2 starts from the observation of survivorship proportions and
attenpts to determine the constant (mortality and) migration rates or survival probabilities that would lead to such proportions.
Unfortunately, the survivorship proportions are observed for a
time period, say five years as in the above numerical illustration, during which the age-specific migration rates are not necessarily constant and may fluctuate greatly. Second, the nature of the Option 2 method does not permit us to estimate mortality/mobility rates and survival probabilities separately for each group. Since equation (141) relates statistics of two consecutive age groups, estimation errors made on a given age group are passed on the the next.
In brief, since migration is a more volatile phenomenon than mortality, (i.e., age-specific outmigration rates, unlike age- specific mortality rates, may present large fluctuations over a short period of time), the Option 2 method does not appear to be as useful a method for constructing a multiregional life table as for constructing a single-region life table.
Evaluation of the Alternative Vari.ants in ~ultiregional ~ i f e Table Construction
As just seen, the choice of the Option 2 method as a way of constructing a multiregional life table must be avoided whenever possible: a multiregional life table is best constructed when using the Option 1 method based on the equalization of life table rates with their observed counterparts.
Moreover, because of the type of mortality and mobility data available, the mixed approach (a combination of the movement and transition approaches) must preferably be chosen among the vari- ations of the Option 1 method. However, the use of the trans- ition approach yields acceptable results in view of the slight discrepancy existing between corresponding movement and transition death rates. That statement would not be true if the movement
approach was used instead. In other words, in contrast to the analysis of life status (Schoen and Nelson, 1974; Schoen, 1975)
*
the study of interregional migration generally requires the choice of the mixed approach which is closely related to the transition approach.
It is clear that the most feasible integration methods to
derive { L 1 are the linear and cubic integration methods. However,
X
in contrast to the linear method that can be easily used whatever the approach chosen (movement, transition, mixed), the cubic method can only be used in the case of the movement approach. Since
movement migration data are sometimes available, we have used this integration method to calculate a multiregional life table of the four region system for the U.S. female population in which the values of the observed transition rates were substituted for those of the movement rates. The age-specific survival probabilities thus obtained (Table 13) were then directly comparable with the ones similarly obtained when using a linear integration method
( Takle 9).
The result is that: a) the cubic integration method does not yield radically different estimates, b) the discrepancy be-
tween the linear and cubic estimates mostly affects the retention I
probabilities and the probabilities of dying, and c) this dis- I
I
crepancy tends to be higher for older ages (see age qroup 75 to 80).
**
The mean durations of transfers implied by the choice of the cubic integration method appear in the bottom part of Table 13.The discrepancies between the linear and cublc integration methods on the one hand, and the linear and the interpolative- iterative methods on the dther hand point in opposite directions.
Whereas the interpolative-iterative method yields higher retention probabilities and smaller probabilities of dying than the linear integration method (as suggested by the comparison of the survival
*The type of data available for the problem studied by Schoen makes the use of the movement approach preferable.
**The estimates of the survival probabilities for age groups 5 to 10 and 80 to 85 were identical in both Tables 9 and 13 since the linear integration method was substituted for the cubic integra- tion method.
probabilities in Tables 3 and 7), the cubic integration method yields smaller retention probabilities and higher probabilities of dying. Also, the interpolative-iterative method yields
(see Table 7) ax coefficients slightly higher than 6 2.5 (except for the first age group), while the cubic integration method leads to a: coefficients much higher than 2 . 5 (see able 13).
If the interpolative-iterative method is assumed to be more accurate than any other method, it then appears that the linear integration method yields estimates of the multistate life table functions which are better than those of the cubic integration method. Then, even if its use is made possible by the type of data available, the cubic integration method will not be prefered to the linear integration method. Moreover, since the interpolative- iterative method yields estimates of the multistate life table
functions only slightly different from those obtained in the linear case, the linear integration method would generally be prefered because of the larger computer time required for the interpolative-
iterative method.
Finally, the mixed approach of the Option 1 method based on a linear integration over {l } for deriving { L ~ } appears as the
Y
best variant in calculating a multiregional life table.*
Migration Rates and the Calculation of a ~ultiregional Life Table Clearly, the accuracy of the columns of a multiregional life table calculated by the Option 1 method depends on the precision of observed mortality and mobility rates' measurement.
Impact of Alternative Measures of ran sit ion Migration Rates Whereas the measurement of movement rates as proposed by (134) and (135) does not raise any particular problem (straight- forward extension of the single region case), the measurement of transition rates suggested in (136) and (137) raises some difficul- ties because the numerators and denon~inators of these definitions
*The present conclusion is indeed limited to the case of a demo- graphic system for which available data are movement mobility data and transition migration data. However, it can be extended to the case of any demographic system, as we will see later.
Table 13. Multiregional l i f e t a b l e based on t h e movement approach, cubic c a s e , United S t a t e s , four region system (1965&1970), females, s u r v i v a l p r o b a b i l i t i e s and mean d u r a t i o n s of t r a n s f e r s (South Region).
SURVIVAL PROBABILITIES