A set of indicators for decomposing the secular increase of life expectancy
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The ongoing increase in life expectancy in developed countries is associated with changes in the shape of the survival curve. These changes can be characterized by two main, distinct components: (i) the decline in premature mortality, i.e., the concentration of deaths around some high value of the mean age at death, also termed rectangularization of the survival curve; and (ii) the increase of this mean age at death, i.e., longevity, which directly reflects the reduction of mortality at advanced ages. Several recent observations suggest that both mechanisms are simultaneously taking place.
We propose a set of indicators aiming to quantify, disentangle, and compare the respective contribution of rectangularization and longevity increase to the secular increase of life expectancy. These indicators, based on a nonparametric approach, are easy to implement.
We illustrate the method with the evolution of the Swiss mortality data between 1876 and 2006. Using our approach, we are able to say that the increase in longevity and rectangularization explain each about 50% of the secular increase of life expectancy.
Our method may provide a useful tool to assess whether the contribution of rectangularization to the secular increase of life expectancy will remain around 50% or whether it will be increasing in the next few years, and thus whether concentration of mortality will eventually take place against some ultimate biological limit.
KeywordsLife Expectancy Survival Curve Middle Panel Premature Mortality High Quantile
Life expectancy almost doubled in developed countries during the 20th century. This increase was initially related to the reduction in infant mortality. After the 1950 s, the increase was related to the decline in mortality in old age . In both phases, the evolution is usually related to the combination of a massive improvement in the social and physical environment and to an improvement in health care. Whether the former improvement had more effect on the decline of mortality than the latter (as suggested by McKeown's theory) is still a matter of debate . No strong, comprehensive theory is yet available to fully understand the past trends, and hence, the current and future evolution of human longevity. However, several paradigms have been proposed to capture the underlying mechanisms at stake. One of them was provided by Fries in 1980 , relating the secular increase of life expectancy to the postponement of age of death. This corresponds to the strong decline (or even the fading) of premature mortality and morbidity, and to the concentration (or compression) of deaths around some fixed, optimal mean age at death value, estimated at 85 years. This evolution was coined as "rectangularization," suggesting that the final stage of the survival curve will be a perfect rectangle.
This paradigm developed by Fries was mainly based on visual inspection. Several authors developed quantitative indicators to capture the shape of the survival curves [4, 5, 6, 7, 8]. Most confirmed a concentration of age at death around high mean values with a decreasing variability. However, some papers showed a slowdown of rectangularization during the last decades, sometimes with a decline resuming after a plateau , while others found an increase in the variability of age at death over 60 years [10, 11]. In the same line, others showed a continuous increase of the maximal age at death in the last century [1, 12], with no signs of a finite life-span limit . In the latter paper, Wilmoth explicitly suggested that rectangularization of the survival curve and increase of longevity are in fact occurring simultaneously, i.e., that the distribution of age at death could become more and more compressed while also shifting further and further to the right. In order to evaluate and monitor this hypothesis, we propose in this paper a set of quantitative, nonparametric indicators, namely a rectangularity index and a longevity index, to quantify and disentangle the respective contributions of rectangularization and longevity increase to the secular increase of life expectancy.
Our approach is based on this indicator, and is developed in three steps as follows.
The higher R, the more rectangular the survival curve. In particular, R = 1 if the survival curve is perfectly rectangular between t0 and t Q .
The second step is to define a "longevity index" as the quantile defined above, t Q . The higher t Q , the more shifted to the right the survival curve.
In particular, for t0 = 0, CLE 0,Q = R·t Q is simply the product of both indexes.
Thus, any difference between two CLE s is a sum of two terms. The first term is the difference of CLE, which would be obtained if the longevity index would remain constant, set at the average of t Q (A) and t Q (B). The second term is the difference of CLE, which would be obtained if the rectangularity index would remain constant, set at the average of R(A) and R(B). In other words, the first term is interpreted as the part of the CLE difference attributable to a change of rectangularity, and the second term as the part attributable to a change of longevity.
In other words, 4.3 years (37% of the overall gain in CLE) are attributable to a difference in rectangularity, while the remaining 7.4 years (63%) are attributable to a difference in longevity. In what follows, we shall refer to "life expectancy difference attributable to rectangularization" (LEAR), the part (in %) of difference in CLE due to a change in rectangularity (in this example, LEAR = 37% ).
The above approach has been applied to Swiss period life tables established by year of death for cohorts of men and women born in Switzerland between 1876 and 2006 . The indicators above are implemented using t0 = 50 years and Q = .9.
Overall, there were few changes over the years, with LEAR fluctuating around means slightly under 50% in both genders. Fluctuations were slightly more marked for men than for women, and more marked before 1920 (likely because of the stronger annual variation of mortality due to epidemics of communicable diseases). Overall, LEAR increased between 1876 and 1920, followed by a plateau and then by a decline since the 1940 s for men and since the 1960 s for women. LEAR was minimal (32%) for men in the 1970 s, and then increased slightly to below 50% at the turn of the century. For women, LEAR was minimal in the 1980 s at 36.5%, then increasing to 43% in 2006.
The same Figure 5 shows the secular trends in the number of years gained in 20 years attributable to rectangularization (top panels) and to longevity increase (middle panels). It is important to note that the rhythm of increase accelerated over the period, although there is a decline and a plateau for the figures for women.
Obviously, the same indicators can be used to compare the survival curves from two populations at a given moment. For example, applying these indicators to compare men and women, one finds that the difference in CLE was 4.3 years in 2006, with about half (2.2 years, LEAR at 51%) attributable to a difference in rectangularity.
This paper introduces the use of indicators characterizing the shape of a survival curve and measuring the impact of this shape on life expectancy. The first two indicators measure the rectangularity (R) and the longevity (t Q ) of the survival curve at a given moment. The third indicator (LEAR) individualizes the impact of rectangularity and longevity on the differences in conditional life expectancy when comparing different moments or different populations. Unlike many others, our indicators are nonparametric and do not rely on any model, such as Gompertz , logistic , or Gompertz mortality change models .
The index of rectangularity is based on the "moving rectangle," an idea explored by Wilmoth and Horiuchi . It depends on two parameters, t0 and Q, which will in turn affect the longevity index t Q . The method can use any value of t0 and Q, although the values of the indicators are sensitive to this choice. When choosing t0 = 40 instead of t0 = 50, for example, for the Swiss mortality data from the previous section, the LEAR index comparing the years 1876 and 2006 is 48% for men and 46% for women instead of 45% and 44%. A similar comparison choosing t0 = 10 results in a LEAR index as high as 56% for men and 55% for women. The choice of the parameter values should relate to the question under study. If, for example, the interest lies in longevity and old age mortality, one should select a value for t0 high enough to remove premature mortality (largely due to accidents). In developed countries, a choice of t0 = 50 might be motivated by the fact that it is usually at that age that degenerative diseases (such as cardiovascular diseases or cancer) start to have some real impact. On the other hand, choosing Q = .9, i.e., removing the most elderly from the analysis, stabilizes the value of the longevity index (the index will not depend on a few persons, neither from the sample size if it is calculated using a sample of data instead of a life table). The resulting conditional life expectancy CLE50,0.9 is thus the mean age at death of those persons alive at 50 and dying before the 90th percentile of age at death for this population. However, we do not exclude the possibility that other authors might wish to apply the method using other values of t0 and Q, depending on their interest. Obviously, one should keep the same values for these parameters when comparing mortality in two different countries, or when evaluating the trend of the LEAR index along the years. Note that the corresponding parameters for the life expectancy at birth would be t0 = 0 and Q = 1.
Other nonparametric indicators have been proposed in the literature to measure the various dimensions of lifespan and high longevity. In their review, Wilmoth and Horiuchi  analyzed 10 indicators to measure either the rectangularity of the survival curve or the variability of the distribution of age at death (coined as "moving rectangle", "fastest decline", "sharpest corner", or Keyfitz's H index), which were all highly correlated, such that these measures capture aspects of the shape of a survival curve that are strongly related. The authors concluded that one can choose one of them on the basis of convenience, and they favored the use of the interquartile range of age at death. Kannisto  further developed the idea, suggesting to consider the shortest age interval in which 50% of deaths take place as the best indicator of variability of age at death. Recognizing the need of having two indicators instead of one, Kannisto  finally proposed the modal age at death as a measure of longevity (which is in fact an alternative to our t Q ), in combination with the standard deviation of the ages at death occurring above it as a measure of variability (an alternative to R). Canudas-Romo  also considered the modal age at death, although in combination with the standard deviation around (not above) the mode.
The indicators proposed by Kannisto  have been recently applied by Cheung et al  to analyze the evolution of life expectancy in Switzerland. With a decreasing standard deviation above modal age over the years, Cheung et al concluded to a significant compression of adult mortality, at least from 1920. However, because modal age at death was still increasing without respite, their analysis did not find any evidence that we are approaching longevity limits in terms of maximum life span. As seen in the Results section, we could find similar conclusions using our indicators.
One possible advantage of considering our indicators R and t Q instead of those of Kannisto is that they can be easily (almost visually) calculated given any survival curve. In contrast, modal age at death is known to be difficult to estimate, and its appropriate determination is crucial [7, 20]. The most substantial single advantage of using our indicators, however, is that they can be naturally combined with each other, allowing the proposed decomposition of a change of life expectancy and hence to separate and to compare the effects of rectangularization and longevity increase to the secular increase of (conditional) life expectancy. This may be useful in the context of the debate initiated by Fries  as discussed in the Introduction.
Analyzing the Swiss mortality data, our indicators suggest that the secular increase of life expectancy since the 1870 s is explained by the increase in both longevity and rectangularity, each responsible for about 50% of the increase. This corroborates Wilmoth's observation that both a rectangularization and an increase of longevity are simultaneously taking place . It will be interesting to see whether the contribution of rectangularization to the secular increase of life expectancy will remain at around 50% or whether it will be increasing in the next few years. Also, a comparison among various industrialized and less industrialized countries with respect to this decomposition, as well as an analysis of the factors associated with the impact of rectangularization, or with the one of longevity increase on the decline of mortality, will be the subject of future research and should be of interest to many demographers.
We are grateful to Prof. Jean-Marie Robine for helpful discussions on an early phase of the manuscript.
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