We set out to compare the merits of six models. We examined the data for 13 countries individually, looking at 6 or 8 separate data sets for each. We plotted their deviations from each of the models on scatter diagrams, and formed some preliminary conclusions.
We then proceeded to test the models further by using pooled data for the 13 countries. We decided to fit the models to the data at ages 80-98, and then extrapolate the models to produce estimates (or "predictions") of mortality rates at ages 99-109, which could then be compared with the observed values.
The investigation in Chapter 6 is somewhat lengthy, but the results can be described concisely. They show quite decisively that in all cases the Gompertz model gave the estimates which are farthest away from the actual values. This was followed closely by the Weibull model and then by the Heligman & Pollard model. These three models all gave estimates of mortality which were far too high above age 100.
The other three models (logistic, Kannisto and quadratic) were all far closer to the observed values, and at first sight it is not easy to choose between them. Formal tests of goodness of fit, using both the likelihood and chi-squared methods, did not give a decisive result. No single model was always best.
However, tests of goodness of fit are not the only consideration. From a theoretical point of view, the logistic model has important advantages. It is more general than the other models and will work in situations where they will not. It also has some explanatory backing. There are theories about why it works and about circumstances in which it might fail.
The Kannisto model is simply
a special case of the logistic model, but it is a useful one. We can see from its
construction that it will only work if the constant 
in the logistic model is small, if 
can be fitted
reasonably well by a function which tends towards 1, and if (as a corollary) the sex ratio
of mortality is close to 1 at very high ages (over 105). As it happens, though, all these
conditions are satisfied reasonably well in our data base at present. The Kannisto model
then has the practical advantage that it has only two parameters, which makes it easy to
fit, and it gives consistent predictions about high age mortality if it is applied in
successive periods.
The quadratic model is pragmatic, but it has little theoretical support. (On the contrary: if extended indefinitely it would imply that the force of mortality will eventually reach zero, and this can only happen if immortality is possible). Essentially, the quadratic model uses a parabola as an approximation to a more general curve. This is a device which can often be very successful, but only over a limited range of ages. The quadratic model can also be sensitive to the choice of the age range which is chosen for fitting, and it gives some highly dubious extrapolations.
All in all, therefore, we conclude that the logistic model and its Kannisto approximation are the best of the original six models.
Extrapolation to age 120
In Chapter 9 we gave estimates of the force of mortality at age 120, obtained by assuming that models which fit the available data from age 80 up to approaching age 110 will continue to apply up to age 120. Is this a reasonable assumption? It could break down if the force of mortality starts to depart from the logistic model somewhere between age 110 and age 120, either by bending upwards or by bending downwards.
It is sometimes said that "no law is always valid". The word "law" was used by Laplace and his successors to describe statistical regularities which can be fitted by a simple mathematical formula. It was therefore much the same as what would now be called a descriptive model. Do all models break down in the end? Even if they do (and not everyone would agree) this does not really answer the immediate question here, which is whether the logistic model will necessarily break down somewhere between age 110 and age 120.
The force of mortality could depart from the logistic model by bending upwards if there is a fixed, definitive limit to the length of human life. However, our investigation has found no evidence of such an upward bend. A fixed limit, if it exits, must be at such an extreme age that it has so far had no visible effect on the observed data.
A downward bend could occur if the population includes even more extraordinarily robust individuals than is assumed by the logistic model. There is nothing inherently impossible in this. However, the only direct evidence that this may be true is the case of Mme Calment, who has already lived longer than is plausible under the logistic model (see Chapter 7(d)).
A common approach in scientific work is to find a working hypothesis which satisfactorily accounts for all the known data, and then continue to use it until further notice: that is to say, until it is contradicted by new observations or until someone thinks of a better hypothesis. The work in the present monograph has identified the logistic model, and its Kannisto approximation, as a working hypothesis which adequately fits all the data in our database except for Mme Calment. This working hypothesis leads to the prediction that the force of mortality at age 120 lies between 0.7 and 1.0. Does Mme Calment affect this prediction ? This is a very legitimate question, but unfortunately the answer depends on whether she is an exceptional, aberrant case, or whether she is a trend-setter. With only one observation above age 116, it is at present impossible to tell.
There are those who are looking for a model which can be applied up to age 120, for purposes such as completing a life table. For such applications, we have identified a model and an approximate model which are the best available at present, and which are certainly far better than many other models in common use.
There may be some who will not wish to extend the model beyond age 115, until it becomes clear whether Mme Calment is a harbinger of change or a unique outlier. This, however, cannot be settled until there are more observations. In the meantime, those who are particularly concerned by this question may prefer to stop their life tables at age 115.