In Chapter 6 we fitted various models to the data for ages 80-98 and then extrapolated them to higher ages, in order to compare each model with the actual data at ages 99-109. There, we were comparing the extrapolations with the data. In this chapter, we shall be comparing the extrapolations with each other.
The reason for doing this is that there are certain practical problems, such as the need to complete life tables which are published at regular intervals, where one is obliged to make extrapolations in successive periods. It is then of interest to know whether one can expect the successive extrapolations to be consistent with each other. This, in turn, will depend on the sensitivity and robustness of the chosen model. If some models are particularly sensitive to fluctuations in the data at ages 80-98, then their extrapolations to higher ages may fluctuate even more. If one is looking for consistent extrapolations, there may sometimes be advantages in choosing a less sensitive model.
A related question, also of interest, is how extrapolations based on period data compare with extrapolations based on cohort data.
The comparisons in this chapter are straight forward. The models, already fitted for use in Chapter 6, have the parameters given in Appendix D. The results are plotted in Figure 8 and we shall comment on these in turn.
Figure 8 (logistic model)
The two top curves for males in Figure 8 show a cross-over between ages 110 and 120 for the extrapolations based on the periods 1960-70 and 1970-80. Admittedly this is a very minor inconsistency, which can be attributed to the effect on the fitted parameters of fluctuations in the data at ages 80-98.
A much more striking feature of Figure 8 is the way in which the cohort curve for males falls below the period curves. This is due to the way which mortality has fallen at high ages during the last few decades. Since the cohorts born in 1871-80 reached age 90 in 1961-70, the cohort curve should intersect the period curve for 1960-70 at about age 90. Similarly, the cohort curve should intersect the period curve for 1970-80 at about age 100. Since mortality for males was falling both at ages 90 and 100, the slope of the cohort curve between these ages will be less than the slope of either of the period curves. As the curves are extrapolated, the difference grows larger. In fact, it eventually grows too large to be credible.
It will be noted, though, that the cohort curve for females does not fall in the same way. This is because the fall in the level of mortality for females followed a different pattern, with larger falls at age 90 than at age 100. As a result, the extrapolated cohort curve does not intersect the extrapolated period curve for 1980-90 at age 110, as it should.
Another feature of the logistic model is that the sex ratio implied by the period extrapolations at age 110 is higher than for the cohort extrapolations, and indeed higher than our best estimate of the actual sex ratio (see Table 7.2). This suggests that the period extrapolations given by the logistic model may be either a little too high for males, or a little too low for females, or both.
Figure 8 (Kannisto model)
It will be seen from Figure 8 and Table 7.2 that the extrapolations given by the Kannisto model are almost completely free from the inconsistencies shown by the logistic extrapolations. This is because the Kannisto model has only two free parameters and its behaviour at the highest ages is constrained.
It is important to note that the Kannisto model is a special case of the logistic model. This means that if both these models are fitted to the same set of data, the Kannisto model cannot possibly give a closer fit than the logistic model, within the range of ages to which they are both fitted. However, the fact that the logistic model fits better within this range does not necessarily guarantee that it will give a better extrapolation to higher ages. This will depend not on the goodness of fit, but also on the extent of fluctuations in the data to which the models were fitted, and on the robustness and sensitivity of the models.
The Kannisto model is simple, easy to fit and gives sensible and consistent extrapolations. For the limited purpose of making extrapolations, these are obviously advantages.
Figure 8 (quadratic model)
Figure 8 shows inconsistencies in the quadratic model extrapolations which are far more serious than those which we detected in the logistic extrapolations. There is a huge gap between the period and cohort extrapolations given by the quadratic model. Indeed, at ages 110-120 the cohort extrapolations are actually falling, while the period extrapolations are rising.
The main reason for this lies
in the very nature of the quadratic model itself. A quadratic in which 
has a negative coefficient is bound to fall
sooner or later. The age at which this will start to happen will depend on the size of the
negative coefficient, which in turn will depend on the changes in the slope of the force
of mortality (i.e. the rate at which the force of mortality increases with age) in the
data to which the model is fitted.
If a model is fitted to period data, the data will all relate to the same moment in time. In a cohort fit, the data for lower ages will relate to an earlier time than the data for higher ages. If mortality at each age is falling with time, but not necessarily at the same rate, this can easily affect the negative coefficient in the quadratic model. This may well be enough to explain the major inconsistencies between the period and cohort extrapolations as simply the effect of fitting quadratic curves to falling levels of mortality.