NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR
HOW CHANGE IN AGE-SPECIFIC MORTALITY AFFECTS LIFE EXPECTANCY
,I
W.
VaupelMarch
1 9 8 5 WP-85-17Working Papers a r e interim reports on work of t h e International Institute for Appli.ed Systems Analysis and have received only limited review. Views or opinions expressed h e r e i n do n o t necessarily represent; those of t h e Institute or of i t s National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
Summary
A t c u r r e n t mortality r a t e s , life expectancy i s most responsive t o c h a n g e in mortality r a t e s a t older ages. Mathematical formulas t h a t describe t h e linkage between c h a n g e i n age-specific mortality r a t e s a n d c h a n g e in life e x p e c t a n c y reveal why. These formulas also s h e d light on how past progress against mortali- t y h a s been t r a n s l a t e d i n t o i n c r e a s e s in life expectancy-and o n t h e impact t h a t f u t u r e progress i s likely t o have. F u r t h e r m o r e , t h e m a t h e m a t i c s c a n be adapted t o s t u d y t h e effect of mortality change in h e t e r o g e n e o u s populations in which t h o s e who die a t some a g e would, if saved, have a different life e x p e c t a n c y t h a n t h o s e who live.
Acknowledgments
I t h a n k Nathan Keyfitz, Kenneth Manton, a n d Anatoli Yashin for t h e i r in- s i g h t s a n d suggestions a n d Carolyn F u h r m a n n a n d S u s a n n e S t o c k for t h e i r h e l p i n preparing t h e m a n u s c r i p t .
HOW CHANGE IN AGE-SPECIFIC MORTALITY AFFECTS LIFT EXPECTANCY
J
W.
Vaupelr n D U C T I O N
Suppose t h e goal is t o increase t h e life expectancy of a population. Or, equivalently, assume t h e objective is t o save as many life years as possible. If 100 deaths could be averted in any decade of life--0 t o 10, say, or 42 t o 52--which decade would be best? The answer is simple--the first decade of life, because children lose the most years of life expectancy. Suppose, however, deaths could be reduced by one percent i n any decade of life. Which decade would then be best?
I t may seem reasonable, a t first thought, t o guess 0 to 10, or 17 to 27, or some other young decade. Using the life table for Swedish males for 1982, how- ever, the correct answer is 67 t o 77. And using t h e life table for Swedish fe- males for 1982 t h e answer is 74 t o 84.
The Swedish life table indicates t h a t out of a synthetic cohort of 100,000 women, 13 girls would die between t h e i r t e n t h and eleventh birthdays. Life ex- pectancy a t age 10.5 is about 69.5 years. The product approximates t h e number of years of life expectancy lost-just over 900 life years. At age 80, more than 3600 women die, losing 7.7 years of life expectancy. The product gives 28,000 life years. More than thirty times a s many years of life expectancy a r e lost a t age 80 than a t age 10.
Large numbers of life years are lost in infancy. In the Swedish female life table, 653 infants under age one die, losing about 79.2 years of life expectancy each., or about 52 thousand years all together. But only 189 additional girls die between ages one a n d 10; t h e total loss of life expectancy between birth and age 10 is about 66 thousand years. Compared with these 842 deaths, nearly 32,000 women die in the decade between age 74 and 84. They lose about 8.3 years of life expectancy on average, or about 260 thousand years all together. Thus, four times as many years of life expectancy a r e lost in the decade between ages 74 and 84 as a r e lost in t h e first t e n years of life.
HOW REDUCXIONS IN
MORTALITY INCREASE LlFE EXPECXANCYLet p ( a , t ) be t h e f o r c e of mortality a n d p ( a , t ) be t h e period survivorship at age a a n d time t ; p a n d p a r e interrelated by the familiar formulas:
and
0
Let e ( a , t ) r e p r e s e n t period life expectancy, given by t h e well-known formula:
where w is an age beyond which no one survives. Note t h a t life expectancy a t birth is given by:
0
How does change in the trajectory of p affect e? Demographers have taken two basic approaches in answering this question. The first, exemplified by Pollard [ I ] and in a United Nations study [2], focuses on how the difference between two alternative trajectories--pl(a) and p 2 ( a ) , say--translates into t h e difference
0 1 0 2
between t h e resulting life expectancies, e and e
.
The second approach, pioneered by Keyfitz [3] and extended here, focuses on how the r a t e or intensity of progress in p. given bytranslates i n t o t h e r a t e of progress in life expectancy, given by
I t follows from (4) t h a t
As shown by Keyfitz [3], if equal progress is being made a g a i n s t mortality a t all ages a t time t , i.e.,
p ( a , t ) = p ( t ) , all a
.
( 8 )t h e n
where
Because
H ( t ) c a n also be c a l c u l a t e d by
As discussed by Demetrius [4,5] a n d Keyfitz [3], H, which is a v a r i a n t of t h e measure known a s e n t r o p y o r information in o t h e r c o n t e x t s , c a n be i n t e r p r e t e d a s a m e a s u r e of t h e homogeneity of a population over age with r e s p e c t t o mor- tality: if H equals zero, t h e n everyone dies at t h e same age; if H equals one, t h e n t h e force of mortality i s equal at all ages. As i n d i c a t e d by (9), H gives t h e p e r c e n t a g e c h a n g e in life e x p e c t a n c y p r o d u c e d by a one p e r c e n t r e d u c t i o n in t h e force of mortality at all ages: if H i s 0.2, say, t h e n a uniform one p e r c e n t de-
0
crease in /I. would increase eo by 0.2 percent. As Keyfitz [6] further explains,
"male mortality is clearly higher t h a n female in most rich countries, usually by
0
50 percent or more a t typical ages. But eo is only about 10 percent g r e a t e r for females
....
The parameter H...
is intended to carry one from t h e 50 percent t o t h e 10 percent".An alternative expression for H is revealing. I t follows from (7) t h a t
Substituting (3) a n d (4) yields
Let
so that q represents t h e total number of years of life expectancy lost by those who die a t age z , divided by life expectancy a t birth. Then
Three special cases a r e of interest:
i) If progress against mortality is uniform a t all ages,
p ( z , t )
=
p(t ), all z , (17) t h e nBy comparing (9) and (18), i t is clear t h a t H (which is, fittingly, t h e symbol for capital q ) is given by
Substituting (15) yields
Because t h e product of p a n d p gives t h e density of deaths a t age z , t h i s formu- l a helps reveal why H is a measure of t h e homogeneity of a population with re- g a r d t o age of death (or lifespan). Furthermore, this formula facilitates under- standing of why H is a measure of t h e percentage increase in life expectancy generated by a one percent decrease in mortality rates. If a death is averted a t
0
age z , t h e n e ( z ) years of life expectancy a r e gained. The numerator of (20) measures t h e total effect of reducing deaths a t all ages; t h e denominator con- v e r t s t h e absolute effect into a relative effect. As suggested t o me by my col- league Anatoli I. Yashin, t h i s implies t h a t H ( t ) gives t h e proportional increase in life expectancy a t birth if everyone's first death were averted. The assump- tion is t h a t each individual a t t h e hour of death is saved and given t h e life ex- pectancy of individuals surviving a t that; age. Thus, if H ( t ) i.s 0.15, staying the h a n d of death once would increase life expectancy by 15 percent. Compared with (20), t h e expressions for H given in (10) and (12) seem less intuitive.
ii) If progress against mortality only occurs between ages a a n d and if t h e r a t e of progress is uniform between these ages, t h e n
This formula was used t o answer t h e question posed a t t h e s t a r t of t h i s paper.
iii) Finally, if progress against mortality only occurs a t a single instantaneous age a,
where
6
is a Dirac function, thenThus, ~ ( a ) is a measure of t h e potential for increasing life expectancy by reduc- ing t h e force of mortality a t age a . Because ~ ( a ) is proportional t o p(a)p(a)
0
a n d e ( a ) , this potential depends both on t h e density of deaths a t age a and on t h e number of years of life expectancy lost by those dying a t age a . The implicit assumption is t h a t the population is homogeneous a t any specific age: those who die would, if they could be saved, have the same life expectancy as those who live. This assumption will be relaxed later in this paper.
THE POTENTIAL
MIR
SAVINGLFE
YENC32 +5
Table 1 presents values of q ( a ) d c for Swedish males and females in
Z
1982*. After infancy, t h e maximum value of 9 for t h e men occurs a t age 72.5; for t h e women, it occurs a t age 60.0. A one percent reduction in t h e force of mortal- ity between ages 75 and 60 would increase male life expectancy by .036 percent a n d female life expectancy .031 percent. A one percent reduction in t h e force of mortality a t all ages would increase male life expectancy by about .15 per- c e n t and female life expectancy by about .13 percent.
*A number of different life tables, from different sources, were used to make the calculation in this paper. The life tables for Sweden from 1780 t o 1850 are from [7]; Swedish life tables after 1950, ex- cept for 1970 and 1982, are from the annual Swedish Statistical Yearbook. These various life tables are baaed on five years of data centered on the year given: the 1910 table, for example, is based on data from 1908 through 1912. The Swedish life tables for 1970 and 1982 were supplied by Professor Ingvar Holmberg of the University of Gothenberg: these tables pertain to a single gear of time. The U S , life table for 1918 is based on the advanced report of final mortality statistics in the Monthly Vi- tal Statistics Report, September 1882; the figures were adjusted by the correction factors given in t h a t Report so that they are consistent with population estimates based on the 1880 census. The U.S.
life table for 1970 is the decennial table based on data from 1969 through 1871, as published in "Unit- ed States l f e Tables: 196971" (National Center for Health Statistics, May 1975). The U.S. life table for ID00 is from [8]. The 1J.S. life tables for 1080 and 2000 are fromVLife Tables for t,he United States:
1800-205W', Actuarial Study No. 87, U.S. Dept. of Health and Human Services, September 1982. All the remaining life tables pertaining to years between 1975 and 1879 are from the United Nations Demo- graphic Yearbook 1980. All other life tables are from [7].
2 +5
Table 1. Values of
J
t)(a)da for Swedish males a n d females i n 1982.2
Age Period Males Females
0- 5 .00853 .00763
5-10 .00085 .00068
10-15 .00060 .00060
15-20 .00226 .00079
20-25 .00289 .00120
25-30 .00344 .00140
30-35 .00341 .00172
35-40 .00434 .00243
40-45 .00530 .00313
45-50 .00736 .00447
50-55 .00942 .00591
55-60 ,01258 .OD779
60-65 .01555 .DO936
65-70 .01788 .01202
70-75 .01869 .01464
75-80 ,01719 .01679
80-85 .01282 .01650
85-90 .00675 .01201
90-95 .00231 ,00545
95-100 .00055 .OO 142
H (i.e., t o t a l for .I5270 .I2622 all a g e s )
In which r o u n d five-year period (e.g., 25-30 or 60-65), n o t counting early childhood from 0 t o 5, i s t h e potential for saving life y e a r s g r e a t e s t ? Table 2 p r e s e n t s t h e answer for an a s s o r t m e n t of c o u n t r i e s a t different times with vary- ing life expectancies. The r o u g h r u l e of t h u m b is t h a t t h e optimal five y e a r period is n e a r t h e life e x p e c t a n c y of t h e population: t h e r u l e holds particularly well for populations with life e x p e c t a n c i e s of 65 y e a r s o r more.
A simple model a n d some e l e m e n t a r y calculus s h e d s some light on t h i s finding. If t h e force of mortality follows a Gompertz curve,
t h e n i t follows from (15), (3) a n d (2) t h a t
Setting t h e derivative of q ( a ) with respect t o a equal to zero yields t h e result t h a t t h e maximum value of q ( a ) occurs a t t h e value of a s u c h t h a t
For a Gompertz curve of mortality, this value of a t u r n s o u t t o be roughly equal to life expectancy a t birth.
Table 2. The five-year period following infancy for which t h e potential for sav- ing lives years is greatest, for various male and female populations with different life expectancies, from different countries, a t different periods.
5-year period following infancy
a +5
for which q ( z ) &
a
0
Country Period Sex is greatest e o
Italy 188 1
Sweden 1780
USA 1900
Taiwan 1920
Taiwan 1920
Chile 1909
Chile 1909
Italy 188 1
England and Wales 1861
USA 1900
Sweden 1780
England and Wales 1861
Japan 1899
Japan 1899
Czechoslovakia 1934
Australia 1911
Costa Rica 1960
Mexico 1975
Czechoslovakia 1934
Australia 1911
Chile 1979-80
Costa Rica 1960
Poland 1960
Australia 1964
Mexico 1975
USA 1970
Japan 1964
Canada 1965
England and Wales 1976-78
USA 1980
USA 2000
Table 2 (continued)
5-year period following infancy
a +5
for which q ( z ) d r a
0
Country Period Sex is greatest e~
Chile 1979-50 F 70-75 6 8
Poland 1960
F
7 1Sweden 1978
M
7 2Japan 1964
F
7 3Japan 1978
M
7 3Sweden 1982
M
7 3Australia 1964
F
74Iceland 1977-78
M
74Canada 1965
F
7 5USA 1970
F
75-80 7 5England and Wales 1976-78 F 7 6
USA 1980 F 7 8
Japan 1978
F
7 8Sweden 1978
F
7 9Sweden 1982
F
79Iceland 1977-78
F
80-85 7 9USA 2000
F
8 1A one percent reduction in t h e force of mortality a t all ages would produce much less increase in life expectancy today t h a n it would fifty years or a centu- ry ago. This decline is, in large measure, a price of t h e progress t h a t has been made in reducing deaths in infancy--the age a t which t h e most years of life ex- pectancy are lost. Another result of this progress is a shift in t h e ages where further progress against mortality would be most effective in increasing life ex- pectancy. Before 1900, most of t h e potential for saving life years was concen- t r a t e d in t h e first five years of chj.ldhood; today, in developed countries, most of t h e potential is in old age. Table 3 shows t h e decline in H and t h e shift; in the profile of q by presenting data based on Swedish life tables from 1800 to 1980.
Keyfitz [3] shows t h e decline in H for U.S. males and females from 1920 to 1960.
As Pollard
[I]
demonstrates, more rapid progress against mortality may be occu.rring a t all ages in one population compared with another, but nonetheless life expectancy may be increasing less rapidly. This seeming paradox is less puzzling when viewed through t h e lens of7.
Let p i ( a ) be t h e r a t e a t which t h e force of mortality is being reduced a t age a i n population i. Supposebut t h a t
r l l ( a ) p l ( a )
<
r l z ( a ) p 2 ( a ) , all a.
Then i t follows from ( 1 6 ) t h a t life expectancy will be increasing less rapidly in t h e first population t h a n in t h e second.
Table 3. The potential for saving life years (H), t h e proportion of this potential below age 5 and above age 65, and life expectancy a t birth for selected Swedish populations.
Males
Females
NOTE: The life tables used before 1900 included n o esti- mate of t h e force of mortality after age 85. For t h e s e
W
tables th.e value of Jq(+)dz was assumed t o equal
7
q(r)dz. This approximation i s based o n t h e life 85 00tables for which mortality rates a r e available after age 85.
Clearly t h e condition in (28) can be relaxed; essentially what is required is t h a t q l be sufficiently smaller t h a n q 2 a t enough ages. As indicated in Table 3 a n d as discussed by Keyfitz [3], t h e value of H, t h e integral of t h e q ' s , tends to fall a s life expectancy increases; as H falls, t h e value of q a t most ages also must fall. Thus a s life expectancy increases progress in reducing age-specific mortality translates into less and less progress in further increasing life expec- tancy. A population with a higher life expectancy than a second population can be making greater progress against mortality a t all ages but nonetheless be making less progress in increasing life expectancy.
RATES
OF PROGRESS AGAIN=MORTALITY
The potential for saving life years is measured by q ; progress against mor- tality is given by p . As indicated by (16), progress in increasing life expectancy, as measured by sr, depends on t h e product of q and p . Thus, even if t h e potential for saving life years is greatest in old age, if little progress is being made in reducing mortality a t older ages then this potential will not translate into life expectancy gains.
Table 4 presents data on q and p for Swedish females in 1982. Progress in reducing mortality is highest in infancy and childhood; afterwards, the ann.ua1 r a t e of progress hovers between one and two percent or so a t most ages. Be- cause of t h e rapid r a t e of progress in t h e childhood years, almost a sixth of t h e life expectancy gains occur before age 20 even though less t h a n a tenth of the potential lies in these years. By age 55, however, potential and actual progress a r e in rough balance: seventy percent of t h e potential for saving life years oc- curs after age 55 a n d seventy percent of t h e actual improvement in. life expec- tancy can be attributed t o progress made in reducing mortality after age 55.
It may seem a bit surprising t h a t progress in reducing mortality r a t e s hovers around roughly t h e same level a t all ages after childhood and t h a t significant improvements a r e being made a t older ages. Table 5 presents d a t a on trends in mortality r a t e s since 1780 for Swedish females and males and since 1920 for U.S. females a n d males. In most cases, p r o g r e s s i n reducing mortality after age 85 is comparable t o t h e progress made between ages 5 a n d 65. Except for Swedish males, progress since 1950 and especially since 1970 against mor- tality in old age h a s been substantial. (For a discussion of r e c e n t U.S. mortality trends, see [9].)
Table 4. Average values of r ) and p in various age categories a n d t h e cumulative percentage of r ) and of t h e product of q p u p through these age categories, for Swedish females in 1982.
Age Category
0
-
0-
(through age a ) q
J
q ( z ) & pJ
q ( z ) p ( z ) &0 0
NOTE: The rate of progress in reducing mortality, i s the average rate from 1970 t o 1982. The formulas used to calculate
5
andp
are:
0 0
-
ndz e z + e z + n 7 =-.
2 / e o , (wherez + n = a ) , n
and
z = ( h ( - L n ( l - n q A ) ) - L n ( - L n ( l - n q z ) ) ) / t ,
where q' i s from the earlier life table and t is the number of years that have elapsed.
Suppose progress against mortality continues. Will. H decline much furth- e r ? Will life expectancy level off a s i t becomes more a n d more difficult t o in- crease life expectancy by decreasing mortality rates? Some insight into these questions can be gained by a simple model. Assume t h a t t h e force of mortality can be described by a Gompertz curve, as given in (24). This is not an unrea- sonable assumption for our purposes here, given t h e low level of mortality in in- fancy and childhood in developed countries. Furthermore, because
Table 5. The average annual r a t e of progress
p
in reducing t h e force of mortali- ty for Swedish females and males from 1780 to 1982 and for U.S. fe- males and males for 1920 t o 1979 for various age categories.Age Category
Population Period 0-5 5-25 25-45 45-65 65-85 Swedish
F
1780-1870 0.4% 0.4% 0.3% 0.1% 0.1%1870-1910 2.1 0.8 0.7 1.1 0.8 1910-1950 3.9 4.9 3.5 1.1 0.2 1950-1970 3.4 2.3 2.1 1.8 1.6 1970-1982 3.0 4.3 1.7 1.3 1.8
Swedish
M
1780-1870 0.4 0.5 0.2 0.0 -0.11870-1910 2.0 1.0 1.0 1.3 0.9 1910-1950 3.6 3.6 2.9 1.0 0.2 1950-1970 3.2 1.7 0.5 0.3 0.4 1970-1982 4.7 3.9 0.6 0.3 0.1
U.S. F 1920-1950 3.9 5.5 4.0 1.8 1.2
1950-1970 2.5 1.5 1.4 1.3 1.2 1970-1979 4.8 2.3 3.6 2.1 2.7
U.S.
M
1920-1950 3.7 1.3 3.5 0.3 0.51950-1970 2.6 -0.1 0.3 0.4 0.2 1970-1979 5.0 1.7 2.3 2.4 1.9 NOTE: The values of
5
were calculated using the formula given in Table 4.0
Table 6. Values of eo and H over time, when t h e force of mortality follows a Gompertz curve and is being reduced one percent per year before age 85, for various assumptions about t h e r a t e of aging,
B,
and t h e r a t e of progress against mortality after age 85.B
Year0.08 0 100 200 300 0.12 0 100 200 300
p
=
0 after 85 p=
0.5% after 85 p=
1% after 85i t is c l e a r t h a t life e x p e c t a n c y is largely d e t e r m i n e d by mortality a t t h e ages between 3 5 a n d 9 0 o r 95 for which a Gompertz c u r v e g e n e r a l l y provides a n ade- q u a t e fit t o t h e f o r c e of mortality.
Table 6 p r o j e c t s life e x p e c t a n c y over a period of 300 y e a r s u n d e r various assumptions. The v a l u e of
p
in t h e Gompertz curve is t a k e n t o be e i t h e r .08 o r .12--values roughly b r a c k e t i n g t h e values observed in developed c o u n t r i e s . Pro- g r e s s in r e d u c i n g m o r t a l i t y before age 8 5 is a s s u m e d t o be o n e p e r c e n t p e r y e a r ; a f t e r a g e 85, p r o g r e s s i s e i t h e r one p e r c e n t p e r y e a r , half of p e r c e n t p e r y e a r , o r zero.Table 6 shows t h a t
H
does c o n t i n u e t o fall a s life e x p e c t a n c y increases.When t h e r e is n o p r o g r e s s a g a i n s t mortality a f t e r a g e 85, H falls t o especially low levels a n d life e x p e c t a n c y does show some signs of leveling off--at a level well above 85. When mortality a t a d v a n c e d ages is r e d u c e d by one p e r c e n t p e r year, life e x p e c t a n c y i n c r e a s e s t o a c e n t u r y or more. And even when t h e r a t e of progress i s only half a p e r c e n t p e r y e a r , life expectancy r i s e s i n t o t h e 90's. A r a t e of p r o g r e s s of one p e r c e n t a y e a r does n o t seem u n r e a s o n a b l e i n light of t h e s t a t i s t i c s p r e s e n t e d in Table 5 a n d given t h e ignorance a n d u n c e r t a i n t y en- veloping o u r u n d e r s t a n d i n g of aging processes
[lo,
111. Indeed, a r a t e of two p e r c e n t p e r y e a r might be plausible, i n which c a s e t h e 300 y e a r s in Table 6 would be r:ompressed i n t o a c e n t u r y a n d a half.P r o g r e s s in i n c r e a s i n g life e x p e c t a n c y is g r e a t e r when
8,
t h e r a t e of aging, is lower. This s u g g e s t s that; r e d u c t i o n s inp
might be f a r more effective t h a n r e d u c t i o n s i n a. If @ i s c u t from .12 t o .08, t h e n d e a t h r a t e s a t all a g e s m u s t be multiplied by a f a c t o r of n e a r l y 14 before life expectancy r e t u r n s t o i t s original level.The d a t a i n Table 6 i n d i c a t e t h a t when progress a g a i n s t mortality is uni- form a t all ages, t h e n t h e g a i n i n life e x p e c t a n c y e a c h c e n t u r y i s roughly t h e same--about 12.5 y e a r s when /3
=
.08 a n d 8.4 y e a r s when @ = .12. Thus, althoughH,
which m e a s u r e s t h e relative o r proportional r a t e of i n c r e a s e in life expec-0
t a n c y , i s d e c r e a s i n g , t h e a b s o l u t e r a t e of i n c r e a s e , given by d e o / d t , s e e m s t o r e m a i n m o r e or l e s s c o n s t a n t . The t r u t h of t h i s i s readily d e m o n s t r a t e d . If mor- t a l i t y r a t e s follow a Gompertz c u r v e a n d if st.eady progress a t r a t e p is being made in r e d u c i n g m o r t a l i t y r a t e s , t h e n
'2 - a - P : ( e @ - l )
=
J e7
d z .0
Elementary methods of differential c a l c u l u s yield t h e r e s u l t
The first i n t e g r a l i s simply t h e integral of t h e density of d e a t h a n d h e n c e equals one; t h e second integral equals life e x p e c t a n c y a t birth. Letting
t h e r e s u l t simplifies t o
If a' i s small, which i t will be for a population with a long life e x p e c t a n c y (be- c a u s e for s u c h a population e i t h e r a will b e small or pt will be large), t h e n t h e change i n life e x p e c t a n c y over time will be roughly c o n s t a n t :
THE IMPACT OF HEI'EXOGENE3TY
The assumption is questionable t h a t t h o s e who die a t some age would, if saved, have t h e same life expectancy a s those who live. For i n s t a n c e , a s dis- c u s s e d i n [12], t h e victim of a s e r i o u s h e a r t a t t a c k or m o t o r vehicle a c c i d e n t might, if d e a t h were averted, be prone t o a n o t h e r h e a r t a t t a c k o r motor vehicle accident. More generally, individuals of t h e same age may differ from e a c h o t h e r i n t h e i r "frailty" o r relative risk of d e a t h [13,14,15]. Let t h e life e x p e c t a n c y of t h o s e who a r e saved a t age a (i.e., t h e average n u m b e r of years, u n d e r c u r r e n t mortality conditions, t h a t t h e s e individuals would live if d e a t h could be a v e r t e d ) b e d e n o t e d by e + ( a ) . In a homogeneous population, t h i s life e x p e c t a n c y would
0
equal e ( a ) ; in a heterogeneous population i t will probably be lower, although it could, conceivably, be higher. Then, ( 1 5 ) becomes
As before,
0
As a simple example, suppose e + were only half e a t all ages. The values of q and the value of H would be half a s great as t h e assumption of homogeneity would indicate. The profile of t h e q ' s would be t h e same--and hence t h e age a t which t h e r e was t h e g r e a t e s t potential for saving life years would not change-- but the impact of a one percent reduction in death r a t e s on life expectancy would be c u t in half.
THE LIFE MPECTANCY OF THE
DEAD
More elaborately, following [ 1 3 ] , l e t z be a measure of frailty or relative risk such t h a t an in&vidual a t some particular age with frailty z is subject to a force of mortality t h a t is z times g r e a t e r t h a n the force of mortality of a "stan- dard" individual of t h e same age who h a s a frailty value of one. Then e + can be calculated if a distribution is specified for z .
As noted by Beard [16] and others, the Gamma distribution is a plausible, tractable, and flexible probability distribution t o use when studying heterogene- ous populations. Vaupel e t al. [ 1 3 ] prove t h a t if z is Gamma distributed a t age a with shape parameter k and scale parameter A, t h e n t h e frailty of those who die a t age a follows a Gamma distribution with t h e same scale parameter A and with shape parameter k + l . Consider t h e two cohorts of those who would ordinarily survive a t age a and those who would have died but a r e saved-and hence die later. If individual frailty does n o t change after age a , if Z ( z ) and ZC(z) a r e t h e mean frailties of t h e surviving members of t h e two cohorts a t age z
>
a , and if f i represents t h e force of mortality for t h e standard individual, then i t can be shown t h a tBecause. as shown in [13], t h e force of mortality among surviving individuals is given by
t h e force of mortality among t h e survivors of the saved cohort will be
If individual frailty is c o n s t a n t a t all ages after birth, t h e n
where a2 is t h e variance in frailty a t birth. Let s ( a . z ) represent t h e proportion of those alive a t age a who a r e surviving a t age z
>
a :Clearly,
Consequently, t h e life expectancy of those whose death is averted a t age a is given by:
Indeed, (44) has a broader interpretation because of t h e remarkable result in (40): t h e force of mortality a t age z is th.e same for all the cohorts of survivors of those saved a t any age before z . Thus, (44) gives t h e life expectancy a t age a of t h e survivors of those whose death was averted a t any age before a.*
*The formulas for
9
and e + in (40) and (44) readily generalize t o cohorts for whom death is averted more than once, e.g., those who would die a t some age but are saved and then would die at some later age and are saved again. Far those who are saved rn times, substitute rn +$ for l+ue.Estimates of u2 a r e available from t h r e e studies. The eight estimates in [17] range from 0.25 t o 1.67; t h e median is 0.37. The eight estimates i n
[la]
range from .12 t o .54; t h e median is .22. Finally, t h e parameters estimated in [19] imply a value of u2 of .22 for t h e model t h e fit t h e data best a n d a value of .03 for a model t h a t fit less well. All these estimates a r e based on data for elder- ly populations a n d on t h e assumption t h a t individual frailty does not change with age a n d follows a Gamma distribution. Although t h e estimates a r e subject t o question, i t seems plausible t h a t a n appropriate value of
c?
may be of t h e ord- e r of magnitude of 0.1 to 1.0 and t h a t a best guess might be 0.25.Table '7 presents life expectancies for those who die a t various ages (if they were saved) and t h e corresponding values of 7 , for five values of
9.
Whenc?
iszero, t h e population is homogeneous and t h e life expectancy of those who die is t h e same as t h e life expectancy of those who live. Note t h a t t h e effect of hetero- geneity is t o reduce life expectancy by a g r e a t e r absolute amount a t younger ages but by a g r e a t e r proportional amount a t older ages. If, for example, u2 is one, t h e n a t age zero the life expectancy of those who die is more t h a n 7 years less, but still about 90% of, t h e life expectancy of those who survive. A t age 90, t h e loss i s only 1.5 years, but this is nearly half of a ninety-year-old's life expec- tancy.
I t is t h e proportional losses t h a t affect 7 and H; consequently, t h e g r e a t e r t h e heterogeneity, t h e lower t h e value of H and t h e more t h e potential for sav- ing lives lies a t younger ages. As t h e table shows, however, even i f u2 were as high as one, H would fall by less t h a n a third, from . I 2 7 t o .091, a n d t h e poten- tial for saving life years by averting deaths above age 65 would only fall from 60 percent t o 54 p e r c e n t of t h e total potential.
A more general model of t h e life expectancy of those saved from death c a n be constructed a s follows. Let &(z) represent t h e force of mortality at age z of those who would have died a t age a
<
z , perhaps from some specified cause, but were saved. Let t h e risk ratio be given bywhere ~ ( z ) is simply the force of mortality a t age z (i.e.. among t h o s e who would n o t have died). Then it follows from. (42) t h a t
Table 7. Life expectancies of those saved from.death and resulting profiles of q for various values of u2. the variance in frailty at birth, for Swedish females in 1978.
e + when m2 =
.25 .5
77 when u2 =
Thus, letting e z ( x ) denote the life expectancy a t age x of those who were saved from death a t age a ,
Various special cases of this general result may be of interest. For in- stance, ya(t ) could be constant for all t , could gradually decline toward one, or could be constant for a decade, say, and t h e n fall to one. If
then (47) implies
The similarity of this formula to (44) means t h a t t h e values in Table 7 for
u2
equal to 0 t o 4 can be interpreted as pertaining to 7, equal to 1 t o 5. For exam- ple, consider a group of 50-year-olds who would have died from a h e a r t attack but were saved. Suppose this group would face a force of mortality, for the rest of their lives, some five times greater than the normal force of mortality. Then their remaining life expectancy would be 18.0 years, rather than the normal 30.7 years.
*
POLTCY WUCATIONS
AND
INSINUATIONSAs discussed by Vaupel [20], nearly all statistics presented in policy- relevant studies are really vectors: they not only summarize a body of data. but they also imply a policy thrust. lmplicational honesty requires some discussion of lurking insinuations t h a t may appear to be simple facts. If mortality rates were reduced by one percent, over 60 percent of t h e life years gained would be gained by averting deaths above age 65. Does this imply t h a t t h e life-saving efforts should be directed toward the elderly population? Not necessarily, for several reasons. First, the 60 percent figure is based on the 1982 life table for Swedish females. For males and for other countries the figure is generally
*The 40 formulas given inthis paper can be readily generalized to specific causes of death, in much the same way that [3] and [I] generalize their formulas.
lower--for Swedish males in 1982 i t is under 50 percent. Heterogeneity, a s dis- cussed above, would reduce this somewhat further.
Second, t h e figure is based on a life table-i.e., on a hypothetical, station- ary population--rather t h a n on t h e actual distribution of a population by age. In most populations t h e r e a r e more young people t h a n implied by t h e life table.
Consequently, t h e goal of increasing life expectancy is not completely congruent with t h e goal of saving as many life years as possible given t h e c u r r e n t population distribution. For instance, for the total U.S. life table for 1979, about 50 p e r c e n t of t h e increase in life expectancy produced by a one per- c e n t reduction in mortality rates can be attributed t o t h e reduction in mortali- t y r a t e s above age 65. However, only about 36 percent of t h e gain in life years produced by a one percent reduction in t h e a c t u a l number of deaths a t all ages would be due t o averting deaths above age 65.
Third, t h e quality of life a t advanced ages may tend, on average, t o be lower t h a n a t younger ages. If the goal is t o save as many quality-adjusted life years a s possible [12], t h e n efforts to avert deaths a t younger ages will appear more favorable. Other goals t h a t might be proposed--e.g., maximize life years saved before t h e Biblical allotment of t h r e e score and ten, maximize economic pro- duction. minimize deaths of parents with young children, or minimize inequali- ties in lifespans--also favor efforts t o reduce early deaths. Vaupel [21,22] ex- amines several c r i t e r i a and concludes t h a t most, of t h e losses due t o death a r e due t o deaths before age 65.
Fourth, it may be easier t o avert deaths before age 65 than afterwards. As t h e data presented on t h e r a t e of progress against mortality show, progress against e a r l y death has generally tended t o be somewhat more rapid t h a n pro- gress against death after age 65.
Offsetting t h e s e considerations a r e many others. As Vaupel a n d Yashin [14]
suggest, t h e t r u e r a t e of progress being made in reducing mortality r a t e s a t ad- vanced ages may be masked by t h e effects of heterogeneity. The quality of life of many of those who die before age 65 may be relatively low, even if t h e quality of life a t younger ages does t e n d t o be higher t h a n t h a t after, say, age 85. Further- more, those who die early may tend t o be the kind of people who, if saved, would have relatively s h o r t life expectancies. Finally, there a r e several appealing ob- jectives t h a t favor life saving a t older ages. I t is desirable t o avert death p e r s e , regardless of life expectancy, and m.ost deaths occur in old age. It is desirable t o have a society t h a t is diverse in its age composition--and in i t s memories and
experiences. P e r s o n s born in t h e 19th c e n t u r y , who experienced t h e world with Kaiser Franz Joseph a n d without radio, a r e n o t only relatively r a r e b u t also con- s t i t u t e , i n a s e n s e , a n e n d a n g e r e d species t h a t will b e e x t i n c t in a few years.
Suppose i t were possible t o save t h e lives of t e n BO-year-olds, giving them, on average, seven additional y e a r s of life. And suppose t h e a l t e r n a t i v e was t o save t h e lives of two 40-year-olds, giving e a c h of t h e m a n e x p e c t e d additional li- fespan of 3 5 years. E i t h e r way, 70 y e a r s of life expectancy a r e gained. Which al- t e r n a t i v e would b e preferable? Recommendations c o n c e r n i n g t h e focus of poli- cies t o save lives depend n o t only o n statistical analyses b u t also on answers t o s u c h difficult value questions.
Beyond this, policy decisions a r e usually made concerning specific lifesav- ing alternatives. Should a n e x t r a million dollars b e devoted t o r e s e a r c h on influenza? Should passive r e s t r a i n t systems for automobiles be required? These decisions depend n o t only on broad value judgments b u t also o n t h e details of t h e specific proposal. How effective i s i t likely t o be? How m u c h will i t cost?
How many v o t e r s will like it?
Nonetheless, t h e methods and findings of t h i s paper may b e of some relevance t o policy discussions. In particular, t h e r e is considerable potential for saving life y e a r s a n d i n c r e a s i n g life expectancy by reducing mortality in old age, more potential t h a n generally realized. F u r t h e r m o r e , because consider- able progress i s being made in reducing mortality among t h e elderly, t h i s poten- tial is being realized. The r e s u l t is a shift in t h e age composition of t h e popula- tion: progress i n reducing mortality r a t e s is adding relatively few life y e a r s among t h e working-age population compared with t h e e x t r a life y e a r s added a f t e r age 65.
References
1. J.H. Pollard, "The Expectation of Life and. Its Relationship t o Mortality,"
J o u r n a l of the h t i t u t e of Actuaries 109(&), pp.225-240 (1982).
2. Uni.ted Nations. Levels a n d B e n d s of Mortality Since 1950, United Nations, New York (1982).
3. N. Keyfitz, Applied Mat.hernatical Demography, Wiley. New York (1977).
L. Demetrius , "Demographic Parameters and Natural Selection," P r o c e e d - i n g s of t h e N a t i o n a l A c a d e m y of .W.ences 71(12), pp.4645-4647 (1974).
L. Demetrius, "Adaptive Value, Entropy, and Survivorship Curves," N a t u r e 275(September 21), pp.213-214 (1978).
N. Keyfitz, "Choice of Function for Mortality Analysis: Effective Forecasting Depends on a Minimum Parameter Representation," Theoretical P o p u l a t i o n Biology 21(3), pp.329-352 (1981).
N. Keyfitz and W. Flieger, World P o p u l a t i o n : An A n a l y s i s of Vital Data, University of Chicago Press, Chicago (1968).
S.H. Preston, N. Keyfitz, and R. Schoen, C a u s e s of Death: Life Tables f o r Nu- t i o n a l P o p u l a t i o n s , Seminar Press, New York (1972).
E.M. Crimmins, "The Changing P a t t e r n of American Mortality Decline, 1940-1977, and Its Implications for t h e Future," P o p u l a t i o n a n d Develop- m e n t R e v i e w 7(2), pp.229-254 (198 1).
K.G. Manton, "Changing Concepts of Morbidity and Mortality in t h e Elderly Population," Milbank Memorial h n d Q u a r t e r l y / H e a l t h a n d S o c i e t y 60, pp. 183-244 (1982).
J.M. Owen and J.W. Vaupel, "Anna's Life Expectancy", WP-85-11, Interna- tional Institute for Applied Systems Analysis, Laxenburg. Austria (1985).
Forthcoming in A m e r i c a n D e m o g r a p h i c s .
R. Zeckhauser and D. Shepard, "Where Now for Saving Lives?," Lavl a n d C o n t e m p o r a r y Pro b.1em.s 40(4), pp.5-45 ( 1976).
J.W. Vaupel, K.G. Manton, a n d
E.
Stallard, "The Impact of Heterogeneity in Individual Frailty on t h e Dynamics of Mortality," D e m o g r a p h y 16, pp.439- 454 (1979).J.W. Vaupel and A.I. Yashin, "The Deviant Dynamics of Death in Heterogene- ous Populations", RR-83-1, International Institute for Applied Systems Analysis (1983). An abridged version is in Nancy Tuma (ed.) Sociological Methodology 1985, Jossey-Bass, San Francisco.
J.W. Vaupel and A.1. Yashin, "Heterogeneity's Ruses: Some Surprising Effects of Selection on Population Dynamics," Forthcoming in 73.e A m e r i - c a n S t a t i s t i c a n , August 1985.
16. R.E. Beard, "A Theory of Mortality Based on Actuarial, Biological a n d Medi- cal Considerations," (1963). Proceedmgs of International Population Conference, New York, 1961, volume 1, pp.611-625
17. K.G. Manton, E. Stallard, and
J.W.
Vaupel. "Methods for Comparing t h e Mor- tality Experience of Heterogeneous Populations," D e m o g r a p h y 18(3), pp.389-410 (1981).18. K.G. Manton, E. Stallard, and J.W. Vaupel, "Alternative Estimates of t h e Heterogeneity of Mortality Risks Among the Aged," Submitted t o t h e Jour- n a l of t h e A m e r i c a n S a t i s t i c a l A s s o c i a t i o n (1984).
19. S. Horiuchi and AJ. Coale, "Age Patterns of Mortality for Older Women: An Analysis Using t h e Age-Specific Rate of Mortality Change with Age," (1983).
P r e s e n t e d a t Population Association of America meeting, Pittsburgh, Pennsylvania, April 14-16, 1983
20. J.W. Vaupel, "Statistical Insinuation," J o u r n a l o f P o l i c y A n a l y s i s a n d M u n w e m e n t 1(2), pp.26 1-263 (1982).
21.
J.W.
Vaupel, "Early Death: An American Tragedy," Law a n d C o n t e m p o r a r y P r o b l e m 40(4), pp.73-121 (1976).22.
