Intercensal life tables consistent with population projections

Transcription

1 Itercesal life tables cosistet with populatio projectios Populatio Associatio of America 2011 Aual Meetig Program Washigto, DC March 31- April 2 Author iformatio: Jeroimo Oliveira Muiz is a Assistat Professor of Sociology at the Federal Uiversity of Mias Gerais (UFMG). He has a Phd i Sociology from the Uiversity of Wiscosi- Madiso, ad a Masters i Demography from Cedeplar, UFMG. For commets suggestios ad further iformatio please cotact jeroimomuiz@gmail.com

2 Itercesal life tables cosistet with populatio projectios Itercesal methods have bee broadly used to estimate mortality i developed ad less developed coutries with deficiet or icomplete data. These methods have several advatages over idirect methods because they do ot require the use of model life tables ad provide sufficietly accurate results eve i the presece of age distortios ad death uder-registratio. The drawback of these methods, however, is that geerated life tables do ot provide projectios of the iitial populatio that are cosistet with the subsequet eumeratio. This article demostrates these icosistecies usig three differet methods ad itroduces a simple procedure to solve this icosistecy ad to provide life tables that are accurate ad compatible with projected populatios. The empirical illustratio demostratig its efficacy draws o data from Vietam, but the method ca be eteded to ay cotet ad time period. I the early 1980s a series of articles demostrated how to estimate mortality usig cosecutive age distributios ad itercesal deaths (Beett ad Horiuchi 1981, Beett ad Horiuchi 1984, Presto ad Beett 1983, Presto ad Coale 1982). I the 1990s, these procedures were further developed to ehace their accuracy ad facilitate their applicability (Merli 1998, Presto et al 1996). These procedures are advatageous because they do ot deped o the use of model life tables - as is the case of idirect mortality methods-, they circumvet the assumptio of stability, ad they are sufficietly accurate eve whe the registry of deaths is icomplete. Itercesal techiques are particularly useful i developig coutries with deficiet registratio of deaths because they allow the estimatio of life tables directly from cesus age distributios ad agespecific growth rates (Presto ad Beett 1983). Alteratively, these age-specific growth rates ca also be used to adjust the reported umber of deaths ad estimate life tables eve i populatios with fairly accurate age reportig (Beett ad Horiuchi 1984, Merli 1998, Presto et al 1996). These methods, however, geerate life tables that are icosistet with projected populatios. The icosistecy lies i the umerical differece betwee the epected populatio ad the populatio projected with survival probabilities from estimated itercesal life tables. The mortality estimatio from two age distributios ad from itercesal deaths overcomes several problems regardig data quality ad availability, but they are iterally icosistet whe it comes to a posteriori cohort-compoet projectios. Usig itercesal survival probabilities to project the populatio by age results i a projected populatio that differs from the oe iitially used the estimate the life table. The goal of this paper is to demostrate the icosistecy i the projected populatio stemmig from itercesal life tables ad to suggest a methodological adjustmet to improve previous procedures that will allow the costructio of life tables that are accurate ad cosistet with projected populatios. The first sectio of this article reviews past itercesal methods of mortality estimatio ad demostrates how much projected ad epected populatios differ whe itercesal probabilities of survival are utilized to project the populatio usig the cohort-compoet method. The secod sectio suggests a ew techique to build life tables cosistet with projected populatios. The third sectio compares the life epectacies, survival ratios ad the demographic projectios geerated by itercesal methods ad evaluates their efficacy i light of actual populatios. Methods We start by describig ad presetig the results of three direct procedures of mortality estimatio. The first oe was itroduced by Presto ad Beett (1983) to estimate life tables

3 usig two successive age distributios. The secod procedure was developed by Beett ad Horiuchi (1984) ad improved by Presto et al (1996) to estimate mortality from a set of registered deaths by age ad age-specific growth rates. The third oe was suggested by Merli (1998) ad uses a iterative mechaism betwee the two aforemetioed methods to overcome some of the limitatios preset i both procedures ad to recocile their results. To assure aalytical comparability we replicate the results from Merli (1998) usig male data from 1979 ad 1989 Vietamese Cesuses. The demographic equatios are preseted i discrete form to reflect its commo usage ad to facilitate its applicatio ad replicatio to alterative empirical data. Cesus-based method The advatage of this procedure, also kow as r-method, is its ability to ifer mortality coditios usig oly two cosecutive age distributios. The method does ot require the use of model life tables ad does ot assume populatio stability. The r-method oly assumes that the average growth betwee populatio couts is costat by age. With this assumptio, the umber of perso-years i the life table ca be estimated as: * S L N e (1) where, N * N ( t1) N ( t2) 1/ = geometric mea of the populatio betwee t1 ad t2 (2) r a a0,5 S 5 r = cumulatio of age-specific growth rates to midpoit of iterval (3) l N ( t2) N ( t1) r[ t1, t2] = age-specific growth rate (4) ( t2 t1) The remaiig fuctios of the life table are the umber of idividuals survivig to age (l ), the umber of perso-years lived above age (T ), the life epectacy of idividuals i age (e o ), ad the survivorship ratios betwee age ad + ( p ). These fuctios are calculated as: 1 l L L (5) 2 T La (6) a o T e (7) l L p (8) L L p (9) L L The survivorship ratios i equatios (8) ad (9), whe multiplied by the baselie populatio i 1979, provide the projected Vietamese male populatio i Whe this ew projected populatio for 1984 is multiplied agai for the respective survivorship ratio it provides the projected populatio for For istace:

4 N Proj( t2) N 2 ( t1) p p Equatio (10) shows how the umber of people at age i 1979 icreased, or decreased, te years later as a cosequece of the combied effect of mortality ad et migratio ito that age group. The mortality of Vietamese males usig the cesus-based method suggested by Presto ad Beett (1983) ad the resultig populatio projectios usig the survivorship ratios i equatios (8) ad (9) are reported i Table 1: [TABLE 1 HERE] The values i colums (11) ad (2) are very similar to each other, but they are ot idetical. The similarity betwee projected ad epected populatios suggests that survivorship probabilities are accurate eough to represet the joit effect of mortality ad migratio o populatio growth, but they are ot eact. 1 Death-distributio method Beett ad Horiuchi (1984) developed a techique to estimate mortality eve whe deaths are udercouted ad age reportig is iaccurate. The method geeralizes the idea itroduced by Presto ad Coale (1982) to o-stable populatios to allow the iferece of the umber of deaths i the life table ( d ) from the umber of deaths recorded i the populatio ( D ) adjusted by age specific growth rates. Presto et al (1996), for eample, used this approach to estimate the mortality rates of Africa Americas above 64 years old, ad Merli (1998) used the method to ifer mortality coditios i Vietam. The method cosists i covertig the distributio of deaths i the populatio ito the correspodig distributio of deaths i the life table usig itercesal age-specific growth rates. The method assumes that uder-registratio is costat by age. I the discrete case the other fuctios of the life table ca be epressed as (Presto et al 1996, Presto et al 2001): ( r r ) d D 2 e (11) d D d d d d, assumig that d0 D0 i the first age group (12) L l d (13) 2 The death-distributio method assumes that the ratio D / D - is costat by age. Whe uderregistratio icreases with age, the ratio D / D - decreases ad life epectacies are uderestimated at all ages bellow the ages at which deaths are omitted (Merli 1998: 352). Moreover, sice Beett ad Horiuchi (1984) s death-distributio method is based o specific growth rates betwee two age distributios, it is ot immue to the presece of differetial cesus coverage, itercesal migratio, ad icomplete death registratio. Nevertheless, the method is less sesitive to these problems tha the cesus-based method (Presto ad Beett (10) 1 The results displayed i Table 1 are slightly differet from Merli (1998: 351) because we costraied the fial age group to 80 ad more istead of 85.

5 1983). 2 The method also has the advatage of providig a estimate for the life epectacy at birth. The mortality estimates based o the death-distributio method are i Table 2. They are slightly differet from those i Merli (1998: 353) ad i Presto et al (2001: 188) because we use a simpler method to calculate the umber of perso-years i the first ad last age groups. 3 [TABLE 2 HERE] The life epectacies show i Table 2 are less sesitive to the age-specific growth rates derived from the Vietamese age structures. Nevertheless, oce agai, projected (colum 21) ad epected populatios are differet. Iterative method To circumvet the bias implicit i the growth rates associated to age distortios, differetial coverage ad itercesal migratio, Merli (1998) developed a two-stage iterative procedure to cociliate the cesus-based ad the death-distributio methods. I the first stage the populatio is projected forward usig the survival probabilities of the life table calculated accordig to Beett ad Horiuchi s death-distributio method. As a result of this eercise we obtai two age distributios. The first oe refers to the populatio of Vietam i 1979, ad the secod oe represets the projected closed populatio te years later, which is ot iflueced by coverage errors or itercesal migratio. From these two distributios we the calculate a ew life table usig Presto ad Beett s cesus-based model. The differece is that while usig the projected populatio to estimate age-specific growth rates we are simultaeously correctig the sesitivity of the cesus-based method to differetial completeess of cesus eumeratio ad residual emigratio. 4 I the secod stage of the iterative method, i order to correct the iitial growth rates estimated i the death-distributio method, a ew roud of forward projectios is coducted, but this time usig ew itercesal growth rates estimated i the first stage of the process through the cesusbased method. 5 This iterative process betwee the two methods cotiues util the life epectacies obtaied i the previous ad i the et iteratio do ot differ at the secod decimal. This covergece i life epectacy occurred after 25 iteratios. At the ed of the iterative process we get two ew life tables. The first table is based o the cesus-based method usig the projected age distributio corrected for differetial completeess of cesus eumeratio ad residual emigratio. The secod life table uses the death-distributio method, but this time employig age-specific growth rates from the iterative process. 2 Beett ad Horiuchi (1984: 222) showed that life epectacy at age five is te times more sesitive to errors i the growth rate i the cesus-based method tha i the death-distributio method. 3 For the first age group we use L 0 = l + a d ad assume a separatio factor equal to 0.78 (See Presto 2001: 188). The fial estimates of life epectacy, however, are ot very sesitive to the separatio factor. I the last age group the umber of perso-years is L80 l80 log10( l80). 4 The impact of migratio could also be uderstood as racial reclassificatio, whe the populatio is disaggregated by race, or as social mobility, whe it is disaggregated by icome, wealth, educatio or occupatio. 5 The uder-10 populatio, 5 N 0 ad 5 N 5, is projected usig the third equatio described i Footote 6 of Merli (1998, p. 356): 5 N55N10ep 2.5( 5r5 5r10) 5 5 L L 5 10

6 [TABLE 3 HERE] The iterative procedure approimates the life epectacies provided by the cesus-based ad death-distributio methods. The resultig projectios are, however, still differet from what we should epect. The differeces betwee projected ad observed populatios i 1989 are show i the last two colums of Table 3. They demostrate that the umber of people betwee 15 ad 25 years old is uderestimated by both projectio methods. The survivorship ratios of the iterative cesus-based ad death-distributio methods uderestimate the projected populatio i these age-groups by about oe millio people. I the et sectio we itroduce a method to solve this icosistecy. Projectio-cosistet method A alterative method to geerate life tables whose projected populatio is compatible with the epected populatio cosists i calculatig a ew set of survivorship ratios, which i the case of five-year age groups are defied as: N5 * p5* (14) N5 ( t1) N ( t2) p* 5 (15) N ( t1) p * 2 L p * L L (16) where, L0 L 0 i Table 1 defied by the r-method (17) L p * L (18) L The fuctio i the ope-age group is defied through mathematical iteratios to make projected ad observed populatios i (t2) to have the same size i the last age-group. The remaiig fuctios of the life table, l *, T *, ad e *, are calculated as described i equatios (5), (6) ad (7). The iputs ad life table fuctios for the Vietamese populatio usig 1979 ad 1989 cesuses are i Table 4. The survivorships of the projectio-cosistet method derive from the age structures preseted i the cesus-based ad iterative methods. The projectio-cosistetmethod, however, could replace the r-method i cases where the two age structures used the procedure are closed to migratio ad i situatios where has good quality. [TABLE 4 ABOUT HERE] Table 4 shows that projected (colum 34) ad epected (colum 25) populatios i 1989 are ow idetical ad solve the iteral cosistecy of previous methods. The survivorship ratios reported i colum (29), whe multiplied by the baselie populatio i colum (1) produce a populatio that is equal i size to what was ideed observed i the cesus. Overall, the life epectacies usig this methodology are similar but slightly higher tha those reported i Tables 1 (r-method) ad Table 3 (iterative method). Compariso betwee mortality estimates ad their demographic projectios

7 A rigorous way to validate the data ad the accuracy of life tables is to use the projectio as a historical simulatio. The idea is to replay past populatio chage by startig the projectio from some past cesus date. As the populatio projectio moves over time from the origial cesus date up to the preset, it should give populatio distributios by age that agree with successive cesuses. If the agreemet is good, the the demographic estimates for mortality i the past are cosistet with the projectio. Figure 1 plots life epectacies ad the survivorship ratios betwee ages ad + accordig to the three variatios of the cesus-based methods of mortality estimatio. It shows life epectacies (o the left ais) ad survivorship ratios (o the right ais) for the origial cesusbased, ad for its iterative ad projectio-cosistet variatios: [FIGURE 1 ABOUT HERE] I compariso to the iterative ad projectio-cosistet variatios, the origial cesus-based method uderestimates life epectacies i ifacy ad overestimates them after age 45, mostly because of its sesitivity to the age populatio age structure recorded betwee the two periods. The iterative variatio of the cesus-based method helps to reduce the ifluece of the populatio age structure over the life epectacy estimates, but it does ot ecessarily provide survivorship ratios that are iterally valid ad cosistet with projected populatios. If we eamie the survivorship probabilities provided by each method, we quickly realize that the cesus-based projectio-cosistet method (gray dashed curve) is the oly oe with survivorship ratios lower tha oe ad without abrupt variatios alog its distributio. Moreover, this is the oly method whose survivorship probabilities provide idetical observed ad projected populatios to reiforce its iteral validity. This is particularly importat because small absolute differeces i survivorship ratios ca be resposible for large differeces i projected populatios. To illustrate this poit, Table 5 shows the absolute size of projected populatios ad compares it to the baselie ad epected populatios i each method. It shows that depedig o the mortality method used the total projected populatio ca be 28 to 34 percet larger tha the populatio baselie. Beett ad Horiuchi s death-distributio method, for istace, overestimates the epected populatio by 4.45 percet, which i absolute terms represets a surplus of more tha oe millio people. The projectio error usig cesus-based methods is smaller tha i the death-distributio method, but the sum of the differece betwee epected ad projected populatios is larger tha half millio people. I relative terms, the differece betwee projected ad epected populatios may ot be relevat, but i absolute terms that represets a sigificat cotiget of people that should be there ad that are ot due to a iteral icosistecy i the death-distributio ad cesus-based methods. The cesus-based projectio-cosistet method is the oly procedure where projected ad epected populatios are idetical. [TABLE 5 ABOUT HERE]

8 The last row of Table 5 reports Keyfitz s, which is a stadard measure of the distace betwee probability vectors represetig the proportio of the populatio i differet ages. 6 It idicates how differet the age structures of projected ad epected cesual populatios are i relatio to each other. Overall, the age structure of the projected ad epected populatios is most similar whe the projectio is geerated by the cesus-based methods tha by the death-distributio method. Nevertheless, the oly procedure where the projectio is fully cosistet with the epected populatio is represeted i the last colum of Table 5. Discussio The compariso betwee projected ad epected populatios, total ad by age group, demostrates that the survivorship ratios derived from the life tables provided by curret itercesal procedures are ot as accurate as they could be to coduct populatio projectios. Usig the male populatio of Vietam, previously described by Merli (1998), we foud that the discrepacy betwee observed ad projected populatios ca be superior to oe millio people. I terms of plaig ad allocatio of resources, this is ot a small figure. I larger coutries such as Chia, Idia, Idoesia, Brazil ad the Uited States, the discrepacies betwee projected ad observed populatios would be eve larger. With these umerical discrepacies comes the realizatio that the curret itercesal procedures suggested by Presto ad Beett (1983), Beett ad Horiuchi (1984), Presto et al (1996) ad Merli (1998) are iterally icosistet with the projectios produced by the cohort-compoet method. The estimated life tables ad life epectacies produced by these procedures are good eough as approimatios to describe the curret mortality status of certai cotets, but they do ot produce projected populatios that are umerically eact ad fully compatible with the observed populatio i the secod cesus eumeratio. To solve this icosistecy we suggest a procedure based o two age distributios ad o the umber of perso-years derived from the survivorship ratios. The method does ot deped o age-specific growth rates, but the estimates of life epectacy will certaily be more accurate if the procedure is used after the last stage of Merli s (1998) iterative process. Oce multiplied by the populatio baselie, the projectio-cosistet method geerates a projected populatio that is idetical to the oe observed i the secod eumeratio. Sice the projectio-cosistet procedure of mortality estimatio builds o cesus-based ad death-distributio methods, it has the same advatages of previous itercesal procedures, but it also has the added virtue of providig projectios that are iterally cosistet with the age distributio used as a iput. The aalysis of data from Vietam shows that employig this refied method results i greater accuracy of the projected populatio, regardless of the completeess of death registratio i destabilized populatios. The results show the relative sesitivity of life tables to differet methods of mortality estimatio ad demostrated the differet projected populatio sizes ad age structures geerated by each oe of them. I particular, they cotribute to advace our kowledge of mortality estimatio ad projectio methods by describig how the future growth ad distributio of populatios may differ uder slightly differet mortality scearios. We show that apparetly small differeces may have large impacts i the projected umber of people. I the case of Vietam, the et impact 6 Keyfitz (1968: 47) proposed a measure equivalet to 1 (, w) i w i, where i is the proportio of the 2 epected populatio i 1989; w i is the proportio of the projected populatio te years after the baselie, ad i subscribes five-year age groups betwee 10 ad 80. i

9 could vary betwee 650,000 ad 1,600,000 people depedig o the set of survivorship ratios used i the cohort-compoet projectio. I log term projectios ad stable populatio aalysis, the error of the projectio builds up ad these discrepacies will be eve larger. It is crucial, therefore, to assure that the life tables defied i the baselie are as accurate as possible i order to miimize the propagatio of the projectio error i the future. The projectio-cosistet method preset here does this. Alterative data ad further empirical validatios cosiderig other time periods ad cotets would, however, be required before makig fial assertios o this matter. REFERENCES: Beett, N. G. ad S. Horiuchi Estimatig the Completeess of Death Registratio i a Closed Populatio. Populatio Ide 17(2), Mortality estimatio from registered deaths i less developed coutries. Demography, v.21, Keyfitz, N Itroductio to the mathematics of populatio. Readig, Mass.: Addiso- Wesley Pub. Co. Merli, G Mortality i Vietam, Demography 35(3), Presto, S. H. ad N. G. Beett A cesus-based method for estimatig adult mortality. Populatio Studies 37(1), Presto, S. H. ad A. J. Coale Age Structure, Growth Attritio ad Accessio: a New Sythesis. Populatio Ide 48: Presto, S. H., P. Heuvelie ad M. Guillot Demography: measurig ad modelig populatio processes. Oford: Blackwell. Presto, S. H., I. T. Elo, I. Rosewaike ad M. Hill Africa-America Mortality at Older Ages: Results of a Matchig Study. Demography 33(2): Table 1. Applicatio of the Cesus-based Method to Vietam s Populatio, Males: Age N N N * r S L l e p N proj. N proj (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 0 3,946,224 4,710,423 4,311, ,506, ,928,795 4,430,179 4,171, ,696, , ,113, ,632,555 3,898,298 3,763, ,443, , ,716,797 3,891, ,954,333 3,427,357 3,182, ,969, , ,244,656 3,319, ,281,171 2,974,282 2,604, ,603, , ,681,953 2,945, ,742,277 2,832,160 2,221, ,707, , ,347,262 2,759, ,177,320 2,361,692 1,667, ,739, , ,757,482 2,367, ,580 1,604,918 1,245, ,773, , ,188,001 1,773, ,291 1,048, , ,488, , ,628 1,098, , , , ,325, , , , , , , ,954, , , , , , , ,013, , , , , , , ,766, , , , , , , ,387, , , , , , , ,701, , , , , , , ,194, , , , , , , , , , ,973

10 Note: The statioary populatio above age (T ) is ot show but it ca be easily derived multiplyig colum (7) by colum (8). Sources: Vietam Geeral Statistical Office (1983); Vietam Cetral Cesus Steerig Committee (1994); Merli (1998: 348) Table 2. Applicatio of the Death-distributio Method to Vietam s Populatio, Males: Age r D D / D - d / d - d L l e p N proj. N proj (4) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) , ,508 2,495, , , ,648 2,456, , ,884, , ,437 2,434, , ,893,619 3,849, , ,514 2,413, , ,601,074 3,859, , ,103 2,389, , ,924,437 3,564, , ,539 2,363, , ,257,142 2,893, , ,551 2,337, , ,722,586 2,231, , ,966 2,298, , ,157,771 1,693, , ,374 2,254, , ,021 1,135, , ,469 2,182, , , , , ,766 2,110, , , , , ,308 1,987, , , , , ,014 1,770, , , , , ,909 1,466, , , , , ,175 1,101, , , , , , , , , , , , , , ,180 Note: The statioary populatio above age (T ) is ot show but it ca be easily derived multiplyig colum (7) by colum (8). Sources: Vietam Geeral Statistical Office (1983); Vietam Cetral Cesus Steerig Committee (1994); Merli (1998: 348) Table 3. Applicatio of the Iterative Method to Vietam s Populatio, Males: Corrected Death-distributoi method Cesus-based method Age r e p N proj. ( ) e p N proj. ( ) N ( )- N ( )- (22) (23) (24) (25) (26) (27) (28) colum (25) colum (28) ,892, , ,862, , ,857, ,780,725 40, , ,868, ,742, , , ,570, ,597, , , ,893, ,980,740-61, , ,229, ,189, , , ,693, ,784,190-88, , ,134, ,214,514-86, , , ,805-36,847-24, , ,464 12,565 96, , ,304 23,579 50, , ,195 48,337-2, , ,099 46,386 43, , ,430 5,307 9, , ,466 23,136 20, , ,324 30,104 35,313

11 Life epectacy (e ) Survivorship ratios ( p ) Table 4. Applicatio of the Projectio-cosistet Method to Vietam s Populatio, Males: , N 5 * = 3,895,337 NN N N proj. p * L l * e * N Proj. * N proj. * Age ( ) ( ) (1) (25) (29) (30) (31) (32) (33) (34) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Figure 1. Life epectacies ad survivorship ratios accordig to three variatios of the cesus-based method: Vietam, Cesus-based Cesus-based (iterative) Cesus-based (iterative ad projectio-cosistet) Age Table 5. Comparisos betwee projected ad epected male populatios of Vietam above age 10

12 Projected populatio above age 10 Cesus-based Deathdistributio 22,672,867 23,771,611 Cesus-based Iterative method Cesus-based projectio cosistet 23,673,564 23,706,057 S epected- projected a 650,063 1,603, ,730 0 Projected/ Baselie Projected/ Epected Keyfitz's b a The epected populatio i the cesus-based ad death-distributio methods are equal to the observed populatio i I the iterative methods the epected populatio is equal to the projected populatio i the last roud of the iterative procedure. b Keyfitz s measure of proimity betwee vectors compares the age structure of projected ad epected populatios. Maimum value is 1 ad miimum is 0 whe the vectors are idetical.