On the decomposition of life expectancy and limits to life.

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On the decomposition of life expectancy and limits to life a

a

Les Mayhew & David Smith a

Cass Business School, City University London Published online: 20 Jan 2015.

Click for updates To cite this article: Les Mayhew & David Smith (2015) On the decomposition of life expectancy and limits to life, Population Studies: A Journal of Demography, 69:1, 73-89, DOI: 10.1080/00324728.2014.972433 To link to this article: http://dx.doi.org/10.1080/00324728.2014.972433

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Population Studies, 2015 Vol. 69, No. 1, 73–89, http://dx.doi.org/10.1080/00324728.2014.972433

On the decomposition of life expectancy and limits to life Les Mayhew and David Smith

Downloaded by [ECU Libraries] at 17:49 25 April 2015

Cass Business School, City University London

Life expectancy is a measure of how long people are expected to live and is widely used as a measure of human development. Variations in the measure reflect not only the process of ageing but also the impacts of such events as epidemics, wars, and economic recessions. Since 1950, the influence of these events in the most developed countries has waned and life expectancy continues to lengthen unabated. As a result, it has become more difficult to forecast long-run trends accurately, or identify possible upper limits. We present new methods for comparing past improvements in life expectancy and also future prospects, using data from five developed, low-mortality countries. We consider life expectancy in 10-year age intervals rather than over the remaining lifetime, and show how natural limits to life expectancy can be used to extrapolate trends. We discuss the implications and compare our approach with other commonly used methods. Supplementary material for this article is available at: http://dx.doi.org/10.1080/00324728.2014.972433

Keywords: decomposition; life expectancy; limits to life; developed countries [Submitted March 2013; Final version accepted May 2014]

1. Introduction—statement of the problem Life expectancy is a measure of how long people are expected to live and is widely used as a measure of human development. Variations in life expectancy reflect not only the process of ageing but also the impacts of such events as epidemics, wars, economic recessions, etc. Since 1950, the influence of these events in the most developed countries has waned and life expectancy continues to lengthen unabated. However, a disadvantage of life expectancy is that it is an aggregate measure that disguises important detail. For example, a population with a constant number of deaths at each age, in which everybody dies before age 100, would have the same life expectancy as a population in which all died on their 50th birthday. A single measure cannot easily distinguish between such cases. This deficiency of the measure poses two practical problems. Firstly, it is unhelpful to the debate on how much variation in life expectancy there should be within a nation, that is, whether it is ‘fair’ that people from different parts of a population should face large differences in expected life, and whether it is connected to systemic social inequalities, for example, in wealth. The second disadvantage, linked to the first, becomes © 2015 Population Investigation Committee

evident when we are trying to forecast future life expectancy. In developed countries there appears to be little evidence that the improvements are slowing significantly. We are unsure if increases are due to the elimination over time of premature deaths, in which case growth should slow down, or if life expectancy is set to increase indefinitely. It is this dilemma that is the basis for the debate about whether or not there is a natural limit to life expectancy (e.g., Olshansky et al. 1998; Oeppen and Vaupel 2002, 2006). The development of the method reported here was prompted by a concern with the future change in life expectancy among developed countries, but from age 30 rather than at birth. One reason for this choice was to eliminate the effect of any reduction in child mortality, which can cause modification to the pattern of increases in life expectancy over time (CanudasRomo 2010). A second reason was to remove differences by sex in life expectancy which arise, in part, because of social or economic factors in young adulthood and, for biological reasons, at birth (Meslé 2004; Trovato 2005; Preston and Wang 2006; Staetsky 2009). We believed that concentrating on the 30-and-over age group would help remove both effects, and so simplify the problem to a degree (Mayhew and Smith 2014b).

Les Mayhew and David Smith

We focused on two aspects of life expectancy: firstly, an improved understanding of historical trends and secondly, on the development of a method that had the potential to produce better forecasts, including ones with any upper limit. Experience clearly shows that past trends are not always a reliable guide to the future and all extrapolation techniques, whether of mortality or life expectancy, are subject to error (Lee and Miller 2000; Bengtsson and Keilman 2003; Shaw 2007). A common reason is that most methods cannot predict turning points, and so ultimately produce implausibly high or low values of life expectancy with no basis in evidence. This is a fundamental problem that is often addressed by demographers subjectively, rather than through analysis (Booth 2006). An example of this problem is shown in Figure 1, which shows the change in life expectancy for men at age 30 in England and Wales (E&W) and Japan between 1950 and 2009. Quadratic functions were fitted to each time series which, as can be seen, track life expectancy in both countries with considerable accuracy. If extrapolated, the chart shows that E&W is forecast to overtake Japan from 2015, but we can also see that in the longer run the divergence is projected to increase indefinitely. It is obvious that if we base forecasts on such projections they soon become implausible. Admittedly, this would be only after a reasonable period of time, and outside typical planning horizons, but nonetheless the position is unsatisfactory from a methodological standpoint. The lack of reliable information on natural limits to life expectancy, other than judgement, is therefore a serious problem (Vaupel 2010). Forecasters

would prefer to use any available evidence to build into their projections, but what should they do if there is none? Given the state of the art, it seems this issue will not be easily resolved, but if the judgemental component involved in making forecasts could be better managed, greater accuracy might be possible. Specifically, if observed data could be partitioned in a way that respected any natural upper or lower bounds, it might be possible to narrow the range of uncertainty. We developed a method of using decomposition techniques for this purpose. These techniques constrain life expectancies within discrete age intervals that are naturally bounded from above, but which can be aggregated to produce estimates of life expectancy over any desired age interval. In this way, we were able to narrow down the inherent uncertainty and margins for judgement and, as a by-product, conduct other types of novel analysis, such as the production of league tables of improvements in life expectancy in fixed age intervals, at different times and in different countries. For example, we calculated and compared 10-year life expectancies between 1950 and 2009 for a range of ages in the following five developed countries, all of which have experienced rapid improvements in life expectancy, albeit at varying rates: E&W; France; Japan; Sweden; and the USA. How accurate are the forecasts of life expectancy produced from decomposition? This is obviously the crucial question. In the case of Japanese women, we produced forecasts of life expectancy at age 30 in 20-year calendar intervals from 1980 to 2100, using data for the following periods: 1950–80, 1950–90,

54 52 50

Life expectancy at age 30


74

48 46 44

Japan 42

England and Wales 40 38 36 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030

Year

Figure 1 Trends in life expectancy of men at age 30 in Japan and England and Wales with ﬁtted quadratic trend Source: Human Mortality Database.

The decomposition of life expectancy 1950–2000, and 1950–2009. We found that: (1) the model accurately re-produced observed life expectancies to 2009 in all cases, regardless of the data set used; and (2) separate forecasts based on a time horizon of 2020–2100 using the same subsets of data were within a year or two of one another. Only when the test was limited to data from 1950 to 1980 did we find that the model performed slightly less well. These results taken together suggest that decomposition has the potential to overcome at least some of the problems encountered in forecasting life expectancy.


1.1. Changes in life expectancy within discrete age intervals It is standard practice to measure life expectancy at different ages, such as birth, age 30, or some other age. For example, life expectancy for males at birth in E&W in 2009 was 78.04 years (source: Human Mortality Database), suggesting that at age 65 there should be 13.04 years of life remaining. In fact, the data show that life expectancy at age 65 was 18.04 years because of survival and selection effects. We therefore sought a method for calculating life expectancy within discrete age intervals that would address this anomaly. It would also enable us to assess prospects for improvements in life expectancy by aggregating component life expectancies for the different age intervals to produce estimates of future life expectancy from any starting age. Hickman and Estell (1969) proposed a similar idea, which they termed ‘partial life expectancies’. They argued that these are relevant to discussions on the economic costs of illness, since ‘partial life expectancies may be related to the ages at which the economic contributions … are usually greater’ (p. 2244). Pollard (1982), posing a different question, showed that the change in expectation of life can be expressed as a function of weighted mortality changes at individual ages plus interaction effects. This discovery makes it possible to calculate the contribution of life expectancy in a given age interval to the overall change in life expectancy, which he calls ‘temporary life expectancy’. Arriaga (1984) further developed this procedure by setting out the basic equations for measuring ‘temporary life expectancies’ using a discrete life table method, rather than the continuous methods of Pollard. We proceeded similarly, but our aims were different. Our definition of life expectancy was similar to Arriaga’s ‘temporary life expectancy’, but our focus was on trends in life expectancy within specified age intervals. The technique we developed, and which

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we present here, gives the expectations of life of a person aged 30, 40, 50, 60, etc. on reaching age 40, 50, 60, 70, etc. For example, if life expectancy at age 40 is 10 years, everyone may expect to survive to age 50; a 2-year life expectancy at age 40 means that a person can expect to live only 2 of a possible 10 years, and so on. It should therefore be easier, by fitting bounded functions to trends in survivorship, to pinpoint in which age intervals future increases in life expectancy are more likely. The results strongly indicate that improvements to life expectancy follow a predictable wave-like path as follows: a period of rapid increase in younger age intervals is observed, followed by a slowing down as life expectancy reaches an asymptotic maximum. Older age intervals replicate this pattern, after a delay, before catching up with earlier age intervals as they approach their asymptotes. For example, in 1950 when life expectancy of men at age 30 in E&W was 40.5 years, men aged 80–90 contributed only 2.9 per cent to this figure, but in 2009, when life expectancy was 49.4 years, they contributed 9.3 per cent. Usually we find that 10-year life expectancy is strongly correlated with the probability of surviving for 10 years. Thus, if life expectancy at age 60 is 8 years, then the probability of survival is 80 per cent; conversely, if the chance of surviving for 10 years is 80 per cent, one can expect to live 8 years, and so on. Because Japanese females are currently the longest lived, their inclusion in the analysis provides a natural benchmark. However, as the data show, it does not follow that they have always had the longest expectations in every age interval. A useful property of decomposition is that we can select countries with the longest life expectancy in each fixed age interval of life at a given time, and create a composite of these ‘best’ countries to give the longest overall life expectancy. The result is what we term the ‘leading edge’ of human survival at the given time (the opposite concept, termed ‘lagging edge’, may be similarly identified). Note, that the word ‘best’ is used loosely here, since we consider neither younger age groups (for which special factors may apply), nor all possible countries. In what follows, Section 2 sets out the method and Section 3 provides results for the five countries under review, the shifting historical patterns of which are analysed in the form of a league table for each decade of life from 1950. In Section 4, we demonstrate how trends in 10-year life expectancies can be used to forecast future life expectancy, and discuss the assumptions and limitations of the method used. An important feature of our method is the exploitation of a ‘natural upper limit’ to the expectation of life within

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an age interval, which is approached asymptotically and fitted statistically, using a logistic function. We use the natural upper limit to compare forecasts for each country up to 2100. In Section 5, we discuss our findings in relation to other methods of forecasting life expectancy in human populations and comment on their approaches, assumptions, and limitations.

We derive the future expectation of life for a life currently aged x1 by calculating the area under the population age curve and dividing by the starting population (i.e., we turn the population age distribution into one of the proportion surviving to each age for a standard member of the population). Figure 2 shows the population at the start of each age interval. Each adjacent pair of lxs is joined by a straight line on the assumption that the probability of survival decreases linearly between the two ages. The standard equation for the expectation of future life at age x1 is: x5 x5 1 X 1 1 X 1 ly ¼ ly þ ex1 ¼ lx1 y¼x1 2 lx1 y¼x2 2 In effect, what we are calculating is a series of areas of rectangles and triangles. Given that we are increasing age by 1 each time, that is, xn+1 – xn = 1, the area of each rectangle is simply the height or number of people alive at age xn+1, and the area of each triangle is (1/2) (lx – lx+1). Therefore, 9 8 1 1 > > > ðl ðl þ l Þ þ l þ l Þ þ> l x x x2 x3 x x3 = 1< 2 2 1 2 2 ex1 ¼ > 1 1 l x1 > > ; : lx4 þ ðlx3 lx4 Þ þ lx5 þ ðlx4 lx5 Þ > 2 2 ( ) x5 x5 1 X 1 1 X 1 ¼ ly þ ðlx1 lx5 Þ ¼ ly þ lx1 y¼x2 2 lx1 y¼x2 2 (because lx5 ¼ 0). lx1 lx2 Population


2. Segmenting life expectancy by age

2.1. Contribution to expected life We can now break this future lifetime into two parts— from ages x1 to x3 and from x3 to x5 and see how much of the total expected future lifetime each section contributes. Define: exn ðxi :xj Þ as the future expected life of someone currently aged xn between the ages of xi and xj. Hence: 1 1 1 lx2 þ ðlx1 lx2 Þ þ lx3 þ ðlx2 lx3 Þ ex1 ðx1 :x3 Þ ¼ lx1 2 2 ( ) x 3 1 X 1 ¼ ly þ ðlx1 lx3 Þ lx1 y¼x2 2 and 1 ex1 ðx3 :x5 Þ ¼ lx1

Hence, we can see that: ex1 ¼ ex1 ðx1 :x3 Þ þ ex1 ðx3 :x5 Þ : We can extend this idea to as many ages as we want. For example, if we have a population that we are studying from age 30, where the population completely dies out at age 110, then: 110 1 X 1 e30 ¼ ly þ l30 y¼31 2 and we can disaggregate this expectation of life into components of expected life in exact 10-year age intervals, that is, those for ages 30–40, 40–50, …, 100–110: ( ) 40 1 X 1 e30ð30:40Þ ¼ ly þ ðl30 l40 Þ l30 y¼31 2 ( ) 50 1 X 1 e30ð40:50Þ ¼ ly þ ðl40 l50 Þ l30 y¼41 2 ( ) 60 1 X 1 e30ð50:60Þ ¼ ly þ ðl50 l60 Þ l30 y¼51 2 .. .

lx3

1 e30ð100:110Þ ¼ l30

(

110 X

1 ly þ ðl100 l110 Þ 2 y¼101

)

and

lx4

e30 ¼ e30ð30:40Þ þ e30ð40:50Þ þ ... þ e30ð100:110Þ : x1

Figure 2

1 1 lx4 þ ðlx3 lx4 Þ þ lx5 þ ðlx4 lx5 Þ 2 2 ( ) x5 X 1 1 ¼ ly þ ðlx3 lx5 Þ : lx1 y¼x4 2

x2

x3 Age

x4

x5

Population curve divided into age intervals

2.2. Another way at looking at contributions When we calculate a term such as e30(40:50) we are calculating the future expected life between ages 40

The decomposition of life expectancy and 50 of someone currently aged 30. Instead of considering e30(40:50), we can consider e40(40:50), that is, the expectation of life over the next 10 years of a person now aged 40 which is as follows: ( ) 50 1 X 1 e40ð40:50Þ ¼ ly þ ðl40 l50 Þ : l40 y¼41 2 We can see that: e30ð40:50Þ ¼

1 l30

¼

l40 l30

(

50 X

)

1 ly þ ðl40 l50 Þ 2 y¼41 ( ) 50 1 X 1 ly þ ðl40 l50 Þ l40 y¼41 2

l40 e40ð40:50Þ : l30 This expression can be interpreted intuitively: the term on the left is the expected life between the ages of 40 and 50 of someone currently aged 30; the term on the right is the expected life between the ages of 40 and 50 of someone currently aged 40, multiplied by the probability that someone currently aged 30 reaches age 40, that is, a conditional probability. Hence: l40 l100 e30 ¼ e30ð30:40Þ þ e40ð40:50Þ þ . . . þ e100ð100:110Þ : l30 l30


¼

2.3. Constructing a population with greatest probability of survival As explained in the introduction, we can also construct a theoretical population to derive the ‘greatest possible life expectancy’ based on the maximum life expectancy already achieved in each age interval. Let us assume that there is more than one population and every 10 years a person is able, if still alive, to move to a new population and experience their mortality rates for the next 10 years. This could either be a totally different population or we could use the same idea to model one population which experiences improvements in their mortality rates every 10 years. A 30-year-old individual can therefore join eight different populations during a lifetime, assuming that in none of the populations does anyone live beyond age 110. Let us assume that this person spends the first 10 years in population A and then moves to population B for the next 10 years and so forth until he or she spends the last 10 years in population H. Ignoring the effects of selection, to which we return later, the expectation of life can be broken down into steps. Define: lxZ as the number of people alive in population Z at age x

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eZx1 ðx1 :x2 Þ as the expectation of life between the ages of x1 and x2 of someone in population Z currently aged x1. Then the expected life of the person aged 30 is: A A B A l40 l40 l50 C l40 B e30 ¼ eA þ e þ e . . . þ 30ð30:40Þ A 40ð40:50Þ A l B 50ð50:60Þ A l30 l30 l30 40

B C D E F G l50 l60 l70 l80 l90 l100 H e : B C lD lE lF G 100ð100:110Þ l40 l50 60 70 80 l90

To maximize the total expected life we therefore want, for each 10-year period, to pick the value of eZx1 ðx1 :x2 Þ that gives the longest expectation of life. The result can thus be imagined as the survival frontier or, as we have previously termed it, the ‘leading edge’ in terms of human survival for any particular population or populations.

3. Results Here we present the results of applying the decomposition techniques to five low-mortality countries: E&W; France; Japan; Sweden; and the USA, between 1950 and 2009. With these decomposition techniques, a wide variety of results is possible by using data for different ages, sexes, calendar years, and countries. The number of possible tables and charts is therefore potentially extensive. The following sections are necessarily selective and concentrate on those aspects which are of greatest potential interest and which demonstrate the application of decomposition in practice. Detailed country-specific tables and other results are provided in the supplementary material (Mayhew and Smith 2014a). We focus first on the historical trends from 1950 to 2009 to identify which countries are either ‘leading’ or ‘lagging’ the group. A ‘leading country’ is defined as the one which had the longest life expectancy for the given decade of life and for the given year from 1950; in similar fashion, a ‘lagging country’ is defined as the one which had the shortest life expectancy for the given decade of life and the particular year from 1950. Using results for men and women, we show that countries meeting these criteria have changed at various times, but that these changes have been systematic and part of a wider trend which has seen the USA fall down the rankings of life expectancy, whilst Europe and Japan have risen.

3.1. Men Table 1(a) gives the expected life over the following decade for a man who has reached age 40, 50, etc. in

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Table 1 Life expectancy of men from age 30 every tenth year: (a) ‘leading country’ by 10-year age interval; (b) 10-year life expectancy for the corresponding country and ‘leading edge’ (a) From age 30

1950

1960

1970

1980

1990

2000

Sweden Sweden Sweden Sweden Sweden Sweden USA USA Sweden

E&W Sweden Sweden Sweden Sweden Sweden USA USA Sweden

E&W E&W Sweden Sweden Sweden Sweden Sweden USA Sweden

E&W E&W Japan Japan Japan Japan USA USA Japan

Japan Japan Japan Japan Japan Japan Japan USA Japan

Sweden Sweden Sweden Sweden Japan Japan Japan Japan Japan

From age 30

1950

1960

1970

1980

1990

2000

2009

Change 1950– 2009 (years)

Maximum potential years remaining

e30 (30:40) e30 (40:50) e30 (50:60) e30 (60:70) e30 (70:80) e30(80:90) e30 (90:100) e30 (100:110) e30 (LE) e30 (LC)

9.92 9.66 9.08 7.73 5.09 1.76 0.20 0.00 43.44 43.28

9.93 9.71 9.20 7.88 5.21 1.86 0.21 0.00 44.00 43.91

9.94 9.72 9.19 7.95 5.39 2.07 0.26 0.01 44.53 44.41

9.95 9.75 9.23 8.11 5.67 2.26 0.33 0.01 45.31 44.35

9.96 9.8 9.38 8.39 6.35 2.89 0.47 0.01 47.25 47.17

9.96 9.83 9.5 8.68 6.81 3.58 0.69 0.02 49.07 48.66

9.96 9.86 9.58 8.89 7.33 4.20 1.00 0.04 50.86 50.39

0.04 0.20 0.50 1.16 2.24 2.44 0.80 0.04 7.42 7.11

0.04 0.14 0.42 1.11 2.67 5.80 9.00 9.96 29.14 29.61

e30 (30:40) e40 (40:50) e50 (50:60) e60 (60:70) e70 (70:80) e80 (80:90) e90 (90:100) e100 (1001:10) e30 (LC)

2009 Sweden Sweden Sweden Sweden Sweden Japan Japan Japan Japan


(b)

Note: LC, leading country; LE, leading edge. Source: As for Figure 1.

the given years for ‘leading countries’. It shows that the best-performing country was Sweden in earlier decades, and Japan in later decades, especially at the older ages. E&W performed best for the younger age groups between 1960 and 1980, although the improvements in the expectations for E&W at these ages were small because the expectations were already close to the 10-year limit. The USA was best in the age range 90 and over. Men reaching this age may well have been a small self-selecting group who were well cared for and lived healthy lives. However, the USA ceded its leading position for the oldest ages to Japan in 1990. Table 1(b) gives the largest contribution to the expectation of life of a 30-year-old man in each 10year age interval for one who notionally moved countries every decade to live in the country with the longest life expectancy. We appreciate this is not feasible in practice but we are merely seeking to identify the boundary of human survival which we have defined as the ‘leading edge’ (LE). We see from Table 1(b) that the largest increases in 10-year life expectancies between 1950 and 2009 occurred in the 70–80 and 80–90 age intervals, with increases of

2.24 and 2.44 years, respectively. It is noteworthy that in none of the ‘leading countries’ is there any ground for assuming further increases in life expectancy after age 100, which raises the important question of whether they will ever occur. The examples given above illustrate the key idea of the paper: the narrowing down of the scope for subjective judgement in assessing the potential for further increases in life expectancy. The penultimate row shows the life expectancy of a ‘leading edge’ man at age 30 for the rest of his life, that is, for one who moved at the start of each decade of life to the country with the longest life expectancy. The ‘rest-of-life expectancy’ can be obtained by summing the 10-year expectancies in the rows above. It shows that the increase has been 7.42 years over the period to 50.86 years. Beneath this table, for comparison, we show the rest-of-life expectancy for a man experiencing the ‘leading country’ (LC) life expectancy, which is the life expectancy of someone who remained in the same country with the highest life expectancy from age 30, that is, the country appearing in the bottom line of Table 1(a). We can see that the difference between


The decomposition of life expectancy the two sets of expectations given in the last two rows of Table 1(b) is never greater than 1 year and peaks in 1980. In more recent years, as expectancies in the youngest 10-year age intervals for the five countries have approached 10, future life expectancies are largely influenced by survivorship at the older ages, where Japan is now the clear leader. A final column shows the potential years remaining in each age interval before the maximum limit is reached, that is, the difference between 10 years, the limit, and life expectancy in 2009. This shows that the ‘leading edge’ has potentially 29.14 years remaining if the maximum age interval is set at 110 years as it is in this table, and that these are mostly concentrated in older age intervals. Taken together these are important shifts in rankings, which are more apparent for men in ‘lagging countries’ (see Appendix A). The largest change has been the gradual slippage of the USA to become the worst-performing country of the five, for all ages up to and including age 80. The Swedish case is noteworthy because it was the best-performing country in 2009 up to and including age 70, but then the worst performing at age 90 and over. The reason may be

79

either the increased variability of Swedish data at older ages because of its smaller population, or selection effects, that is, more people reaching this age than in other countries. Even the ‘lagging edge’ experienced considerable increases in life expectancy in the older age intervals, whilst the rest-of-life expectancy at age 30 grew by 10.43 years to 47.47 years, suggesting that the difference between the ‘leading edge’ and ‘lagging edge’ is narrowing.

3.2. Women Tables 2(a) and (b) refer to women and are constructed in the same way as Table 1(a) and (b) for men. Table 2(a) shows that, as for men, there has been a shift in rankings. Until 1960, Swedish women were the longest lived in each decade of life up to age interval 70–80 and US women the longest lived at higher age intervals. In 1980, Sweden gave way to Japan while the USA maintained a slight lead from age interval 90–100. From 2000, Sweden re-emerged to replace Japan in the 30–40 and 40–50 age intervals, but at other ages, Japanese women lived the longest. Although from 1990 there was some

Table 2 Life expectancy of women from age 30 every tenth year: (a) ‘leading country’ by 10-year age interval; (b) 10-year life expectancy for the corresponding country and ‘leading edge’ (a) From age 30

1950

1960

1970

1980

1990

2000

Sweden Sweden Sweden Sweden Sweden USA USA USA Sweden

Sweden Sweden Sweden Sweden Sweden USA USA USA Sweden

Sweden Sweden Sweden Sweden Sweden Sweden USA USA Sweden

Japan Japan Japan Japan Japan France USA USA France

Japan Japan Japan Japan Japan Japan USA USA Japan

Sweden Sweden Japan Japan Japan Japan Japan Japan Japan

From age 30

1950

1960

1970

1980

1990

2000

2009



e30(30:40) e30(40:50) e30(50:60) e30(60:70) e30(70:80) e30(80:90) e30(90:100) e30(100:110) e30(LE) e30(LC)

9.93 9.72 9.24 8.16 5.67 2.29 0.35 0.01 45.37 44.92

9.96 9.80 9.42 8.54 6.28 2.73 0.43 0.01 47.17 46.76

9.96 9.82 9.49 8.75 6.87 3.42 0.66 0.02 48.99 48.68

9.97 9.85 9.58 8.96 7.29 4.00 0.93 0.04 50.62 49.87

9.97 9.89 9.68 9.21 8.00 4.91 1.28 0.07 53.01 52.67

9.98 9.90 9.71 9.33 8.39 5.92 1.92 0.11 55.26 55.23

9.98 9.91 9.75 9.43 8.67 6.55 2.52 0.20 57.01 56.96

0.05 0.19 0.51 1.27 3.00 4.26 2.17 0.19 11.64 12.04

0.02 0.09 0.25 0.57 1.33 3.45 7.48 9.80 22.99 23.04

e30(30:40) e40(40:50) e50(50:60) e60(60:70) e70(70:80) e80(80:90) e90(90:100) e100(1001:10) e30(LC)

2009 Sweden Sweden Japan Japan Japan Japan Japan Japan Japan

(b)

Note: LC, leading country; LE, leading edge. Source: As for Figure 1.


80


increase in life expectancy after age 100, the increase was small and confined to Japan. Compared with the results in Table 1 for men, those for women differ in two respects: first, ‘leading edge’ life expectancy at age 30 increased faster, thus widening the gap between women and men from 1.93 to 6.15 years over the period; second, the increases were considerably larger in the 70–100 age intervals. As was the case for men, the difference between the ‘leading edge’ life expectancy and the leading country’s life expectancy was small, peaking in 1980. For the years 2000 and 2009, because Japan’s life expectancy leads at nearly all ages, the difference between Japanese life expectancy and ‘leading edge’ is only 0.03 and 0.05 years. The corresponding final column for women shows 22.99 potential years remaining, compared with 29.14 years for men in Table 1(b), but as with men this is concentrated at older age intervals. The main feature of the life expectancies for women for the ‘lagging countries’, shown in Appendix B, has been the replacement of the worst-performing country, Japan, by the USA, so that by 2009 the USA had the shortest life expectancy in the entire group of countries up to age 80. E&W had a temporary spell in the years 1980 and 1990 as the country with the shortest life expectancy between the ages of 70 and 90, before returning to the middle of the country ranking. However, the emergence of Sweden in 2009 as the country with the shortest life expectancy for women at age 90, is, as it was for men, a surprise result. Again, the explanation may be the greater volatility of Swedish data, or a selection effect. In the case of ‘lagging countries’, there were appreciable gains in life expectancy for women in the 70–80 and 80–90 age intervals of 3.17 and 3.47 years. Meanwhile, the ‘lagging edge’ rest-of-life expectancy for women at age 30 grew by 11.73 years to 51.90 years, only slightly less than in the ‘leading edge’, but still larger than the 10.43 years for men in the ‘lagging edge’. The results show that the potential for future increases in life expectancy are nearly all at the older ages. To give an example, based on the life expectancy of men in the ‘leading countries’ in 2009, the maximum future possible addition to life expectancy between the ages of 30 and 70 in 2009 is 1.71 years (0.93 years for women). However, if the age interval 70–80 is included, potential life expectancy would be extended by a further 2.67 years (1.33 years for women). Including the age interval 80–90 would add a further 5.80 years (3.45 years for women), and including 90-year-olds would add a further 9.00 years (7.48 years). A key finding, therefore, is that there is considerable scope for further improvements in life

expectancy beyond current levels without considering a future contribution from centenarians. The foregoing analysis assumes, of course, that the limit on the life expectancies within 10-year age intervals is, in fact, 10 years and not a period intermediate between the current longest observed expectation and ten, in which case conclusions would differ. However, if current trends persist, nearly all the increases in life expectancy in the next decade or so are likely to occur at ages older than 70. A further implication is that, if future increases in life expectancy were to follow the current trend, the human survival curve would become rectangularized (see, for example, Wilmoth and Horiuchi 1999; Canudas-Romo 2008; Thatcher et al. 2010). This outcome would result from an increasing number of survivors in each age interval, and a concentration of deaths around the modal age of death. This development will occur unless there are comparable increases in life expectancy at ages over 100, but, as can be seen from Tables 1 and 2, no such trends have yet been observed.

4. Future trends in life expectancy by 10-year intervals As Section 3 has consistently shown, the leading population in life expectancy is that of Japanese women, especially after 1980, and it is clear that the leading position of this population is now well established. It is therefore important to know if their longevity is approaching a natural limit, since their progress will inform expectations about future improvements in other countries. However, as we showed in Section 1, a quadratic function does not have the right mathematical properties to project existing trends because it is unbounded. Consequently, a different mathematical function and procedure were required. The chosen function needed to be non-decreasing (if we are modelling Western populations whose life expectancy we expect to continue to increase). We also needed to ensure that there was a natural limit to life expectancy, so that we did not project infinite life spans. By working with 10-year age intervals we knew there was a definite limit within each age interval and we could therefore concentrate on age intervals within which there was still scope for an increase in life expectancy. Since increasing trends were well established we assumed it would be possible to identify functions that fit the data well, and also approach this natural upper limit.

To this end, life expectancy for Japanese women in 10-year age intervals was calculated and graphed for each calendar year from 1950 to 2009. The results are presented in Figure 3, in which each set of data points represents the contribution to life expectancies at age 30 within each 10-year age interval, starting at age 30, for each calendar year since 1950. The dotted extensions are trend lines based on the functions outlined and justified below. For each age interval we have assumed that the maximum life expectancy of ten will be eventually reached, although clearly the implication would be the elimination of all deaths if projected over a long enough period. For younger age intervals, starting at ages 30, 40, and 50, the impact of the limit is minimal as the life expectancy is already at, or close to, ten, and has been for many years. For older age intervals the actual limit may, in reality, be less than ten, and projections will therefore be uncertain, but the impact is likely to be small since only short to medium-term projections will typically be used in day to day practice. It can be seen from Figure 3 that the expectations for 60-yearolds have already started to plateau, with life expectancy within the age interval being only 0.6 years below the limit; at age 70 it is 1.3 years below. At older ages there have been rapid increases in expectancy, which supports the argument for retaining the current limit of 10 years. A further complication is that because deaths are increasingly occurring at older ages, there is a concentration in the ages at death. The model is currently constructed to require everyone to die before they reach the age of 110, but a reduction in mortality is occurring at younger ages, albeit shifting to the right. Because the trend in the age interval 100–110 shows

Life expectancy in 10-year intervals (women)


The decomposition of life expectancy

81

little improvement (an extra 0.2 years since 1950), projections at the oldest ages cannot be taken as seriously as those for younger ages because clearly they are more uncertain. As summarized in Table 3, the most steeply rising trends, and where most recent gains have been made, are in the age intervals 70–80 (4.30 years since 1950) and 80–90 (5.15 years). This trend of increasing contributions to life expectancy has also occurred in the age interval 90–100 with an increase of 2.41 years since 1950. However, as the two final columns of Table 3 show, the contribution to life expectancy at age intervals 90–100 and 100–110, if extrapolated to 2050 based on the trends shown in Figure 3, could be as high as 8.14 and 4.93 years, respectively.

4.1. Basic projection model Long-range projections of this kind are obviously sensitive both to the assumptions on which they are based and the projection method. As noted above, we sought a function with an unbreachable limit, rather than one which may have no limit or one which is inappropriate such as a quadratic. We discussed the problem of demographic institutions imposing arbitrary limits to life expectancy which subsequently prove to be too low in an earlier paper (Mayhew and Smith 2013). We showed that by allowing the data to determine the trend curve, forecasts could have been improved, though we also saw that our predictions became unrealistic after 2030, as life expectancy increased at an increasing rate. Our chosen function is based on a natural limit within each age interval. Using this method, we are

10 9 8 7

30–40 40–50 50–60 60–70 70–80 80–90 90–100 100+

6 5 4 3 2 1

0 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

Year

Figure 3 Trends in life expectancy for Japanese women from age 30 for every ten-year age interval from 1950. Dotted line shows extrapolated trends based on function described in text Source: As for Figure 1.

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Table 3 Decomposition of life expectancy for Japanese women at age 30 at 10-year intervals From age 30

1950

1960

1970

1980

1990

2000

2009

2030

2050

e30(30:40) e30(40:50) e30(50:60) e30(60:70) e30(70:80) e30(80:90) e30(90:100) e30(100:110) e30

9.75 9.20 8.40 6.95 4.36 1.40 0.11 0.00 40.17

9.91 9.64 9.10 7.94 5.41 1.89 0.17 0.00 44.06

9.95 9.77 9.37 8.45 6.22 2.51 0.27 0.00 46.54

9.97 9.85 9.58 8.96 7.29 3.67 0.55 0.01 49.88

9.97 9.89 9.68 9.21 8.00 4.86 1.04 0.03 52.68

9.98 9.90 9.71 9.32 8.38 5.92 1.92 0.11 55.24

9.98 9.90 9.74 9.42 8.66 6.55 2.52 0.20 56.97

10.00 9.97 9.89 9.75 9.41 8.37 5.67 1.27 64.33

10.00 9.98 9.94 9.87 9.70 9.24 8.14 4.93 71.80


Source: As for Figure 1.

able to combine the expectations of life for each age interval to obtain a projection of aggregate life expectancy. The results in Figure 3 have been derived using this procedure based on 10-year age intervals, and are shown in the bottom row of Table 3 as the rest-of-life expectancy at age 30. The logistic function was found to be the most useful because of its convenient properties and the ease with which its parameters can be estimated; it can be written as follows: Ae fi ðt;nÞ 1 þ e fi ðt;nÞ where fi(t, n) is a polynomial of order n which generally takes the value of one or two. That is: fi ðtÞ ¼ ai þ bi t yi ¼

or fi ðtÞ ¼ ai þ bi t þ ci t 2 where ai, bi, and ci are parameters to be estimated, t the calendar year, and y the life expectancy in age interval i. Here A is a constant, equal to the size of the age interval selected. In this case it is set to 10 years. Re-arranging yields: yi ln ¼ fi ðtÞ ¼ ai þ bi t Ay or

yi ln ¼ fi ðtÞ ¼ ai þ bi t þ ci t 2 : Ay

In these equations, ai, bi, and ci are estimated using multiple linear regression. Note that higher-order polynomials (n > 2) could also be considered depending on the patterns observed in practice. By fitting both functions to data in each age interval from 1950 to 2009 and selecting the best fit, we analysed how future life expectancy might develop, assuming a continuation of underlying trends. In low-mortality countries, such as those considered here, we found that a linear equation (first-order polynomial) generally performed best for women and a quadratic (second-order polynomial) for men.

Second- and higher-order polynomials have the useful property that life expectancy can fall and then rise again before progressing towards a limit. This characteristic is appropriate for countries, such as some in Eastern Europe, which are emerging from a period of declining life expectancy. By fitting the function to data in each age interval and from 1950 to 2009, we analysed how life expectancy might be expected to change, assuming a continuation of the underlying trend. Table 4 gives the estimated regression parameters for Japanese women. All coefficients in each age interval, except for age 30–40, are statistically significant at the 1 per cent level and since the estimate of the limit is close to ten for the 30–40 age interval, the result is of no particular consequence. For older ages, the fitted trend lines have values of R2 that are between 0.92 and 0.99, indicating very good statistical fits. Corresponding results for other countries in the group are given in the supplement (Mayhew and Smith 2014a).

4.2. Future prospects The extrapolated trends suggest that life expectancies for Japanese women could continue to grow rapidly in the coming decades. Although speculative, Figure 3 suggests that this rapid growth will continue until at least the turn of the century in 2100. Table 4 Fitted values of the parameters for the projection model for Japanese women a

b

R2

5.2789 −63.2477 −59.0748 −61.9846 −71.4010 −86.3251 −122.2067 −194.6224

0.0023560 0.0339573 0.0313364 0.0323309 0.0365338 0.0433296 0.0603322 0.0949237

0.68 0.92 0.94 0.97 0.99 0.99 0.98 0.96

Age interval 30–40 40–50 50–60 60–70 70–80 80–90 90–100 100+

Source: As for Figure 1.

Arguably, it is the forecasts of life expectancy for the coming decades which are of more practical value. It is of interest, for example, to compare how the trends shown in Figure 3 compare with quadratic or linear trends fitted to the data. Such a comparison is presented in Figure 4, which shows past and projected life expectancies at age 30 fitted by (A) quadratic, (B) logistic, and (C) linear functions. Conventional statistical measures show that the goodness of fit to the data from 1950 to 2009 for each function is very similar and there is nothing to suggest that one is superior to the others. From 2010 onwards, the logistic function gives slightly higher values than the quadratic, and the difference widens as the upper limit of 80 is approached in 2100 (horizontal line D). The quadratic trend peaks in 2130 at 70 years, at a value which is lower than predicted by the logistic function. The linear function reaches the natural upper limit soonest, and subsequently exceeds it around 2090.

4.3. Model limitations Our extrapolations are based on an assumption that Japanese women can live to a maximum of 110 years (i.e., 30 + 80 = 110), but we know that a small number have already survived beyond this age. Whether it is possible for much larger numbers to survive beyond age 110 must remain an open question, with some arguing that there is no limit (e.g., Wilmoth 2000; Vaupel 2010). The uncertainty is obviously greater at older ages, at which life expectancy has increased least, so contributing very little to total future life expectancy. For example, in 2009, the contribution to

83

life expectancy in age interval 80–90 was 6.55 years, and at 90–100, 2.52 years, but at 100–110 it was only 0.20 years (Table 3). The extrapolation of existing trends is not possible owing to the small numbers at risk. The Human Mortality Database (University of California 2009), the data source, provides only the number of survivors at ages above 110 but these are not broken down by age. However, it can be shown that the probability of a Japanese woman age 30 surviving to 110 is only 0.0006, so that any contribution to life expectancy at older age intervals will be trivially small. It is therefore important to regard the upper age interval in our model as flexible and able to be updated in the light of any new evidence of people living beyond 110. Note also that if the question is ‘what is the life expectancy in the age interval 30 to 110?’ rather than ‘what is the upper limit?’, information on survivorship beyond age 110 is not needed. Because of the paucity of data for ages 110 and over, this procedure is a more reliable way of comparing future life expectancy between different countries (see below). This arbitrary cut-off does not appear to affect the accuracy of the predictions unduly, at least not for the immediately ensuing decades. To determine their accuracy, we made three forecasts over the same period using data for Japanese women for the periods 1950–80, 1950–90, and 1950–2000. We then compared the estimated life expectancies based on the three forecasts up to 2009 and found that all three gave estimates of life expectancy that were very close to actual outcomes, showing the closeness of fit and hence robustness of the method. We then fitted the same function to life

100 90

Life expectancy at 30 (women)



D

80

B

C

70 60

A

50 40 30 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

Year Key: A – Quadratic trend; B – Logistic function; C – Linear trend; D – Asymptote in the logistic model

Figure 4 Comparison of trends in life expectancy for Japanese women at age 30 using three forecasting methods Source: As for Figure 1.

84


expectancy at age 30 from 1950 to 2009 and used it alongside the three previous forecasts to predict future values of life expectancy from 2020 to 2100. The results are shown in Table 5, in which we see that the forecasts are very similar, regardless of which of the data sets are used. Only when data from 1950 to 1980 were used did the results deviate by more than 2 years from the other sets of forecasts. All four forecasts show that by 2100, life expectancy for Japanese women at age 30 will be close to the maximum of 80 years, based on an upper bound of 110 years. We also estimated life expectancy in a smaller age range 30–100 to see if our findings would differ, but in fact they were similar.

Country (a) Men England and Wales France Japan Sweden USA (b) Women England and Wales France Japan Sweden USA

2009

2020 2040

2060 2080 2100

49.4

53.6

62.6

68.5

69.7

69.9

48.8 50.4 50.2 47.8

52.3 53.8 53.3 51.0

59.1 59.1 59.4 57.4

65.2 64.0 65.0 63.7

68.6 67.2 68.6 67.9

69.6 68.7 69.7 69.5

53.2

55.7

61.7

67.2

69.4

69.9

55.1 57.0 53.8 52.1

57.5 60.5 55.9 53.6

61.4 65.2 59.1 56.1

64.7 68.0 62.2 58.7

67.1 69.2 64.9 61.1

68.5 69.7 66.9 63.3

Note: Numbers in bold are actual values. Source: As for Figure 1.

4.4. Other countries Downloaded by [ECU Libraries] at 17:49 25 April 2015

Table 6 Predicted life expectancies between ages 30 and 100 from 2009 to 2100

Similar analyses were also made for the four other countries, all of which, like Japan, had falling mortality and ageing populations. For each country, a long time series of demographic data is available, an essential requirement of the successful application of the decomposition technique. The results, which can be seen in the supplementary material (Mayhew and Smith 2014a), show that decomposition works equally well for most age intervals and for both sexes. A key reason for this good fit is that life expectancy for men in middle age is increasing more rapidly than for women, albeit from an initial shorter life expectancy, and a quadratic better reflects this trend. As with Japan, the main uncertainty is the improvement in life expectancy beyond age 100. Owing to the small numbers, accepting the quadratic prediction of rapid growth at the oldest ages could be misleading. For greater consistency we therefore restricted forecasts to the age interval 30–100 (i.e. a Table 5 Actual and predicted life expectancies for Japanese women at age 30 from four different sets of data from the database Base data

1980 2000

2020 2040

2060 2080 2100

1950–80 1950–90 1950– 2000 1950– 2009 Actual

49.7 49.9 49.8

54.6 55.4 55.3

59.2 60.8 61.1

63.6 66.4 67.6

67.9 72.3 74.5

72.1 77.3 78.5

76.2 79.3 79.6

49.7

55.1

60.0

68.0

75.2

78.7

79.6

49.9

55.2

–

–

–

–

–

Note: Numbers in bold are actual values. Source: As for Figure 1.

maximum of 70 years); the results are shown in Tables 6(a) and (b) for men and women. The results show that the life expectancies for men and women between the ages 30 and 100 are predicted to converge in 2100, at just under the possible maximum of 70 years. Between now and 2100, the life expectancy of men in E&W will overtake that for men in the other countries, and the life expectancy of women in E&W will match that of Japanese women by 2080. A key finding is that, if current trends continue, the life expectancy of women in the USA will fall increasingly behind that of other countries, and that the life expectancy of men in the USA will exceed that of US women after 2060. These implications are remarkable. Obviously, they are less reliable than forecasts to, say, 2040, but are nevertheless interesting and noteworthy.

5. Discussion Life expectancy is a widely used and understood measure of human development. While it is useful when comparing one country with another, the measure has proved of much less use for the projection of trends by more than a few years using standard extrapolation methods, whether deterministic or stochastic. The alternative practice of using age-specific mortality rates is widely used in the insurance industry, but because mortality rates are expressed in different units to life expectancy they are difficult to compare. We therefore sought a method that would enable us to disaggregate life expectancy into smaller age intervals, rather than


The decomposition of life expectancy only one to the end of life, and to relate these component contributions to changes in overall life expectancy. We find that splitting life expectancy into 10-year age intervals is helpful in comparing the results for each age group between different countries and different years. Results show that ‘leading edge’ and ‘lagging edge’ countries have alternated many times, but Japan and Sweden are the countries that have consistently had the longest life expectancies, while the USA has increasingly ranked among the shortest. Possible reasons for the USA’s deteriorating record have been discussed extensively in Preston et al. (1999), Kulkarni et al. (2011), National Research Council (2011a, 2011b), and Olshansky et al. (2012). We have shown that decomposition offers a potentially easier way to track and project changes in survivorship. When life expectancies in the different age intervals are combined it is possible to make an overall forecast of the life expectancy at a particular age, and one which has been constrained to an achievable limit within each age interval. The maximum life expectancy at age 30 in the present case is 80 years, that is, 10 years for each decade of life between 30 and 110. Our forecasts using variants of the logistic function are promising, as illustrated by the Japanese case. In tests using data from 1950– 80, 1950–90, and 1950–2000 to forecast 2009 life expectancies, we found the accuracy to be similar for the different trend functions, suggesting the method is robust for countries with many years of goodquality demographic data. There is considerable interest in how long people will live in the future, and the life expectancy of Japanese women, the longest-lived in the world, provides several insights. We have shown that life expectancy should continue to increase in the next decades with most of the increase occurring between the ages of 70 and 90, and not as a result of reductions in the mortality of centenarians. At age 100 and over, longevity is much less established and may not increase before 2050. Extrapolations suggest that a limit to the increase in life expectancy will be reached, perhaps towards the end of this century; however, this is not a firm forecast. An implication of our findings is that we can expect an increasing concentration of mortality in the 90–100 age interval. The possibility of more people living to 110 and beyond does not affect our analysis because their numbers are currently so few. Evidently, further work is needed to discover and test which functions are appropriate in different circumstances, but also the number and width of

85

age intervals to use. The results for four other countries have been briefly reviewed and found to be consistent in the goodness of fit and accuracy of predictions. Our limited examples have confirmed established facts and have also identified other, new, features. For example, life expectancy is forecast to improve more slowly for women than for men, except for Japan. Mayhew and Smith (2014b) found convergence in life expectancy between the sexes to be especially rapid in E&W, largely owing to the fall in men’s tobacco consumption. Currently, decomposition produces a single forecast for each country and age interval. This deterministic approach could be supplemented using regression confidence intervals to show range of uncertainty, which obviously would be most useful for the older age intervals. Stochastic formulations are another possible direction for development. Because our method is new, it is too early to compare its forecasting potential with that of the methods based on either life expectancy or mortality described in the expanding literature on demographic forecasting (for reviews, see Booth 2006 or Raftery et al. 2013). Conceptually, however, there are clear differences of approach. Our model is based on trends in what we termed ‘partial life expectancy’, from which mortality rates may be derived as required. For best results, a long time series of data is required as well as unabridged life tables. In the widely used Lee–Carter model (Lee and Carter 1992), it is the other way round, with trends in mortality rates the starting point. Our model is deterministic, whereas Lee–Carter’s is stochastic, but our model is able to predict turning points using the logistic function and the natural limit found in each age interval. This difference is important since, as Bongaarts (2005) notes, the invariance in the rate of decline of mortality assumed by Lee–Carter produces implausible results after only a few decades. Using the logistic function, Bongaarts was able to project the ‘force of mortality’ more accurately than the Lee–Carter method. Nevertheless, drawing general conclusions on the relative merits of each approach, and comparing them with our own is somewhat complicated, and also arguably premature, because of the many Lee–Carter variants available (e.g., see Renshaw and Haberman 2003, 2006, or the ‘rotation’ approach in Li et al. 2013). In contrast to the Lee–Carter method, deterministic models are widely used for forecasting life expectancy; the United Nations (UN) uses them to produce life expectancies by country, for the world (UN 2009). More recently, Raftery et al. (2013) have


86


proposed a stochastic variant of the method used by the UN for which superior accuracy is claimed. Both the UN and Raftery et al. base their forecasts on the double logistic function. The assumption is that life expectancy will decline over a period, before continuing asymptotically. Our method also uses a logistic function to forecast life expectancy but within discrete 10-year age intervals that are upper-bounded. In other words, we do not consider the possibility of ever-increasing life spans nor do we rule them out. By working with, in this case, 10-year age intervals, we know there is a definite limit within each age interval, and we can therefore concentrate on age intervals within which life expectancy is still increasing. Put another way, decomposition can address the question ‘what is life expectancy in an arbitrary age interval, say, between 30 and 100?’ rather than ‘what is the upper limit to life expectancy?’ The maximum attainable is 70 years in this particular example. Although the method is flexible in this respect, and works equally well for men and women, the availability of data from which to establish trends at the oldest age interval is clearly more problematic. Under our method, users can decide whether to include them or not.

Notes 1. Les Mayhew and David Smith are at the Faculty of Actuarial Science and Insurance, Cass Business School, City University London, 106 Bunhill Row, London EC1Y 8TZ, UK. E-mail: [email protected]

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Appendix A: Lagging countries and 10-year life expectancy (men) (a) Lagging country From age 30

1950

1960

1970

1980

1990

2000

e30(30:40) e40(40:50) e50(50:60) e60(60:70) e70(70:80) e80(80:90) e90(90:100) e100(100:110) e30(LgC)

Japan Japan Japan Japan Japan Japan Japan Japan Japan

Japan USA USA USA Japan Japan Japan Japan Japan

USA USA USA USA E&W Japan Japan Japan USA

USA France France E&W E&W E&W Japan Japan France

USA France France USA E&W E&W E&W France USA

USA France France USA E&W E&W Sweden Sweden USA

2009 USA USA USA USA USA Sweden Sweden Sweden USA


(b) Ten-year life expectancy and lagging edge From age 30

1950

1960

1970

1980

1990

2000

2009



e30(30:40) e30(40:50) e30(50:60) e30(60:70) e30(70:80) e30(80:90) e30(90:100) e30(100:110) e30(LgE) e30(LgC)

9.73 9.09 8.06 6.13 3.21 0.78 0.04 0.00 37.04 37.04

9.88 9.49 8.57 6.74 3.83 0.97 0.06 0.00 39.54 40.06

9.88 9.48 8.58 6.79 3.95 1.16 0.09 0.00 39.93 40.53

9.90 9.58 8.80 7.26 4.48 1.46 0.14 0.00 41.62 42.58

9.88 9.57 8.96 7.65 5.17 1.96 0.24 0.00 43.43 44.17

9.92 9.66 9.12 8.03 5.86 2.63 0.38 0.01 45.61 45.91

9.92 9.69 9.19 8.21 6.44 3.43 0.58 0.01 47.47 47.79

0.19 0.60 1.13 2.08 3.23 2.65 0.54 0.01 10.43 10.75

0.08 0.31 0.81 1.79 3.56 6.57 9.42 9.99 32.53 32.21

Note: LgC, lagging country; LgE, lagging edge.


89

Appendix B: Lagging countries and 10-year life expectancy (women) (a) Lagging country From age 30

1950

1960

1970

1980

1990

2000

2009

e30(30:40) e40(40:50) e50(50:60) e60(60:70) e70(70:80) e80(80:90) e90(90:100) e100(100:110) e30(LgC)



USA USA USA USA Japan Japan Japan Japan Japan

USA USA USA USA E&W E&W Japan Japan E&W

USA USA USA USA E&W E&W E&W Sweden E&W

USA USA USA USA USA USA E&W Sweden USA

USA USA USA USA USA USA Sweden Sweden USA


(b) Ten-year life expectancy and lagging edge From age 30

1950

1960

1970

1980

1990

2000

2009



e30(30:40) e30(40:50) e30(50:60) e30(60:70) e30(70:80) e30(80:90) e30(90:100) e30(100:110) e30(LgE) e30(LgC)

9.75 9.20 8.40 6.95 4.36 1.40 0.11 0.00 40.18 40.18

9.91 9.64 9.10 7.94 5.41 1.89 0.17 0.00 44.06 44.06

9.94 9.70 9.20 8.21 6.05 2.45 0.27 0.00 45.82 46.54

9.95 9.79 9.38 8.52 6.64 3.26 0.51 0.01 48.06 48.21

9.96 9.81 9.46 8.66 6.92 3.77 0.79 0.02 49.40 49.73

9.96 9.82 9.50 8.76 7.19 4.28 1.02 0.03 50.54 50.57

9.96 9.82 9.51 8.88 7.53 4.87 1.29 0.04 51.90 52.07

0.21 0.62 1.11 1.93 3.17 3.47 1.18 0.04 11.73 11.89

0.04 0.18 0.49 1.12 2.47 5.13 8.71 9.96 28.10 27.93

Note: LgC, lagging country; LgE, lagging edge.

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