The purpose of the scatter diagrams presented in this chapter is to show the degree of consistency with which the data for each of the different countries adhere to or depart from each of the six selected models. It is thought that the closer the data conform to a given model, and the less scattered the diagram, then the more sure are the conclusions drawn regarding the adequacy of the model.
Each of the six models was
fitted to each of the eight data sets for each of the countries, separately, using the
maximum likelihood method. These fitted models provide the expected value of 
. The ratios of the
observed to the expected value of 
were then calculated in all cases where the observed value was
based on at least 100 deaths. The resulting ratios are plotted on the scatter diagrams in Figure 5.
The twelve countries included in the test belong to the group of thirteen with the most reliable and complete information. The one excluded is Iceland, the country of lowest mortality in old age, for the reason that the small numbers of observations there would have predicated an early cut-off age.
When examining the results,
it should be kept in mind that the three decades represent different levels of mortality.
Between 1960-70 and 1980-90 there were very considerable falls in 
. Moreover, in the
case of females these falls were larger at age 80 than at age 100, with the result that
the rate of increase of 
with age became somewhat steeper. However, the fitted models adapted to these
changes. It will be seen from the figures that the pattern of departures from the
models was not greatly affected. For each of the models there are four figures for males
and four figures for females. The pattern is sometimes different between the sexes; but
within each sex, there is no great difference between the patterns for 1960-70, 1970-80
and 1980-90.
Gompertz model
All eight diagrams for the Gompertz model display strong curvature which is convex. This reflects the well-known fact that observed death rates do not rise as rapidly at high ages as the model would have it. In the case of males, this departure from the model is relatively slight at first but becomes increasingly noticeable after about age 95. Though the dots become more widely scattered, the general downward deviation is more and more obvious.
In the diagrams for females, the curvature is much more pronounced and is clearly visible already at age 80. This indicates deceleration of the mortality rise over this whole age span. Consistency between countries is also greater and has increased slightly over time. The 1980-90 diagram is a convincing proof of the deviation of death rates observed in modern low-mortality countries from the Gompertz model.
It will be noted that in the figures, the cohort data are more scattered and have a less pronounced form than the period data.
Weibull model
The period diagrams for the Weibull model for males show a good fit until about age 95, after which they become more scattered and with a distinct tendency to fall below the model. A slight curvature appears in the cohort diagram, which may perhaps be attributable to the fact that the data for older ages are more recent than for the younger ages.
The Weibull model does not fit the female data nearly so well, a fact which has become progressively more obvious over time. Even so, the fit is somewhat better than with the Gompertz model.
Heligman & Pollard model
The Heligman & Pollard model fits the male data quite well over the range from age 80 to age 95, but after that the deviations increase and show a definite downward bias.
The female data display a picture closely similar to that of the Weibull model. The age at which half the dots fall below unity is approximately the same for both models: 91, 92 and 93 years respectively for the three successive decades. This retardation over time implies a widening arch but not an improving fit. In the Heligman & Pollard model, the downward deviations at the highest ages have become ever more consistent. For both sexes, the cohort diagrams are more blurred and the features are less distinct.
Quadratic model
The quadratic model was designed by Coale & Kisker (1990) to start at age 85. It was therefore fitted to the data for ages 85 and over. However, the scatter diagrams also show the implied ratios at ages 80-84, if the fitted model is extrapolated backwards. The first thing to be said is that the fit at these ages is very poor indeed, with a large scatter and often a bias as well. Indeed, at age 80 the deviations of the male cohorts from the quadratic model as fitted are the largest in the entire study. Thus this is not a proper model of mortality which can be applied at ages from 80 upwards. Instead it must be regarded (as its designers intended) as a device to fit the data over a limited range from age 85 upwards.
Judged in these terms, the fit for both males and females is good at ages 85-95, but beyond that the scatter widens a great deal, but without any clear tendency either upwards or downwards.
Logistic model
All the period diagrams for males show that the logistic model gives a very good fit up to at least age 90 and sometimes age 95, with little scatter. After that, there is scatter but it is quite well balanced above and below unity. The cohorts give a looser fit with some very slight undulations and, at the highest ages, predominantly upward deviations.
For females, a very good fit indeed is seen up to about age 95. The increased scatter around and above 100 years was in the first two decades mainly upward, in the last decade well balanced. The fit of the logistic model to the data for females in 1980-90 is perhaps the best in the entire exercise. Also the cohort observations conform satisfactorily to the logistic model, though not as well as the period data, and reveal at the oldest ages an upward bias which is similar to that for males.
Kannisto model
The fit is good for males to just past 90 years, though with a slight tendency to concavity. After 90, upward deviations predominate, particularly in the 1970-80 period. In the cohort diagram these various tendencies are fused and the fit, though looser, has no noticeable bias.
For females the fit is very good up to around age 95 but, while the male diagram showed some concavity, we find here a slight tendency to convexity. At the highest ages, the dots have a slight but increasing tendency to fall below unity. As with the males, the cohort diagram shows a looser fit but, on the whole, is roughly on target.
Discussion
The first important observation is that the fit of any particular model for either sex remained largely similar over the three decades, in spite of large changes in the underlying mortality and its age pattern. The deviations often decreased because of the increasing numbers of observations at any given age, but the general picture was not significantly modified. The strengths and weaknesses of the various models thus did not depend on the level or age pattern of observed mortality.
The adherence of individual countries to each model, calculated separately for each country, was also more uniform than their underlying mortality. In 1980-90 the difference between the highest and lowest national death rates was 13.7 percent for males and 17.5 percent for females, while in the best-fitting models the range of deviations was only a few percent at ages with large numbers of observations.
Secondly, as the number of observations increases over time, chance variation is reduced and the cut-off point moves to a higher age. Consequently, the latest period measures the performance best. This fact, however, has been of relatively minor importance.
Thirdly, while period analysis showed usually very satisfactory consistency over time, and in many cases a close fit to the model, cohort analysis tended to display a wider scatter and in some cases a general form slightly different from the period data. This is undoubtedly because the cohorts experience their age-specific mortality under different mortality regimes. It is often said that a period life table is a hybrid of many generations and therefore not quite "real", but it can be said with equal justification that a cohort life table is a hybrid of many periods. It has been demonstrated that the recent changes in old age mortality have been overwhelmingly produced by period factors, not cohort factors (Kannisto, 1994 and 1996). Analysis by period may therefore seem to be more appropriate than analysis by cohort for testing mortality models, but nevertheless we shall use both methods.
As a final remark, it is fair to say that all six of the models are relatively robust in adapting themselves to different and changing mortality levels and patterns.
Ranking of the models according to goodness of fit
Based on observation of the scatter diagrams we would rank the six models for their validity in the following order, starting with the best. It needs to be pointed out, however, that the order between the third and fifth positions is difficult to establish, because the respective merits of these models are roughly equal within the range of ages shown in the diagrams.
1. Logistic model
This model is on all accounts the most successful of the six in describing the trajectory of mortality in old age in low-mortality populations. The fit is good in individual countries, the deviations small. The scatter, besides being narrow, shows no noticeable tendency to deviate to either side of the theoretical line. Even the cohort diagrams, which in some models display a bias of their own, adhere well to the logistic model.
2. Kannisto model
The description of mortality given by this simple two-parameter model is also satisfactory, but not quite as good as the one by the logistic model. The male dots around or above 100 years show an upward bias, and the female diagram of 1980-90 a slight but definite convex form.
3. Weibull model
This model considerably overestimates the ascent of mortality with advancing age, the discrepancy with observed data widening progressively. This tendency appears earlier and is stronger among the females.
4. Heligman & Pollard model
The performance of this two-parameter model is closely similar to that of the Weibull model, at the ages considered here, showing only a slightly greater downward bias of ratios for men at high ages.
5. Quadratic model
This model gives a very good fit at ages 85-90 and, after that, an essentially neutral scattering as far as age 103; but it fails to describe even approximately the values below 85 years.
6. Gompertz model
It is well known that this classical model overestimates the rise of mortality with age and it is therefore not surprising that the other models, developed much later, are improvements on it.