3. Probability of dying

The basic life table parameter q_x is in the present study obtained through follow-up of a closed group of persons as they move from one age to the next. The database gives for each country, sex and cohort the number of persons who reached the exact age x and the number of them who died without reaching the age x + 1. The proportion of the latter among the former or D_x/N_x indicates the probability q_x of dying at age x. The direct follow-up ensures strict correspondence between numerator and denominator which is a great advantage with the small numbers observed at high ages. It also allows easy measurement of chance variation.

        The use of cohort histories is not predicated by a belief that cohort is the main or even an important factor in mortality but actually serves to eliminate the effect of the often uneven size of adjacent cohorts. The one-year observations of exposed-to-risk and the corresponding deaths are used as building blocks in the construction of period life tables with only the peculiarity that tables for two consecutive periods overlap in one calendar year. At high ages where observations are few, meaningful numbers can only be assembled by joining a number of calendar years together in which case the exact chronology becomes less important while the need for accuracy of age remains paramount.

        The annual probability of dying may be converted into a central death rate m_x in which case it maintains its property of internal consistency while in an m_x calculated from the mean population, deaths are not necessarily included among the exposed-to-risk of the same age, with the result that rates may become volatile. An annual probability of dying is also convertible to force of mortality µ_x.

        The probability of dying q_x and its complement p_x = 1 - q_x will in some connections below be extended to wider age bands. For 5-year age groups, central values of q have been calculated as geometric means of single-age probabilities.

        Table 2 gives a general account of the probability of dying in old age in today's low-mortality societies. Its thirteen component countries have experienced the recent remarkable progress in reduction of old age mortality. The table gives the results for each of the four post-war decades with an overlap in years ending in zero. The 1950-60 period includes the persons who reached age x during 1950-59 and follows them to age x + 1 which they would reach in 1951-1960. The survivors from 1959 to 1960 are then included in the risk population of age x + 1 in 1960-70 and successively at each age which they reach in that or the following decades. This procedure and this designation of overlapping time periods was already used by VINCENT (1951) and DEPOID (1973) in their pathbreaking work.

Figure 1 depicts in logarithmic scale the course of the risk of dying as age advances and shows that it continues to increase as far as it is today measurable, namely close to age 110, without indicating a point where it might peak or stabilize. The increase is, however, less than exponential and displays a curvature well known to demographers and actuaries.

        The risk of dying is at all ages higher for men than women. The apparently narrowing gap between the two curves might lead one to expect the two lines eventually to merge or intersect but the absolute difference in the risks does not decline. It actually grows slightly in the early 80s and then remains stationary at least a few years past age 100 after which the picture becomes uncertain due to small numbers. Table 24 in Chapter 11 shows, however, that the female advantage in survival continues till age 110 at least.

        Table 3 and Figure 2 give the corresponding results for the high-, medium- and low-mortality groups to which the most reliably documented countries were divided. The curves are drawn only to age 99 because the data for the first group do not go beyond it. All curves display the same general form and the logical relative positions but the differences between the low and medium groups which together form the group of thirteen, are small.

        Finally, the mortality decline can be seen in Figure 3. The curves of successive decades move gradually to lower positions while maintaining the same typical form. The decennial differences show a tendency to increase indicating an accelerating mortality decline, particularly between the 1970s and 1980s. At the same time, the curves for females have become slightly steeper while the male ones have stayed roughly parallel. These phenomena will be elaborated in later chapters.

Updated by V. Castanova, 1 November 1999