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
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?
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
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
αx
risk of death
A – Makeham term or background mortality
R e αx – age-dependent mortality; x - age
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
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
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
How can the GompertzMakeham law be used?
By studying the historical
dynamics of the mortality
components in this law:
μ(x) = A + R e
Makeham component
αx
Gompertz component
Historical Stability of the Gompertz
Mortality Component Before the 1980s
Historical Changes in Mortality for 40-year-old Swedish Males
1.
2.
3.
Total mortality, μ40
Background
mortality (A)
Age-dependent
mortality (Reα40)
Source:
Gavrilov, Gavrilova, “The
Biology of Life Span” 1991
Predicting Mortality Crossover
Historical Changes in Mortality for
40-year-old Women in Norway and Denmark
1.
2.
3.
4.
Norway, total mortality
Denmark, total
mortality
Norway, agedependent mortality
Denmark, agedependent mortality
Source: Gavrilov, Gavrilova,
“The Biology of Life Span”
1991
Predicting Mortality Divergence
Historical Changes in Mortality for
40-year-old Italian Women and Men
1.
2.
3.
4.
Women, total
mortality
Men, total mortality
Women, agedependent mortality
Men, age-dependent
mortality
Source: Gavrilov, Gavrilova,
“The Biology of Life
Span” 1991
A Broader View on the
Historical Changes in Mortality
10-1
Log (Hazard Rate)
1925
1960
1980
2000
Swedish Females
10-2
Data source: Human
Mortality Database
10-3
10-4
0
20
40
Age
60
80
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)
Factor Analysis of Mortality Trends
Swedish Females
Mortality factor score
4
Makeham-like factor 1
("young ages")
Gompertz-like factor 2
("old ages")
“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.
2
0
Data source for the
current slide: Human
Mortality Database
-2
1900
1920
1940
1960
Calendar Year
1980
2000
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
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.
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)
Compensation Law of Mortality
Convergence of Mortality Rates with Age
1
2
3
4
– India, 1941-1950, males
– Turkey, 1950-1951, males
– Kenya, 1969, males
- Northern Ireland, 19501952, males
5 - England and Wales, 19301932, females
6 - Austria, 1959-1961, females
7 - Norway, 1956-1960, females
Source: Gavrilov, Gavrilova,
“The Biology of Life Span” 1991
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
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)
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
Our analysis of
the life table for
4,650 male house
flies published by
Rockstein &
Lieberman, 1959.
hazard rate, log scale
0.1
Source:
0.01
Gavrilov & Gavrilova.
Handbook of the
Biology of Aging,
Academic Press,
2005, pp.1-40.
0.001
0
10
20
Age, days
30
40
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.
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 63:12.
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.
Testing the “Limit-to-Lifespan” Hypothesis
Source: Gavrilov L.A., Gavrilova N.S. 1991. The Biology of Life Span
Actuarial Implications
There is no fixed “ω” –
the last old age when
mortality tables could be
closed “for sure”
Latest Developments
Was the mortality deceleration
law overblown?
A Study of the Real Extinct Birth
Cohorts in the United States
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
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
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 19372004
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
Mortality at Advanced Ages by Sex
What are the explanations of
mortality laws?
Mortality and aging theories
Aging is a Very General Phenomenon!
Stages of Life in Machines and Humans
The so-called bathtub curve for
technical systems
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.
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.
What Is Reliability Theory?

Reliability theory is a general
theory of systems failure.
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.
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)
Reliability Model
of a Simple Parallel System
Failure rate of the system:
( x) =
d S ( x)
nk e
=
S ( x ) dx
1
kx
(1
e
kx n
(1
e
kx n
)
1
)
 nknxn-1 early-life period approximation, when 1-e-kx  kx
 k
late-life period approximation, when 1-e-kx  1
Elements fail
randomly and
independently
with a constant
failure rate, k
n – initial
number of
elements
Failure Rate as a Function of Age
in Systems with Different Redundancy Levels
Failure of elements is random
Standard Reliability Models Explain


Mortality deceleration and
leveling-off at advanced ages
Compensation law of mortality
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
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.
Reliability structure of
(a) technical devices and (b) biological systems
Low redundancy
Low damage load
High redundancy
High damage load
X - defect
Model of organism
with initial damage load
Failure rate of a system with binomially distributed
redundancy (approximation for initial period of life):
n
(x ) Cmn (q k )
where
x0 =
qk
q
1
qk
q
1
n
+ x
1
=
n
(x 0 + x )
1
Binomial
law of
mortality
- the initial virtual age of the system
The initial virtual age of a system defines the law of
system’s mortality:
 x0 = 0 - ideal system, Weibull law of mortality
 x0 >> 0 - highly damaged system, Gompertz law of mortality
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.
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
Life Expectancy and Month of Birth
7.9
In Real U.S. Birth Cohorts
life expectancy at age 80, years
1885 Birth Cohort
1891 Birth Cohort
Source:
L.A. Gavrilov, N.S.
Gavrilova. (2005).
"Mortality of
Centenarians: A Study
Based on the Social
Security Administration
Death Master File"
7.8
7.7
Presented at the 2005
Annual Meeting of the
Population Association
of America.
7.6
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month of Birth
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
For More Information and Updates
Please Visit Our
Scientific and Educational Website
on Human Longevity:
 http://longevity-science.org
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.