisner, Sheard (1935), Schlettwein-Gsell
(1970)
Mortality of Wistar rats
males


females
Actuarial estimate of hazard rate with 50-day age intervals
Data source: Weisner, Sheard, 1935
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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