Gender Specific Effects of Early-Life Events on Adult Lifespan

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

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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.

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Life Table

• Cohort life table as a simple example
• Consider survival in the cohort of fruit flies
  born in the same time

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Number of dying, d(x)
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Number of survivors, l(x)
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Number of survivors at the beginning of the next
age interval
l(x1) l(x) d(x)
Probability of death in the age interval
q(x) d(x)/l(x)
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Probability of death, q(x)
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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)
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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
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Life expectancy, e(x)

• T(x) L(x) L?
• where L? 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, ,?

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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

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

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Life table number of survivors, l(x), for men in
Russia and USA. 2005.
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Life table number of dying, d(x), for men in
Russia and USA. 2005
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Life expectancy, e(x), for men in Russia and USA.
2005
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Trends in life expectancy for men in Russia, USA
and Estonia
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Trends in life expectancy for women in Russia,
USA and Estonia
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A growing number of persons living beyond age 80
emphasizes the need for accurate measurement and
modeling of mortality at advanced ages.
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What do we know about late-life mortality?
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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.
  Engineerings reliability theory explains human
  aging. IEEE Spectrum. 2004.

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(No Transcript)
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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

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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.

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Mortality Deceleration in Animal Species

• 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)

• 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)

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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 organisms 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)

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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
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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)

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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

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Social Security Administrations Death Master
File (SSAs 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

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What Is SSAs 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. SSAs 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.

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SSAs 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

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Social Security Administrations 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
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Monthly Estimates of Mortality are More
AccurateSimulation 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
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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
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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
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Women have lower mortality at advanced ages
Hence number of females to number of males ratio
should grow with age
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Observed female to male ratio at advanced ages
for combined 1887-1892 birth cohort
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Age of maximum female to male ratio by birth
cohort
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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

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Fitting mortality with logistic and Gompertz
models
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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
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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
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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)

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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)

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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
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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
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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
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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)

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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 100-110

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Simulation study of Gompertz mortalityCompare
Sacher hazard rate estimate and probability of
death in a yearly age interval
Sacher estimates practically coincide with
theoretical mortality trajectory Probabil
ity of death values strongly undeestimate
mortality after age 100
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Simulation study of Gompertz mortalityCompare
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
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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
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Accuracy of hazard rate estimates
Relative difference between theoretical and observed values, Relative difference between theoretical and observed values, 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

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Mortality of 1894 birth cohortMonthly and Yearly
Estimates of Hazard Rates using Nelson-Aalen
formula (Stata)
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Sacher formula for hazard rate estimation(Sacher,
1956 1966)
Hazard rate
lx - survivor function at age x ?x age
interval
Simplified version suggested by Gehan (1969) µx
-ln(1-qx)
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Mortality of 1894 birth cohort Sacher formula
for yearly estimates of hazard rates
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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

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

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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)

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Mortality of mice (log scale) Miller data
males
females

• Actuarial estimate of hazard rate with 10-day age
  intervals

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Laboratory rats

• Data sources Dunning, Curtis (1946) Weisner,
  Sheard (1935), Schlettwein-Gsell (1970)

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Mortality of Wistar rats
males
females

• Actuarial estimate of hazard rate with 50-day age
  intervals
• Data source Weisner, Sheard, 1935

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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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