The multiregional life table is a device for exhibiting the mortality and mobility history of an artificial population, called a cohort. Methods for constructing such a life table are treated in detail in Rogers (1975a, Chapter 3).
The cohort we deal with is a birth cohort, or radix. It represents a group of people born at the same moment in time and in the same region. Their life history is of special interest because it provides the necessary input information for numerical computations with multiregional demographic growth models. In multiregional demography, it is convenient to work with unit radices, i.e., birth cohorts of single persons. This allows a separation of the calculation of life table and other demographic statistics from the radix problem. Unless stated otherwise, the figures presented in this report will be per unit radix.
The computation of the multiregional life table begins with the estimation of age-specific death and outmigration probabili- ties. The probabilities are derived from observed schedules or rates of mortality and migration. The procedure is described at the end of this section. The probabilities of dying and outmi- grating of the female population of the two-region system of
Yugoslavia are given in Table 2.1. Note that they differ slightly from the probabilities presented in Rogers (1975a, p. 66), due to a small difference in the estimation method. As a consequence, all life-table statistics deviate sliqhtly from those in Roqers
(1975a). Probabilities and the two-region life table, consistent with Rogers', are given in Appendix B.
Probabilities of dying and migrating are the inputs for cal- culating life table statistics. The following statistics are computed by the program and are reviewed in the subsequent sec- tions:
1. life history of a regional birth cohort, 2. number of survivors at exact age x,
Table 2.1. Probabilities o f dying and outmigrating.
3 . number o f y e a r s l i v e d b e t w e e n two c o n s e c u t i v e a g e s , o r ,
The quantities j O R i ( ~ ) , j O R i d (x) and j O R i k ( ~ ) may also be expressed per unit born, i.e. for a cohort of a single person.
They then may be interpreted as probabilities. For instance,
?,
R . (x) is the probability that a j-born person is in region i j0 1
at exact age x, and jOiik(x) is the probability that a j-born person changes his residence from i to k between ages x and x + 5. The relation between, for example, jo 'i 9 (x) and 1 0 .
R .
1 (x)is straightforward:
The probability-interpretation will be particularly useful in fertility and mobility analyses of stationary and stable populations.
The life history of the cohorts is derived by the consecu- tive multiplication of the birth cohort by the mortality and migration probabilities. For example, of the 100,000 babies born in Slovenia (region I), 3081 will die before they reach age 5, i.e.,
and 1310 will move to the Rest o f Yugoslavia (region 2),
The rest, i. e.,
r e m a i n i n S l o v e n i a , a n d a r e t h e r e a t e x a c t a g e 5 . T h e r e f o r e , o f t h e f e m a l e s b o r n i n S l o v e n i a , o n l y 9 5 . 6 5 w i l l s t i l l b e t h e r e 5 y e a r s l a t e r .
Of t h e 1 0 0 , 0 0 0 f e m a l e s b o r n i n S l o v e n i a , 9 6 , 9 1 9 w i l l s t i l l b e a l i v e a t e x a c t a g e 5 . A t o t a l o f 9 5 , 6 0 8 w i l l s t i l l b e i n S l o v e n i a a n d 1 , 3 1 0 w i l l b e i n t h e R e s t o f Y u g o s l a v i a . Of t h e s e 9 5 , 6 0 8 , t h e n u m b e r o f g i r l s d y i n g b e f o r e r e a c h i n g a g e 1 0 i s
a n d t h e n u m b e r m i g r a t i n g t o t h e R e s t o f Y u g o s l a v i a i s
T h e r e s i d u a l i s t h e n u m b e r o f g i r l s who w e r e i n S l o v e n i a a t a g e 5 a n d a r e s t i l l t h e r e a t a g e 1 0 :
Note that
10R16
(5) =10R16
(5)/10R1 (0) = 0.00207 is the probability that a girl born in Slovenia dies in that region between ages 5 and 10. An analogous interpretation may be given to10112
(5) and(5). Expressing the life histories per unit born yields a set of unconditional probabilities.
What happens to the 1310 migrants born in Slovenia, but who are in the Rest of Yugoslavia at exact age 5? They die, move back to Slovenia or stay in the Rest of Yugoslavia. If one assumes that the mortality and migration behavior depends on the region of residence at the beginning of the interva1,t then
girls die before reaching age 10, and
move back to Slovenia, while
remain in the Rest of Yugoslavia.
Pursuing this procedure until the last age group, we obtain a detailed description of the life history of the people born in Slovenia. The last age group is open-ended; therefore all people who reach age 85 are expected to die in that age group, i.e. qi(85) = 1.0, and hence
tThis is the Markovian assumption. It is a fundamental hypo- thesis underlying multiregional and other- increment-decrement life tables.
Note t h a t t h e t o t a l number o f d e a t h s i s e q u a l t o t h e t o t a l number o f b i r t h s . F o r example, o f t h e 100,000 b a b i e s b o r n i n S l o v e n i a , 84,721 d i e i n S l o v e n i a and 15,279 d i e i n t h e R e s t o f Y u g o s l a v i a .
An a n a l o g o u s p r o c e d u r e i s f o l l o w e d t o d e r i v e t h e l i f e h i s t o r y o f t h e f e m a l e s b o r n i n t h e R e s t o f Y u g o s l a v i a ( T a b l e 2 . 2 ) .
2.2 E x p e c t e d Number o f S u r v i v o r s a t E x a c t Age x
T a b l e 2 . 3 i s a n a g g r e g a t i o n o f T a b l e 2.2. We n o t e d e a r l i e r t h a t of t h e 100,000 g i r l s b o r n i n S l o v e n i a , t h e r e a r e 1310 who a t e x a c t a g e 5 r e s i d e i n t h e R e s t o f Y u g o s l a v i a . T h i s number may a l s o b e found i n T a b l e 2.3. Of t h e p e o p l e b o r n i n S l o v e n i a and r e s i d i n g i n t h e R e s t o f Y u g o s l a v i a a t a g e 1 0 , f o r example, some were t h e r e a l r e a d y a t a g e 5 and s t a y e d t h e r e , w h i l e o t h e r s moved i n from S l o v e n i a , i . e .
where j O R i ( ~ ) i s t h e number o f p e o p l e i n r e g i o n i a t e x a c t a g e x , who were b o r n i n r e g i o n j . T h i s e x p r e s s i o n i s e q u i v a l e n t t o :
The t o t a l o f 2392 i s g i v e n i n T a b l e 2 . 3 , i t s components may be found i n T a b l e 2.2
T a b l e 2 . 3 g i v e s t h e number of p e o p l e by p l a c e o f b i r t h and p l a c e of r e s i d e n c e . Hence, i t m e a s u r e s t h e a g e s t r u c t u r e o f t h e l i f e t a b l e p o p u l a t i o n , a l t h o u g h o n l y p e o p l e a t e x a c t a g e s a r e c o n s i d e r e d . A more c o m p l e t e e x p r e s s i o n o f t h e a g e s t r u c t u r e i s g i v e n i n t h e n e x t s e c t i o n .
T a b l e 2 . 2 . L i f e h i s t o r y o f i n i t i a l c o h o r t .
Table 2.2. (cont'd)
initial region of cohort r.yudos.
t l t l l l l l l i i l i l l l l l l l l l l l l l l l i i i i i i
1 . - r e g i o n o f residence s l o v e n i a d e a t r ~ s m i g r a n t s t o
a g e s l o v e n i a r . y u g o s .
t o t a l 1 7 1 3 . 1 4 2 0 3 . 1 2 6 .
2 . - r e g i o n o f r e s i d e n c e r . y u g o s . d e a t h s m i g r a n t s t o
a g e s l o v e n i a r . y u g o s .
T a b l e 2 . 3 . Expected number o f s u r v i v o r s a t e x a c t a g e x i n e a c h r e g i o n .
2:
i n i t i a l r e g i o n o f c o h o r t s l o v e n i a111111111~1111111*11111i1&1111~111
t o t a l s l o v e n i a r . y u g o s .
2:
i n i t i a l r e g i o n o f c o h o r tr .
y u g o s .1 1 1 1 1 1 1 1 1 1 1 t t 1 1 1 N 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 N 1
t o t a l s l o v e n i a r . y u g o s .
The c o m p u t a t i o n of t h e e x p e c t e d number of s u r v i v o r s a t e x a c t a g e x i n a m u l t i r e g i o n a l s y s t e m i s more c o n v e n i e n t l y p e r -
formed u s i n g m a t r i x n o t a t i o n . For o u r two-region example, l e t
Note t h a t L ( 0 )
-
i s a d i a g o n a l m a t r i x w i t h t h e r e g i o n a l r a d i c e s i n t h e d i a g o n a l . The m a t r i x a n a l o g u e o f e q u a t i o n ( 2 . 3 ) i s t h e nFor x = 5 , we have
A s b e f o r e , we may e x p r e s s t h e l i f e h i s t o r y of t h e hypo- t h e t i c a l p o p u l a t i o n i n t e r m s o f u n i t b o r n . T h i s y i e l d s a s e t o f p r o b a b i l i t i e s . F o r example, t h e p r o b a b i l i t y t h a t a p e r s o n b o r n
A
i n r e g i o n j be i n r e g i o n i x y e a r s l a t e r i s s i m p l y j O R i ( ~ ) = j O I i ( ~ ) / j O R j ( 0 ) ' which i s e a s i l y d e r i v e d from T a b l e 2.3. The p r o b a b i l i t y o f s u r v i v i n g t o a g e x i s t h e p r o d u c t o f c o n d i t i o n a l p r o b a b i l i t i e s :
The p r o b a b i l i t y o f s u r v i v i n g f r o m x t o x
+
n i s a l s o e a s i l y computed from T a b l e 2 . 3 . I t i s e q u a l t o t h e p r o d u c t~t f o l l o w s f r o m ( 2 . 6 ) t h a t
Hence.
The p r o b a b i l i t y t h a t a n i n d i v i d u a l i n r e g i o n i a t a g e x w i l l b e i n j, n y e a r s l a t e r , i s t h e r e f o r e g i v e n by
w h e r e t h e e n t r i e s o f
-
!L ( x+
n ) and 1-
( x ) a r e f o u n d i n T a b l e 2 . 3 and( X
+
n ) a n d ( x ) a r e t h e e n t r i e s d i v i d e d by t h e r e g i o n a l r a d i c e s .- -
F o r e x a m p l e , i f o n e knows t h e d i s t r i b u t i o n o f p e o p l e a t t h e t i m e t h e y e n t e r t h e l a b o r f o r c e o r m a r r i a g e , a g e 2 0 s a y , and d e n o t e t h i s by f w ( 2 0 )
-
1 , t h e n t h e i r d i s t r i b u t i o n a t r e t i r e m e n t a g e , 6 0 s a y , i s g i v e n byThe p r o b a b i l i t y t h a t a n i n d i v i d u a l i n S l o v e n i a a t a g e 20 w i l l b e i n t h e R e s t o f Y u g o s l a v i a a t r e t i r e m e n t a g e i s q u i t e h i g h ,
a l m o s t o n e - s e v e n t h .
2 . 3 D u r a t i o n o f R e s i d e n c e a n d Age C o m p o s i t i o n o f t h e L i f e T a b l e P o p u l a t i o n
The k n o w l e d g e o f t h e p r o b a b i l i t y t h a t a p e r s o n b o r n i n a g i v e n r e g i o n s u r v i v e s t o a g e x and i s t h e n i n a n o t h e r g i v e n r e g i o n l e a d s u s t o a s k : how l o n g w i l l t h e p e r s o n s t a y i n t h a t r e g i o n ? T h i s d u r a t i o n - o f - r e s i d e n c e q u e s t i o n may b e a n s w e r e d f o r p e r s o n s b o r n i n a g i v e n r e g i o n and f o r p e r s o n s l i v i n g i n a
s p e c i f i c r e g i o n a t a g e x .
a . D u r a t i o n o f R e s i d e n c e by P l a c e o f B i r t h
The number o f y e a r s i n d i v i d u a l s a t a g e x may e x p e c t t o l i v e i n t h e n e x t f i v e y e a r s , o n t h e a v e r a g e , i s
w h e r e i n t h e t w o - r e g i o n c a s e
w i t h L . ( x ) b e i n g t h e e x p e c t e d number o f p e r s o n - y e a r s l i v e d i n j 0 1
r e g i o n i b e t w e e n x a n d x
+
5 , by a n i n d i v i d u a l b o r n i n r e g i o n j . I t d e n o t e s t h e a v e r a g e d u r a t i o n of r e s i d e n c e i n r e g i o n i by a j - b o r n p e r s o n and d e p e n d s on two c o m p o n e n t s : ( i ) t h e p r o b a b i l i t y o f s u r v i v i n g t o a g e x a n d ( i i ) t h e a v e r a g e t i m e s p e n t i n r e g i o n i i n a 5 - y e a r i n t e r v a l by a p e r s o n o f a g e x a t t h e b e g i n n i n g o f t h e i n t e r v a l .The numerical approximation of (2.9) has given rise to a number of variants of life table construction (Keyfitz, 1968, p. 228). A simple approximation of L(x] is a linear combination
-
of the probabilities of surviving to exact ages x and x
+
5:In the computer program, a is set equal to 0.5. here fore,
For example, L(10) given in Table 2.4 is computed from Table 2.7
.
as follows:
The terminal age interval in a life table is a half-open interval: z years and over. The probability of dying in this interval therefore is unity. Since the length of the interval is infinite, (z
- +
5) is not available and (2.11) cannot be used to compute L(z).-
The number of years lived in the last age group is given by:where M(z) is a matrix with observed regional death and migration
-
rates of the last age group (see Section 2.7).
The duration of residence or person-years-lived interpreta- tion of L(x) is one of several possible peyspectives. It also
-
may be viewed as a measure of the age composition of the multi- regional life table population. In this perspective, an element
T a b l e 2 . 4 . Number of y e a r s l i v e d i n e a c h r e g i o n b y a u n i t b i r t h
L.(x) denotes the number of j-born people in region i of age j0 1
x to x
+
5, per unit born. The product . L. (x)X R . (0) is the 10 1 j0 Itotal number of j-born people living in region i and x to x
+
5years old. Note that L(x) represents the relative population
-
distribution by place of residence and place of birth. Instead of being expressed in percentages (fractions of the total), or in some other manner, the population is given in unit births.
This is a logical procedure in demography since it separates the fertility component from the survivorship (mortality and migration) component. This will be seen to be a very convenient way of "scaling" in spatial population analysis.
b. Duration of Residence by Place of Residence
As mentioned above, the duration of residence in each region depends on two components: (i) the probability of surviving to age x, and (ii) the average time spent in each region during the 5-year interval by a person of age x at the beginning of the interval. The latter component is the person-years lived between x and x
+
5 by region of residence at age x and is equal toNote that
Lr
(x) is a conditional measure, since it gives the duration of residence in each region between ages x and x+
5,given that the person reaches age x and is in a specific region at that time. Using the linear approximation of L(x) we may
-
reduce this expression to
The number of years lived in the last age group is
w h i c h i s s i m p l y
N u m e r i c a l v a l u e s f o r L ( x ) a r e g i v e n i n T a b l e 2 . 5 .
-
r2 . 4 T o t a l Yumber o f Y e a r s L i v e d Beyond Age x
The t o t a l number o f y e a r s n e w l y b o r n b a b i e s may e x p e c t t o l i v e b e y o n d a g e x i s
w h e r e z i s t h e o l d e s t a g e g r o u p . F o r e x a m p l e , t h e v a l u e o f T ( 1 0 ) i n T a b l e 2 . 6 i s
The number o f y e a r s t h a t a g i r l , j u s t b o r n i n S l o v e n i a , may e x p e c t t o l i v e b e y o n d a g e 10 i s 6 2 . 7 1 . From t h i s t o t a l , 5 5 . 2 6 y e a r s a r e e x p e c t e d t o b e l i v e d i n S l o v e n i a a n d 7 . 4 5 y e a r s i n t h e R e s t o f Y u g o s l a v i a . S i m i l a r l y , a new-born S l o v e n i a n b a b y g i r l h a s 1 0 T ( 6 0 ) o r 1 5 . 7 4 y e a r s o f r e t i r e m e n t t o l o o k f o r w a r d t o , 2 . 4 8 y e a r s o f w h i c h w i l l b e s p e n t i n t h e R e s t o f Y u g o s l a v i a .
2 . 5 E x p e c t a t i o n o f L i f e
The m o s t i m p o r t a n t l i f e t a b l e s t a t i s t i c i s t h e l i f e e x p e c - t a n c y . The e x p e c t a t i o n o f l i f e a t a g e x i s t h e number o f y e a r s a n i n d i v i d u a l may e x p e c t t o l i v e b e y o n d a g e x , g i v e n t h a t h e
Table 2 . 5 . Number o f y e a r s l i v e d i n e a c h r e g i o n by a person
T a b l e 2 . 6 . T o t a l number of y e a r s l i v e d beyond age x.
i n i t i a l r e g i o n o f c o h o r t s l o v e n i a
...
t o t a l s l o v e n i a r . y u g o s .
age
* *
i n i t i a l r e g i o n o f c o h o r t r . y u g o s ....
t o t a l s l o v e n i a r . y u g o s .
r e a c h e s a g e x . I n m u l t i r e g i o n a l demography, two t y p e s o f l i f e e x p e c t a n c i e s may be d i s t i n g u i s h e d : l i f e e x p e c t a n c y by p l a c e of r e s i d e n c e a n d l i f e e x p e c t a n c y by p l a c e o f b i r t h .
a . L i f e E x p e c t a n c y by P l a c e o f R e s i d e n c e
The l i f e e x p e c t a n c y by p l a c e o f r e s i d e n c e g i v e s t h e e x p e c t a - t i o n o f l i f e a t a g e x o f a p e r s o n r e s i d i n g i n a s p e c i f i c r e g i o n a t t h a t a g e . I t i s computed a s f o l l o w s :
w h e r e i n t h e t w o - r e g i o n c a s e
and i x e j ( x ) i s t h e a v e r a g e number o f y e a r s l i v e d i n r e g i o n j beyond a g e x by a n i n d i v i d u a l r e s i d i n g i n r e g i o n i a n d x y e a r s o f a g e ( w h a t e v e r t h e r e g i o n o f b i r t h ) . The l i f e e x p e c t a n c y a t e a c h a g e e x c e p t t h e f i r s t i s h i g h e r t h a n T ( x ) , s i n c e
-
i t i s a c o n d i t i o n a l m e a s u r e . Note t h a t f o r t h e l a s t a g e g r o u p e ( z ) =I r ( z ) = [ M ( z ) I - ~
. -
The e x p e c t a t i o n s o f l i f e by p l a c e o f r e s i d e n c e f o r 10-year o l d g i r l s , f o r e x a m p l e , a r e ( T a b l e 2 . 7 )
T a b l e 2 . 7 . E x p e c t a t i o n s o f l i f e by p l a c e o f r e s i d e n c e .
a g e
**
r e g i o n o f r e s i d e n c e a t a g e. . .
x s l o v e n i a t o t a l s l o v e n i a r . y u g o s .a g e
* * *
r e g i o n o f r e s i d e n c e a t a g e x r . y u g o s .* * * * * * n * * * * n * * * * * * * * * * * * * * * * * * * * * * * * * m t o t a l s l o v e n i a r . y u z o s .
A 10-year o l d g i r l , l i v i n g i n S l o v e n i a , may e x p e c t t o l i v e an- o t h e r 64.87 y e a r s . Of t h i s , 6.30 y e a r s w i l l b e s p e n t i n t h e R e s t o f Y u g o s l a v i a , i . e . 10%. A g i r l o f t h e same a g e i n t h e R e s t of Y u g o s l a v i a may e x p e c t t o s p e n d 0.77 y e a r s i n S l o v e n i a .
b . L i f e Expectancy by P l a c e o f B i r t h
T h i s measure g i v e s t h e e x p e c t a t i o n o f l i f e a t a g e x by r e g i o n o f b i r t h o f t h e p e r s o n . The r e g i o n o f r e s i d e n c e a t a g e x i s n o t t a k e n i n t o a c c o u n t . D e f i n e t h e d i a g o n a l m a t r i x ( x )
,
w i t h t h eA
-
e l e m e n t s o f t h e v e c t o r { l ) ' l ( x ) i n t h e d i a g o n a l ( { I ) '
- - -
i s a row v e c t o r o f o n e s ) . F o r t h e t w o - r e g i o n c a s e t h i s g i v e sThe m a t r i x o f l i f e e x p e c t a n c i e s by p l a c e o f b i r t h i s o b t a i n e d a s f o l l o w s :
L i f e e x p e c t a n c i e s o f 10-year o l d g i r l s a r e ( T a b l e 2 . 8 )
A g i r l b o r n i n S l o v e n i a may e x p e c t t o l i v e a n o t h e r 64.85 y e a r s ,
when
r e a c h i n g 10 y e a r s o f a g e . Of t h i s , 7.70 y e a r s w i l l b e s p e n tin the Rest of Yugoslavia, i,e. 12%. At age 65, however, 2.34 years of the future lifetime of 14.47 years will be spent in the Rest of Yugoslavia, i.e. 16% (Table 2.81.
It is the special feature of the multiregional life table that the demographic measure of the expectation of life is de- composed according to where that life is spent. It introduces the spatial dimension into classical demographic analysis.
2.6 Survivorship and Outmigration Proportions
A useful application of the multiregional life table is found in multiregional population projection. The assumption is that the survivorship and migration behavior exhibited by the station- ary life table population adequately represents the survivorship and migration experience of the empirical population for which the life table was developed.
The necessary information for the projection of age groups beyond the first one is given by age-specific matrices of sur- vivorship proportions. The number of people in age group (x
+
5,x
+
10) in the stationary population iswhere, in the two-region case,
with sij(x) being the proportion of individuals aged x to x
+
4who survive to be x
+
5 to x+
9 years old 5 years later, by new places of residence.T a b l e 2 . 8 . E x p e c t a t i o n s o f l i f e by p l a c e o f b i r t h .
F o r e x a m p l e , t h e number o f p e o p l e i n t h e R e s t o f Y u g o s l a v i a a t a g e s 1 5 t o 1 9 , who were b o r n i n S l o v e n i a , p e r u n i t r a d i x i s
( T a b l e s 2 . 4 a n d 2 . 9 )
T h e c o m p u t a t i o n o f L ( x ) i n t h e l i f e t a b l e
-
i s n o t p e r f o r m e d u s i n g ( 2 . 2 1 ) b u t b y ( 2 . 1 1 ) . I n ( 2 . 2 1 ) , t h e unknown i s S ( x ) ;-
t h e r e f o r e
F o r x = 10 i n t h e Y u g o s l a v i a n e x a m p l e , S ( x )
-
i sT h e number 0 . 0 1 6 3 1 , f o r i n s t a n c e , i s t h e p r o p o r t i o n o f t h e q i r l s r e s i d i n g i n S l o v e n i a a n d 10 t o 14 y e a r s o l d t h a t w i l l b e a l i v e a n d i n t h e R e s t o f Y u g o s l a v i a 5 y e a r s f r o m now.
W i t h t h e s u r v i v o r s h i p p r o p o r t i o n s , a l l t h e l i f e t a b l e s t a t i s - t i c s a r e d e r i v e d . T h e y a r e s u m m a r i z e d i n T a b l e 2 . 1 0 . t
t A summary t a b l e i s p r o d u c e d b y t h e c o m p u t e r f o r a s y s t e m o f t w o r e g i o n s o n l y .
Table 2.9. Survivorship proportions.
r e g i o n s l o v e n i a
M M M M M M M M M M M M M M M M
t o t a l s l o v e n i a r.yugos.
r e g i o n r.yugos.
...
M l t M . M . ~ . . l t . M lt o t a l s l o v e n i a r.yugos.
Table 2.10. Multiregional (two-region) life table option 3." tColumn variables are defined on the next page.
T a b l e 2.10 ( c o n t i n u e d )
2 . 7 E s t i m a t i o n o f A g e - S p e c i f i c O u t m i g r a t i o n a n d D e a t h
It can be shown that the probability matrix P(x) is (Rogers and
..
Ledent, 19761
where, for a two-region model,
with pij(x) being the probability that an individual in region i at exact age x will survive and be in region j five years later.
The off-diagonal elements are migration probabilities analogous to transition probabilities in Markov theory. The diagonal ele- ment pii(x) denotes the probability of surviving and remaining in
(or returning to) region i. The elements of each column in P(x)
+
do not sum up to unity since the effects of mortality are included.
Rather, P(x) is analogous to the transition matrix of an absorbing
-
Markov chain. Note than an element p (x) does not denote the i j
probability of making a move from i to j by a person living in i at the beginning of the transition period. !ghat it represents is the probability that an individual in region i at the beginning of the time period is in region j at the beginning of the next period. During the period, several moves may have been made.
For example, the matrix of probabilities at age 10 is (Table 2.1)
0.007381 0.996834 0.998512 0.997615
The probability that a female in Slovenia at age 10 will survive to age 15 is 0.998512. The probability that she will be in the Rest of Yugoslavia at age 15 is 0.007381.
The probabilities of dying are found by subtraction. The probability that an individual in region i at age x dies before reaching x
+
5 isThe probability of dying in the next five years for a 10 year old in Slovenia is (Table 2.1)
Note that (2.25) is analogous to the single region formula
Formula (2.27) is equivalent to equation (1.1.9) of Keyfitz (1968, p. 14) and Keyfitz and Flieger (1971, p. 135). The probability of dying is then
b. Estimation Under Option 1
On the assumption of no multiple transitions, the outmigra- tion probability pij (x) is given by (Rogers, 1975a, p. 82)
5 Mi. (x) p . . (x) =
1 I 5 5
1
+ 2
Mi& (x)+
- 21
j #i
Mij (XI
The probability of dying in region i is
The probability of surviving and remaining in the region is found as a residual
Probabilities computed by this method are given in Appendix B as the first three columns of the two-region life table. The matrix of probabilities at age 10 for our example is
For a single region case, p .
.
(x) = 0, and formula (2.25) re- 1 3duces to (2.27). The distinction between multiple transitions and no multiple transitions is irrelevant in a single-region situation, since one can die only once.
The assumption of multiple versus no multiple transitions affects not only the probabilities directly, but also the person- years lived in the last open-ended age group. Recall (2.12),
Under the assumption of no multiple transitions, people cannot migrate and die during the same time-interval. Since all people
die in the last age group, the off-diagonal elements of M ( z )
-
are zero and the diagonal consists of regional death rates. HenceA
L . (z) = . L . (z)/Mi6(z) (Rogers, 1475a, p. 64).
j 0 1 l o 1
2.8 Aggregated Life Table Statistics
The life table statistics considered thus far refer to a multiregional system. The life table functions are basically matrix equations and give regional statistics. In order to aggre- gate the regional measures to yield the life table statistics for the whole system (country), regional weights must be introduced.
The weights are the regional radices specified by the user.
The life table of the aggregate system is a single-region table derived from a set of age-specific mortality rates, which are computed as follows:
with RR!') the radix ratio in the life table or stationary popu- 3
lation
The d e a t h r a t e o f the l a s t a g e g r o u p i s
T a b l e 2.11 g i v e s t h e a g g r e g a t e d l i f e - t a b l e s t a t i s t i c s f o r Y u g o s l a v i a . E q u a l r a d i c e s a r e s p e c i f i e d f o r b o t h r e g i o n s . I n i n t e r p r e t i n g t h e r e s u l t s , o n e m u s t k e e p i n mind t h a t u n l e s s re- g i o n a l r a d i c e s a r e s e t i n p r o p o r t i o n t o a n e s t i m a t e o f a p p r o p r i a t e l i f e t a b l e b i r t h s , t h e a g g r e g a t e d l i f e t a b l e v a l u e s w i l l b e i n - c o r r e c t . S e t t i n g a l l r a d i c e s e q u a l i m p l i e s t h a t r e g i o n a l b i r t h s i n t h e l i f e t a b l e p o p u l a t i o n a r e a l l e q u a l i n number. I f i n t h e o b s e r v e d p o p u l a t i o n t h e y a r e n o t , t h e n o b v i o u s l y t h e l i f e t a b l e s t a t i s t i c s i n t h e a g g r e g a t e d t a b l e a r e n o t r e a l i s t i c .
N o t e t h e d i f f e r e n c e b e t w e e n T a b l e 2.11 a n d T a b l e 1.6b. The l a t t e r i s d e r i v e d f r o m a s e t o f a v e r a g e n a t i o n a l a g e - s p e c i f i c d e a t h r a t e s . R e g i o n a l d i f f e r e n c e s a r e n o t a c c o u n t e d f o r a n d i n - t e r n s 1 m i g r a t i o n i s n o t c o n s i d e r e d . T a b l e 2 . 1 1 , o n t h e o t h e r h a n d , i s a g g r e g a t e d from a m u l t i r e q i o n a l l i f e t a b l e , w h i c h e x p l i c i t l y c o n s i d e r s r e g i o n a l l y d e v i a t i n g m o r t a l i t y a n d m i g r a t i o n . I f mor- t a l i t y i s t h e same i n a l l r e g i o n s , t h e n T a b l e 2.11 a n d T a b l e 1 . 6 b c o i n c i d e .
- - - -
o n m m o m G
N L - ~ O ~ ~ - T U ~ T
= 0 0 0 0 - - - N - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
. . .
C O O O O O O O O