In Chapter 6, we found that the logistic, Kannisto and quadratic models fitted the data decisively better than the Gompertz, Weibull and Heligman & Pollard models. In Chapter 8, we concentrated on the three best-fitting models and examined their properties when they were extrapolated to higher ages in successive periods. It appeared that the Kannisto model gave the most stable extrapolations, because the parameters are constrained. The logistic model came next. The quadratic model, though often fitting the data at ages 99-109 rather well, produced less stable extrapolations to age 120. These extrapolations were all based on the models when fitted to the age range 80-98 (or 85-98 in the case of the quadratic model).
However, unless a model fits perfectly throughout the whole range from age 80 to the highest age in the data set, it is always possible that the fitted parameters (and hence the extrapolations to age 120) will depend, to some extent, on the particular range of ages which is used for the fitting process. At least, we need to investigate this possibility. In this chapter, we shall examine the results obtained when the three best-fitting models are fitted to alternative age ranges, first to ages 80-98, then to ages 85-98, then to all ages 80 and over, and finally to all ages 99 and over.
This final range, all ages 99 and over, is in a different category from the others. It totally ignores all the data in the age ranges 80-98, which we originally chose because they are believed to be the most reliable. Against this, by not starting until age 99, it is not necessary to extrapolate the model nearly so far. Also, if there are any systematic departures from the model which do not show up until high ages, they will be given more weight in the models fitted to ages 99 and over than they are in the models fitted to ages 80 and over. These are definite advantages, but they need to be weighed against the disadvantage that the fits to ages 99 and over are based on more limited and less reliable data.
In order to produce the
extrapolations to age 120, some special computer programmes were used. These shifted the
origin of the hazard function to age 120. In this way, 
becomes a parameter in the transformed hazard function and its value and
standard error are estimated automatically in the course of the fitting process.
Estimates of 
If one is trying to make estimates of the force of mortality at age 120,
an important question is whether models which fit the data reasonably well up to ages 100
or 110 can be expected to continue to hold as far as age 120. Could there be some
biological process which causes the model to change between age 110 and age 120? We shall
return to this question later, but in the present chapter we shall simply present the
estimates of 
which are found by
extrapolating the existing models.
Tables
9.1 and 9.2 show the estimates of 
and their standard errors, found by
extrapolating the three best models when fitted to four selected age ranges in each of
four datasets. The results are illustrated visually in Figure 9.1.
The 48 different entries in Tables 9.1 and 9.2 give a range of possible
estimates of 
which is rather wide.
However, we can (and should) narrow the range a little by ignoring some particular
estimates which are believed to be less reliable than others.
The outstandingly unreliable
figure in Table 9.1 is the estimate 0.242 which is found by
fitting the quadratic model to the ages 99 and over to the data set for males in the
period 1980-90. It is natural to wonder how such an estimate can possibly have come about.
The answer is very instructive and can be seen by looking at Figure
9.2. Although this actually plots 
rather than 
, the principle is the
same. The dotted line shows the quadratic model fitted to ages 99 and over by the method
of maximum likelihood. This method places particularly high weights on ages 99-101, which
is where we earlier found evidence of age heaping. The result is a fitted curve which
bears no relationship to the actual data below age 99 and which looks totally implausible
above age 110. This is not a method which can be recommended for making estimates of 
. The same problems affects, though to a
lesser degree, all the other estimates found by fitting models to the data for males at
ages 99 and over. Together, these make 6 dubious cases.
The problem is not nearly so acute for the fits to the data for females at ages 99 and over. In only one case (the logistic model fitted to the cohort data) is the fitted model seriously different from the actual data below age 99. Curiously, the quadratic model, and also the logistic model fitted to the period data, are not affected. Thus there is only one dubious case in this group.
The final group of estimates
which are believed to be less reliable than others is the group found when the quadratic
model is fitted to ages 80-98 and to all ages 80 and over. Horiuchi and Coale (1990) found
a "hump" in 
at ages 75-85
which was consistent with the logistic model but not with the quadratic model. When Coale
and Kisker (1990) used the quadratic model, they did not start it until age 85. There are
8 cases in this category.
Even if we ignore all of the
15 dubious cases which are identified above, we are still left with 33 remaining separate
point estimates of 
. These all lie in
the range from 0.750 to 1.048 for males, and from 0.718 to 1.019 for females. These
correspond to estimates of 
ranging
from 0.528 to 0.649 for males, and from 0.512 to 0.639 for females.
To summarise, the methods in
this chapter lead to the estimates that 
is between about 0.7 and 1.0, and that 
is between about 0.5 and 0.65, for both
males and females.