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

μ(x) = A + R e αx

A – Makeham term or background mortalityR e αx – age-dependent mortality; x - age

Death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age.

risk of death

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 Gompertz-Makeham law be used?

By studying the historical dynamics of the mortality components in this law:

μ(x) = A + R e αx

Makeham component Gompertz component

Historical Stability of the Gompertz

Mortality Component Before the 1980s

Historical Changes in Mortality for 40-year-old Swedish Males

1. Total mortality, μ40 2. Background

mortality (A)3. 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. Norway, total mortality

2. Denmark, total mortality

3. Norway, age-dependent mortality

4. Denmark, age-dependent 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. Women, total mortality

2. Men, total mortality3. Women, age-

dependent mortality4. Men, age-dependent

mortality

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

A Broader View on the Historical Changes in

Mortality

Swedish Females

Data source: Human Mortality Database

Age

0 20 40 60 80

Lo

g (

Ha

zard

Ra

te)

10-4

10-3

10-2

10-1

1925196019802000

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

Calendar Year

1900 1920 1940 1960 1980 2000

Mo

rtal

ity

fact

or

sco

re

-2

0

2

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.

Data source for the current slide: Human Mortality Database

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 – India, 1941-1950, males 2 – Turkey, 1950-1951, males3 – Kenya, 1969, males 4 - Northern Ireland, 1950-

1952, males5 - 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

Compensation Law of Mortality in Laboratory

Drosophila1 – 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.

Source: Gavrilov & Gavrilova.

Handbook of the Biology of Aging, Academic Press, 2005, pp.1-40.

Age, days

0 10 20 30 40

ha

zard

ra

te,

log

sc

ale

0.001

0.01

0.1

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 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

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 with redundancy accumulates damage (aging)

System without redundancy dies after the first random damage (no aging)

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

( )x =dS( )x

S( )x dx=

nk e kx( )1 e kx n 1

1 ( )1 e kx n

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):

x0 = 0 - ideal system, Weibull law of mortality x0 >> 0 - highly damaged system, Gompertz law of mortality

( )x Cmn( )qk n 1 q

qkx +

n 1

= ( )x0 x + n 1

where - the initial virtual age of the systemx0 =1 q

qk

The initial virtual age of a system defines the law of system’s mortality:

Binomial 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

Month of Birth

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

life

exp

ecta

ncy

at

age

80, y

ears

7.6

7.7

7.8

7.9

1885 Birth Cohort1891 Birth Cohort

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.

AcknowledgmentsThis 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.