Anna's Life Expectancy

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NOT FOR QUOTATION WITHOUT PERMISSION

OF AUTHORS

ANNA'S LIFE EXPECTANCY

J.M. Owen J.W. Vaupel

March 1985 WP-85-

11

Working Papers a r e interim reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do n o t necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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We t h a n k Bradley Gambill, Nathan Keyfitz, Thomas C. Schelling, Sergei Scherbov, Michael Stoto, a n d Anatoli I. Yashin for t h e i r suggestions.

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ANNA'S LIFX

EXPECTANCY

J.M. Owen a n d J.W. Vaupel

Every y e a r t h e Office of t h e Actuary of t h e

U.S.

Department of Health a n d Human Services r e l e a s e s an e s t i m a t e of t h e life expectancy for a newborn child: t h e most r e c e n t figure for a female was 78 y e a r s , a n d for a male, 71 y e a r s [I]. But t h i s number assumes t h a t over t h e c o u r s e of t h e child's life t h e r e will b e n o f u r t h e r progress against mortality, a n improbable supposition indeed. The a c t u a r i e s who c a l c u l a t e official life expectancy e s t i m a t e s assume a baby born in a given y e a r will experience t h e mortality r a t e s for every age of t h a t y e a r t h r o u g h o u t h e r life.

On March 20 of 1984 Anna Bodil Vaupel was born. The Off'ice of t h e Actuary would

0

give h e r a life expectancy, or what demographers denote by eo, of n o t much over 78.

A.B., however, should live much longer t h a n t h a t . Throughout t h i s c e n t u r y g r e a t pro- g r e s s h a s b e e n made a g a i n s t mortality in medicine, public h e a l t h , s a n i t a t i o n , n u t r i - t i o n , safety, a n d o t h e r a r e a s , a n d i t seems implausible t o suppose s u c h progress will come to a h a l t now. There a r e several ways of thinking a b o u t t h e c o n t i n u a t i o n of t h i s

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progress, a n d h e r e we shall look a t t h r e e : c o n t i n u e d progress a t t h e same r a t e s , a c c e l e r a t e d progress, a n d cause-specific, or disease-by-disease, progress. We will base o u r e s t i m a t e s on rough calculations t h a t a r e hopeful in t h a t we ignore c a t a s t r o p h e s s u c h as n u c l e a r war.* Although t h e estimat,es a r e approximate, t h e message is clear:

Anna's life e x p e c t a n c y a n d t h e life expectancies of o t h e r newborn girls a n d boys is probably many y e a r s longer t h a n t h e official estimates.

SEADY PROGRESS

If progress a g a i n s t mortality a t each age c o n t i n u e s a t t h e same r a t e s a s in t h e 1970s, how long c a n Anna B. expect t o live? I t t u r n s o u t t h a t if we do assume s u c h steady progress against mortality, h e r expected life s p a n is 90 y e a r s , fully 12 years more t h a n official e s t i m a t e s give her. If corlditions improve a t t h e same r a t e s indefin- itely and Anna h a s a d a u g h t e r a t age 25, h e r d a u g h t e r c a n expect t o live t o be 94. If t h e d a u g h t e r h a s a d a u g h t e r u n d e r t h e same conditions, Anna's g r a n d d a u g h t e r h a s a n expected lifespan of 97.

ACCEIXRATED F'ROGRESS

There is n o n e e d t o r e s t r i c t t h e calculations t o assumptions of a c o n s t a n t r a t e of progress, however. A look a t progress against m.ortality in t h e previous decades of t h e t.wentieth c e n t u r y reveals t h a t in t h e 1970s t h e r a t e s of s u c h progress were g r e a t e r t h a n i n previous decades a t most ages. This acceleration may c o n t i n u e , a n d i t is n a t u r a l t o wonder what i t s c o n t i n u a t i o n would m.ean for p e r s o n s being born now.

In f a c t , when extrapolations a r e made based on t h e acceleration of mortality pro- g r e s s from t h e 1960s t o t h e 1970s, Anna's life expectancy i n c r e a s e s t o 9 3 years. When

*The methods used in this paper are discussed in the Appendfix.

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we a c c o u n t for acceleration in figuring male life expectancy, t h e r e s u l t i s 97 years,

0

which would finally give males a higher e o t h a n females. Clearly this means t h a t pro- g r e s s a g a i n s t male mortality a c c e l e r a t e d g r e a t l y in t h e 1970s; little progress was made a t most a g e s in t h e 1960s, a n d a t many ages mortality r a t e s i n f a c t increased.

OTHER

RATES OF PROGRESS

A r e c e n t r e p o r t of t h e Office of t h e Actuary does make some conser-vative projections of f u t u r e progress against mortality, although even t h i s progress i s riot t a k e n i n t o a c c o u n t when figuring official estimates of life expectancy [2,3]. Since t h e a c t u a r i a l figures do not go p a s t t h e y e a r 2050, we will assume mortality r a t e s a t all ages remain c o n s t a n t a f t e r 2050. Under t h e s e conditions A.13.'~ life expectancy i s 83, some five y e a r s h i g h e r t h a n t h e official estimate.

Mortality r a t e s improved a t an average of almost 2 p e r c e n t p e r y e a r in t h e 1970s.

If t h e 2 p e r c e n t improvement r a t e i s applied uniformly t o all baby A.B.'s ages, h e r e x p e c t e d lifespan i s 102 years. The g r e a t difference between t h i s number a n d t h e 90 which r e s u l t s from extrapolating t h e c u r r e n t r a t e s of progress, which average 2 per- c e n t , is a t t r i b u t a b l e t o t h e fact t h a t more people die a t older a g e s t h a n a t younger ages, whereas t h e r a t e of progress against mortality t e n d e d t o less t h a n 2 p e r c e n t a t older ages a n d g r e a t e r t h a n 2 p e r c e n t a t younger ages. Even if progress against mortality i s only 1 p e r c e n t p e r y e a r a t all ages, Anna c a n e x p e c t t o become a n octo-

0

g e n a r i a n with an eo of 87 years.

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Table 1. Various estimates of the life expectancy of a newborn.

0 0

Circumstances Baby A.B.'s e o Baby Boy's e o

Official prediction 7 8 7 1

Continued progress against

mortality as in 1970s 90 8 1

Progress continues t o accelerate

a s i t did between 1960 and 1980 93 97

Progress slows down as a c t u a r y projects 8 3 7 5 Progress is 1 per c e n t every year

a t all ages 87 79

Progress is 2 p e r c e n t every y e a r

a t all ages 102 94

DISEASE-gY-DI!XWE PROGRESS

But where specifically will t h i s progress against mortality be made? Again, it seems reasonable to gain insight from extrapolating r e c e n t r a t e s of progress against

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t h e several causes of death to s e e what A.B. will probably die from. Anna's eo t u r n s out t o be 86 y e a r s after t h i s extrapolation.

As s e e n in Table 2, A.B. would be more likely t o die from h e a r t disease t h a n any o t h e r cause if c u r r e n t mortality r a t e s persisted. If t h e r a t e s of improvement against t h e s e causes continue, a s t h e right columns of t h e table show, c a n c e r will overtake all o t h e r causes of d e a t h and become t h e leader.

Some of these numbers may be surprising, but t h e y a r e explained by t h e ages a t which people die from t h e s e diseases today and t h e ages a t which improvement is being made. For example, although many more people die from h e a r t disease than from c a n c e r today, a n d some progress is being made against c a n c e r , most of t h a t progress is made a t younger ages, where t h e chances of A.B. dying a r e relatively low; in t h e case of h e a r t disease, progress is being made a t older ages, when t h e chances of h e r dying a r e high. In fact, while in r e c e n t years total average annual progress

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Table 2. What will a newborn eventually die from?

Probability of dying from various c a u s e s Assuming n o progress Continued progress

Cause Female Male Female Male

H e a r t disease 42% 40X 11% 9 %

Cancer

Vascular disease Violence

Respiratory c a u s e s Congenital c a u s e s Digestive d i s e a s e s Diabetes mellitus Liver c i r r h o s i s Other c a u s e s Total

a g a i n s t h e a r t d i s e a s e for females was close t o 3 p e r c e n t , i t was slightly negative for c a n c e r - - t h a t is, c a n c e r m o r t a l i t y a c t u a l l y i n c r e a s e d . As i s a p p a r e n t from t h e even m o r e e x t r e m e n u m b e r s f o r male m o r t a l i t y , c a n c e r t r e n d s h a v e b e e n still more dismal f o r males, averaging n e a r l y a 1 p e r c e n t a n n u a l i n c r e a s e .

The probability of Anna dying from violent c a u s e s , which i n c l u d e a c c i d e n t s , homi- cides, a n d suicides, i s only a r o u n d 1 p e r c e n t a n d may be u n r e a l i s t i c a l l y low, a l t h o u g h t h e p o i n t is debatable. T h e f a c t i s t h a t g r e a t p r o g r e s s h a s b e e n m a d e a g a i n s t d e a t h by violence a t most ages, especially o l d e r ages. The a c c i d e n t d e a t h r a t e among t h e eld- e r l y is a good m e a s u r e of how h e a l t h y t h e y a r e , a n d s o t h i s p r o g r e s s may b e a sign of improvenlents in h e a l t h among t h e elderly. Extrapolating t h i s p r o g r e s s over A.B.'s life yields t h e low v a l u e of 1 p e r c e n t . B u t if all disease were eliminated, everyone would h a v e t o e v e n t u a l l y die from violence, which would r a i s e A.B.'s probability of dying from violence t o 100 p e r c e n t .

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Models a r e sometimes more u s e f u l f o r counter-prediction t h a n for prediction. If w e view a 1 p e r c e n t probability of dying from violence a s highly unlikely, t h e n we must believe t h a t t h e r a t e of improvement cannot continue a s it did in t h e 1970s.

IF

CANCER

IS ELlMINATED

Since c a n c e r becomes such a major cause of d e a t h , i t is n a t u r a l to wonder how t h e eradication of c a n c e r would affect life expectancy. The conventional answer has been t h a t curing c a n c e r would only increase expected lifespan by little more t h a n two years [4]. This slight prediction of a cancer cure's impact is due t o t h e fact t h a t a t current r a t e s only about a fifth of males and females can expect t o die from cancer.

Like t h e conventional life expectancy estimates, however, this projection changes if we assume continued r a t e s of progress against mortality.

If we take t h e cause-specific death r a t e s projected above, most of which improve each year, and s u b t r a c t t h e c a n c e r mortality r a t e s from each year of Anna's life, A.B.'s

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eo i n c r e a s e s t o 96 years. This substantial increase of a fu1.l decade stems from t h e high c h a n c e s , 44 percent, of Anna dying from c a n c e r if c u r r e n t progress against t h e various causes of d e a t h continues. Among males, where c a n c e r becomes even more dominant, accounting for two-thirds of all deaths, i t s eradi-cation raises life expec- t a n c y from 74 t o 88.

Eradicating today's number one killer, h e a r t disease, only increases k B . ' s life expectancy from 86 t o 87. A continuation of present; t r e n d s in progress against mortality would mean t h a t h e a r t disease can be expected t o become much less important, a n d t h u s its total elimination would do little t o increase expected life span. Getting

0

rid of all cardiovascular disease, in fact, only raises Anna's e o by two years.

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THE PROSPECIS FOR INCREASING

LIFE

EXPECTANCY

The projections made h e r e for baby AB.'s lifespan may seem overly optimistic, b u t t h e y may eventually t u r n out to be conservative if t h e predictions of c e r t a i n biomedi- cal scientists on f u t u r e progress against mortality r a t e s , a n d even t h e imminence of a breakthrough against aging itself, t u r n out t o be c o r r e c t . According t o biologist Robert A. Weinberg of M.I.T., for example, t h e r e c e n t a d v a n c e s i n g e n e t i c engineering will make g r e a t s t r i d e s possible against s u c h ailments a s atherosclerosis, c a n c e r , a n d diabetes both in o u r h o w l e d g e of what c a u s e s them a n d in o u r ability t o t r e a t t h e m [4]. And U.C.L.A. gerontologist Roy Walford writes t h a t his field is showing t h e early signs of a scientific revolution, with several equally credible competing hypotheses about t h e aging process being offered [5].

One s e t of t h e o r i e s about aging assumes t h a t t h e process is c a u s e d by cellular damage. A n u m b e r of medical s c i e n t i s t s have done r e s e a r c h i n t o t h e oxygen f r e e r a & - cals which form during normal metabolic processes a n d which play a newly discovered role in cellular injury a n d p e r h a p s in aging itself (e.g., [6, 7,8] ) . Other s c i e n t i s t s a r e experimenting with increasing t h e r a t e of cellular r e p a i r , which t h e y say will i n c r e a s e life span substantially.

An a l t e r n a t e s e t of t h e o r i e s i s based on t h e paradigm t h a t aging is programmed i n t o cells. One holds t h a t t h e expression and repression of c e r t a i n g e n e s a t c e r t a i n ages is responsible for aging, a n d manipulation of this expression a n d repression could e x t e n d lifespan. Another says t h a t t h e body r e l e a s e s a d e s t r u c t i v e hormone a t a p r e d e t e r m i n e d time; t h i s hormone may soon be identified. A t h i r d t h e o r y holds t h a t i t is t h e breakdown of t h e immune syst.em which holds t h e key t o extending life expec- t a n c y [5].

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If a revolution in t h e biological s c i e n c e s does t a k e place, by o n e of t h e s e t h e o r i e s or by a s y n t h e s i s of t h e m , t h e demographic impact will be d r a s t i c , with e n t i r e l y new c h a l l e n g e s p r e s e n t i n g t h e m s e l v e s t o d e m o g r a p h e r s a n d t o society i n g e n e r a l . Imagine t h e social, economic, a n d political c o n s e q u e n c e s if Anna's life e x p e c t a n c y was 150 o r even 200 y e a r s .

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Use of t h e approximate formula for life expectancy,

where q ( a ) is t h e mortality r a t e a t age a , p ( a ) is t h e probability of being alive a t age a , a n d w is t h e age beyond which n o one lives, shows what needs t o be c a l c u l a t e d t o

0

find baby A.B.'s e o . In all of o u r calculations r e p o r t e d h e r e , we took w t o be 200 y e a r s . In n o c a s e would t h e use of a h i g h e r w (of, s a y , 250 y e a r s ) make any difference in t h e r e s u l t s ; i n nearly all cases, use of a lower w (of even 120) would n o t c h a n g e t h e r e s u l t s by more t h a n a fraction of a year.

Linear Progress Against Mortality:

Using mortality r a t e s for age 0 t h r o u g h age 119 i n 1970 a n d comparing t h e m with t h e i r 1980 c o u n t e r p a r t s [ 2 , 3 ] allows extrapolat.ion of t h e 1970s r a t e s of progress over

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Anna's life. First each q ( a ) for Anna m u s t be modified according t o t h e r a t e s of pro- g r e s s observed in t h e 1970s. If R ( a ) equals t h e progress multiplier, or t h e number by which we multiply an old mortality r a t e t o g e t a new mortality r a t e , a n d

sy

( a ) equals t h e mortality r a t e a t age a in y e a r y , t h e n

The resulting number r e p r e s e n t s progress over an e n t i r e decade; i t will be more useful t o find progr-ess over a single year. Since t h e r a t e of progress over a decade is given simply by t h e a n n u a l r a t e s within t h a t decade multiplier t o g e t h e r , t h e average a n n u a l progress multiplier is found a s follows:

To figure e a c h s ( a ) for Anna B., simply u s e e a c h R ( a ) r a i s e d t o t h e appropriate power:

The exponent m u s t be ( a + 4 ) because Anna was born i n 1984 r a t h e r t h a n 1980.

Next t h e t a s k is t o figure t h e probability Anria will be alive a t a g e a . The probability of h e r living t o age a equals t h e probability t h a t s h e lived t.o t h e previous age times t h e mortality r a t e for t h e previous age:

where p (0)

=

1.

Now equation ( 1 ) yields Anna's life expectancy.

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Accelerated Progress:

Just a s before, dividing t h e r a t e s of progress against mortality for a later decade by those of a n earlier decade a n d finding t h e t e n t h root of t h e r e s u l t gives t h e average a n n u a l r a t e of progress for each age. This average is probably n o t exactly t r u e for any o n e year, b u t we can approximately r e p r e s e n t i t a s applying t o t h e midpoint of each decade:

For most ages, R 7 5 ( a ) is less t h a n R g i j ( a ) ; th a t is, q B O ( a ) is less t h a n by a g r e a t e r p e r c e n t a g e t h a n is less t h a n ~ ~ ~ ( a ) . To find out by how much progress a c c e l e r a t e d a t each age, divide t h e second decade's progress multiplier by t h e first decade's. As 1975 minus 1965 equals 10 years, raise t h e acceleration r a t e to t h e one- t e n t h power:

A t several ages, progress against mortality actually decelerated in t h e 1970s. We ignored t h i s decelerati.on by changing t h e acceleration r a t e from t h e negative number signi.fy-ing deceleration t o 0, signifying no acceleration.

The n e x t task is t o find t h e mortality r a t e s , q , for e a c h of Anna's ages. Baby A.B.'s q ' s must be based on t h e 1980 q ' s a s modified by R75(a) a n d ~ ( a ) . The mortality r a t e a t age a for Anna is equal to t h e mortality rat,e for t h a t age in 1980 times t h e r a t e of progress for t h a t age between 1980 a n d 1981, .times t h e r a t e of progress between 1981 a n d 1982, a n d s o f o r t h , u n t i l t h e y e a r in question is reached:

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The answer t o what R B l ( a ) and RBZ(a), etc., a r e , lies in ~ ( a ) , t h e a c c e l e r a t i o n multiplier. The r a t e of progress between 1980 a n d 1981 equals t h e r a t e of progress in 1975, our most r e c e n t a n d a c c u r a t e figure, times t h e acceleration multiplier raised t o t h e fifth power ( t o a c c o u n t for t h e passage of five years). Likewise, RB1(a) is

~ 7 5 b ) X A ( a )6.

Thus:

It follows t h a t

q ( a ) = qao(a )x[R75(a ) ( a + 4 ) ] x [ ~ ( a ) Z l 8

where z is t h e sum of t h e values of all exponerlts of A ( a ) . To find x , we use t h e r u l e Gauss discovered a s a schoolboy for summing consecutive numbers. Note t h a t

which equals [(4x6)

+

(6x7)]/ 2. More generally, t h e value of x is given by

0

The same equatiorls for p ( a ) a n d eo a r e used as before.

Using the Actuary k Estimates:

The d a t a on f u t u r e mortality in t h e actuary's r e p o r t is by decade r a t h e r t h a n by year. Since mortality r a t e s improve every y e a r r a t h e r t h a n every t e n years, we must t a k e t h e a c t u a r i a l figures For each decade a n d i n t e r p o l a t e yearly improvement. The projected r a t e of progress against mortality must be found, using t h e equations above.

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As a n example, a s s u m e we a r e given q ( 3 0 ) for t h e y e a r 2010 a n d q ( 3 0 ) for 2020, b u t n o t for a n y y e a r s in between. The yearly p r o g r e s s multiplier is t h e n

R

=

[ ( q 2 0 ( 3 0 ) / 9 10(30))0'11

.

Suppose we want q ( 3 0 ) in t h e y e a r 2014, t h e y e a r Anna is 30. Now, q ( 3 0 ) in 2014 e q u a l s q ( 3 0 ) in 2010 t i m e s t h e yearly p r o g r e s s multiplier r a i s e d t o t h e f o u r t h power, t o a c c o u n t f o r t h e passage of four y e a r s . Thus Anna's mortality r a t e is:

C a u s e - @ e c i f i c M o r t a l i t y R a t e s :

We u s e d cause-specific period d a t a r e l e a s e d by t h e Office of t h e Actuary on mor- t a l i t y r a t e s in 1977 a n d a v e r a g e a n n u a l r a t e s of improvement between 1968 a n d 1978.

Although t h e s e d a t a a r e p e r i o d data, o r slices of time, we c a n u s e t h e m t o figure o u t Anna's cause-specific mortality r a t e s . All we n e e d do is improve all t h e mortality r a t e s for e a c h disease t h r o u g h all of k B . ' s life, using t h e same a n n u a l e p e r c e n t a g e improve- m e n t s a s were e x p e r i e n c e d in 1968-1978, t h e n apply t h e s e r a t e s t o Anna's lifetime.

The d a t a we u s e only i n c l u d e s mortality r a t e s u p t o t h e c a t e g o r y 85+; for h i g h e r ages u p t o 119, we u s e d total d e a t h r a t e s subdivided, b a s e d o n t h e proportions in t h e c a t e g o r y 85+. i n t o t h e t e n c a u s e s ; a n d a f t e r age 119, we e x t r a p o l a t e d t h e c a u s e - specific m o r t a l i t y r a t e s based on t h e average a n n u a l r a t e of i n c r e a s e i n m o r t a l i t y for a g e s 100 to 119.

We e n c o u n t e r e d a problem with t h e c a n c e r r a t e s , which becomes g r e a t e r t h a n 1 before a g e 200 b e c a u s e of t h e c o n t i n u e d n e g a t i v e p r o g r e s s a g a i n s t t h e m . Therefore we worked with p, t h e force of mortality, aft,er a g e 119, b e c a u s e i t is a c c e p t a b l e for p t o become g r e a t e r t h a n 1. The formula we u s e d t o c o n v e r t q ' s t o p ' s is:

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Before we figure life e x p e c t a n c y , we c o n v e r t t h e p ' s b a c k to q ' s , using t h e formula:

q ( a ) = I - e - ~ ( = ) ,

EZiminating Cancer:

Clearly, t h e f i r s t s t e p is t o s u b t r a c t t h e c a n c e r d e a t h r a t e s from t h e total d e a t h r a t e s . But h e r e a g a i n i s a complication: by eliminating c a n c e r , we m u s t a c c o u n t for t h e i n c r e a s e d exposure t o t h e o t h e r diseases AB. experiences e a c h y e a r since s h e did n o t die from c a n c e r . In o t h e r words, t h e d e a t h r a t e a f t e r c a n c e r eradication equals t h e total d e a t h r a t e m i n u s t h e c a n c e r death r a t e plus t h e additional d e a t h r a t e c a u s e d by slightly i n c r e a s e d exposure t o o t h e r c a u s e s of death.

Since people die on t h e average on July 1, we m u s t allow for a n additional half year of exposure t o n o n - c a n c e r o u s c a u s e s of d e a t h . Therefore, l e t t i n g q, d e n o t e c a n c e r mortality r a t e s a n d q ' denote t h e t o t a l mortality r a t e s a f t e r c a n c e r is e r a d i - c a t e d ,

q ' ( a >

=

s ( a >

-

q , ( a > + s , ( a > x [ t g ( a ) - q c ( a ) ) / 2 1 (12)

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1. S a t i s t i c a l E h l l e t i n o f M e t r o p o l i t a n L i f e , 1983.

2. J.F. Faber, U . S . P o p u l a t i o n F o r e c a s t s 1980, Actuarial Study No. 82. U.S.

Department of Health and Human Services SSA Pub. No. 11-11529, 1980.

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87. U.S. Department of Health a n d Human Services Pub. No. 11-11534, 1982.

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