The trajectory of mortality in old age, plotted in logarithmic scale, is a convex curve. Its level varies between populations and over time, a great deal but its form very little. The remarkable regularity of this form has led generations of actuaries and demographers to search for a formula which would express it neatly and be universally valid. Leaving this question aside, we shall simply give indications of the steepness of the arch as observed in the countries of the database. We shall call it slope as we can do about a hill without implying that it is linear.
This increase by advancing age can be measured by the parameter k defined by HORIUCHI and COALE (1990) as the relative derivative of µ (x):
The force of mortality is obtained by the approximate formula:
where p = 1 - q. For the greater stability we averaged the p (x) values
which were already smoothed by 5-year moving averages, over 5-year age periods by:
A value for the force of mortality was this way calculated for the approximate ages 83, 88, 93 and 103, and from them, the values of k for the intervals as shown in Table 13 for the aggregate of thirteen countries. The same are illustrated in Figure 17.
Certain characteristics of the mortality curve (expressed in qx,
mx, or µx) and of the k curve which is defined as the slope of
ln (µx), correspond in the following way:
Mortality curve |
k curve |
|
| 1) | ascends with age | positive values |
| 2) | decelerates in log. scale | declines |
| 3) | regular convex form | linear decline |
| 4) | steeper over time or vice versa | higher over time or vice versa |
Though the first two characteristics are obvious in both presentations, the features are generally better perceived and become measurable in the k curve. Correspondence 3) is meaningful only from k to mortality curve. Altogether, k is a good analytical tool which facilitates seeing and measuring facets that are imbedded but not very apparent in the mortality curve.
For the thirteen countries, the k values of the last three decades (the somewhat irregular 1950-60 data are excluded) in Figure 17 show the principal features of the slope of mortality in the post-war era. The slope of the female curve is steeper (k is larger) than that of the male over most of the age range but approximates it in the proximity of 100 years. Its steepness has further increased noticeably from one decade to the next while the male curve shows only the slightest tendency in this direction. All lines are fairly liniear which can be interpreted as a certain basic regularity in the well-known curvature that characterizes the ascent of mortality.
The 15-year age interval 83-98 summarizes the situation and the development best because, in spite of double smoothing, the interval 98-103 is not entirely free from irregularity. This summary measure is given in the last column in Table 13. We will note for males a near-complete immobility of k but for females a steady and accelerating increase from decade to decade, namely .0026, .0048 and .0068. This aggravation of the slope of female mortality is of course a consequence of greater reduction in death rates of the relatively younger old.
For individual countries, the k function was calculated for the 83-98 year interval by the same method which makes
In the ensuing international comparison, presented in Table 14, it is noteworthy that the slope is found in a relatively narrow range: for men mostly between .07 and .09 for women between .08 and slightly over .10. In all cases, with one insignificant exception, k has been larger for women that men, a feature which is closely associated with the lower mortality level of women.
The countries with steepest slopes for men are Iceland, Switzerland, France and Sweden while those where the slope has increased most are Japan, France and Finland, all three of them countries with a large mortality decline for men. Lack of a uniform tendency by men is shown by the fact that in eleven countries, k has increased, in nine others decreased. In the criss-crossing movements among the 20 countries, the unweighted mean of k as well as the standard deviation remained virtually unchanged.
Among women, the slope became steeper with only two exceptions in Eastern Europe. The magnitude of the change varied widely between countries and both the mean and the standard deviation increased significantly, the former from .0860 to .0938 and the latter from .0048 to .0071. The steepest slopes for women are found in France, Japan, Sweden, Denmark and Switzerland which all boast low death rates. The increase in k in the past 20 years has been largest in Japan, Sweden, France, Finland and Denmark.
It can thus be observed that countries with lowest mortality tend to exhibit relatively steep slopes in it. This is confirmed by respective correlation coefficients which are for males +0.75 and for females +0.73, both of them significant at more than 99 percent level.The regression lines of k by e(80) in Figure 18 are almost parallel for both sexes but that for women lies on a higher level. Even with equal life expectancy the female slope of mortality tends to be slightly steeper. For the most part, however, the steeper slopes for women are associated with their higher life expectancy.
A close look at the changes of the last 20 years in Figure 19 reveals different patterns of development between men and women. Though both show a high positive correlation between increase in life expectancy and increase in steepness (for males, r = +0.87, for females +0.74), the regression lines intersect. The slope has changed in the female populations more uniformly while the male populations are vey sharply divided, and low overall progress has been accompanied by an actual reduction in the slope. Such reduction has not been caused by unusually large gains at highest ages but by stagnation or slow progress among the relatively younger.