Mortality Tables and Laws: Biodemographic Analysis and Reliability Theory Approach

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Mortality Tables and Laws Biodemographic
Analysis and Reliability Theory Approach

• Dr. Natalia S. Gavrilova, Ph.D.
• Dr. Leonid A. Gavrilov, Ph.D.
• Center on Aging
• NORC and the University of Chicago
• Chicago, Illinois, USA

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Questions of Actuarial Significance

• How far could mortality decline go?
• (absolute zero seems implausible)
• Are there any biological limits to human
  mortality decline, determined by reliability of
  human body?
• (lower limits of mortality dependent on age,
  sex, and population genetics)
• Were there any indications for biological
  mortality limits in the past?
• Are there any indications for mortality limits
  now?

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How can we improve the actuarial forecasts of
mortality and longevity ?

• By taking into account the mortality laws
  summarizing prior experience in mortality changes
  over age and time
• Gompertz-Makeham law of mortality
• Compensation law of mortality
• Late-life mortality deceleration

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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 ax
• A Makeham term or background mortality
• R e ax age-dependent mortality x - age

risk of death
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Gompertz Law of Mortality in Fruit Flies

• Based on the life table for 2400 females of
  Drosophila melanogaster published by Hall (1969).
  
• Source Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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Gompertz-Makeham Law of Mortality in Flour Beetles

• Based on the life table for 400 female flour
  beetles (Tribolium confusum Duval). published by
  Pearl and Miner (1941).
• Source Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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Gompertz-Makeham Law of Mortality in Italian
Women

• Based on the official Italian period life table
  for 1964-1967.
• Source Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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How can the Gompertz-Makeham law be used?

• By studying the historical dynamics of the
  mortality components in this law
• µ(x) A R e ax

Makeham component
Gompertz component
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Historical Stability of the Gompertz Mortality
Component Before the 1980sHistorical Changes in
Mortality for 40-year-old Swedish Males

• Total mortality, µ40
• Background mortality (A)
• Age-dependent mortality (Rea40)
• Source
• Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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Predicting Mortality Crossover Historical
Changes in Mortality for 40-year-old Women in
Norway and Denmark

• Norway, total mortality
• Denmark, total mortality
• Norway, age-dependent mortality
• Denmark, age-dependent mortality
• Source Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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Predicting Mortality Divergence Historical
Changes in Mortality for 40-year-old Italian
Women and Men

• Women, total mortality
• Men, total mortality
• Women, age-dependent mortality
• Men, age-dependent mortality
• Source Gavrilov, Gavrilova, The Biology of Life
  Span 1991

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A Broader View on the Historical Changes in
Mortality
Swedish Females Data source Human Mortality
Database
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Extension of the Gompertz-Makeham Model Through
the Factor Analysis of Mortality Trends

• Mortality force (age, time)
• a0(age) a1(age) x F1(time) a2(age) x
  F2(time)

• Where
• ai(age) a set of numbers each number is fixed
  for specific age group
• Fj(time) factors, a set of standardized
  numbers each number is fixed for specific moment
  of time (mean 0 st. dev. 1)

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Factor Analysis of Mortality Trends Swedish
Females
Factor analysis of the time series of mortality
confirms the preferential reduction in the
mortality of old-aged and senile people in
recent years Gavrilov, Gavrilova, The Biology
of Life Span, 1991. Data source for the current
slide Human Mortality Database
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Actuarial Implications
Mortality trends before the 1950s are useless or
even misleading for the current mortality
forecasts because all the rules of the game has
been changed dramatically
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Preliminary Conclusions

• There was some evidence for biological
  mortality limits in the past, but these limits
  proved to be responsive to the recent
  technological and medical progress.
• Thus, there is no convincing evidence for
  absolute biological mortality limits now.
• Analogy for illustration and clarification There
  was a limit to the speed of airplane flight in
  the past (sound barrier), but it was overcome
  by further technological progress. Similar
  observations seems to be applicable to current
  human mortality decline.

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Compensation Law of Mortality(late-life
mortality convergence)

• Relative differences in death rates are
  decreasing with age, because the lower initial
  death rates are compensated by higher slope of
  mortality growth with age (actuarial aging rate)

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Compensation Law of MortalityConvergence of
Mortality Rates with Age

• 1 India, 1941-1950, males
• 2 Turkey, 1950-1951, males
• 3 Kenya, 1969, males
• 4 - Northern Ireland, 1950-1952, males
• 5 - England and Wales, 1930-1932, females
• 6 - Austria, 1959-1961, females
• 7 - Norway, 1956-1960, females
• Source Gavrilov, Gavrilova,
• The Biology of Life Span 1991

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Compensation Law of Mortality in Laboratory
Drosophila

• 1 drosophila of the Old Falmouth, New Falmouth,
  Sepia and Eagle Point strains (1,000 virgin
  females)
• 2 drosophila of the Canton-S strain (1,200
  males)
• 3 drosophila of the Canton-S strain (1,200
  females)
• 4 - drosophila of the Canton-S strain (2,400
  virgin females)
• Mortality force was calculated for 6-day age
  intervals.
• Source Gavrilov, Gavrilova,
• The Biology of Life Span 1991

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Actuarial Implications
Be prepared to a paradox that higher actuarial
aging rates may be associated with higher life
expectancy in compared populations (e.g., males
vs females)
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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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M. Greenwood, J. O. Irwin. BIOSTATISTICS OF
SENILITY
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Mortality Leveling-Off in House Fly Musca
domestica

• Our analysis of the life table for 4,650 male
  house flies published by Rockstein Lieberman,
  1959.
• Source
• Gavrilov Gavrilova. Handbook of the Biology of
  Aging, Academic Press, 2005, pp.1-40.

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Non-Aging Mortality Kinetics in Later Life
Source Economos, A. (1979). A non-Gompertzian
paradigm for mortality kinetics of metazoan
animals and failure kinetics of manufactured
products. AGE, 2 74-76.
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Classic Actuarial Publications on Late-Life
Mortality Deceleration

• Perks, W. 1932. On some experiments in the
  graduation of mortality statistics. Journal of
  the Institute of Actuaries 6312.
• Beard, R.E. 1959. Note on some mathematical
  mortality models. In The Lifespan of Animals.
  Little, Brown, Boston, 302-311
• It became clear in the early part of this
  century that it Gompertz-Makeham law was not
  universally applicable, particularly at the older
  ages where accumulating data suggested a slowing
  down of the rate of increase with age. Beard,
  In Biological Aspects of Demography, 1971.

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Testing the Limit-to-Lifespan Hypothesis

• Source Gavrilov L.A., Gavrilova N.S. 1991. The
  Biology of Life Span

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Actuarial Implications
There is no fixed ? the last old age when
mortality tables could be closed for sure
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Latest Developments

• Was the mortality deceleration law overblown?
• A Study of the Real Extinct Birth Cohorts in the
  United States

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Challenges in Death Rate Estimation at Extremely
Old Ages

• Mortality deceleration may be an artifact of
  mixing different birth cohorts with different
  mortality (heterogeneity effect)
• Standard assumptions of hazard rate estimates may
  be invalid when risk of death is extremely high
• Ages of very old people may be highly exaggerated

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U.S. Social Security Administration Death Master
File Helps to Relax the First Two Problems

• Allows to study mortality in large, more
  homogeneous single-year or even single-month
  birth cohorts
• Allows to study mortality in one-month age
  intervals narrowing the interval of hazard rates
  estimation

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What Is SSA DMF ?

• SSA DMF is a publicly available data resource
  (available at Rootsweb.com)
• Covers 93-96 percent deaths of persons 65
  occurred in the United States in the period
  1937-2004
• Some birth cohorts covered by DMF could be
  studied by method of extinct generations
• Considered superior in data quality compared to
  vital statistics records by some researchers

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Mortality at Advanced Ages by Sex
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What are the explanations of mortality laws?

• Mortality and aging theories

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Aging is a Very General Phenomenon!
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Stages of Life in Machines and Humans
Bathtub curve for human mortality as seen in the
U.S. population in 1999 has the same shape as the
curve for failure rates of many machines.
The so-called bathtub curve for technical systems
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Non-Aging Failure Kinetics of Industrial
Materials in Later Life(steel, relays, heat
insulators)
Source Economos, A. (1979). A
non-Gompertzian paradigm for mortality kinetics
of metazoan animals and failure kinetics of
manufactured products. AGE, 2 74-76.
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What Is Reliability Theory?

• Reliability theory is a general theory of systems
  failure.

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Reliability Theory

• Reliability theory was historically developed
  to describe failure and aging of complex
  electronic (military) equipment, but the theory
  itself is a very general theory.

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Redundancy Creates Both Damage Tolerance and
Damage Accumulation (Aging)
System without redundancy dies after the first
random damage (no aging)
System with redundancy accumulates damage
(aging)
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Reliability Model of a Simple Parallel System

• Failure rate of the system

Elements fail randomly and independently with a
constant failure rate, k n initial number of
elements
? nknxn-1 early-life period approximation,
when 1-e-kx ? kx ? k late-life
period approximation, when 1-e-kx ? 1
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Failure Rate as a Function of Age in Systems
with Different Redundancy Levels
Failure of elements is random
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Standard Reliability Models Explain

• Mortality deceleration and leveling-off at
  advanced ages
• Compensation law of mortality

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Standard Reliability Models Do Not Explain

• The Gompertz law of mortality observed in
  biological systems
• Instead they produce Weibull (power) law of
  mortality growth with age
• µ(x) a xb

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An Insight Came To Us While Working With
Dilapidated Mainframe Computer

• The complex unpredictable behavior of this
  computer could only be described by resorting to
  such 'human' concepts as character, personality,
  and change of mood.

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Reliability structure of (a) technical devices
and (b) biological systems
Low redundancy Low damage load
High redundancy High damage load
X - defect
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Model of organism with initial damage load

• Failure rate of a system with binomially
  distributed redundancy (approximation for initial
  period of life)

Binomial law of mortality
- the initial virtual age of the system
where
The initial virtual age of a system defines the
law of systems mortality

• x0 0 - ideal system, Weibull law of mortality
• x0 gtgt 0 - highly damaged system, Gompertz law of
  mortality

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People age more like machines built with lots of
faulty parts than like ones built with pristine
parts.

• As the number of bad components, the initial
  damage load, increases bottom to top, machine
  failure rates begin to mimic human death rates.

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Actuarial Implications
If the initial damage load is really important,
then we may expect significant effects of
early-life conditions (like season-of-birth) on
late-life mortality
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Life Expectancy and Month of Birth In Real
U.S. Birth Cohorts
Source L.A. Gavrilov, N.S. Gavrilova. (2005).
"Mortality of Centenarians A Study Based on the
Social Security Administration Death Master
File" Presented at the 2005 Annual Meeting of the
Population Association of America.
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Acknowledgments

• This study was made possible thanks to
• generous support from the National Institute on
  Aging, and
• stimulating working environment at the Center
  on Aging, NORC/University of Chicago

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For More Information and Updates Please Visit Our
Scientific and Educational Website on Human
Longevity

• http//longevity-science.org

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• Gavrilov, L., Gavrilova, N. Reliability theory
  of aging and longevity. In Handbook of the
  Biology of Aging. Academic Press, 6th edition,
  2005, pp.1-40.