Table of Contents Introduction...1. Life tables...34 Period life tables...35 Cohort life tables...39 Multi-year cohorts...41

Transcription

1 as Revsed: May 7 Verso 5 Mehods Proocol for he Huma Moraly aabase J.R. Wlmoh K. Adreev. Jdaov ad.a. Gle wh he asssace of C. Boe M. Bubehem. Phlpov V. Sholov P. Vacho Table of Coes Iroduco... Geeral prcples... oao ad ermology for age ad me... es dagram... Sadard cofguraos of age ad me...4 Female / male / oal...6 Perods ad cohors...6 Adjusmes o raw daa...7 Forma of daa fles...7 Seps for compug moraly raes ad lfe ables...7 Commo adjusmes o raw daa...9 srbug deahs of uow age...9 Splg deah cous o es ragles... Splg 5 deah cous o daa...4 Splg deah cous ope age ervals o es ragles...4 Populao esmaes Jauary s...5 ear erpolao...5 Iercesal survval mehods...6 Specfc eample...6 Geeralzg he mehod... Pre- ad poscesal survval mehod...5 Iercesal survval wh cesus daa -year age groups...6 Ec cohors mehods...7 Survvor rao...9 eah raes... fe ables...4 Perod lfe ables...5 Cohor lfe ables...9 Mul-year cohors...4 Ths docume grew ou of a seres of dscussos held varous locaos begg Jue. The four dvduals lsed as auhors wroe he orgal verso ad/or have acvely corbued o subseque versos cludg hrough he developme of addoal mehods. Several ohers asssed wh he creao of hs docume hrough her acve parcpao meegs ad ogog dscussos va emal. The auhors are fully resposble for ay errors or ambgues. They ha Georg Helma for hs asssace wh he graphs. - -

2 as Revsed: May 7 Verso 5 Almos-ec cohors...4 Abrdged lfe ables...44 Apped A. ear model for splg deah cous...45 Apped B. Compuaoal mehods for fg cubc sples...5 Splg daa o forma...5 Splg perod-cohor parallelograms coverg cohors...54 Apped C. Mehod for splg deahs a ope age erval...55 Correco for uusual flucuaos deahs...56 Correco for cohor sze...59 Apped. Adjusmes for chages populao coverage...6 Brh cous used splg deahs...6 Ec cohor mehods...65 Iercesal survval mehods...67 ear erpolao...7 Perod deah raes aroud he me of a erroral chage...7 Cohor moraly esmaes aroud he me of a erroral chage...7 Oher chages populao coverage...7 Apped E. Compug deah raes ad probables of deah...7 form dsrbuo of deahs...7 Cohor deah raes ad probables...7 Perod deah raes ad probables...75 Apped F. Specal mehods used for seleced populaos...78 Refereces...8 s of Fgures Fgure. Eample of a es dagram... Fgure. Illusrao of es ragles...4 Fgure. Iercesal survval mehod eample...7 Fgure 4. Iercesal survval mehod geeral... Fgure 5. Pre- ad poscesal survval mehod...6 Fgure 6. Mehods used for populao esmaes...8 Fgure 7. Illusrao of eco rule wh l 5 ad ω Fgure 8. Survvor rao mehod a age ω - wh m 5... Fgure 9. aa for perod deah raes ad probables... Fgure. aa for cohor deah raes ad probables...4 Fgure. Illusrao of fve-year cohor assumg o mgrao...4 Fgure. fe able calculaos for almos-ec cohors...4 Fgure. fe able calculaos for almos-ec cohors aged o 4 year...44 Fgure A-. Proporo of male fa deahs lower ragle...48 Fgure A-. Proporo of male age 8 deahs lower ragle...49 Fgure C-. Frs dffereces deahs Wes Germa females

3 as Revsed: May 7 Verso 5 Fgure C-. Eample depcg he procedure o correc for cohor sze...6 s of Tables Table A-. ear models of he proporo of lower-ragle deahs...5 Table E-. Implcaos of assumg uform dsrbuo of deahs wh es ragles a age

4 Mehods Proocol for he HM Iroduco The Huma Moraly aabase HM s a collaborave projec sposored by he versy of Calfora a Bereley ed Saes ad he Ma Plac Isue for emographc Research Rosoc Germay. The purpose of he daabase s o provde researchers aroud he world wh easy access o dealed ad comparable aoal moraly daa va he Iere. Whe complee he daabase wll coa orgal lfe ables for aroud 5-4 coures or areas as well as all raw daa used cosrucg hose ables. 4 The raw daa geerally coss of brh ad deah cous from val sascs plus populao cous from perodc cesuses ad/or offcal populao esmaes. Boh geeral documeao ad he dvdual seps followed compug moraly raes ad lfe ables are descrbed here. More dealed formao for eample sources of raw daa specfc adjusmes o raw daa ad commes abou daa qualy are covered separaely he documeao for each populao. We beg by descrbg cera geeral prcples ha are used cosrucg ad preseg he daabase. e we provde a overvew of he seps followed coverg raw daa o moraly raes ad lfe ables. The remag secos cludg he Appedces coa dealed descrpos of all ecessary calculaos. The corbuo of C Bereley o hs projec s fuded par by a gra from he.s. aoal Isue o Agg. A hrd eam of researchers based a Rocefeller versy ew Yor Cy s also worg drecly o hs projec. I addo he projec depeds o he cooperao of aoal sascal offces ad academc researchers may coures. The HM s accessble hrough eher of he followg addresses: ad 4 By desg populaos he HM are resrced o hose wh daa boh val sascs ad cesus formao ha cover he ere populao ad ha are very early complee. We have o esablshed precse crera for cluso sce we are sll learg abou he sascal sysems of may coures. Mmally he HM wll cover almos all of Europe plus Ausrala Caada Japa ew Zealad ad he ed Saes. Ousde hs group here are oly a few coures or areas he world ha may possess he d of daa requred for he HM e.g. Chle Cosa Rca Tawa Sgapore. everheless oher regos ad coures are sll beg cosdered ad we do o ow ye he eac ls of populaos ha wll eveually be cluded he HM. We are cocered however abou he eed o mprove access o moraly formao for coures ha do o mee he src daa requremes of he HM. Therefore addo o hs projec we are also assemblg a large colleco of lfe ables cosruced by oher orgazaos or dvduals. Ths colleco s ow as he Huma feable aabase H ad wll clude daa for may coures o covered by he HM. The H s avalable a - -

5 Mehods Proocol for he HM Geeral prcples oao ad ermology for age ad me Boh age ad me ca be eher couous or dscree varables. I dscree erms a perso of age or aged has a eac age wh he erval [. Ths cocep s also ow as age las brhday. Smlarly a eve ha occurs caledar year or more smply year occurs durg he me erval [. I should always be possble o dsgush bewee dscree ad couous oos of age or me by usage ad coe. For eample he populao aged a me refers o all persos he age rage [ a eac me or o Jauary s of caledar year. ewse he eposure-o-rs a age year refers o he oal perso-years lved he age erval [ durg caledar year. es dagram The es dagram s a devce for depcg he soc ad flow of a populao ad he occurrece of demographc eves over age ad me. For our purposes s useful for descrbg boh he forma of he raw daa ad varous compuaoal procedures. Fgure shows a small seco of a es dagram ha has bee dvded o cells.e. oe year of age by oe year of me. Each 45-degree le represes a dvdual lfeme whch may ed deah deoed by les c ad e or ou-mgrao deoed by a sold crcle le b. A dvdual may also mgrae o he populao deoed by a ope crcle les d ad g. Oher lfe-les may merely pass hrough he seco of he es dagram uder cosderao les a ad f. Suppose we wa o esmae he deah rae for he cell ha s hghlghed Fgure.e. for age o ad me o. If he eac coordaes of all lfe-les are ow he he eposureo-rs perso-years ca be calculaed precsely by addg up he legh of each le segme wh he cell of course he acual legh of each segme mus be dvded by sce lfe-les are 45 from he age or me aes. Followg hs procedure he observed deah rae for hs cell would be.9 whch s he umber of deahs hs case oe dvded by he perso-years of eposure abou.. Ths s he - -

6 Mehods Proocol for he HM bes esmae possble for he uderlyg deah rae ha cell.e. he deah rae ha would be observed a ha age a very large populao subjec o he same hsorcal codos. Fgure. Eample of a es agram Age a d o e b c f g - o - Tme However eac lfe-les are rarely ow sudes of large aoal populaos. Isead we ofe have cous of deahs over ervals of age ad me ad cous or esmaes of he umber of dvduals of a gve age who are alve a specfc momes of me. Cosderg aga he hghlghed cell Fgure he populao cou a age s a me les b ad c ad a me le e. Gve oly hs formao our bes esmae of he eposure-o-rs wh he cell s merely he average of hese wo umbers hus.5 perso-years. sg hs mehod he observed deah rae would be.5.67 whch s lower ha he more precse calculao gve above because he acual eposureo-rs has bee overesmaed. The esmao of deah raes s evably less precse he absece of - -

7 Mehods Proocol for he HM formao abou dvdual lfe-les alhough esmaes based o aggregae daa usg such a procedure are geerally que relable for large populaos. eah cous are ofe avalable by age year of deah.e. perod ad year of brh.e. cohor. Such cous ca be represeed by a es ragle as llusraed Fgure. eah cous a hs level of deal are used may mpora calculaos he HM. Oe of he mos mpora seps compug he deah raes ad lfe ables for he HM s o esmae deah cous by es ragle f hese are o already avalable he raw daa. Fgure. Illusrao of es ragles Age - cohor -- cohor - - Tme Sadard cofguraos of age ad me For all daa hs colleco age ad me are arraged - 5- ad -year ervals. The cofgurao of a mar of deah raes or some oher quay s deoed by 5 5 ec. I hs oao he frs umber always refers o he age erval ad he secod umber refers o he me - 4 -

8 Mehods Proocol for he HM erval. For eample deoes a cofgurao wh sgle years of age ad -year me ervals. I he HM deah raes ad lfe ables are geerally preseed s sadard cofguraos: ad 5. Furhermore he daabase cludes esmaes of deah cous by es ragle ad of populao sze o Jauary s by sgle years of age mag possble for he sophscaed user o compue deah raes ad lfe ables ay cofgurao desred. 5 All rages of age ad me descrbe clusve ses of oe-year ervals. For eample he age group -4 eeds from eac age up o bu o cludg eac age 5 ad he me perod desgaed by begs a he frs mome of Jauary 98 ad eds a he las mome of ecember 984. I addo he followg coveos are used hroughou he daabase for orgazg formao by age ad me: 5-year me ervals beg wh years edg or 5 ad fsh wh years edg 4 or 9 ; -year me ervals beg wh years edg ad fsh wh years edg 9 ; complee 5- or -year me ervals are cluded preseaos of deah raes or lfe ables f daa are avalable for a leas years a eher he begg or he ed of he seres; for raw daa daa oe-year age groups are always provded up o he hghes age avalable followed by a ope age erval oly f more dealed daa are o avalable; for all daa o coury pages oe-year age groups sop a age 9 wh a fal caegory for ages ad above; for 5-year age groups he frs year of lfe age s always separaed from he res of s age group ages -4 ad he las age caegory s for ages ad above. Thus a 5 cofgurao coas daa for sgle years of me wh ypcally he followg age ervals: I fuure versos of hs daabase we hope o add a eracve compoe ha would allow a user o reques deah raes or lfe ables a wder varey of age-me cofguraos

9 Mehods Proocol for he HM I s mpora o oe ha he daa show o coury pages by sgle years of age up o are somemes he produc of aggregae daa e.g. fve-year age groups ope age ervals whch are spl o fer age caegores usg he mehods descrbed here. Alhough here are some obvous advaages o maag a uform forma he preseao of deah raes ad lfe ables s mpora o o erpre fcous daa lerally. I all cases he user mus ae resposbly for udersadg he sources ad lmaos of all daa provded here. Female / male / oal I hs daabase lfe ables ad all daa used her cosruco are avalable for wome ad me separaely ad ogeher. I mos cases a sgle fle coas colums labeled female male ad oal oe ha hs s alphabecal order. However he case of lfe ables whch already coa several colums of daa for each group daa for hese hree groups are sored separae fles. Raw daa for wome ad me are always pooled pror o mag oal calculaos. I oher words deah raes ad oher quaes are o merely he average of he separae values for females ad males. For hs reaso all oal values are affeced by he relave sze of he wo sees a a gve age ad me. Perods ad cohors Raw daa are usually obaed a perod forma.e. by he year of occurrece raher ha by year of brh. eahs are somemes repored by age ad year of brh bu he sascs are ypcally colleced publshed ad abulaed by year of occurrece. Alhough raw daa are preseed here a perod forma oly deah raes ad lfe ables are provded boh formas f he observao perod s suffcely log o jusfy such a preseao. eah raes are gve a cohor forma.e. by year of brh f here are a leas cosecuve caledar years of daa for ha cohor. Cohor lfe ables are - 6 -

10 Mehods Proocol for he HM preseed f here s a leas oe cohor observed from brh ul eco. 6 I ha case lfe ables are provded for all ec cohors ad for some almos-ec cohors as well. 7 Adjusmes o raw daa Mos raw daa are o oally clea ad requre varous adjusmes before beg used as pus o he calculaos descrbed here. The mos commo adjusme s o dsrbue persos of uow age eher deah or cesus cous across he age rage proporo o he umber of observed dvduals each age group. Aoher commo adjusme s o spl aggregae daa o fer age caegores he case of deah cous from 5 o daa ad from daa o es ragles. These wo commo procedures are descrbed laer hs docume. Forma of daa fles Raw daa for hs daabase have bee assembled from varous sources. However all raw daa have bee assembled o fles coformg o a sadardzed forma. There are dffere formas for brhs deahs cesus cous ad populao esmaes. The raw daa fles o he web page are always preseed oe of hese sadardzed formas. Oupu daa such as eposure esmaes deah raes ad lfe ables are also preseed sadardzed formas. Seps for compug moraly raes ad lfe ables There are s seps volved compug moraly raes ad lfe ables for he core seco of he HM. Compuaoal deals are provded laer secos of hs docume cludg he appedces. Here s jus a overvew of he process:. Brhs. Aual cous of lve brhs by se are colleced for each populao over he loges possble me perod. A a mmum a complee seres of brh cous s eeded for he me perod over whch moraly raes ad perod lfe ables are compued. These cous are used maly for 6 A ec cohor s oe whose members are assumed o have all ded by he ed of he observao perod. A rule for defyg he mos rece ec cohor s gve laer. 7 A smple decso rule s used o deerme whe s accepable o compue lfe ables for almosec cohors. I such cases deah raes for ages o ye observed are based o he average eperece of prevous cohors. A dealed descrpo of hese procedures s gve a laer seco

11 Mehods Proocol for he HM esmag he sze o Jauary s of each year of dvdual cohors from brh ul he me of her frs cesus ad for oher adjusmes based o relave cohor sze.. eahs. eah cous are colleced a he fes level of deal avalable deally by dvdual ragles of he es dagram. Somemes however deah cous are avalable oly for es squares or 5 es recagles. Before mag subseque calculaos deahs of uow age are dsrbued proporoaely across he age rage ad aggregaed deahs are spl o fer age caegores. Addoal adjusmes or ad hoc esmaos may be ecessary depedg o he characerscs of he raw daa for a parcular populao ay such adjusmes are descrbed he documeao for ha populao.. Populao sze. Below age 8 esmaes of populao sze o Jauary s of each year are eher obaed from aoher source mos commoly offcal esmaes or derved usg ercesal survval. I mos cases all avalable cesus cous are colleced for he me perod over whch moraly raes ad lfe ables are compued. The mamum level of age deal s always reaed he raw daa ad used subseque calculaos. Whe ecessary persos of uow age are dsrbued proporoaely o oher age groups before mag subseque calculaos. Above age 8 populao esmaes are derved by he mehod of ec geeraos for all cohors ha are ec see below for eco rule ad by he survvor rao mehod for o-ec cohors who are older ha age 9 a he ed of he observao perod. For o-ec cohors aged 8 o 9 a he ed of he observao perod populao esmaes are obaed eher from aoher source or by applyg he mehod of ercesal survval. 4. Eposure-o-rs. Esmaes of he populao eposed o he rs of deah durg some age-me erval are based o aual Jauary s populao esmaes wh a small correco ha reflecs he mg of deahs durg he erval. 5. eah raes. For boh perods ad cohors deah raes are smply he rao of deah cous ad eposure-o-rs esmaes mached ervals of age ad me

12 Mehods Proocol for he HM 6. fe ables. Perod deah raes are covered o probables of deah by a sadard mehod. Cohor probables of deah are compued drecly from raw daa bu hey are relaed o cohor deah raes a cosse way. These probables of deah are used o cosruc lfe ables. Commo adjusmes o raw daa I hs seco we gve formulas for four commo adjusmes o raw daa: redsrbug deahs of uow age splg deahs cous o es ragles splg 5 deahs cous o daa ad 4 splg deahs cous ope age ervals o daa. 8 srbug deahs of uow age The mos commo adjusme o raw daa volves dsrbug observaos eher deahs or cesus cous where age s uow o specfc age caegores. I geeral such observaos are dsrbued proporoally across he age rage. For eample suppose ha deah cous are avalable for dvdual ragles of he es dagram bu ha age s uow for some umber of deahs. Formally le umber of lower-ragle deahs recorded amog hose aged [ year ; umber of upper-ragle deahs recorded amog hose age [ year ; K umber of deahs of uow age year ; ad TOT oal umber of deahs year K. [ ] 8 I rece years some aoal sascal offces have begu reporg deahs by year of occurrece as well as by year of regsrao whch may dffer f regsrao was delayed. I such cases we abulae deahs accordg o he year whch hey occurred. If daa are o avalable by es ragle we spl hem o ragles usg he mehods descrbed hs docume. If deahs ha were regsered lae are avalable he same forma as oher deahs we sum he wo ses of daa frs ad he spl hem o ragles

13 Mehods Proocol for he HM The he followg par of equaos redsrbues deahs of uow age proporoally across upper ad lower es ragles over he full age rage: TOT [ ] K TOT K ad TOT [ ] K TOT K for all ages year. 9 Obvously hese calculaos ypcally resul o-eger deah cous for dvdual ages ad es ragles. I fac such umbers are o loger rue cous bu raher esmaed cous. However sce hey are our bes esmaes of acual deah cous s approprae o use hem all subseque calculaos. I all formulas gve below s assumed ha deahs of uow age have bee dsrbued proporoally f eeded ad he superscrp used hs seco s suppressed for sae of smplcy. Whe raw deah cous are avalable a or 5 forma deahs of uow age f ay are dsrbued across he esg age groups before splg he raw cous o es ragles as descrbed below. oe however ha he fal resul of hese calculaos would o chage f aggregae daa were frs spl o fer age caegores before redsrbug deahs of uow age. I oher words he orderg of hese procedures does o maer. e deah cous cesus abulaos may coa persos of uow age. If eeded a smlar adjusme s made before proceedg wh he calculaos used for esmag populao o Jauary s as descrbed a laer seco. 9 Ths adjusme reflecs a assumpo ha he probably of age o beg repored s depede of age self. - -

14 Mehods Proocol for he HM Splg deah cous o es ragles eah cous are ofe avalable oly es squares ad o es ragles. Sce may of our subseque calculaos are based o es ragles s ecessary o devse a mehod for splg deah daa o ragles whe ecessary. I geeral he proporo of deahs lower ad upper es ragles vares wh age as show by he regresso model preseed laer hs seco see also Vall 97. everheless a adequae procedure may cases s smply o assg half of each deah cou o he correspodg lower ad upper ragles sce errors of overesmao for oe ragle a lower-upper par are ypcally balaced by errors of uderesmao for he oher ragle almos all subseque calculaos. Ths smple procedure was appled successfully o he aalyss of moraly above age 8 he Kaso-Thacher daabase Adreev. However for a colleco of moraly daa boh perod ad cohor formas coverg he ere age rage a more complcaed procedure s eeded for a leas wo reasos: deahs he frs year of lfe are heavly coceraed he lower ragle ad should o be spl half ad a ay age he dsrbuo of deahs across he wo ragles s affeced by he relave sze of he wo cohors ad somemes by hsorcal eves as well. The secod po s especally mpora suaos where here are rapd chages cohor sze due o mared dscoues he brh seres as occurs mes of rapd socal chage for eample a he begg or ed of a major war. Oce he procedure for splg deahs s modfed o ae hese maers o accou s oly a small sep furher oward a complee model ha adjuss for several facors ha are ow o affec he dsrbuo of deahs by es ragle. For hese reasos we have developed a regresso equao for use splg deahs o es ragles. The equao s based o a mulple regresso aalyss of daa for hree coures whch Somemes deah cous are avalable oly by perod-cohor parallelogram.e. holdg caledar year ad brh cohor cosa bu coverg more ha oe age year. Wh each sgle year brh cohor hese deahs are smply spl half o he wo respecve es ragles. Smlarly deahs cous may be avalable by age-cohor parallelogram.e. age ad brh cohor are cosa bu he parallelogram covers more ha oe caledar year whch case we also spl he deahs half o es ragles for year ad year. - -

15 Mehods Proocol for he HM s descrbed more fully Apped A. The equao s epressed erms of he proporo of deahs ha occur he lower ragle. I geeral we deoe hs proporo as follows: π d. Whe he values of ad are o ow our as s o derve a esmaed proporo he lower ragle deoed ˆ π. From hs quay we compue esmaes of lower- ad upperragle deahs: ˆ ˆ d d ˆ ˆ ˆ π π ad [ ] where s he observed umber of deahs he es square. The equao for esmag π dffers by se. For wome he equao s as follows: d d F ˆ π.47 ˆ α d [ π.5].5 I 98.7 I 99. log IMR.688 log IMR I.68 log IMR I b [ log IMR log.] I I IMR <.. 4 I hs equao log refers o he aural logarhm. The dcaor fuco I. equals oe f he logcal saeme wh pareheses s rue ad zero f s false. ummy varables for years 98 ad 99 are cluded o reflec he srog mpac of he worldwde Spash flu epdemc o he dsrbuo of deahs wh hose wo years. The esmaed age effecs F αˆ for he female verso of he equao are gve Table A-a Apped A uder he colum for Model VI. Ecep for ages ad he same age coeffce s used for more ha oe sgle-year age wh a broader age group ad he coeffce for he age group -4 s used for all ages above years. The brh proporo π b - -

16 Mehods Proocol for he HM s defed formally as follows: B π b 5 B B where B s he umber of brhs sees combed occurrg year he same populao. Wherever he avalable brh seres s complee we se π. 5. The fa moraly rae sees combed s foud usg a mehod proposed by Pressa 98: b IMR B B. 6 oe ha he fa moraly rae ca be compued hs maer before splg deahs o ragles. If B ad are ow bu B s uow he we se B B o calculae IMR. I geeral he hsorcal decle fa moraly has bee assocaed wh a hgher proporo of deahs he lower ragle relave o he upper ragle across he age rage ecep a age. A age he decle fa moraly s assocaed wh a rapdly creasg cocerao of deahs wh he lower ragle ul he IMR falls below oe perce. Below ha level he hsorcal red reverses self ad he proporo of fa deahs he lower es ragle eds o fall. For me he equao for esmag π s as follows: d ˆ π.488 ˆ α d.59 log IMR I.67 M.699 [ π.5].78 I 98.5 I log IMR.745 log IMR I b [ log IMR log.] I I IMR <.. 7 The brh proporo provdes formao abou he relave sze of wo successve brh cohors who boh pass hrough he age erval [ durg caledar year. More precsely epresses he orgal sze of he youger cohor passg hrough he lower ragle of a es square as a proporo of he oal brhs for he wo cohors. Alhough hs umber measures he relave sze of he wo cohors a brh ca also serve as a useful dcaor of her relave szes a laer ages. I he case of a coury or area ha has udergoe erroral chages s mpora o adjus he brh seres so ha refers always o he same populao. See Apped for a geeral dscusso of how we deal wh chages populao coverage. - -

17 Mehods Proocol for he HM I hs equao π ad IMR are he same as he female equao sce each s based o he b oal populao. However he age coeffces as well as all oher coeffces are dffere ad are gve Table A-b Apped A uder Model VI. Splg 5 deah cous o daa eah cous a 5 cofgurao are spl o daa usg cubc sples fed o he cumulave dsrbuo of deahs wh each caledar year. I prcple he same or a smlar mehod could be appled o ay cofgurao of deah cous by age. The mehod used here requres oly ha he raw daa clude deah cous for he frs year of lfe ad for he frs fve years of lfe. Oher ha hese wo resrcos does o maer wheher he raw daa are srcly fve-year age groups afer age fve or some oher cofgurao. Also here ca be a ope age erval above 9 or some oher age. The sple mehod s used o spl deah cous for all ages below he ope age erval. eals of he compuaoal mehods are gve Apped B. Splg deah cous ope age ervals o es ragles I some cases he raw daa provde o age deal o deah cous above a cera age. Isead we ow oly he oal umber of deahs hs ope age erval for some caledar year whch we deoe. I hese suaos we eed a mehod for splg o fer age caegores. Oe possbly would be o spl deah cous he ope age erval o daa ad he o apply he mehod descrbed earler for splg deah cous o es ragles. However he mehod for splg he ope age erval self s evably arbrary ad mprecse ad seems ha lle would be gaed by such a -sep procedure. Therefore our mehod spls mmedaely o es ragles. For some populaos we have deah cous by perod-cohor parallelograms coverg fve cohors e.g. deahs year for he -9 o -5 brh cohors who wll complee ages 5-9 year. I hs case we use he cubc sple mehod descrbed here o spl hese deahs o sgle brh cohors see Apped B for more deals

18 Mehods Proocol for he HM I order o dsrbue deahs he ope age erval we f he Kaso model of old-age moraly Thacher e al. 998 o deah cous for ages ad above where s he lower boudary of he ope age group e.g. 8 9 hus reag deah cous wh a perod as hough hey pera o a closed cohor. We he use he fed model o erapolae deah raes by es ragle wh he ope age erval ad use hose raes o derve he umber of survvors a each age. For deals see Apped C. Populao esmaes Jauary s We descrbe four mehods for dervg age-specfc esmaes of populao sze o Jauary s of each year: lear erpolao ercesal survval ec cohors ad 4 survvor raos. For mos of he age rage we use eher lear erpolao of populao esmaes from oher sources 4 or ercesal survval mehods. A ages 8 ad older we use populao esmaes compued usg he mehods of ec cohors ad survvor raos ecep for hose cohors who are youger ha age 9 a he ed of he observao perod. We descrbe he four mehods separaely. I case of erroral chages or oher chages populao coverage durg he me perod covered by HM adjusmes o hese mehods are descrbed Apped. ear erpolao I some cases he avalable populao esmaes from oher sources are for some dae oher ha Jauary s e.g. md-year esmaes. Whe he perod bewee oe populao esmae ad he e or a populao esmae ad a cesus cou s oe year or less we use lear erpolao o derve he Jauary s populao esmae. 5 Whe he perod bewee populao cous s greaer ha oe year e.g. cesus cous we employ ercesal survval. 4 The ma crera for usg populao esmaes from aoher source are ha hey are avalable ad ha hey are beleved o be relable. 5 We calculae he populao as of Jauary s of year as a weghed average of he esmaes years ad - where he weghs are based o he proporo of he year bewee Jauary s ad he dae of he avalable esmae. For eample f we have Ocober s esmaes he he Jauary s populao a age s calculaed as: P.. YYYY.75 P.. YYYY -.5 P.. YYYY

19 Mehods Proocol for he HM Iercesal survval mehods Iercesal survval mehods provde a covee ad relable meas of esmag he populao by age o Jauary s of every year durg he ercesal perod. There are wo cases: pre-esg cohors.e. hose already alve a he me of he frs cesus ad ew cohors.e. hose bor durg he ercesal erval. We develop formulas for hese wo suaos separaely by frs cosderg he smple case of a coury ha coducs cesuses every fve years o Jauary s. We he propose a more geeral mehod ha ca be used for cesuses occurrg a ay me of he year ad for ercesal ervals of ay legh. Specfc eample Suppose ha a coury coducs cesuses every fve years ad suppose ha each cesus occurs o Jauary s. Therefore populao esmaes by sgle years of age are avalable a fve-year ervals bu o comparable esmaes are avalable for erveg years.. Pre-esg cohors The es dagram Fgure a depcs a cohor who s already alve a he me of he frs cesus. The cohor aged a me s followed hrough me for 5 years. Suppose ha all deahs he populao are recorded wh a relavely hgh level of deal such ha for each year he ercesal perod deah cous are avalable by boh age ad year of brh. Thus s ow wh some precso how may lfe-les eded by deah each of he small ragles show hs fgure. A he begg or ed of he daa seres we cao use lear erpolao because here are o wo daa pos e.g. he las populao esmae he seres s for July s of year. I hese cases we use pre-cesal or pos-cesal esmao see p. 5 o derve he Jauary s esmae.e. by addg or subracg deahs for each cohor

20 Mehods Proocol for he HM Fgure. Iercesal survval mehod eample a pre-esg cohors Age 6 P P 4 5 Tme The formao represeed by Fgure a ca be used o esmae he sze of he cohor o Jauary s of each ercesal year. The smples procedure cosss merely of subracg deah cous from he al cesus cou o oba cohor populao esmaes o Jauary s of each succeedg year. foruaely he fal sep of such a compuao usually yelds a esmae of cohor sze a me 5 ha dffers from he umber gve by he correspodg cesus. Ths cossecy s caused by wo facors: mgrao ad error. Alhough boh of hese facors ed o be small relave o cohor sze a leas for aoal populaos as a maer of prcple hey should o be gored. The sadard mehod cosss of dsrbug mpled mgrao/error uformly over he parallelogram show Fgure a. The esmaes of cohor sze for ercesal years are foud by subracg from he al cesus cou boh he observed deah cous ad a esmae of e mgrao/error. Formally he procedure ca be descrbed as follows. e C equal he cesus cou for - 7 -

21 Mehods Proocol for he HM persos aged [ o Jauary s of year. Assumg ha here s o mgrao or error oe ha [ ] C. 8 Ths formula resembles oe ha s used for esmag populao szes a older ages he ec cohor mehod see below. sg cesus formao abou he sze of a cohor a me we ca esmae s sze a he me of he e cesus 5 by he followg formula: 4 5 C C ˆ. 9 [ ] However f here s ay mgrao o or ou of hs cohor durg he ercesal perod or ay error he recordg of cesus or deah cous hs esmae wll dffer from he acual cou a he me of he e cesus C 5. By defo oal mgrao/error s equal o he observed cohor sze a he secod cesus mus s esmaed sze C ˆ 5. We call hs dfferece : ˆ C 5 C 5. Assumg ha mgrao/error s dsrbued uformly across he parallelogram show Fgure a he esmaed populao sze o Jauary s of each year s as follows: P C [ ] 5 for K 5. By desg whe or 5 hese populao esmaes mach cesus cous eacly:. ew cohors P C ad P 5 5 C 5. The above formula apples oly o cohors who are already alve a he me of he frs cesus. For cohors bor bewee he wo cesuses ercesal populao esmaes are obaed by subracg he umber of deahs occurrg before he secod cesus from he umber of brhs for he cohor. For a - 8 -

22 Mehods Proocol for he HM cohor bor year j wh he ercesal erval [ 5 le K legh of he erval [ j 5 age a las brhday of he cohor bor year j a he me of he secod cesus 4 j ; ad B umber of brhs year j. j A al esmae of populao sze for he cohor bor year j a he me of he secod cesus s K K B j j C ˆ 4 [ j j ] ad he dfferece bewee hs esmae ad he acual populao cou s ˆ C K C. 5 j K Thus he esmaed sze of he cohor o Jauary s of each year from brh ul he secod cesus s: P j B j j K [ j j ] j 6 for K K. As before populao esmaes a me 5 mach he cous he secod cesus eacly: P K 5 C K. For eample cosder he cohor bor year. Thus j ad K 4 j. I oher words he cohor bor year wll be aged a he me of he secod cesus as llusraed Fgure b. Populao esmaes for hs cohor o Jauary s of each year ul 5 are as follows: P B 7 5 P 4 B [ ] B P. 9 ad [ ] - 9 -

23 Mehods Proocol for he HM Fgure. Iercesal survval mehod eample b ew cohors Age 5 4 P5 B 4 5 Tme Geeralzg he mehod The argumes above mae he eplc assumpo ha he wo cesuses boudg he ercesal perod each occur o Jauary s ad are eacly fve years apar. However realy s ypcally more complcaed. I hs seco we geeralze he mehod o allow for cesuses ha occur o ay dae of he year ad for ercesal ervals of ay legh.. Pre-esg cohors e ad Fgure 4a depcs a ercesal perod bouded by wo cesuses ha occur o arbrary daes. be he mes of he frs ad he las Jauary s wh he ercesal erval. Thus equals he umber of complee caledar years bewee he wo cesuses. e f be he fraco of caledar year before he frs cesus ad le f be he fraco of caledar year before he - -

24 Mehods Proocol for he HM secod cesus. Thus he wo cesuses occur a mes f ad f ad he oal legh of he ercesal perod s f f. Fgure 4. Iercesal survval mehod geeral a pre-esg cohors Age - c d f C -f - C a b - - Tme The hghlghed cohor Fgure 4a s of age o Jauary s of year. Ths cohor was aged or a he me of he frs cesus ad wll be aged or a he me of he secod cesus. If dvduals are uformly dsrbued across her respecve age ervals a each cesus eumerao he szes of hs cohor a he begg ad ed of he ercesal erval are C f C f C ad C f C f C - -

25 Mehods Proocol for he HM respecvely. Alhough he assumpo of a uform dsrbuo across age ervals s obvously correc errors of eaggerao wll ed o be balaced by hose of udersaeme yeldg suffcely accurae esmaes mos cases. Smlarly assumg a uform dsrbuo of deahs wh es ragles deahs o hs cohor year afer he frs cesus eumerao wll be composed of wo compoes: a f ad b f. ewse uder he same assumpo deahs o hs cohor year eumerao wll be c before he secod cesus f 4 d ad f f. 5 sg hese umbers alog wh deah cous durg complee caledar years of he ercesal erval we esmae he sze of he hghlghed cohor a he me of he secod cesus as follows: C ˆ C [ ] a b c d. 6 The dfferece bewee he acual cesus cou ad hs esmae C Ĉ represes he oal ercesal mgrao/error for hs cohor. Fally he sze of he cohor o each Jauary s of he ercesal erval s esmaed as follows: f [ ] f f P C 7 a b for K.. Ifa cohor The above formulas are applcable for cohors ha are aged or more o he frs Jauary s of he ercesal erval. For he cohor aged o hs dae Fgure 4b ad for ew cohors bor durg - -

26 Mehods Proocol for he HM he ercesal erval Fgure 4c dffere formulas are eeded. For he fa cohor he followg modfcaos o he above formulas are ecessary: C f B f 8 C C ˆ C [ ] a c d ad C [ ] f f 9 P a f for K where C Ĉ. oe he followg four dffereces bewee hese formulas ad hose gve earler: dsappears from he laer wo equaos sce ; he frs formula C s replaced by B he umber of brhs durg he caledar year of he frs cesus; he laer wo formulas replaced by f ualered. b s abse as s udefed; ad 4 he las erm of he hrd equao f boh umeraor ad deomaor. The formulas for a c s d ad C are - -

27 Mehods Proocol for he HM Fgure 4. Iercesal survval mehod geeral b fa cohor Age c C d - f -f a C - - Tme. ew cohors asly we cosder he case of a cohor bor durg complee caledar years of he ercesal erval. A cohor bor year j wll be aged K j o he las Jauary s before he secod cesus. efg f c ad d as before he followg equaos are used o esmae he sze of ew cohors o Jauary s of each year from brh ul jus before he secod cesus: K C ˆ B j j [ j j ] c d P j B j j K f ad [ j j ] j for K K where j C Ĉ

28 Mehods Proocol for he HM Fgure 4. Iercesal survval mehod geeral c ew cohors Age -j -j- c C d f B j j j Tme Pre- ad poscesal survval mehod For a shor perod before he frs cesus or afer he las cesus populao esmaes ca be derved smply by addg or subracg deahs from populao cous a cesus or for ew cohors from brh cous. The formulas are smlar o hose preseed earler alhough hey lac a correco for mgrao/error. Therefore populao esmaes for rece years ha are derved hs maer mus be cosdered provsoal. They wll be replaced by fal esmaes oce aoher cesus s avalable o close he ercesal erval. The purpose of such esmaes s o allow moraly esmao durg rece years or for a shor perod before a early cesus whe approprae deah cous are avalable durg a ope cesus erval. Eamples of pre- ad poscesal survval esmao are show Fgure 5. The sze of he cohor bor year o Jauary s of years ad s esmaed as follows: P C a b ad P C a b

29 Mehods Proocol for he HM To esmae he sze of he same cohor o Jauary s of years ad we have: P C c d 5 ad P C c d. 6 I hs oao a b c ad d are he complemes of a b c ad d respecvely. Tha s he sum of each par of deah cous equals he umber of deahs a es ragle. For eample comparg Fgures 4a ad 5 we see ha Age a a. Fgure 5. Pre- ad poscesal survval mehod C c -f f P P d a P-- C - b - P Tme Iercesal survval wh cesus daa -year age groups The above dscusso assumes ha cesus daa are avalable sgle-year age groups. However for may hsorcal cesuses he avalable cous refer o -year age groups where s ofe 5. I hese cases we mus frs spl he daa o oe-year age groups before compug populao esmaes usg - 6 -

30 Mehods Proocol for he HM he mehod of ercesal survval. We employ a smple mehod for hs purpose. We assume ha a more rece cesus s avalable whch coas populao cous by sgle years of age. sg he age dsrbuo a he me of he laer cesus plus deah cous he ercesal erval we esmae he age dsrbuo of he earler cesus by he mehod of reverse survval. However hese esmaes may o sum o he oal or sub-oals gve he earler cesus. Therefore we use oly he esmaed dsrbuo of he populao by age a he me of he earler cesus whch s appled o he observed cous wh -year age ervals as a meas of creag fer age caegores. Thus all cous coaed he earler cesus are preserved he process of mag hese calculaos. Ec cohors mehods The mehod of ec geeraos ca be used o oba populao esmaes for cohors wh o survvg members a he ed of he observao perod. Wh hs mehod he populao sze for a cohor a age s esmaed by summg all fuure deahs for he cohor whch ca be wre as follows: [ ] P. 7 Ths mehod assumes ha here s o eraoal mgrao afer age for he cohor queso whch s a reasoable assumpo oly for advaced ages. We use he mehod of ec geeraos o esmae populao szes for ages 8 ad above oly as llusraed Fgure 6. Pror o applyg he mehod of ec cohors s ecessary o deerme whch cohors are ec. For hs purpose we adop a mehod proposed by Väö Kaso ad used already he Kaso-Thacher oldes-old moraly daabase Adreev. We say ha a cohor s ec f has aaed age ω by ed of he observao perod assumed o occur o Jauary s of year. Thus we eed o fd ω or equvalely ω- he age of he oldes o-ec cohor

31 Mehods Proocol for he HM Fgure 6. Mehods used for populao esmaes Age ω 9 B C 8 A A A - Offcal esmaes / ercesal survval B - Ec cohors C - Survvor rao SR9 Tme Cosder a cohor aged a he ed of he observao perod where s some very hgh age le. We eame he mos rece l cohors from a smlar po her lfe hsores. Specfcally we cosder he observed deahs for hese cohors from Jauary s of he year whe hey were aged ul he ed of he observao perod see llusrao Fgure 7 where l 5 ad ω. For hese cohors over he specfed ervals of age ad me we compue he average umber of deahs: l j ~ l [ j j ] 8 l j ~ wh l 5. For very hgh ages l wll be close o zero. We defe ω o be he lowes age such ~ ~ ha l. 5. Equvalely ω- s he hghes age for whch l >

32 Mehods Proocol for he HM Fgure 7. Illusrao of eco rule wh l 5 ad ω - Age ω ω- -5 Tme Survvor rao The survvor rao mehod s used o esmae populao szes above age 8 for almos-ec cohors see Fgure 6. The mehod s appled o cohors ha are a leas age 9 a he ed of he observao perod bu o ye ec accordg o he rule gve above. 6 Varous versos of hs mehod have bee proposed ad suded prevously see dscusso Adreev 999. We use he verso ha proved mos relable a earler comparave sudy Thacher e al.. efe a survvor rao o be he rao of survvors alve a age o Jauary s of year o 6 We mae a ecepo for he small umber of coures ha have relable Jauary s populao esmaes by sgle year of age o he mamum age ω for he las year of observao.e. Swede emar orway Flad ad Icelad. For hese coures we use he offcal populao esmaes for ages 9 ad older o Jauary s of year ad derve populao esmaes earler years for each cohor by addg observed deah cous bac o age 8 le for he ec cohor mehod

33 Mehods Proocol for he HM - - hose he same cohor who were alve years earler: P P R. 9 Assumg ha here s o mgrao he cohor over he erval hs rao ca also be wre: P P R 4 where [ ]. Solvg hs equao for P we oba: R R P. 4 The survvor rao for he oldes o-ec cohor aged ω- a me s llusraed Fgure 8. Ths survvor rao s uow sce we do o ow he sze of he cohor P ω a he ed of he observao perod. However comparable survvor raos.e. wh age ω- he umeraor for all prevous cohors are avalable sce populao sze ca be esmaed usg he mehod of ec cohors. Suppose ha a survvor rao has appromaely he same value for he cohor queso ad for he prevous m cohors. Tha s suppose ha m P m P P P P P R. 4 The we ca esmae R by compug he pooled survvor rao for he m prevous cohors: m m P P R. 4 If boh R ad are avalable for a gve cohor we ca esmae P as follows: R R P ~. 44

34 Mehods Proocol for he HM Fgure 8. Survvor rao mehod a age ω - wh m 5 Age ω ω- Pω ω- ω- ω-4 ω-5 ω-6 Pω Tme I he smples verso of he survvor rao mehod hs procedure s used o oba P ω ad he he sze of hs cohor prevous years s esmaed by addg observed deah cous bac o age 8 a fasho smlar o he ec cohor mehod. I s he possble o apply he same mehod recursvely o oba P ω P ω ec. dow o some lower age lm e.g. 9 years. Ths mehod wors well f s fudameal assumpo s o volaed ha s f he survvor raos for successve cohors are early equal. A commo occurrece however s ha hese survvor raos crease over me as a resul of moraly decle. Therefore o uderesmae P. R eds o uderesmae R ad P ~ eds These cosderaos movae a modfed verso of he survvor rao esmae: ~ Pˆ c P R c R

35 Mehods Proocol for he HM where c s a cosa ha mus be esmaed. Whe moraly s declg/creasg/cosa c should be greaer ha/less ha/equal o oe. The problem obvously s how o choose he proper value of c. Followg Thacher e al. we choose a value of c such ha ω 9 P ˆ P9 46 where P 9 s a offcal esmae of he populao sze he ope erval aged 9 ad above a he ed of he observao perod. Ths verso of he survval rao mehod s ow as SR9 ad s used for he HM wh m 5 all cases where P 9 s avalable ad s beleved o be relable. 7 Oherwse we use he smpler verso of he survval rao mehod.e. wh c. eah raes eah raes coss of deah cous dvded by he eposure-o-rs. I he case of a oe-year age group ad a sgle caledar year.e. a perod deah rae we have he followg formula: where p p p M 47 E p 48 ad p E s he eposure-o-rs he age erval [ durg caledar year. The eposure-o-rs s always measured erms of perso-years ad for perods s compued by he followg formula see Apped E for a dervao: E p [ P P ] [ ] The daa used for compug hese quaes are llusraed Fgure 9. 7 For some populaos offcal populao esmaes are avalable oly for age 85. I such cases we use SR85 ad oe hs modfcao he geeral commes see coury-specfc documeao for deals. - -

36 Mehods Proocol for he HM Fgure 9. aa for perod deah raes ad probables Age P P Tme Cohor formulas are oly slghly dffere. A cohor deah rae s c c c M 5 E where deah cous ad eposure esmaes ecep a age are defed as follows: c 5 c E P [ ]. 5 The daa used for compug hese quaes are llusraed Fgure. The eposure esmae a age s a ecepo. Because he cohor lfe able deah rae c m s derved dfferely a age ha a oher ages see p.4 we defe c c c E order o esure ha m c c M m. - -

37 Mehods Proocol for he HM Fgure. aa for cohor deah raes ad probables Age P Tme For broader ervals of age ad/or me wheher me s defed by perods or cohors deah raes are always foud by poolg deahs ad eposures frs ad he dvdg he former by he laer. Throughou he res of hs dscusso we wll refer eher o oe-year or fve-year deah raes.e. or 5 M. For smplcy of oao we wll o specfy a parcular me erval because he formulas for compug probables of deah ad/or lfe ables are he same for ay erval of me. Also we wll o use a p or c superscrp o dsgush bewee perod ad cohor deah raes sce he dfferece should always be appare from he coe. fe ables fe able calculaos do o deped o he orgazao of he daa over me. For ay me erval he same mehods are used for compug lfe ables from a se of age-specfc deah raes. However he mehods used here are slghly dffere for perod ad cohor lfe ables. Perod ables are compued by coverg deah raes o probables of deah. Before hs coverso deah raes a older ages are smoohed by fg a logsc fuco. For cohor lfe ables we compue probables of deah drecly from he daa ad perform o smoohg a older ages. As dscussed Apped E cohor M - 4 -

38 Mehods Proocol for he HM probables of deah compued hs maer are fully cosse wh he cohor deah raes descrbed he prevous seco. For boh perods ad cohors we beg by compug complee lfe ables.e. sgle-year age groups usg our fal esmaes of deah cous by es ragle ad populao sze o Jauary s by sgle years of age. The abrdged ables.e. fve-year age groups are eraced from he complee ables. ervg abrdged ables from complee oes raher ha compug hem drecly from daa fve-year age ervals esures ha boh ses of ables coa decal values of lfe epecacy ad oher quaes. Perod lfe ables Whereas a cohor lfe able depcs he lfe hsory of a specfc group of dvduals a perod lfe able s supposed o represe he moraly codos a a specfc mome me. However observed perod deah raes are oly oe resul of a radom process for whch oher oucomes are possble as well. A older ages where hs here radomess s mos oceable s well jusfed o smooh he observed values order o oba a mproved represeao of he uderlyg moraly codos. Thus for perod lfe ables by sgle years of age we beg by smoohg observed deah raes a older ages by fg a logsc fuco o observed deah raes for ages 8 ad above separaely for males ad females. 8 Suppose ha we have deahs ad eposure E for ages 8 8 for coveece we defe for he ope caegory above age. We smooh observed deah raes M by fg he Kaso model of old-age moraly Thacher e al. 998 wh a asympoe E 8 I s a commo acuaral pracce o f a curve o deah raes a older ages he process of compug a lfe able. We use he logsc fuco because a rece sudy cocluded ha such a curve fs he moraly paer a old ages a leas as well as ad usually beer ha ay oher moraly model Thacher e al Fg he value of he asympoe a oe smplfes hese calculaos ad avods cera aomales ha may occur as a resul of radom flucuaos. I ay eve esmaes of hs asympoe have bee aroud oe mos prevous sudes

39 Mehods Proocol for he HM equal o oe o esmae he uderlyg hazards fuco µ : 9 where we requre a ad b 8 ae µ b b 8 a 5 ae b. Assumg ha ~ Posso E 5 a. b µ we derve parameer esmaes â ad bˆ by mamzg he followg log-lelhood fuco: [ log a b E a b ] cosa log a b µ.5 µ Subsug â ad bˆ o equao 5 yelds smoohed deah raes Mˆ where Mˆ ˆ µ µ aˆ ˆ. I hs model specfcao â ad bˆ are cosraed o be posve so ha.5.5 b smoohed deah raes cao decle above age 8. For he res of he calculaos descrbed here fed deah raes replace observed deah raes for all ages a or above Y where Y s defed as he lowes age where here are fewer ha deahs bu s cosraed o 8 Y 95. Thus complee perod lfe ables for males ad females are cosruced based o he followg vecor of deah raes: M M M Y Mˆ Y ˆM 9 ˆM. Afer obag smoohed deah raes for males ad females we calculae he smoohed raes for he oal populao as a weghed average of hose for males ad females: Mˆ T w Mˆ w Mˆ 55 F F F M 9 Ths smoohg procedure s used oly f here are a leas wo M > a ages 8 ad older. If here are fewer ha wo o-zero observed deah raes he we assume he deah rae s cosa for ages [ ϖ ] where ϖ s he oldes age where M >. I order o sasfy he cosras o he parameers a ad b we f he model erms of a ad b a b where a e ad b e. I oher words we used he fed deah raes for all ages a or above he greaer of 8 or he lowes age where here are fewer ha male deahs because a older ages here are ypcally fewer male deahs ha female deahs ad for all ages a or above age 95 regardless of he umber of deahs. We beg usg fed deah raes a he same age for boh males ad females

40 Mehods Proocol for he HM where superscrps T F ad M represe oal female ad male respecvely ad wegh for females aged bu hese mus sll be deermed. eposure: F w represes he For observed deah raes he aalogous weghs equal he observed he proporo of female F F M F T F E E π. 56 E E E For smoohed raes as well such weghs could be calculaed from observed eposures bu due o radom flucuaos such values a older ages he resulg seres of deah raes for he oal populao would o be as smooh as hose for males ad females. Cosequely we smooh followg model by he mehod of weghed leas squares: F π self by fg he We drop observaos where F F π z log π l β β β F. 57 π F E M E or boh equal such cases udefed ad for fg equao 57 use weghs equal o follows: T E. F π or ad hus he log s The fed values are obaed as zˆ zˆ ˆ ˆ ˆ F F e β β β ad w πˆ zˆ Fally he smoohed oal deah raes are calculaed as:. 58 e Mˆ T πˆ Mˆ πˆ Mˆ. 59 F F F M We he assume ha deah raes he lfe able equal deah raes he populao.e. ha m M for ages Y- ha m Mˆ for Y Y 9 ad ha m ˆM for he For fg he model equao 57 heorecally he correc weghs would be π ˆ πˆ E bu would requre a erave procedure because he weghs deped o he fed values hemselves. Sce F F T T here s relavely lle varably π ˆ πˆ compared o E over he observed rage usg E as he weghs should provde reasoable accuracy ad s much more covee. F F T - 7 -

41 Mehods Proocol for he HM ope age erval above. Ths assumpo s correc oly whe he age srucure of he acual populao s decal o he age srucure of a saoary.e. lfe able populao wh each age erval for more eplaao see Keyfz 985 or Preso e al.. I mos suaos however devaos from hs assumpo are lely o be small ad umpora for oe-year age ervals. e we cover he lfe able deah raes average umber of years lved wh he age erval [ m o probables of deah q. e a be he for people dyg a ha age. We assume ha a for all sgle-year ages ecep age see below. We he compue q from accordg o he formula m ad a q m a m 6 for 9. For he ope age erval we se a ad m q. For fas we adop he formulas for a suggesed by Preso e al. : 48 whch are adaped from he Coale-emey model lfe ables Coale ad emey 98. Thus f m. 7 : O he oher had f m <. 7 : For a lfe able wh sees combed we compue a as follows:. 5 for females a. 6. for males.5.8 m for females a m for males a T F F F M M M a a 6 where he superscrps F M ad T deoe values for he female male ad oal populaos respecvely ad where refers o all deahs a age zero boh lower ad upper ragles for populao

42 Mehods Proocol for he HM To complee he lfe able calculao le p be he probably of survvg from age o. Therefore p 64 q for all ages. e he rad of he lfe able be l. The he umber of survvors ou of a age s l p The dsrbuo of deahs by age he lfe-able populao s l. 65 d l q 66 for 9. For he ope age caegory d l. The perso-years lved by he lfe-able populao he age erval [ are l a d 67 for 9. For he ope age caegory l a. The perso-years remag for dvduals of age equal T 9 68 for 9. For he ope age caegory T. Remag lfe epecacy a age s for. Cohor lfe ables We ow descrbe he mehod used o compue q a ad T l e o 69 m for cohors. These are foud usg deah cous es ragles wheher observed hs forma or esmaed from or 5 daa as descrbed above ad populao esmaes by sgle years of age ad me. Oce hese values are avalable a complee cohor lfe able s calculaed usg he same formulas as he case of perod ables