VOLUME 13, ARTICLE 5, PAGES PUBLISHED 05 OCTOBER DOI: /DemRes

Transcription

1 Demographi Researh a free, expedited, online journal of peer-reviewed researh and ommentary in the population sienes published by the Max Plank Institute for Demographi Researh Konrad-Zuse Str. 1, D-157 Rostok GERMANY DEMOGRAPHIC RESEARCH VOLUME 13, ARTICLE 5, PAGES PUBLISHED 5 OCTOBER 25 DOI: 1.454/DemRes Researh Artile Changing mortality and average ohort life expetany Robert Shoen Vladimir Canudas-Romo 25 Max-Plank-Gesellshaft.

2 Table of Contents 1 Introdution Aggregate measures of mortality ACLE: A new measure of longevity 12 4 Cohort and period models of hanging mortality ACLE in England and Wales, Norway, and Switzerland Disussion and onlusions Aknowledgments 136 Referenes 137 Appendix: Deomposing the hange over time in ACLE 139

3 Demographi Researh: Volume 13, Artile 5 a researh artile Changing mortality and average ohort life expetany Robert Shoen 1 Vladimir Canudas-Romo 2 Abstrat Period life expetany varies with hanges in mortality, and should not be onfused with the life expetany of those alive during that period. Given past and likely future mortality hanges, a reent debate has arisen on the usefulness of the period life expetany as the leading measure of survivorship. An alternative aggregate measure of period mortality whih has been seen as less sensitive to period hanges, the rosssetional average length of life (CAL) has been proposed as an alternative, but has reeived only limited empirial or analytial examination. Here, we introdue a new measure, the average ohort life expetany (ACLE), to provide a preise measure of the average length of life of ohorts alive at a given time. To ompare the performane of ACLE with CAL and with period and ohort life expetany, we first use population models with hanging mortality. Then the four aggregate measures of mortality are alulated for England and Wales, Norway, and Switzerland for the years 18 to 2. CAL is found to be sensitive to past and present hanges in death rates. ACLE requires the most data, but gives the best representation of the survivorship of ohorts present at a given time. 1 Department of Soiology, Pennsylvania State University, University Park, PA 162; shoen@pop.psu.edu 2 Population Researh Institute, Pennsylvania State University, University Park, PA 162; anudas@pop.psu.edu 117

4 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany 1. Introdution A life table desribes the mortality of a hypothetial birth ohort, and an summarize that experiene in terms of life expetany at birth (LE), i.e. the average number of years lived by the members of the life table ohort. When a life table is based on ohort data, it an reflet the survivorship of an atual group of people. In most ases however, life tables are based on mortality rates observed in a given year (or period). The period life expetany from suh a table an reflet the long term impliations of reent behavior, but does not relate to the experiene of any real ohort, and typially varies from year to year as death rates hange. The question addressed here is What is the average life expetany of ohorts alive in a given year? It is neither the period life expetany nor the ohort life expetany, and indeed is a quantity that has not previously been given a preise definition. Reently a signifiant paper by Bongaarts and Feeney (22) opened a new debate on how to interpret period life expetany when rates of death vary over time. They argue that the period LE exhibits a tempo bias when a rising (falling) mean age of persons at the ourrene of an event results in a temporary deline (inrease) in numbers of events during the period of hange (p. 2). To orret for that bias, Bongaarts and Feeney (22) advane an alternative measure that, in other work (Brouard 1986; Guillot 23a), has been termed the ross-setional average length of life (CAL, a translation from the Frenh dure e de vie moyenne atuelle). Vaupel (22) and Guillot (23b) have ritiized that approah, iting its ambiguous definition of mortality tempo effets. Yet there is no doubt that the period LE an be seriously misleading if it is interpreted as the life expetany of any atual group of persons. Here we do not attempt to adjust for tempo effets, but seek to advane the area by proposing a new measure that gives the average lifetime of a period population. Before presenting that measure, we disuss existing summary measures at greater length. 2. Aggregate measures of mortality The most ommonly known measures of mortality are the ohort and period life expetany. Period life expetany at age a and time t is alulated as the person-years lived above age a divided by the number surviving from birth to age a. For example, the period life expetany at birth at time t an be expressed as 118

5 Demographi Researh: Volume 13, Artile 5 l p e p (, t) = l ( a, t) da p (, t), (1) where l ( a, t) is the period survivorship funtion at age a at time t and is the highest p age attained. Letting the radix of the table be one, i.e. l (, t) = 1, we see that l ( a, t ) is the period life table probability of surviving from birth to age a. Denoting the fore of mortality at age a and time t by µ ( a, t ), we an write the life table probability of surviving from birth to age a as a l p ( a, t) = exp( µ ( x, t) dx). (2) The subsript p in equations (1) and (2) is used to denote period measures. In the rest of the text, subsript will be used to identify ohort measures. For example, l ( a, t a ) denotes the life table probability of surviving from birth to age a for the ohort born at time t-a, i.e. a l ( a, = exp( µ ( x, t a + x) dx). (3) That ohort s life expetany at birth, e (, is given by p p e (, = l l ( x, dx (,, (4) Sine at exat age a and time t the ohort and period fores of mortality are idential, µ ( a, t ) does not have a subsript. The ross-setional average length of life, CAL, inorporates the past and present mortality of all ative ohorts. Consider a population that has a onstant number of births every year, but where mortality is free to vary over age and time. The number of persons in that population at time t is CAL(t). Mathematially, it an be expressed as 119

6 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany CAL( t) = l ( a, da. (5) In the stationary population of the life table, with its onstant age-speifi mortality, CAL equals life expetany. In general, however, the size of CAL(t) differs from a population s period LE at time t. Beause in most ontemporary populations mortality in past periods is generally greater than mortality at reent time t, the survival probabilities that are summed to produe CAL(t) are usually lower than those that are summed to yield the period LE. As a result, the size of CAL typially understates the urrent level of period mortality, that is the level refleted by the period LE. 3. ACLE: A new measure of longevity At its ore, the ritiism of period life expetany is that it fails to provide an aurate measure of the atual longevity of the persons in the population being examined. At present, however, no measure in demography does that, inluding CAL (whih does not inorporate mortality after the speified period). Here we fill that gap by presenting a new measure that does apture the average LE of ative ohorts. Our approah is straightforward: we speify a weighted average of the life expetanies at birth of all ative ohorts. Weights are needed beause it is not reasonable to give equal emphasis to newborns and to the few who survive to high ages. At every age, the weight we propose is the atual ohort probability of survival to that age. In effet, we let ohort survivorship provide the weights for our measure of average ohort survival. Let ACLE(t) denote this Average Cohort Life Expetany. We an then write e (, l ( a, da ACLE( t) = = l ( a, da or e (, l CAL( t) ( a, da, (6a) ACLE( t) = e (, CCAL( a, da, (6b) 12

7 Demographi Researh: Volume 13, Artile 5 where C CAL ( a, denotes the density distribution of the ohort survival probabilities, C CAL l ( a, ( a, = CAL( t). Bongaarts (24) defined an alternative measure ACLE p, using period deaths instead of ohort survivors as weights. Suh a measure is likely to yield values quite different from ACLE, and will in some (though not all) ases approximate CAL. There are two reasons why we believe that alternative is not attrative. First, it seems muh more logial to weight a measure of ohort survivorship by ohort survivors rather than by deaths. Seond, some populations may have deaths ourring only at several high ages. If deaths were used as weights, only those few older ohorts would determine average ohort survivorship, defeating the purpose of having a measure that reflets the longevity of all living ohorts. The value of our ACLE(t) is the average of the life expetanies of the ohorts alive in year t, with those expetanies weighted by the CAL population. When mortality is onstant over time, period life expetany, ohort life expetany, ACLE, and CAL are all equal. When mortality is delining, ACLE will be less than the LE of the youngest ohort, but greater than period LE whih in turn is greater than CAL. To find ACLE, we must know the ultimate life expetany of all ative ohorts, information not available for at least a entury after the time in question. Yet the need for that ohort data is inesapable, sine the purpose of the measure is to determine average ohort life expetany. Three points should be made. First, ACLE provides a preise and oneptually lear measure of average ohort LE. When it an be alulated, it provides a gold standard, i.e. a lear, oneptually based measure that best aptures the idea of an average ohort life expetany. When ACLE annot be alulated, it identifies what needs to be estimated. Seond, in the ontext of dynami population models it an be examined under different mortality senarios and its performane ompared to that of alternative measures. Third, given the ontinuing interest in ohort mortality and future mortality trends, the ACLE standard helps to fous attention on how best to estimate future mortality and to assess the errors in suh estimates. To gain a better appreiation for how ohort LE, period LE, CAL and ACLE differ from eah other, the following setion examines their values in the ontext of models with hanging mortality

8 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany 4. Cohort and period models of hanging mortality Model populations provide a useful way to examine the sensitivity of aggregate measures to known, patterned hanges in period and ohort mortality. To begin, we assume a simple Gompertz form for the fore of mortality and write µ(a,t) = exp[ A + ba f(t) ] (7) where f(t) is a known funtion of time. To start with the simplest dynami model, we assume that f(t)= for t, and f(t)=k for t>, where k an be any onstant. Figures 1a and 1b show values of period and ohort LE, ACLE, and CAL for k=1 and k= 1. In the Gompertz model, we take b=.1, the onventional value for the pae of mortality inrease over age, and set A= so µ(5,)=.2, a reasonable value for a ontemporary low mortality population. With b=.1, putting k=1 is equivalent to a mortality set bak of 1 years, as a person aged x beomes subjet to the mortality risk that haraterized a person aged x 1. Similarly, k= 1 is equivalent to a mortality set ahead of 1 years. Figure 1 shows that at time, when mortality shifts, the ohort LE ompletes its gradual transition from the old to the new level, while the period LE abruptly moves from the old life expetany at birth to the new. At t=, ACLE has ompleted most, but not all of its transition. CAL begins its rise at time, when deaths depart from 1, and is the last measure to omplete the transition

9 Demographi Researh: Volume 13, Artile 5 Figure 1a: Cohort and period life expetany, ACLE and CAL for a Gompertz mortality model with A= , b=.1 and, beginning at time, k = Values of mortality measure Cohort LE Period LE ACLE CAL Year / ohort year of birth Figure 1b: Cohort and period life expetany, ACLE and CAL for a Gompertz mortality model with A= , b=.1 and, beginning at time, k = -1 Values of mortality measure Cohort LE Period LE ACLE CAL Year / ohort year of birth 123

10 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Figure 2a shows the total number of deaths ourring in the population eah year, given one birth a year. Before time, there is one death eah year. At t= and k=1, there is an immediate fall to.38 deaths, with the annual number of deaths then inreasing over time and eventually returning to one. Eah year, CAL grows by the amount that deaths are less than 1. Wahter (24) showed that CAL(t) is a weighted average of period life expetanies for year t and preeding years. In this simple ase, with only 2 period life expetany values, it is possible to determine the weights involved. Speifially, we have CAL(t) = w(t) LE( ) + [1 w(t)] LE(+) (8) Where LE( ) and LE(+) are the old and new life expetanies, respetively, and w(t) is the time t frational weight exerted by LE( ). Figure 2b shows that after t=, w(t) delines to in a roughly exponential fashion, with larger values of k assoiated with slower delines. The deline in w(t) need not be rapid. When k=1 and t=2, the rates prevailing 2 or more years earlier still exert a weight of.24. Let us now turn to an extension of the Gompertz model of mortality that traes a onstant rate of deline over time. The basi model has been disussed by Vaupel (1986) and extended by Shoen, Jonsson and Tufis (24). In that model, the fore of mortality at age a and time t is defined as A+ ba t µ ( a, t) = e, (9) where A and b retain their previously speified values. The annual rate of mortality improvement at all ages,, is examined at two values:.8, less than the reent rate of mortality deline in the West, and.15, a loser approximation to many reent delines

11 Demographi Researh: Volume 13, Artile 5 Figure 2a: Total number of deaths in a population with one birth eah year and Gompertz mortality shifting by k at time 3, 2,5 Number of deaths 2, 1,5 1, k = 1 k = -1,5, Year Figure 2b: Values of the weighting funtion w(t) in the equation of CAL(t) for a Gompertz mortality model with a shift in mortality by k at time ; CAL(t) = w(t) LE(-) + [1-w(t)] LE(+) 1,2 1, Values of the weighting funtion,,6,4 k = 3 k = 1 k = -1 k = -3,2, Year 125

12 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Figure 3 shows ohort and period LE, CAL, and ACLE for times through 2 years based on equation (9) and our hosen parameter values for A, b, and. All four measures are represented by virtually straight lines that appear to be parallel. At any given year t, the ohort life expetany for those born in that year is the highest of the four measures, followed by ACLE, period life expetany and CAL. The differene between CAL and period life expetany is partiularly small (e.g. about.8 years for =.8), and the LE/CAL ratio is about 1+. The slope of the urves depends on parameters b and. For period LE and CAL, the slope is lose to /b. For ohort LE, it is about /(b ), so over time the ohort LE line diverges from CAL and period LE. ACLE is roughly the arithmeti mean of the ohort and period life expetanies, and its slope approximates the average of the slopes of those expetanies. The ratio of ACLE to the ohort LE is lose to onstant over time, at a value of 1 /(2b). Goldstein and Wahter (25) studied gaps and lags between ohort and period life expetanies. They found that in populations with steady mortality delines the gap between ohort and period life expetanies at any point in time remains roughly onstant, while the time lag inreases. The time lag, λ, indiates the number of years between the time that ohort LE reahes a given level and the time period LE attains that level. In the model of Figure 3a, λ ranges from 74 years earlier at time to 9 years earlier at time 2, while in the model of Figure 3b those lag values are 74 and 13 years, respetively. Over those years, however, the gap only inreases from 6.3 to 7.7 years in Figure 3a and from 12.8 to 18.1 years in Figure 3b. Now, let us modify the mortality model in equation (9) to allow ylial flutuations over time. The new expression for the fore of mortality an be written µ ( a, t) A+ ba ( t+ d sin( tθ )) = e, (1) where θ = 2π / 6 (about.1472) yields a yle (from peak to peak) of 6 years, and d=.25 is a onstant that moderates the amplitude of the osillations

13 Demographi Researh: Volume 13, Artile 5 Figure 3a: Cohort and period life expetany, ACLE, and CAL for a ontinually delining mortality model with A= , b=.1 and = Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth Figure 3b: Cohort and period life expetany, ACLE, and CAL for a ontinually delining mortality model with A= , b=.1 and = Values of mortality measure Cohort LE Period LE ACLE CAL Year / ohort year of birth 127

14 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Figure 4a presents values of the ohort and period LE, CAL, and ACLE alulated on that basis. Those trajetories ontrast markedly with those shown in Figure 3. The ohort LE now yles modestly, though it is still the highest of the four measures. However, CAL and the period LE alternate as the lowest measure, as they yle with different phase shifts and the period LE has a slightly larger amplitude. The smoothest of the four measures is ACLE. As an average of ohort LEs, it yles with an amplitude markedly smaller than that of the ohort LE. The ontinually delining mortality model in Figure 3, whih is based on equation (9), will yield similar results (though with a different rate of hange over age) if rates hange over ohorts instead of periods. That is not the ase with Figure 4a, whih is based on equation (1). To examine patterns when mortality hanges ylially over ohorts, we an write µ ( a, t) A+ ( b+ ) a ( t+ d sin(( t a) θ )) = e. (11) The oeffiient of the age variable is (b+) beause age and time move together in the ohort perspetive. Figure 4b shows the results when ohort rates flutuate sinusoidally as desribed by equation (11). The results are similar to those in Figure 4a, but here the ohort LE yles with the largest amplitude. Period LE and CAL alternate in the last position, though CAL is usually lower. Again, ACLE is the seond highest measure and shows the smallest amount of flutuation

15 Demographi Researh: Volume 13, Artile 5 Figure 4a: Cohort and period life expetany, ACLE, and CAL for a sinusoidally delining mortality model with period ylial flutuations of 6 years and parameters A= , b=.1 and = Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth Figure 4b: Cohort and period life expetany, ACLE, and CAL for a sinusoidally delining mortality model with ohort ylial flutuations of 6 years and parameters A= , b=.1 and = Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth

16 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Guillot (23b:2) suggested that tempo distortions in mortality be defined by adapting the approah used by Ryder (1964) in speifying timing effets in fertility. Aordingly, tempo effets on period measures are defined as those produed by hanges in the timing of deaths within ohorts that do not affet the ohort LE. Following that definition, Figure 5 illustrates timing effets by introduing a tempo hange, but no quantum hange, in ohort mortality. The initial 25 and last 125 birth ohorts in those 2 ohorts are subjet to the fore of mortality µ(a,t+a) = exp[ A + ba] (12) where A and b have the same values as before. However, the ohorts born in years 26 through have different parameters, speifially A*=1e A = and b*= As intended by that deliberate hoie of b*, those parameter hanges inrease mortality at younger ages, derease mortality at older ages, but leave ohort LE fixed at years for all ohorts. Figure 5: Cohort and period life expetany, ACLE, and CAL for a model with tempo hanges in ohort mortality between years 25 and but a fixed ohort LE Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth 13

17 Demographi Researh: Volume 13, Artile 5 As a result, Figure 5 shows that ohort LE is a horizontal line. As evident from equations (6), a onstant ohort LE produes ACLE values onstant at that level. It follows that tempo distortions, as defined here, do not affet ACLE. However, those hanges in timing substantially impat both the period LE and CAL. During years - 12, both period LE and CAL derease in size by more than 2, while during years 14-17, both inrease by more than 2. The flutuation in CAL was only a bit less than that in the period LE, indiating that in some irumstanes CAL is nearly as suseptible to tempo effets as is period life expetany. Mortality trends and flutuations have long haraterized human demographi history. As shown in the above models, under those onditions ohort LE, period LE, and CAL an differ substantially from ACLE, the average life expetany of the ohorts present in a given year. Next we examine those four measures of mortality in the ontext of three Western ountries with a long history of data on mortality. 5. ACLE in England and Wales, Norway, and Switzerland The following illustrations are based on data derived from the Human Mortality Database (24). Estimates were made to alulate the average ohort life expetany for years where full data on all ative ohorts were not available. To omplete ohort experiene for years before the first available data year, we used the rates of the earliest available year. For ohorts not extint by the latest data year, two senarios were used. Senario A ompleted ohort experiene using death rates from the last available period. Senario B ompleted ohort experiene by assuming that, at all ages, the death rates of the latest period delined by =.5 annually. Hene, if the last data year is 2, then the alulations begin with the age-speifi death rates for that year, µ(a,2). For age a and for year t > 2, the fore of mortality is given by ( t 2 ) ( a, t) = µ ( a,2) e µ. (13) Figures 6a, 6b, and 6 present the results for onstant rate Senario A, showing the ohort and period LE, CAL and ACLE for England and Wales, Norway and Switzerland. Similar patterns are observed in all three ountries. The ohort LE has the highest value of all four aggregate measures. With the last available year used for future mortality, ohort and period LE onverge to the same value. If that onstant rate pattern ontinues, all four measures will onverge. Initially, the period LE and ACLE alternate as the seond largest value, with CAL having the lowest value. More reently, as mortality delines ontinued, ACLE has onsistently been between the period LE and CAL

18 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Figure 6a: Cohort and period life expetany, ACLE, and CAL for England and Wales from 18 to 1998, with ohorts ompleted preserving the mortality observed in the last period Values of mortality measures Cohort LE ACLE Period LE CA L Year / ohort year of birth Figure 6b: Cohort and period life expetany, ACLE, and CAL for Norway from 18 to 2, with ohorts ompleted preserving the mortality observed in the last period Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth 132

19 Demographi Researh: Volume 13, Artile 5 Figure 6: Cohort and period life expetany, ACLE, and CAL for Switzerland from 18 to 2, with ohorts ompleted preserving the mortality observed in the last period Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth Figures 7a, 7b, and 7 present the results for delining future mortality Senario B, showing our four measures for England and Wales, Norway, and Switzerland. With mortality ontinuing to improve at.5% per year, the ohort LE and ACLE show higher levels in Figure 7 than in Figure 6 from 192 through the latest year shown. Consequently, ACLE exeeds the period LE during the last deades of the entury. Differenes between Figures 6 and 7 are fairly modest in all three ountries. Nonetheless, with mortality steadily delining by more than.5% in reent deades, Figure 7 suggests that urrent life expetanies in many Western ountries may understate the average life expetany of living ohorts

20 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Figure 7a: Cohort and period life expetany, ACLE, and CAL for England and Wales from 18 to 1998, ohorts ompleted by assuming a ontinually delining mortality model with =.5 Values of mortality measure Cohort LE 6 55 ACLE 55 5 Period LE 5 45 CAL Year / ohort year of birth Figure 7b: 9 Cohort and period life expetany, ACLE, and CAL for Norway from 18 to 2, ohorts ompleted by assuming a ontinually delining mortality model with =.5 9 Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth 134

21 Demographi Researh: Volume 13, Artile 5 Figure 7: Cohort and period life expetany, ACLE, and CAL for Switzerland from 18 to 2. The ohorts have been ompleted following a ontinually delining mortality model with = Values of mortality measure Cohort LE ACLE Period LE CAL Year / ohort year of birth 6. Disussion and onlusions Reent disussions of mortality dynamis have noted that period life expetanies may not provide an aurate measure of the longevity of the population living in any given year. To provide suh a measure, this paper presents ACLE, a new aggregate measure of survivorship. As derived, ACLE is a weighted average of the life expetanies of the ohorts present in a given period, with eah ohort weighted by its probability of survival to that given year. Using population models with hanging mortality and data from three Western European ountries, we ompared ACLE with period and ohort life expetanies and with CAL, a measure ombining period and ohort mortality that has been advaned as an index of survivorship. ACLE is relatively insensitive to both period and ohort flutuations in death rates, and is unaffeted by hanges in the timing of mortality that do not hange ohort life expetany. In ontrast, CAL was found to be more sensitive to ohort and period flutuations and to be quite suseptible to distortions from mortality timing effets. In models with ontinually delining mortality, and in ontemporary Western populations with a long history of mortality delines that an reasonably be expeted to ontinue, ACLE is larger than period life expetany

22 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany However, CAL is onsistently less than period LE. If interpreted as a measure of tempo distortion, CAL implies a downward level adjustment in the ontext of sustained inreases in longevity. The value of ACLE is that it provides a quantitative, oneptually lear, and methodologially sound definition for the intuitive but vague idea of the average longevity of ohorts alive in a given year. ACLE demands data that span over two enturies, but those data are essential given what it seeks to measure. The neessary data are in fat available for a number of populations, and are fully knowable in the ontext of population models. Calulations for ontemporary populations require estimates of the future ourse of mortality, but that is an ative area of researh and, provisionally, alternative senarios an be used to assess the range of plausible outomes. Cohort LE only represents one ohort, period LE is sensitive to timing hanges, and CAL is not only sensitive to tempo effets but generally understates the period level of longevity. None of these problems are seen in ACLE, whih reflets atual trends while minimizing flutuations. 7. Aknowledgements The seond author gratefully aknowledges support from the DeWitt Wallae postdotoral fellowship awarded by the Population Counil. This is a revised version of the paper presented at the November 18-19, 24 New York Workshop on Tempo Effets on Mortality sponsored by the Max Plank Institute for Demographi Researh and the Population Counil

23 Demographi Researh: Volume 13, Artile 5 Referenes Bongaarts, John. 24. Six period measures of longevity: A omment. Note presented at the Workshop on Tempo Effets on Mortality, New York, November Bongaarts, John and Griffith Feeney. 22. How long do we live? Population and Development Review 28(1): Brouard, Niolas Struture et dynamique des populations. La pyramide des anne es a vivre, aspets nationaux et exemples re gionaux, Espaes, Populations, Soie te s 2(14-15) : Canudas-Romo, Vladimir. 23. Deomposition Methods in Demography. Rozenberg Publishers: Amsterdam, The Netherlands. Available at: Goldstein, Joshua R. and Kenneth W. Wahter. 25. Gaps and lags: Relationships between ohort and period life expetany. Unpublished manusript, Prineton University and University of California, Berkeley. Guillot, Mihel. 23a. The ross-setional average length of life (CAL): A rosssetional mortality measure that reflets the experiene of ohorts. Population Studies 57(1): Guillot, Mihel. 23b. Does period life expetany overestimate urrent survival? An analysis of tempo effets in mortality. Paper presented at the PAA 23 held in Minneapolis, Minnesota. Human Mortality Database. University of California, Berkeley (USA), and Max Plank Institute for Demographi Researh (Germany). Available at or (data downloaded on [5/1/4]). Ryder, Norman B The proess of demographi translation. Demography 1(1): Shoen, Robert, Stefan H. Jonsson and Paula Tufis. 24. A population with ontinually delining mortality. Working Paper 4-7, Population Researh Institute, Pennsylvania State University, University Park. Vaupel, James W How hange in age-speifi mortality affets life expetany. Population Studies 4:

24 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Vaupel, James W. 22. Life expetany at urrent rates vs. urrent onditions: A reflexion stimulated by Bongaarts and Feeney s How long do we live? Demographi Researh 7: Vaupel, James W. and Vladimir Canudas-Romo. 22. Deomposing demographi hange into diret vs. ompositional omponents. Demographi Researh 7: Wahter, Kenneth W. 24. Tempo and its tribulations. Paper presented at the Workshop on Tempo Effets on Mortality, New York, November

25 Demographi Researh: Volume 13, Artile 5 Appendix: Deomposing the hange over time in ACLE A further analysis of the dynamis of ACLE and its deomposition show that typially the main omponent of hange in this measure reflets hanges in ohort life expetanies, not the survivorship weights used. The hange over time in ACLE an be deomposed using the method introdued by Vaupel and Canudas-Romo (22). It is possible to apply this deomposition beause ACLE(t) an be written as an average of ohort life expetanies weighted by ohort probabilities of survival, i.e. by ACLE( t) = e (, l ( a, da = e ( ). (14) l ( a, da t The hange in this average is due to two omponents: first the average hange in ohort life expetanies, or the diret hange, and seond hanges in the distribution of the probabilities of survival, or the ompositional hange. Following the Vaupel and Canudas-Romo (22) notation of a bar and dot over a variable to denote the average over age and the derivative over time, respetively, we obtain ACLE t = = e + Cov( e, r ), (15) e where the average hange in ohort life expetanies is expressed as e ( t) = e (, l t ( a, da l ( a, da. (16) 139

26 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany The expression r ( a, denotes ohort age-speifi growth rates, i.e. ln[ l r ( a, = ( a, ] t, and the ovariane term is alulated as e(, r ( a, l ( a, da r ( a, l ( a, da Cov( e, r ) = ACLE( t). (17) l ( a, da l ( a, da Table A-1: Change over time in ACLE and its deomposition: England and Wales Year ACLE ACLE Change in Average Covariane Equation (11) t t-5 t+5 ACLE hange [1] [2] [1+2] * *The data only go to 1998, therefore the last period is for eight instead of ten years 14

27 Demographi Researh: Volume 13, Artile 5 Table A-2: Change over time in ACLE and its deomposition: Norway Year ACLE ACLE Change in Average Covariane Equation (11) t t-5 t+5 ACLE hange [1] [2] [1+2] Table A-3: Change over time in ACLE and its deomposition: Switzerland Year ACLE ACLE Change in Average Covariane Equation (11) t t-5 t+5 ACLE hange [1] [2] [1+2] Disrepanies between the hange in ACLE and the results in the last olumn arise beause disrete data over a 1-year period are used to approximate derivatives (see Canudas-Romo, 23) 141

28 Shoen & Canudas-Romo: Changing mortality and average ohort life expetany Tables A-1 to 3 show the annual hange in ACLE for eah deade of the twentieth entury, and its deomposition, for three ountries: England and Wales, Norway and Switzerland. The first olumn shows time t, the midpoint of the 1 year interval examined. The next two olumns give values of ACLE at the beginning (time t-5) and the end (t+5) of the interval. Column four is the annual hange in ACLE. That hange is then deomposed into two omponents: the average hange (or the diret hange) and the ovariane (or the ompositional hange). They are alulated from equations (16) and (17), respetively. Those two omponents sum to the figure shown in the last olumn, whih gives the value that follows from equation (15). That final olumn does not exatly reprodue the Change in ACLE olumn beause of inonsistenies between the ontinuous equations and the disrete data (for a disussion and tehniques for minimizing disrepanies, see Canudas-Romo 23). For the three ountries onsidered, Tables A-1 to 3 show that the diret hange explains most of the hange in average ohort life expetanies over time. The ontribution of the ompositional omponent is minor and almost invariably negative, partially offsetting the inrease in ACLE. The pre-eminene of the diret hanges in ohort life expetany over time and the relative unimportane of the weights used was onfirmed by an analysis using an alternative weighting funtion. Instead of using ohort survival probabilities, period survival probabilities were used as weights and an average ohort survival measure, ACS, was reated. The deomposition of the ACS reinfored the importane of the diret effet. Although ACS gives results similar to ACLE, the rationale for using those weights was weaker, and the ACS was abandoned