Targetting Lifesaving: Demographic Linkages Between Population Structure and Life Expectancy

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHORS

Targetting Lifesaving:

Demographic ljnkages Between

Population Structure and Life Expectanc

y

James W. k u p e l A n a t o l i I. Y a s h i n

November

1 9 8 5 WP-85-78

Working Ftzpers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o not necessarily r e p r e s e n t t h o s e of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE

FOR

APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

Acknowledgments

W e thank Jan Hoem, Nathan Keyfitz, Thomas Schelling, and Michael Stoto f o r helpful suggestions, and Susanne Stock f o r prompt, meticulous typing.

Life expectancy in a heterogeneous population c a n b e i n c r e a s e d by lowering mortality rates o r by a v e r t i n g d e a t h s a t d i f f e r e n t a g e s , f r o m d i f f e r e n t causes, for d i f f e r e n t groups, as well as by changing t h e p r o p o r t i o n s of individuals in v a r i o u s r i s k groups, p e r h a p s by a l t e r i n g t h e transition rates between groups. Under- standing how s u c h c h a n g e s in population s t r u c t u r e a f f e c t s life e x p e c t a n c y i s useful in evaluating a l t e r n a t i v e lifesaving policies.

Targetting Lifesaving:

Demographic Ihkages Between Population Structure and Life Expectancy

James W. Vaupel a n d A n a t o l i I. Yashin

The individuals comprising t h e typical population of men, mice, or machines f a c e differing mortality chances. This heterogeneity a r i s e s , in p a r t , from indivi- dual c h a r a c t e r i s t i c s t h a t c h a n g e or c a n b e changed, like a g e , b e h a v i o r , occupa- tion, or residence. Alteration of t h e a g e composition, occupational s t r u c t u r e , or o t h e r p a t t e r n of h e t e r o g e n e i t y in a population, p e r h a p s as t h e r e s u l t of some poli- c y intervention, will c h a n g e t h e distribution of mortality c h a n c e s a n d h e n c e change t h e life expectancy of t h e population. In t h i s p a p e r w e develop some formu- l a s f o r analyzing how v a r i o u s kinds of changes in population s t r u c t u r e will a f f e c t l i f e expectancy.

Change in life e x p e c t a n c y i s a measure of t h e number of y e a r s of l i f e saved ( o r l o s t ) by a n a l t e r a t i o n in population s t r u c t u r e and hence i s a useful m e a s u r e for policy analysis. In p a r t i c u l a r , t h i s measure is a p p r o p r i a t e f o r what might b e called t a r g e t analysis. If limited r e s o u r c e s are available f o r lifesaving interven- tions, how should t h e r e s o u r c e s b e t a r g e t e d ? How effective would p r o g r a m s b e t h a t are d i r e c t e d toward d i f f e r e n t a g e groups, diseases, r i s k g r o u p s (like c i g a r e t t e smokers), regions, e t c ? A complete t a r g e t analysis would h a v e

to

include consideration of how difficult i t is

to

focus a n intervention on a p a r t i c u l a r g r o u p a n d how r e s i s t a n t t h e g r o u p i s

to

change. Nonetheless, understanding t h e benefits of a change, if achieved, in l i f e e x p e c t a n c y gained or life-years saved i s c l e a r l y a key component of any t a r g e t analysis.

In addition

to

such policy applications, t h e methods and formulas p r e s e n t e d in t h i s p a p e r are useful in gaining a d e e p e r demographic understanding of how

m o r -

tality r a t e s , d e a t h s , r i s k g r o u p s , a n d life e x p e c t a n c y are i n t e r r e l a t e d . How, for instance, d o mortality

rates

c h a n g e if some d e a t h s are a v e r t e d ?

Four d i f f e r e n t analytical a p p r o a c h e s are used in t h e p a p e r

to

analyze t h e demographic linkages between population s t r u c t u r e and life expectancy: t h e comparative-statics a p p r o a c h , t h e dynamics a p p r o a c h , computer simulation, and a novel method t h a t w e call t h e "second-chance" a p p r o a c h . The p a p e r provides some discussion and illustration of t h e s t r e n g t h s , weaknesses, and interrelationships among t h e s e a l t e r n a t i v e methods of demographic analysis.

LIFE AND DEATH RATES

Consider, f i r s t , a g e

structure

^as c h a r a c t e r i z e d by t h e s u r v i v o r s h i p function

where p ( z ) r e p r e s e n t s t h e f o r c e of mortality at a g e

I.

^(Formula (1) and the r e s u l t s t h a t follow c a n b e i n t e r p r e t e d as pertaining

to

e i t h e r p e r i o d

or

c o h o r t cal- culations.) A change in p will change this age s t r u c t u r e and h e n c e life expectancy

at

b i r t h :

where o is a n a g e beyond which no o n e lives.

The effect of a c h a n g e in p on e o c a n b e analyzed by e i t h e r of two a p p r o a c h e s . In t h e comparative-statics a p p r o a c h , t h e t r a j e c t o r y of p i s assumed

to

change

to

p', where

t h e analyst relates t h e c h a n g e b ( z )

to

t h e change in eo, p e r h a p s a s measured by:

In t h e dynamics a p p r o a c h , t h e r e i s some r a t e of change in p ( z , t ) o v e r time t :

t h e analyst relates t h i s

rate

of change p ( z , t )

to

t h e

rate

of c h a n g e in eo(t ):

Both a p p r o a c h e s are informative and w e will c o n s i d e r b o t h . F o r notational simpli- c i t y , w e will d r o p t h e a r g u m e n t

t

t h r o u g h o u t a n d w r i t e A x ) r a t h e r t h a n CL(z,t) a n d e o r a t h e r t h a n e o ( t ) .

If p ( z ) i s c o n s t a n t o v e r a n i n t e r v a l of time of l e n g t h T, t h e n

Combining t h i s r e s u l t with (3) yields t h e relationship between p a n d 6:

If 6 i s small, t h i s r e d u c e s t o

Hence, r e s u l t s c o n c e r n i n g p ( x ) c a n b e d e r i v e d f r o m r e s u l t s c o n c e r n i n g 6 ( z ) and visa-versa: t h e c o m p a r a t i v e - s t a t i c s a p p r o a c h a n d t h e dynamics a p p r o a c h comple- ment e a c h o t h e r . Note t h a t p ( z ) c a n b e a r b i t r a r i l y l a r g e , as long as T i s small enough.

A c o m p a r a t i v e - s t a t i c s r e l a t i o n s h i p c a n r e a d i l y be d e r i v e d f r o m (1)-(4):

s

In t h e c a s e of a uniform c h a n g e in mortality

at

a l l a g e s , b ( z ) = 6 , a l l z , formula (9) c a n b e r e w r i t t e n as

0

F o r small 6 ,

Hence

where

In t h e limit, a s b a p p r o a c h e s z e r o , formula (12) holds exactly. Consequently, i t i s a p p a r e n t t h a t

where p i s t h e uniform

rate

of p r o g r e s s in reducing mortality

rates:

d t

P

= ,

all

z .

d z

Thus, for small changes in & t h e comparative-statics a p p r o a c h yields t h e same formulas

as

t h e dynamics a p p r o a c h . Keyfitz (1977) d e r i v e d (14) and noted t h a t H i s

a

measure of a g e h e t e r o g e n e i t y ; a s Demetrius (1979) indicated, H c a n b e i n t e r - p r e t e d

as

t h e e n t r o p y of t h e a g e composition of t h e population.

THE SECOND-CHANCE APPROACH

Interventions

to

r e d u c e mortality ( o r equipment f a i l u r e ) work by saving lives, i.e. by a v e r t i n g t h e s c y t h e of d e a t h . Suppose t h a t for some p r o p o r t i o n

6

of a c o h o r t ( p e r h a p s a s y n t h e t i c p e r i o d "cohort"), d e a t h is a v e r t e d once. Let 1 ( z ) r e p r e s e n t t h e p r o p o r t i o n of t h e c o h o r t

at

a g e

z

t h a t is alive and h a s n o t b e e n s a v e d and l e t t + ( z ) r e p r e s e n t t h e p r o p o r t i o n of t h e r e s u s c i t a t e d who

are

a l i v e

at

a g e

z .

Since t h e p r o p o r t i o n of t h e c o h o r t surviving

at

a g e

z

i s given by

t h e new life e x p e c t a n c y , 8 ;

,

i s given by

The r e l a t i v e change in life e x p e c t a n c y is simply

An e x p r e s s i o n f o r l + ( z ) i s readily developed. Assuming t h a t t h e resuscitated f a c e t h e same f o r c e of mortality

as

t h o s e who have not been saved, t h e probability of survival

to

a g e

z

f o r t h o s e whose lives

were

saved

at

a g e w i s given by

where

T

r e p r e s e n t s t h e time of death. Because t h e distribution density of w i s CL(w)l(w),

Substituting (20) in (18) yields

Note t h a t t h e H in (21) d e n o t e s t h e same expression as Keyfitz's H in (12) and (14). Hence, (21) p r o v i d e s

a

t h i r d i n t e r p r e t a t i o n of H

as a

measure of t h e p r o p o r - tional i n c r e a s e in life e x p e c t a n c y if everyone's life

were

saved once, o r a l t e r n a - tively,

as

t h e p r o p o r t i o n a l i n c r e a s e in

a

randomly chosen individual's life span if t h a t individual's life i s saved. For Swedish males in 1982, H

was

.15 and s o

was

72 y e a r s . Consequently,

at

1982 p e r i o d mortality r a t e s , a v e r t i n g t h e d e a t h of

a

Swed- ish male would give t h e r e s u s c i t a t e d a b o u t 11 y e a r s of life expectancy.

The formula f o r beo/ e o in ( 2 1 ) holds exactly f o r any 6, whereas t h e analogous formula in ( 1 2 ) only holds approximately, f o r s m a l l 6. The r e a s o n c a n b e under- stood by considering some simple diagrams. The model where death i s only a v e r t e d o n c e can b e r e p r e s e n t e d as:

Individuals

are

a l l initially in t h e l e f t box. A proportion 6 of those who would h a v e died

are

saved. but just once: t h e r e s u s c i t a t e d e x p e r i e n c e t h e o r i g i n a l force of mortality A z ) . On t h e o t h e r hand, t h e model where mortality

rates

are d e c r e a s e d by 6 can b e r e p r e s e n t e d as:

T H E RESUSCITATED ORIGINAL

COHORT

Because t h e force of mortality in a n y

state

i s (1

-

6 ) p ( z ) , t h e o v e r a l l f o r c e of mortality must also b e (1

-

6 ) p ( z ). What t h e decomposition into a n infinite

stream

of

states

r e v e a l s i s t h a t a reduction in mortality

rates

may r e s u l t in

some

people's

'I 'I

6 p ( z )

*

bp(z)

t m m m

-

t v ^t

(1

-

^{6)P(r) (1}

-

~ ) P ( z ) (1

-

~ ) P ( Z )

SAVED TWICE 8 P ( l

C

-

THOSE SAVED ONCE ORIGINAL

POPULATION

8 ~ ( r rn

lives being saved s e v e r a l times.

L e t r e p r e s e n t t h e e x p e c t e d life y e a r s lived by a n individual in t h e i ' t h s t a t e , i.e., by a n individual whose life has been saved i and only t times:

where ( z ) d e n o t e s t h e probability t h a t a newborn individual i s alive and in

state

i at a g e

z .

Note t h a t T: i s equal

to

eo, t h e original life e x p e c t a n c y b e f o r e t h e 11- fesaving intervention. Clearly,

When

6

i s small, i t i s unlikely t h a t anyone will gain much life expectancy by being saved more t h a n once, i.e., t h e

terms

T:, 703, and so on

are

unimportant. (We p r o v e and expand o n t h i s intuitively plausible r e s u l t elsewhere, in Vaupel and Y a s h h (1985).) Hence,

In t h e

two-state

model, where d e a t h i s only a v e r t e d once,

The similarity between (29) a n d (25) sheds light on why Keyfitz's

H

in (12) i s identi- cal

to

t h e

H

in (21).

I t i s sometimes e a s i e r

to

analyze t h e

two-state

model t h a n t h e many-state model. Since t h e

two

models h a v e equivalent implications f o r life expectancy in t h e limit f o r small 6, t h e

two-state

model may provide a convenient line of a t t a c k . W e exploit this, and t h e relationship between

6

and p discussed e a r l i e r , in s e v e r a l subsequent d e r i v a t i o n s in t h i s p a p e r . W e call t h e method involving t h e

two-state

model t h e "second-chance" a p p r o a c h , in c o n t r a s t with t h e comparative-statics ap- proach and t h e dynamics a p p r o a c h . Although in t h i s p a p e r t h e second-chance ap- p r o a c h i s only used

to

analyze changes in life expectancy. i t h a s more g e n e r a l ap- plications

to

any situation, including marriage, divorce, abortion, unemployment, t h e r e p a i r of equipment, e t c . , where changing

some rate

c a n b e considered as equivalent

to

giving

some

individuals

a

second chance.

Suppose, as above, t h a t some proportion

6

of d e a t h s

are

a v e r t e d once. How will t h e t r a j e c t o r y of mortality r a t e s ,

as

given by p ( z ) , c h a n g e ? In brief, how d o e s saving lives a f f e c t mortality r a t e s ? Substituting (20) in (16), taking log deriva-

tives, and t h e n simplifying yields:

A t a g e z e r o , when I ( z ) i s one, t h e formula simplifies

to

A s s u r v i v o r s h i p d e c r e a s e s , however, p ' ( z ) a p p r o a c h e s p ( z ) . Thus, reducing d e a t h s by some proportion b

at

all a g e s r e d u c e s t h e f o r c e of mortality by less t h a n

b

at

all a g e s a f t e r b i r t h . The distribution of death times, as given by

Az)L

( z ) , c h a n g e s

to

so

t h a t a reduction in d e a t h s by b l e a d s

to a

new distribution of d e a t h times s h i f t e d

to

o l d e r ages. S i n c e d e a t h , a s S h a k e s p e a r e put it. "is c e r t a i n

to

all", i t i s c l e a r t h a t

a

d e a t h a v e r t e d today i s

an

additional d e a t h

tomorrow.

The mathematics of t h i s adjustment i s c a p t u r e d b y (26) a n d (28).

IF

THE RESUSCITATED A R E DWPERENT

The formulas a n d calculations a b o v e assume t h a t

a

r e s u s c i t a t e d p e r s o n would face t h e

same force

of mortality o v e r t h e

rest

of h i s

or

h e r life as a p e r s o n whose l i f e had not b e e n saved. To g e n e r a l i z e t h e formula, i t i s useful

to

c o n s i d e r t h e fol- lowing variation o n t h e model discussed above:

Note t h a t now individuals who

are

saved e x p e r i e n c e

a

mortality t r a j e c t o r y given by p + ( z ) , ra t h e r t h a n by

d z ) .

Let r ' ( 2 ) b e t h e remaining life e x p e c t a n c y

at

a g e

THE RESUSCITATED ORIGINAL

COHORT

.

b p ( z

*

z

of t h e r e s u s c i t a t e d :

w h e r e

Because t h e density

at

a g e

z

of t h e distribution of ( f i r s t ) d e a t h i s given by I.l(z)l(z), t h e value of beo must b e given by

Hence

w h e r e

If p + ( z ) e q u a l s A z ) , so t h a t individuals are, in e f f e c t , saved from d e a t h once, t h e n

H +

equals H. If p + ( z ) equals ( 1

-

6 ) p ( z ) ,

so

t h a t d e a t h

rates are

r e d u c e d uniformly for e v e r y o n e , r e g a r d l e s s of whether t h e y h a v e b e e n r e s u s c i t a t e d

or

not,

H+

will b e close in value to H

as

long as b i s small. Consequently,

w h e r e

This expression

for

H, which i s equal in value

to

Keyfitzss e x p r e s s i o n f o r H,

was

d e r i v e d by Vaupel (1986) d i r e c t l y from Keyfitz's formula. The e x p r e s s i o n c l e a r l y

indicates how t h e e f f e c t of saving lives on life expectancy depends on t h e number of d e a t h s

at

various a g e s and on t h e number of additional y e a r s of life

a

resusci- t a t e d p e r s o n might have.

SAVING

THE

OLD BEBORg

THE

_YOUNG

A s Vaupel (1986) discusses

at

length, if d e a t h

rates

a r e reduced by some pro- portion

6

between a g e s

a

and @, then for small

6,

where

Correspondingly, if p r o g r e s s is being made

at a rate

p against mortality between a g e s

a

and @, t h e n

The values of H a p for various five y e a r a g e categories f o r Swedish males and f e males in 1982 are given in Table 1. Remarkably, i t is f o r males 70 t o 75 and f o r f e males 75

to

80 t h a t H a p is largest. A o n e p e r c e n t reduction in mortality in t h o s e a g e c a t e g o r i e s would i n c r e a s e life expectancy

at

b i r t h by

more

t h a n twice

as

much

as a

o n e p e r c e n t reduction in mortality in infancy and e a r l y childhood.

AVERTING NEOPLASTIC DEATH IN VENICE

Let p C ( z ) r e p r e s e n t t h e f o r c e of mortality from c a n c e r , o r more generally any specified c a u s e of death. Suppose t h a t f o r some proportion

d

of individuals who would h a v e died from c a n c e r , t h i s ( f i r s t ) d e a t h from c a n c e r is a v e r t e d . Furth-

er

suppose t h a t t h e s e r e s u s c i t a t e d individuals t h e n h a v e t h e

same

remaining life expectancy

as

o r d i n a r y individuals. Using t h e second-chance a p p r o a c h and t h e

same

kind of reasoning employed

to

d e r i v e formulas (31)-(33), i t i s c l e a r t h a t

Table 1. Values of

Ha,,

f o r Swedish males and females in 1982.

Age Period Males Females

H

(i.e.,

total

f o r .I5270 all a g e s )

SOURCE:

Vaupel (1986).

If

6

i s small, i t is unlikely t h a t a n individual would b e saved from c a n c e r d e a t h more than once. Hence, (39) holds approximately f o r

a

reduction

6

in c a n c e r mor- tality r a t e s

as

long as

6

i s small. I t follows t h a t

where p i s t h e

rate

of p r o g r e s s in reducing c a n c e r mortality

If c a n c e r is independent of o t h e r causes of death, then i t is possible

to

derive a n alternative expression f o r Hc t h a t is similar t o Keyfitz's formula f o r H in (13).

Let l:(z) r e p r e s e n t t h e proportion of people in t h e population who

are

alive

at

a g e

z

and who have been saved once from c a n c e r death (at any a g e p r i o r

to

2 ) . By analogy

to

(20), letting w denote t h e a g e

at

which c a n c e r death w a s a v e r t e d , i t fol- lows t h a t

where l c ( z ) can b e i n t e r p r e t e d as t h e survival function when c a n c e r is t h e only cause of death

Hence, by t h e same logic used

to

d e r i v e (21).

Keyfitz (1977) d e r i v e s formula (44) using a different a p p r m c h . In addition, h e p r e s e n t s some illustrative examples. For instance, f o r Italian females in 1964, Hc f o r d e a t h s from neoplasms

was

0.0300, compared with

a total

H of 0.1631. Thus,

a

o n e p e r c e n t reduction in c a n c e r mortality would increase life expectancy

at

b i r t h by about t h r e e p e r c e n t of one p e r c e n t , o r by about 8 days given Italian fe- m a l e life expectancy of 72.9 y e a r s in 1964. By way of comparison, Hc f o r d e a t h s from cardiovascular diseases

w a s

0.0564, almost twice

as

high

as

t h e Hc f o r d e a t h s from c a n c e r , whereas Hc f o r deaths from influenza, pneumonia and bronchitis

w a s

0.0122, o r less than half

as

g r e a t

as

t h e Hc f o r d e a t h s from c a n c e r .

MALES

GO

FLRST

Consider now a population t h a t i s s t r u c t u r e d according

to

r a c e , s e x , socio- economic s t a t u s , region or some o t h e r classification. Adopting t h e line of a t t a c k of t h e second-chance a p p r o a c h , suppose t h a t a proportion d i of t h e f i r s t d e a t h s in g r o u p i

are

a v e r t e d . What will t h e e f f e c t b e on t h e life e x p e c t a n c y of t h e e n t i r e population? Letting & (z), Li (z), and ei (z) denote t h e f o r c e of mortality,

sur-

vivorship function, and remaining life expectancy a t a g e

z

of t h e i -th group, t h e n

w h e r e mi (0) i s t h e initial p r o p o r t i o n of t h e population

in

t h e g r o u p i . Hence.

a n d

w h e r e

and

The

U.S.

male population, for example, might b e classified as white and nonwhite. The value of Hi f o r

U.S.

nonwhite males in 1950 w a s a b o u t 0.038. S o reducing nonwhite male mortality by

one

p e r c e n t would a d d a b o u t 9 d a y s

to

t h e o v e r a l l

U.S.

male life e x p e c t a n c y of 65.5 y e a r s . By comparison, t h i s r e d u c t i o n in nonwhite

male

mortality would a d d a b o u t 75 d a y s

to

nonwhite

m a l e

life e x p e c t a n c y . The d i f f e r e n c e i s l a r g e l y explained b y t h e proportion of nonwhites

at

b i r t h , a b o u t 12.6 p e r c e n t .

The U.S. population as a whole can b e divided into male and female groups.

The value of H f o r males

at

¹⁹⁸⁰ mortality r a t e s w a s 0.193, t h e value f o r females w a s 0.155. If t h e two g r o u p s are given equal weight, t h e n H f o r t h e e n t i r e popula- tion is 0.179 and Hi i s 0.096 f o r m a l e s and 0.077 f o r females. Suppose t h e r e

are

t h r e e a l t e r n a t i v e interventions. The f i r s t reduces m a l e mortality by 2 p e r c e n t , t h e second r e d u c e s female mortality by 2 p e r c e n t , and t h e t h i r d r e d u c e s t o t a l mor- tality by 1 p e r c e n t . The male s t r a t e g y would save a b o u t 11 p e r c e n t more l i f e y e a r s t h a n t h e

total

s t r a t e g y which, in t u r n , would save a b o u t 15 p e r c e n t more life y e a r s t h a n t h e female s t r a t e g y .

Suppose t h a t

a

population consists of

t w o

subpopulations with a g e s p e c i f i c mortality

rates

k ( z ) a n d @(z), where @(z)

>

k ( z ) and where t h e

t w o

g r o u p s might b e r e s i d e n t s of u r b a n vs. r u r a l a r e a s , smokers vs. non-smokers, blue-collar workers vs. w h i t e c o l l a r workers. people in t h e south of

a

country vs. people in t h e north, people who

are

overweight vs. people who a r e not, e t c . How will changes

in

t h e mix of t h e population between t h e s e

t w o

g r o u p s a f f e c t life expec- tancy?

Consider a n intervention t h a t changes n(z), t h e p r o p o r t i o n of t h e population in t h e high-risk g r o u p , by

some

p r o p o r t i o n

6 at

all a g e s a f t e r s o m e initial a g e

zo:

I t i s convenient

to

c o n s i d e r a g e

z o

t h e a g e

at

"birth", s o t h a t e o r e f e r s

to

remain- ing life expectancy

at

a g e

z o

and

z

r e f e r s

to

y e a r s of a g e s i n c e

zo.

The f o r c e of mortality f o r t h e population as

a

whole is given by

and

Hence.

I t follows t h a t

If d is small,

--

tit _{e 0}

_{- p H ,} .

where

and

A s a n example of t h e u s e of t h e s e formulas, suppose t h a t t h e population con- s i s t s of non-smokers a n d smokers, a n d t h a t t h e population i s being studied s t a r t i n g

at

a g e 35 (so t h a t e o r e f e r s

to

life expectancy

at

a g e 35). F u r t h e r suppose t h a t t h e f o r c e of mortality f o r non-smokers i s . 0 0 1 s . ~ , ( t being a g e minus 35). t h a t t h e f o r c e of mortality f o r s m o k e r s i s twice

as

high, and t h a t half t h e population smokes

at

a g e 35. Remaining life e x p e c t a n c y f o r non-smokers in t h i s c a s e i s a b o u t 40.8 y e a r s and remaining life e x p e c t a n c y f o r smokers about 34.2 y e a r s . Then

H I

t u r n s o u t

to

equal 0.077. If t h e p r o p o r t i o n of t h e population t h a t smokes i s r e d u c e d by 1 p e r c e n t , t h e n life e x p e c t a n c y ( a t a g e 35) will i n c r e a s e by 0.077 p e r c e n t ,

or

by a b o u t 11 days, given t h e a v e r a g e remaining life expectancy f o r t h e population as

a

whole of 37.5 y e a r s .

More generally, i t is i n t e r e s t i n g t o investigate t h e values of HI, and of ex- p e c t e d d a y s of life saved,

at

d i f f e r e n t s t a r t i n g a g e s , i.e.,

at

different a g e s of in- tervention. Table 2 p r e s e n t s some sample calculations. Note t h a t H1 i n c r e a s e s with age: a reduction in smoking yields a g r e a t e r proportional i n c r e a s e in life ex- pectancy

at

t h e a g e s with t h e highest mortality

rates.

The absolute i n c r e a s e in life expectancy, however, as measured by days added, falls off with age. Because i t falls off slowly,

at

l e a s t b e f o r e a g e 55

or

65, it may b e optimal

to

t a r g e t anti- smoking interventions toward o l d e r people-if i t i s relatively e a s i e r

to

induce old- e r smokers

to

quit. The calculations in Table 2

are

merely illustrative, b u t some empirical analysis of t h i s

sort

could s h e d light on t h e effectiveness of t a r g e t i n g v a r i o u s kinds of h e a l t h p r o g r a m s toward individuals in different a g e classes.

Table 2. Values of HI, life expectancy. and days added

to

life expectancy if t h e p r o p o r t i o n of a population t h a t smokes i s reduced by o n e p e r c e n t ,

at

various ages.

Days added

to total

life e x p e c t a n c y Remaining life e x p e c t a n c y (in y e a r s ) f o r : if p r o p o r t i o n t h a t

smoke i s r e d u c e d Age H1 Non-smokers Smokers Total population by o n e p e r c e n t

35 .077 40.8 34.2 37.5 10.5

45 .095 31.4 25.2 28.3 9.9

55 .I20 22.6 17.1 19.9 8.7

65 .I50 14.9 10.5 12.7 7.0

75 . l a 4 8.8 5.6 7.2 4.8

85 .215 4.5 2.7 3.6 2.8

INHIBITING IM.BIBITIONS

The r e s u l t s in t h e p r e v i o u s s e c t i o n c a n b e g e n e d i z e d

to

^{t h e} case where t h e population consists of

N

subpopulations with age-specific mortality

rates

~4 ( 2 ) a n d in p r o p o r t i o n s ni ( z ). where

a n d

A s before, l e t t h e f i r s t subpopulation b e t h e healthiest, % ( z )

<

& ( z ) for a l l

z

and f o r all

t >

^1, and l e t

6(

denote t h e change in proportions

at

s t a r t i n g a g e 0:

Clearly,

For simplicity, assume

6 , = d , all i > l

.

Then i t i s not difficult ta show

-

^'

~ ( 2 )

-22) =

d(*(z)

- 2 z ) )

This formula i s identical ta (53). Consequently,

h e 0

-

_{w d H l}

_,

f o r s m a l l d ,

=

⁰

and

where HI i s defined as b e f o r e by (57) and

P

=

8t , a l l i > l

.

~ ( 2

That HI i s t h e same

as

b e f o r e may,

at

f i r s t glance, seem puzzling but, on c l o s e r thought, i t i s r e a s o n a b l e because t h e assumptions group t h e sub-populations into

t w o

p a r t s . Other formulas can b e readily derived f o r o t h e r special cases.

A s

an

illustration of t h e use of (66), consider

a

population of males with a high prevalence of alcoholism.

In

p a r t i c u l a r , assume t h a t 5 0 p e r c e n t of t h e population d r i n k moderately

or

not

at

all, t h a t 30 p e r c e n t drink heavily, and t h a t t h e remain- ing 20 p e r c e n t d r i n k v e r y heavily. F u r t h e r , assume t h a t t h e heavy d r i n k e r s h a v e twice t h e mortality and t h e v e r y heavy d r i n k e r s have f o u r times t h e mortality of t h e f i r s t group. Finally,

as

in t h e previous example, suppose t h a t t h e population i s

being considered s t a r t i n g

at

a g e 35 and t h a t t h e f o r c e of mortality follows a Gom- p e r t z c u r v e with a

=

0.001 and b

=

0.1 for t h e h e a l t h y subpopulation.

Remaining l i f e e x p e c t a n c y

for

t h e t h r e e g r o u p s t u r n s out

to

b e 40.8, 34.2, and 27.9 y e a r s and, f o r t h e population

as

a whole, 36.2 y e a r s . The value of

Hi

i s 0.108;

a o n e p e r c e n t r e d u c t i o n in t h e p r o p o r t i o n of heavy and of v e r y heavy d r i n k e r s , would add

t w o

weeks

to

t h e population's life e x p e c t a n c y .

STARTING STOPPING

Now c o n s i d e r a population t h a t consists of v a r i o u s subpopulations, with indlvi- duals making t r a n s i t i o n s f r o m o n e subpopulation

to

a n o t h e r , such t h a t t h e transi- tion

rates are

changing

or can

b e changed. F o r instance, t h e population may con- s i s t of s m o k e r s and non-smokers, with

some

s m o k e r s who s t o p and some non- smokers who start. If e i t h e r of t h e s e t r a n s i t i o n s could b e influenced, what would t h e e f f e c t b e o n life e x p e c t a n c y ? This question i s similar

to

t h e question con- s i d e r e d in t h e p r e v i o u s

t w o

sections, e x c e p t now t h e policy l e v e r

or

c o n t r o l p a r a m e t e r i s n o t t h e p r o p o r t i o n of t h e population who smoke, but t h e t r a n s i t i o n

rates

between t h e non-smoking and smoking

states.

Changing t h e transition

rates

will change t h e p r o p o r t i o n s a n d h e n c e life e x p e c t a n c y .

F o r a c o h o r t , t h e c h a n g e in t h e p r o p o r t i o n of individuals in

state

( o r group) j

at

a g e

r

i s given by t h e equation:

where

Xu

( r )

are

t h e t r a n s i t i o n

rates

from

state

i

to state

j a t a g e r , with t h e ini- t i a l p r o p o r t i o n s rrj (0) given.

In t h e simplest

case

^of a

two-state

population with mortality

rates

h ( r ) and

& ( r ) a n d t r a n s i t i o n

rate

X ( r ) from

state

1

to state

^2, U l e p r o p o r t i o n n ( r ) of indi- viduals in

state

2 i s t h e solution of t h e following equation

with rr(0) given. Let t h e

rate

of p r o g r e s s in r e d u c i n g X(r) b e given by p ( r ) :

Straight-forward calculations show

w h e r e q ( z ) is t h e solution of t h e d i f f e r e n t i a l equation

with q ( 0 )

=

0. Note t h a t t h i s equation h a s

to

b e solved t o g e t h e r with equation ( 7 0 ) f o r n ( z ) . If t h e

rate

of p r o g r e s s in d e c r e a s i n g X(z) d o e s not depend o n a g e , t h e n ( 7 2 ) r e d u c e s

to

w h e r e

SIMULATION AS A SLEDGEHAMMER

Solving ( 7 5 ) for

H A

i s n o t e a s y , s i n c e q ( p ) i s t h e solution of a d i f f e r e n t i a l equation (73) t h a t d e p e n d s o n a n o t h e r differential equation ( 7 0 ) . When mathemati- c a l solutions g e t as complicated

as

this, t h e y may not only l o s e elegance but a l s o usefulness for e i t h e r insight

or

computation. I t may t h e n b e f r u i t f u l

to

t a k e a dif- f e r e n t t a c k and r e l y o n numertcal, computer simulation.

C o n s i d e r , f o r i n s t a n c e , t h e following i l l u s t r a t i v e model:

N O N S M O K E R S .

k(i:

*(i:

1

^*(1:

1

( d e a t h ) X12(z

*

The population i s divided i n t o t h r e e groups-non-smokers. s m o k e r s , a n d q u i t t e r s . T h e s t a r t i n g p o i n t of t h e a n a l y s i s i s a g e 10:

z

r e p r e s e n t s a g e minus 10. F o r non- s m o k e r s , t h e f o r c e of mortality i s given by

f o r s m o k e r s i t i s

S M O K E R S

&

a n d f o r q u i t t e r s ,

& ( z )

=

1 . 5 h ( z )

.

To b e g i n with a l l individuals

are

non-smokers:

nl(0)

=

1 ,

n2(0)

=

n3(0)

=

0

.

The t r a n s i t i o n i n t e n s i t i e s

are

xl,(z)

=

.o6r-Sh ,

=.028.O* ,

X32(2 )

=

^{.5 8 -aoa} ,

A Z 3 ( ~ )

C

*

X32(z

QUITTERS

These transition intensities imply:

-

_{about 6} p e r c e n t of non-smokers

start

smoking a t a g e 10. a b o u t 2 p e r c e n t

at

a g e 20, and l e s s t h a n 1 p e r c e n t

at

a g e 30;

-

t h e p r o p o r t i o n of s m o k e r s who quit smoking r i s e s f r o m about 2 p e r c e n t p e r y e a r

at

a g e 1 0 to 1 0 p e r c e n t p e r y e a r

at

a g e 50 and 22 p e r c e n t p e r y e a r

at

a g e 70;

-

t h e recidivism

rate

of q u i t t e r s resuming smoking falls from 5 0 p e r c e n t p e r y e a r at a g e 1 0

to

33 p e r c e n t

at

a g e 30 and 15 p e r c e n t

at

a g e 70;

-

1 0 p e r c e n t of q u i t t e r s become non-smokers e a c h year. implying t h a t i t takes t e n y e a r s , on a v e r a g e , for a f o r m e r smoker

to

r e t u r n

to

t h e health s t a t u s of a non-smoker.

The following formulas a n d approximations c a n b e used

to

analyze t h i s model:

where

where

where t h e pi

Cf),

t h e p r o p o r t i o n s of t h e original c o h o r t t h a t are in

state

i

at

time

1 ,

are given by

where

and

With t h e p a r a m e t e r values given above, remaining life expectancy a t a g e 10 t u r n s o u t t o b e 61.5 y e a r s . The p r o p o r t i o n of t h e surviving population t h a t smokes r i s e s t o 33 p e r c e n t a t a g e 30 and t h e n f a l l s off

to

23 p e r c e n t

at

a g e 50 a n d 6 p e r - c e n t a t a g e 70.

The model c a n b e used

to

e x p l o r e various kinds of interventions. If no o n e e v e r smoked, o r if t h e health h a z a r d s of smoking

were

eliminated, life e x p e c t a n c y would i n c r e a s e b y 1.4 y e a r s . If t h e

rate at

which people began

to

smoke

were

c u t in half, life e x p e c t a n c y would i n c r e a s e b y 0.6 y e a r s . If t h e

rate at

which people gave up smoking doubled, t h e gain would b e 0.4 y e a r s . If t h e

rate

of recidivism could b e c u t in half, 0.3 y e a r s would b e gained; if recidivism could b e eliminated, t h e i n c r e a s e in l i f e e x p e c t a n c y would b e 0.7 y e a r s . If t h e duration of t h e lingering e x c e s s r i s k s f a c e d b y f o r m e r smokers could b e c u t from a n a v e r a g e of 10 y e a r s

to

a n a v e r a g e of 5 y e a r s , 0.3 y e a r s would b e added

to

life expectancy. Finally, if t h e e x c e s s r i s k of smoking

were

c u t In half, so t h a t ~ 1 2 equaled 1.5~. r a t h e r t h a n 2 p , about half

a

y e a r would b e gained.

This example p r o v i d e s a simple illustration of how micro-simulation c a n s h e d light o n models t h a t

are

difficult

to

analyze formally. More e l a b o r a t e , more r e a l i s - t i c models f o r t a r g e t analysis c a n b e handled in t h e

same

g e n e r a l way.

CONCLUSION

The life e x p e c t a n c y of individuals ( o r units) in a heterogeneous population c a n b e i n c r e a s e d b y numerous s t r a t e g i e s , including

-

lowering o v e r a l l mortality ( o r f a i l u r e ) rates.

-

reducing mortality

rates

in s p e c i f i c a g e c a t e g o r i e s ,

-

a v e r t i n g d e a t h s ,

-

lessening mortality

rates

f r o m some c a u s e ,

-

diminishing mortality

rates

in some region

or

f o r

some

population g r o u p ,

-

decreasing t h e p r o p o r t i o n of individuals in high-risk g r o u p s , and

-

changing t r a n s i t i o n rates between r i s k groups.

A s t h e various formulas d e r i v e d ln t h i s p a p e r i l l u s t r a t e , t h e s e changes a f f e c t life expectancy in d i f f e r e n t ways.

The formulas, a n d v a r i o u s extensions or adaptations of them, may b e useful

to

policymakers in t a r g e t analyses of t h e benefits of a l t e r n a t i v e interventions intend- e d

to

s a v e lives. In addition, t h e formulas d e s c r i b e t h e linkages t h a t e x i s t between population s t r u c t u r e and life expectancy. Individuals d i f f e r o n numerous dimen- sions t h a t are r e l a t e d

to

mortality chances. including a g e ,

sex. race.

s o c i e economic s t a t u s . occupation. p l a c e of residence. and personal behavior. A c h a n g e in population s t r u c t u r e along any of t h e s e dimensions will change life expectancy.

Four d i f f e r e n t a p p r o a c h e s

were used to

analyze t h e impact of

a

change in po- pulation s t r v c t u r e o n life expectancy: t h e comparative-statics a p p r o a c h . t h e dynamics a p p r o a c h . t h e method w e called t h e "second-chance" a p p r o a c h . and com- p u t e r simulation. The f i r s t t h r e e a p p r o a c h e s yield analytical solutions t h a t

are

g e n e r a l a n d t h a t may f a c i l i t a t e insight. In t h e limit. when d i s small. t h e t h r e e ap- p r o a c h e s p r o d u c e equivalent formulas,

so

which a p p r o a c h

to

a d o p t i s

to some

ex- t e n t

a matter

of

taste

a n d convenience. The t h r e e a p p r o a c h e s . however. may not b e equivalent when d i s not small. a n d e a c h a p p r o a c h may yield

a

d i f f e r e n t insight a n d provide

a

d i f f e r e n t p e r s p e c t i v e . Computer simulation is useful in a t t a c k i n g complex models t h a t d o not yield

to

t h e o t h e r t h r e e a p p r o a c h e s . The answers p r o - duced by simulation p e r t a i n

to

p a r t i c u l a r realization of

a

model in which t h e coef- f i c i e n t s

are

specified: t h e answers

are

t h u s not g e n e r a l o r elegant. b u t t h e y

are

answers.

REFERENCES

Demetrius, Lloyd (1979) Relations Between Demographic P a t t e r n s . Demography 2(16):329-338.

Keyfitz. Nathan (1977) Applied Mathematical Demography. N e w York: John Wiley a n d Sons.

Yaupel, James W. (1986) How Change in A g e S p e c i f i c Mortality Affects Life Expec- tancy. P o p u l a t i o n S t u d i e s , March.

Vaupel, James W. a n d Anatoli I. Yashin (1985) How Many Times Has Your Life Been Saved? Forthcoming Working P a p e r . Laxenburg. Austria: International Insti- t u t e for Applied Systems Analysis.