Collaborative Paper
A SIMPLE METHOD OF MEASURING THE INCREASE OF L I F E EXPECTANCY WHEN A F I X E D PERCENT OF DEATHS FROM CERTAIN CAUSES ARE ELIMINATED
Z . Nanjo
December 1 9 8 0 C P - 8 0 - 3 5
International Institute for Applied Systems Analysis
A-2361 Laxenburg, Austria
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
A SIMPLE METHOD OF MEASURING THE INCREASE OF LIFE EXPECTANCY WHEN A FIXED PERCENT OF DEATHS FROM CERTAIN CAUSES ARE ELIMINATED
Z. Nanjo
December 1980 CP-80-35
C o Z Z a b o r a t i v e P a p e r s report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
FOREWORD
The principal aim of health care research at IIASA has been to develop a family of submodels of national health care systems for use by health service planners. The modeling work is proceeding along the lines proposed in the Institute's cur- rent Research Plan. It involves the construction of linked submodels dealing with population, disease prevalence, resource need, resource allocation, and resource supply.
In this paper, Professor Nanjo, from the Fukushima Medical College, Japan, generalizes Keyfitz's method for measuring the increase of life expectancy due to a marginal reduction in any one cause of death. He relaxes Keyfitzts assumption that the number of deaths in each age group is decreased at a fixed rate and goes on to derive a mathematical formulation that leads to an improved approximation.
Recent publications in the Health Care Systems Task are listed at the end of this report.
Andrei Rogers Chairman
Human Settlements and Services Area
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
ACKNOWLEDGMENTS
I am v e r y g r a t e f u l t o P r o f . N . K e y f i t z f o r h a v i n g r e a d t h e d r a f t o f t h i s p a p e r a n d h a v i n g made some k i n d comments o n i t . F u r t h e r , I am much o b l i g e d t o D r . A. R o g e r s who h a s b e e n s o k i n d a s t o p r o p o s e some c h a n g e s i n t h e d r a f t a n d make some u s e f u l s u g g e s t i o n s o n i t . T h i s p a p e r c o u l d n e v e r h a v e b e e n c o m p l e t e d w i t h o u t t h e i r i n v a l u a b l e h e l p .
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
ABSTRACT
The effect that one type of medical improvement will have on life expectancies is often computed using a life table. In classical methods, such as Greville's, the increase in life expectancy has been dealt with by assuming that deaths from a particular cause have been eradicated. Keyfitz derived a parameter that measures the increase in life expectancy by a marginal reduction in any cause of death. The parameter is
additive in several causes and useful for various studies of causes of death.
This paper is a generalization of Keyfitz's idea and deals with a case where some percent of the deaths from a particular cause are eliminated, not necessarily uniformly in all age intervals.
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
CONTENTS
1 . INTRODUCTION
2. GENERALIZATION OF KEYFITZ'S RESULTS 3. OUR METHOD OF COMPUTATION
4. RELATION BETWEEN OUR RESULTS AND THE RESULTS OF KEYFITZ
5. AN APPLICATION AND COMMENTS 6. CONCLUSION
REFERENCES APPENDIX
RECENT PUBLICATIONS IN THE HEALTH CARE SYSTEMS TASK
J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,
A SIPlPLE METHOD OF MEASURING THE INCREASE OF
LIFE EXPECTANCY WHEN A FIXED PERCENT OF DEATHS
FROM CERTAIN CAUSES ARE ELIMINATED
1 . INTRODUCTION
L i f e t a b l e s a r e o f t e n u s e d i n t h e a n a l y s i s o f t h e i n c r e a s e i n l i f e e x p e c t a n c y when c e r t a i n d e a t h c a u s e s a r e e r a d i c a t e d . W i t h t h i s t o o l w e a r e a b l e t o o b t a i n t h e d i f f e r e n c e b e t w e e n
t h e l i f e e x p e c t a n c y f o r a l l d e a t h c a u s e s a n d t h e o n e c a l c u l a t e d on t h e a s s u m p t i o n t h a t d e a t h s f r o m a c e r t a i n d e a t h c a u s e h a v e b e e n e l i m i n a t e d . T h e r e a r e s e v e r a l m e t h o d s f o r c a l c u l a t i n g t h e s e l i f e t a b l e s , among t h e m G r e v i l l e ' s m e t h o d s ( G r e v i l l e , l 9 4 8 a n d 1 9 5 4 ) , a n d t h o s e o f P r e s t o n e t a 1 . ( 1 9 7 2 ) a r e w e l l - k n o w n . However, when w e u s e t h e s e t r a d i t i o n a l m e t h o d s , t h e f o l l o w i n g p o i n t s a r e q u e s t i o n e d .
( a ) U s u a l l y
when d A , d B , a n d dAcB d e n o t e , r e s p e c t i v e l y , t h e i n c r e a s e o f l i f e e x p e c t a n c y a s s u m i n g t h a t d e a t h c a u s e s A , B, a n d A+B h a v e b e e n e r a d i c a t e d . I n t h i s c a s e e q u a l i t y d o e s n o t h o l d . T h a t i s t o s a y , t h e i n c r e a s e o f l i f e e x p e c t a n c y i s n o t a d d i t i v e w i t h t h e two c a u s e s of d e a t h .
( b ) G r e v i l l e ' s a n d o t h e r t r a d i t i o n a l m e t h o d s d i s c u s s t h e c a s e i n w h i c h d e a t h s f r o m a c e r t a i n c a u s e h a v e b e e n e r a d i c a t e d . W e a r e more c o n c e r n e d h e r e w i t h t h e c a s e i n w h i c h some p e r c e n t o f t h e d e a t h s f r o m s e v e r a l
c a u s e s a r e d e c r e a s e d . I n t h e f o r m e r c a s e , f o r e x a m p l e , o n e h u n d r e d t h o f a n i n c r e a s e i n l i f e e x - p e c t a n c y , a s s u m i n g t h a t d e a t h s f r o m c a u s e A h a v e b e e n e r a d i c a t e d , c a n n o t b e u s e d a s t h e i n c r e a s e o f l i f e e x p e c t a n c y when o n e p e r c e n t o f d e a t h s f r o m d e a t h c a u s e A h a s b e e n e l i m i n a t e d .
( c ) The t r a d i t i o n a l m e t h o d s a s s u m e t h a t s e v e r a l d e a t h c a u s e s a r e i n d e p e n d e n t .
I n o u r c a s e t h e a s s u m p t i o n i n ( c ) c a n n o t b e a v o i d e d . P o i n t s ( a ) a n d ( b ) , h o w e v e r , h a v e b e e n d i s c u s s e d by K e y f i t z
( 1 9 7 7 a ) . H e h a s d e r i v e d a p a r a m e t e r t h a t m e a s u r e s how much l i f e e x p e c t a n c y i s i n c r e a s e d d u e t o a m a r g i n a l r e d u c t i o n i n a n y c a u s e o f d e a t h . T h i s p a r a m e t e r i s a d d i t i v e when s e v e r a l c a u s e s o f d e a t h a r e c o n s i d e r e d .
Our m e t h o d i s a g e n e r a l i z a t i o n o f K e y f i t z ' s i d e a . When some p e r c e n t o f t h e d e a t h s f r o m c e r t a i n c a u s e s a r e e l i m i n a t e d , n o t n e c e s s a r i l y u n i f o r m l y i n a l l t h e a g e i n t e r v a l s , w e c a n e a s i l y g e t t h e l i f e e x p e c t a n c y , b a s e d o n a g i v e n l i f e t a b l e , by u s i n g t h e s e t s o f p a r a m e t e r s t h a t h a v e b e e n o b t a i n e d b e f o r e - h a n d . Our m e t h o d i s a l s o a d d i t i v e f o r s e v e r a l c a u s e s a n d h a s i n t e r e s t i n g a p p l i c a b i l i t y t o t h e s t u d y o f c a u s e s o f d e a t h .
2 . G E N E R A L I Z A T I O N OF KEYFITZ'S RESULTS
To b e g i n w i t h , I w i l l e x p l a i n b r i e f l y K e y f i t z ' s i d e a
( K e y f i t z , 1 9 7 7 a , b )
.
I f t h e c h a n c e o f d y i n g i n t h e t i m e i n t e r v a l d x o f a y e a r f o r a p e r s o n who h a s r e a c h e d a g e x h a s b e e n y ( x ) d x , s u p p o s e t h a t t h i s i s c h a n g e d t o y'
( x ) d x = y ( x ) ( 1 + 6 ) d x . I n t h i s c a s e , 6 w i l l b e a s m a l l n e g a t i v e q u a n t i t y , t y p i c a l l y - 0 . 0 1 ,r e p r e s e n t i n g 1 p e r c e n t improvement i n a l l c a u s e s a t a l l a g e s . The p r o b a b i l i t y o f l i v i n g t o a g e x t h e n c h a n g e s f r o m
1' ( x ) = e x p [ - i x u ( x ) ( 1 + 6 ) d x ] = 1 ( x )
0
t h i s b e i n g a p p r o x i m a t e l y
i n t h e n e i g h b o r h o o d o f d = 0 by T a y l o r ' s e x p a n s i o n . The l i f e e x p e c t a n c y a t b i r t h c h a n g e s f r o m
0 e; = j w l ' ( x ) d x
0
T h e r e f o r e w e h a v e
=-6 [ j W - 1 ( x ) l n 1 ( x ) d x / j W l ( x ) d x ] = - & H I
0 0
where
H = - j w l ( x ) l n 1 ( x ) d x / j w l ( x ) d x
0 0
T h i s e x p r e s s i o n shows t h a t t h e e f f e c t o f t h e e l i m i n a t i o n o f 1 p e r c e n t o f d e a t h s o n t h e l i f e e x p e c t a n c y w i l l i n c r e a s e
so
byH p e r c e n t .
S i m i l a r l y , s u p p o s e t h e a g e s p e c i f i c d e a t h r a t e f r o m t h e c a u s e i c h a n g e s f r o m
The p r o b a b i l i t y o f l i v i n g t o a g e x t h e n c h a n g e s f r o m 1 ( x ) t o
1 ( x ) [1 ( i ) ( x ) 1 1 ( x ) [ I
+
61n 1 ( i ) ( x ) 1w h e r e l ( i ) ( x ) i s t h e p r o b a b i l i t y o f l i v i n g t o a g e x i n t h e f a c e o f r i s k s f r o m t h e c a u s e i a l o n e . T h e r e f o r e , i n t h e same way a s w e m e n t i o n e d a b o v e , w e h a v e
0 0 0
( e '
-
e ) / e o = -6H ( i0 0
w h e r e
H ( i ) = - j L Y l ( X I l n 1 ( i ) ( X I d x / i w l ( X I d x
0 0
T h i s s h o w s t h a t t h e e f f e c t o f t h e e l i m i n a t i o n o f 1 p e r c e n t o f d e a t h s f r o m t h e i - t h c a u s e on t h e l i f e e x p e c t a n c y w i l l i n c r e a s e
0 e 0 b y H ( ~ ) p e r c e n t .
K e y f i t z ' s i d e a i s b a s e d on a n a s s u m p t i o n t h a t t h e number o f d e a t h s a t e a c h a g e g r o u p i s d e c r e a s e d a t a f i x e d r a t e . T h i s a s s u m p t i o n i s , h o w e v e r , o f t e n u n s u i t a b l e f o r r e a l s i t u a t i o n s . H e r e , t h e r e f o r e , l e t u s make a g e n e r a l a s s u m p t i o n t h a t t h e r a t e o f d e c r e a s e i n d e a t h s a t e a c h a g e g r o u p i s n o t n e c e s s a r i l y f i x e d .
A c c o r d i n g t o G r e v i l l e , i t i s known t h a t t h e p r o b a b i l i t y o f l i v i n g n y e a r a f t e r a g e x i n a l i f e t a b l e f r o m w h i c h t h e c a u s e i i s c o m p l e t e l y e l i m i n a t e d i s c l o s e l y a p p r o x i m a t e d f r o m
w h e r e P r e l a t e t o t h e l i f e t a b l e w i t h a l l c a u s e s p r e s e n t , n x
a n d
i n w h i c h nDx a n d
n ~ h i )
r e p r e s e n t r e s p e c t i v e l y t h e number o f d e a t h s f r o m a l l t h e c a u s e s a n d f r o m t h e c a u s e i a l o n e i n t h e a g e g r o u p ( x , x + n - 1 ) . I n t h i s c a s e , i f t h e number o f d e a t h s f r o m t h e c a u s e i i s n o t e r a d i c a t e d b u t d e c r e a s e d b y - 1 0 0 6 x % , i t c a n b e shown t h a t t h e p r o b a b i l i t y o f l i v i n g i n t h e age i n t e r v a l i sa s i n G r e v i l l e ' s e x p r e s s i o n ( G r e v i l l e , 1 9 4 8 ) .
T h e r e f o r e , i f t h e d e a t h s f r o m t h e c a u s e i a r e d e c r e a s e d a t t h e r a t e o f - 8 , - 8 , g = g , - 6
,
-6 f o r t h e a g e i n t e r v a l 0 - 4 , 5 - 9 ,...,
0 80 8 5
8 0 - 8 4 , 85+ r e s p e c t i v e l y , t h e p r o b a b i l i t y o f l i v i n g t o a g e x i s
1+5.0
'i'
6.1[
liiY5 (i) 65][
(i),
[5?0 5P5
. . .
5 P x-5 l+5Y,-5 x-51
1' (x) =
T h i s i s e q u i v a l e n t t o K e y f i t z ' s e x p r e s s i o n , 1 ( x ) ( 1 ( i ) ( x ) ] 6
,
i f A 0 = 6 5 - -...
= 6 . Here, u s i n g T a y l o r ' s t h e o r e m a s a b o v e t o h o l d l i n e a r i t y , w e h a v eT h i s i s a c l o s e l y approximated e x p r e s s i o n when e a c h 6 i s s m a l l . k
T h u s
Z b -
0 e o = / W l 1 ( x ) d x- rWl
( x ) d x0 0 ( 3 )
c a n b e e x p r e s s e d i n t h e l i n e a r f o r m
( i ) c a n b e c a l c u l a t e d f r o m e q u a t i o n ( 2 )
.
T h e r e f o r e , w h e r e Cxf r o m t h i s e x p r e s s i o n w e c a n o b t a i n t h e q u a n t i t y cf i n c r e a s e
0 0
i n t h e l i f e e x p e c t a n c y a t b i r t h eh- e o a t o n c e , i f w e h a v e ( i )
c o e f f i c i e n t s C o
, . . .
,C:k) f o r t h e c a u s e o f d e a t h i b e f o r e h a n d .3 . OUR METHOD OF COMPUTATION
F o r x = 0 , 5 , l 0 , .
. .
w e d e n o t e 1 ( x ),
1'
( x ) b y l X , 1; r e s p e c t i v e l y , a n d t h e n w r i t eL b e t h e t o t a l p e r s o n - y e a r s l i v e d by t h e s t a t i o n a r y popu- Let n x
l a t i o n i n t h e a g e - i n t e r v a l x t o x + n . Then
is the average number of years lived in the age-interval x to x+n by those who die in it. Using this nAx, we estimate ,LVx, as is often done, by
Here for the sake of brevity, we use A x , d x , d ~ , (i)
I Yx Lx and
Lx in place of 5Axf 5d;, 5 ~ x (i) 5Lx, and 5L;. In the last
age-interval, we use the age-interval of ages over
x.
For example, L l O 0 -
- wL1oo
.
Then we haveSimilarly
In the last age-interval
This is equivalent to Greville's expression (Greville, 1954) used to yet the life expectancy
g x
at age x in the last age- interval ( x , ~ ) when a death cause has been eradicated. Thisis approximately equal t o
Therefore we have
and
Sum up b o t h s i d e s o f t h e e x p r e s s i o n s ( 1 2 ) t o ( 1 6 ) a n d d i v i d e t h e t o t a l by l o . Then i f w e u s e
w e o b t a i n
I f w e w r i t e
a n d i f w e r e f e r t o t h e e x p r e s s i o n ( 5 ) , w e o b t a i n
( i )
I n t h i s e x p r e s s i o n , i f Cs5
+ c : ; ) + . .
.+C ( i ) i s d e n o t e d by C85 ( i ) 1 0 0w e o b t a i n t h e e x p r e s s i o n ( 3 a ) i n s e c t i o n 2 . I n t h e c a s e o f
O f 0
e60- e60 ' a s i n t h e case of a g e 0 , w e s t a r t f r o m 160 f o r a g e 60 a n d o b t a i n
Therefore if we write
we obtain
(i) are computed for x = 0,5,.
. .
,85,Now if the coefficients Cx
or x = 60,65,
...,
85, it seems to serve our purpose sufficiently.However, there is some doubt about the expression in the treat- ment of the last coefficient:
Therefore we computed cLi)to as advanced an age as possible, that is, to 100 years of age as shown above. In this case, it should be noted that C 1 (i) 00 is extremely small. In Japan, the data necessary for this computation are available.
In its practical use, we can use the table that summed up the figures of ages 85 and over form the table calculated in the above-mentioned way (cf. Table in Section 5 and Appendix tables)
.
4. RELATION BETWEEN OUR R E S U L T S AND THE RESULTS O F K E Y F I T Z
~ c c o r d i n g to Keyfitz (1968, p.342), it is known that the effect on OeO of A 5% - - 5q1x
-
5qx when the probability of dying in the age-interval (x,x+4) changed from 5q to 5q1x isapproximately
And if we take out only an age-interval (x, x+4) from the ex- pression (1 7)
,
we obtainThe expression (24) is one which was obtained by quite a different idea from (23), but both of them give much the same result. This is shown in the following way. I f dxyii)is small
1+bxyx (i) Therefore, if we denote P
5 x
-
5 x P by A5Px, we obtain from the expression (25)t h e r e f o r e
-lXA59, = SXyAi) l o g 5Px 1x+5 - - Bx1x+5
s o w e g e t o u r r e s u l t .
5 . AN APPLICATION AND COMMENTS
( i A s a n e x a m p l e , l e t m e show p a r t o f t h e table (Table 1) of Ck f o r m a l e s , w h i c h i s t a k e n o u t o f t h e T a b l e A2 ( A p p e n d i x ) computed by o u r m e t h o d , u s i n g t h e l i f e t a b l e a n d t h e s t a t i s t i c s o f mor-
t a l i t y o f J a p a n f o r 1 3 7 3 .
T a b l e 1 . The c o e f f i c i e n t s C L i ) f o r m a l i g n a n t n e o p l a s m s a n d c e r e b r o v a s c u l a r d i s e a s e o n J a p a n e s e m a l e s , 1970
Start of age Causes of death
interval coefficients ~ 1 9 " B30 b
T o t a l C 2.02632
$19 : Malignant Neoplasms
$30: Cerebrovascular Disease
A c c o r d i n g t o t h e t a b l e a b o v e , i f t h e number o f d e a t h s o r d e a t h r a t e f r o m B19 a t a l l a g e s i s d e c r e a s e d by 3 p e r c e n t , w e h a v e o n l y t o m u l t i p l y C by 0 . 0 3 t o g e t t h e i n c r e m e n t o f 0 eo:
2.02632 x 0 . 0 3 = 0 . 0 6 1 . A s a n o t h e r e x a m p l e , i f t h e d e a t h r a t e f r o m B19 i n 1970 i s d e c r e a s e d by 4 p e r c e n t a t a g e s o v e r 50 a n d by 2 p e r c e n t a t t h e o t h e r a g e s , 0 . 0 2 x ( C O + . . . + C 4 5 )
+
0.04 x( C
+...+
C B 5 ) = 0 . 0 7 2 , t h a t i s , 0 e o w i l l increase by 0.072 of a y e a r . 50And i f t h e d e a t h r a t e from B30 d e c r e a s e s by 3 p e r c e n t a t a g e s o v e r 6 0 ,
t h a t i s , O e 0 w i l l i n c r e a s e by 0.054 of a year. I f w e w a n t t o t a k e t h e s e two c a s e s t o g e t h e r , w e h a v e o n l y t o sum u p t h e t w o r e s u l t s a b o v e , s o w e h a v e
0 . 0 7 2
+
0 . 0 5 4 = 0.126 y e a rI n o u r c o m p u t a t i o n w e a l s o assume t h e i n d e p e n d e n c e o f d e a t h
c a u s e s a s d o t h e t r a d i t i o n a l m e t h o d s . [ A s s u m p t i o n (c) in s e c t i o n 1 1 . And t h e e f f e c t o f r e d u c t i o n o f d e a t h s f r o m a d e a t h c a u s e upon t h e l i f e e x p e c t a n c y i s e x t r e m e l y s m a l l . I f t h e a s s u m p t i o n o f i n d e p e n d e n c e i s n o t b u i l t u p , t h e e f f e c t w i l l become s t i l l s m a l l e r . I n f a c t , h o w e v e r , t h e i n d e p e n d e n c e d o e s n o t e x i s t .
K e y f i t z ' s a n d o u r m e t h o d s s h o u l d b e u s e d f o r t h e c a s e o f m a r g i n a l r e d u c t i o n b u t n o t f o r t h a t o f t h e e r a d i c a t i o n o f
d e a t h s f r o m a p a r t i c u l a r c a u s e . However, i f t h e s e m e t h o d s a r e t o b e u s e d f o r t h e l a t t e r c a s e , t h e e f f e c t o f t h e e r a d i c a t i o n o f d e a t h s f r o m a c a u s e upon t h e l i f e e x p e c t a n c y o b t a i n e d by
K e y f i t z ' s method w i l l b e a l i t t l e s m a l l e r t h a n t h a t o f G r e v i l l e l s n e t h o d , a n d t h e e f f e c t o b t a i n e d by o u r m e t h o d w i l l b e v e r y much s m a l l e r t h a n t h a t o f K e y f i t z ' s m e t h o d . I f w e t a k e t h e assump- t i o n o f i n d e p e n d e n c e i n t o a c c o u n t , we c a n c o n s i d e r o u r r e s u l t t o b e t h e u p p e r l i m i t o f t h e e f f e c t o f t h e e l i m i n a t i o n o f d e a t h s f r o m a c a u s e upon t h e e x p e c t a t i o n of l i f e .
6. CONCLUSION
~t h a s b e e n s a i d t h a t t h e t r a d i t i o n a l methods a r e i n a d e - q u a t e f o r m e a s u r i n g t h e e f f e c t o f t h e m a r g i n a l r e d u c t i o n o f d e a t h s f r o m a p a r t i c u l a r c a u s e upon t h e e x p e c t a t i o n o f l i f e . W e g e n e r a l i z e d K e y f i t z ' s method, which was d e v i s e d t o i m -
p r o v e t h e s e m e t h o d s . By o u r method w e c a n e a s i l y c a l c u l a t e t h e e f f e c t o f some p e r c e n t e l i m i n a t i o n o f d e a t h s f r o m a d e a t h c a u s e i n a n y a g e - i n t e r v a l upon l i f e e x p e c t a n c y . And t h e a p - pended t a b l e s computed f o r t h i s w i l l b e o f much u s e f o r t h e s t u d y o f main d e a t h c a u s e s . By means o f o u r method, a l s o , w e c a n e a s i l y g e t t h e p a r a m e t e r s which a r e e q u i v a l e n t t o K e y f i t z ' s p a r a m e t e r H ( i ) f o r a g e s o v e r 0 a n d Hso ( i ) f o r a g e s o v e r 60.
REFERENCES
Greville, T . N . E . (1 948) M o r t a l i t y T a b l e s A n a l y z e d b y Cause o f D e a t h . Record of the American Institute of Actuaries 37:283-294.
Greville, T . N . E . (1954) On t h e Formula f o r t h e L - F u n c t i o n i n a S p e c i a l M o r t a l i t y T a b l e E l i m i n a t i n g a G i v e n C a u s e o f
D e a t h . Transactions of the Society of Actuaries 6(1):1-5.
Keyfitz, N. (1968) I n t r o d u c t i o n t o t h e M a t h e m a t i c s o f Popu-
l a t i o n . Addison-Wesley, Reading, Mass.
Keyfitz, N. ( 1 9 7 7 a ) What D i f f e r e n c e Would I t Make if C a n c e r Were E r a d i c a t e d ? An E x a m i n a t i o n o f t h e T a e u b e r P a r a d o x . Demography 14:411-418.
Keyfitz, N. ( 1 9 7 7 b ) A p p l i e d M a t h e m a t i c a l Demography. John Wiley & Sons, N c w York.
Preston, S.H., N. Keyfitz, and R. Schoen (1972) C a u s e s o f D e a t h : L i f e T a b l e s f o r N a t i o n a l P o p u Z a t i o n s . Studies in Population Series, Seminar Press, N e w York.
APPENDIX
Table Al. A list of 12 causes of death according to the inter- national classification of diseases.
Enteritis and other diarrhoea1 diseases Gastritis, duodenitis and chronic
gastro-enteritis
Tuberculosis of respiratory system Other tuberculosis, including late effects
Malignant neoplasms, including neoplasms of lymphatic and haematopoietic tissue Chronic rheumatic heart disease
Ischaemic heart disease Other forms of heart disease
B 27 400-404 Hypertensive disease
B 30 430-438 Cerebrovascular disease
Pneumonia Bronchitis
Acute bronchitis and bronchiolitis
B 37 571 Cirrhosis of liver
B 38 580,584 Nephritis and nephrosis
B 45 a 794 Senility without mention of psychosis Motor vehicle accidents
All other accidents
B E 49 E9501E959 Suicide
N o t e s o n T a b l e s A 2 , A 3 . A 4 , and A 5
1. Data are Complete Life Tables and Death Statistics published by Department of Statistics and Information, Ministry of Health and Welfare, Japan.
2. Causes of death are based on International B List Number.
(cf. Table Al)
3. Each figure in the row of Total is the sum of figures for age 0 to 100+ in the corresponding column.
4. Each figure in the row of 85+ is the sum of figures for age 85 to 100+ in the corresponding column.
5. Each figure in the row of G-Method shows the increment of
0 eo by Greville's method in case of a cause being eradicated, and each figure sliould be multiplied by 10 2
.
6. Each figure in the last column is the sum of figures in the corresponding row and should be used for a small quantity of change in the death rate from all causes of death.
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RECENT PUBLICATIONS IN THE HEALTH CARE SYSTEMS TASK
Shigan, E.N. ,ed. ( 1 9 7 8 ) S y s t e m s M o d e l i n g i n H e a l t h C a r e . Proceedings of a n IIASA Conference, November 2 2 - 2 4 ,
1 9 7 7 (CP-78-12)
.
Gibbs, R.J. ( 1 9 7 8 ) T h e I I A S A H e a l t h Care R e s o u r c e s A l l o c a t i o n s u b - ~ o ' d e l s : Mark 1 (RR-78-08)
.
Gibbs, R.J. ( 1 9 7 8 ) A D i s a g g r e g a t e d H e a l t h Care R e s o u r c e A l l o c a t i o n Mode Z (RM-78-01)
.
Kaihara, S., N. Kawamura, K. Atsumi, and I. Fujimasa ( 1 9 7 8 ) A n a l y s i s and F u t u r e E s t i m a t i o n o f M e d i c a l Demands U s i n g A H e a l t h Care S i m u l a t i o n Model: A Case S t u d y o f Japan
(RM-78-03).
Fujimasa, I., S.,Kaihara, and K. Atsumi ( 1 9 7 8 ) A M o r b i d i t y Submodel o f I n f e c t i o u s D i s e a s e (Rid-78-1 0 )
.
Propoi, A. ( 1 9 7 8 ) M o d e l s f o r E d u c a t i o n a l and Manpower P l a n n i n g : A Dynamic L i n e a r Programming A p p r o a c h (RM-79-20).
Klementiev, A.A., and E.N. Shigan ( 1 9 7 8 ) A g g r e g a t e Model f o r E s t i m a t i n g H e a l t h Care S y s t e m R e s o u r c e R e q u i r e m e n t s ( A M E R I
(RM-78-21)
.
Hughes, D.J. ( 1 9 7 8 ) T h e I I A S A H e a l t h Care R e s o u r c e A l l o c a i t o n Sub-Model Mark 2 : T h e A l l o c a t i o n o f Many D i f f e r e n t
R e s o u r c e s (-RM-78-50)
.
Hughes, D.J. ( 1 9 7 8 ) T h e I I A S A H e a l t h Care R e s o u r c e A l l o c a t i o n S u b m o d e l : E s t i m a t i o n o f P a r a m e t e r s (RM-78-67)
.
Hughes, D.J. (1979) A Model o f t h e E q u i l i b r i u m B e t w c c n D i f f e r e n t L e v e l s o f T r e a t m e n t i n t h e H e a l t h Care S y s t e m s : P i l o t
v e r s i o n (WP-79-15)
.
Fleissner, P. (1979) C h r o n i c I l l n e s s e s and S o c i o - E c o n o m i c C o n d i t i o n s : T h e F i n l a n d Case 1964 and 1968 (WP-79-29).
Shigan, E.N., D.J. Hughes, P. Kitsul (1979) H e a l t h Care S y s t e m s Mode l i n g a t I I A S A : A S t a t u s R e p o r t (SR-79-4)
.
Rutten, F.F.H. (1979) P h y s i c i a n B e h a v i o u r : The Key t o M o d e l i n g 1lealt.h Care S y s t e m s f o r Government P l a n n i n g (WP-79-60).
A Committee Report (1979) to IIASA by the.participants in an Informal Meeting on H e a l t h D e l i v e r y S y s t e m s i n D e v e l o p i n g C o u n t r i e s (CP-79-10)
.
Shigan, E.N., P. Aspden, and P. Kitsul (1979) M o d e l i n g H e a l t h Care S y s t e m s : J u n e 1979 Workshop P r o c e e d i n g s (CP-79-15)
.
Hughes, D.J., E. Nurminski, and G. Royston (1979) N o n d i f f e r e n t i - a b l e O p t i m i z a t i o n P r o m o t e s H e a l t h Care (WP-79-90).
Rousseau, J.M., R.J. Gibbs (1980) A Model t o A s s i s t P l a n n i n g t h e P r o v i s i o n o f H o s p i t a l S e r v i c e s (CP-80-3).
Fleissner, P., K. Fuchs-Kittowski, and D.J. Hughes (1980)
A S i m p l e S i c k - L e a v e Model u s e d f o r I n t e r n a t i o n a l C o m p a r i s o n (WP-80-42)
.
Aspden, P., R. Gibbs, and T. Bowen (1980) DRAM B a l a n c e s Care (WP-80-43).
Aspden, P., and M. Rusnak (1980) The I I A S A H e a l t h Care R e s o u r c e A l l o c a t i o n Submocle I : Model C a l i b r a t i o n f o r Data from
C z s c h o s l o v a k i a (WP-80-53)
.
Kitsul, P. (1980) A Dynamic Approach t o t h e E s t i m a t i o n o f M o r b i d i t y (WP-80-71)
.
Shigan, E.N., and P. ::itsul (1 980) A l S e r n a t i v e A p p r o a c h e s t o M o d e l i n g H e a l t h C a r e Demand and S u p p l y (WP-80-80)
.
Hughes, D.J., and A. Wierzbicki (1980) DRAM: A Model o f I I e a l t h Care R e s o u r c e A 2 l o c a t i o n (RR-80-23)
.
Aspden, P. (1980) The I I A S A I I e a l t h Care R e s o u r c e A l l o c a t i o n Submode 2: DRAM C a l i b r a t i o n f o r Data from t h e S o u t h
k 7 e s t H e a l t h R e g i o n , U K (WP-80-115)
.
Mayhcw, L., and A. Taket (1980) RAMOS: A Model o f H e a l t h Care R e s o u r c e A l l o c a t i o n i n S p a c e (WP-80-125)
.
Mayhew, L. (1980) T h e R e g i o n a l P l a n n i n g o f H e a l t h Care S e r v i c e s : RAMOS and R A M O S - I (WP-80-166).
