model life tables for china

5/13/2018 Model Life Tables for China

1/46

MODEL LIFE TABLES FOR CHINA

Jiang Zhenghua

Paper number 0035

Population Research InstituteXi'an liaotong UniversityXi' an, Shaanxi ProvincePeople's Republic of China

1990


2/46


3/46

Preface

M odel life tables provide an im portant tool for studying dem ographic problem s andforecasting population developm ent as patterns of m ortality change. In in te rn atio na l d emo -graphic studies. m ortality in the population of C hina h as re ceiv ed v ery little a tten tio n.Neither the m odel life tables developed by the United Nations in the 1950s nor the regionalm odel life tables developed by Professor A .J. C oale have been applied to the life tables ofChina. Fragmentary m ortality data from the populations of Taiwan and Hong Kong arenowhere near enough to represent most of the People's Republic of China. A lthough them odel life tables for developing countries constructed by the U nited N ations in rec en t y ea rshave incorporated m ortality data from the Far East areas. none of these life tables are fromthe m ainland of C hina. It is im portant to include China, with more than one fifth of theW orld 's population, in such analyses.

Before the 1950s C hina did not collect dem ographic m onality data ina nat ionw idecoordinated m anner. Since the founding of the People's Republic of China, som e age-specific m ortality data have been available through registration of vital data. Ingeneral,m ost of these data are believable, although the num ber of deaths at som e ages, especially atth e y oun gest ag es, are o bv io usly un der-reg istered Inadd itio n. th e data w ere not fullyrepresentative for all sub-regions. From 1973 to 1975, a large scale survey of deaths dueto cancer w as conducted by the Public H ealth Ministry to investigate the causes of specificm ortality rates for the w hole of C hina. Statistical data w hich w ere quite representative ofthe general m ortality characters for each region w ere collected for the first tim e. T hesesurvey data had deficiencies, how ever, and cenain reservations w ere expressed by m edicaland public health statisticians on the reliability of the results. N evertheless, incomparisonw ith the previous situation, these data obviously constituted som e progress.

T he p op ulatio n cen su s co nd ucted in 1 98 2 p ro vid ed v aluab le statistical m aterialconcerning m ortality. This was the first time inChina that inform atio n o n d eaths in th e

- 1 -


4/46

who le co un try w as collected . S pecific ap pen dices w ere u sed to record detailed p erso nalm aterials relating to the d eath s. A fter the cen su s, a sam ple su rv ey w as co nd ucted to te st th eq uality o f th e data an d pro ved th at th e reliab ility of the morta l ity data w as very high. Thisp ro vid ed a so lid b asis fo r stu dyin g d em og rap hic ch an ges an d so cio -eco nomic d ev elo pm en tin C hina and also for filling in a void in inform ation available to i nt er na ti onal s chol ar s o fdemography. The C hin a C lass ified (R eg io nal) Mode l L ife T ab le Compila tio n Comm itteewas established in 1984. M r. Chengrui Li, who was the d irecto r o f th eadm inistrative office of the third population census under the State C ouncil and head of theS tate S tatistical B ureau , w as th e ch airm an o f th e comp ilatio n comm ittee, w hich co nsisted o fex perts and p erson nel in relev an t d ep artm en ts. F irst of all, accord in g to a u nifiedp ro ced ure co ntro lled b y th e th ird p op ulatio n cen su s o ffice u nd er th e S tate C ou ncil o fC hina, 3136 original life tables (half for m ales and half for fem ales) w ere established bylo cal p op ula tio n ce nsu s o ffic es a nd elec tro nic computer c en ters o f v ario us p ro vin ce s,a utonomous r eg ions a nd munic ip alities . U sin g th e material from th e cen su s, s amplesu rv ey a nd v ita l re gis tra tio ns . P ro fe sso r Z ben ghua Jian g, director o f th e Popu la ti onRese arc h In stitu te o f X i' a n Jia oto ng Univ ersity , a nd Mr. Jiy u L uo , asso ciate ch ief en gin eerof the ele ctro nic c ompute r ce nte r o f th e S ic hu an Provincial P la nn in g E conomic Comm is-s io n were incharg e o f th e comp ilatio n tech niqu e fo r th e m od el life tab les. T he P op ulatio nRese arc h In stitu te o f X i'a n J ia oto ng Univ ersity a nd th e comp uter cen ter o f S ich uan P ro vin -cial P lan nin g E conomic Comm is sio n c oope rated inthe de ta iled ca lcula tions . In the proces so f compilatio n. th e o rig inal life tab les fo r each reg io n o f C hin a were giv en a q uality assess-m ent and careful screening. 1926 tables (half for m ales and half fo r fem ales) p assed th eev aluatio n an d b ecam e th e b asis fo r th e classified (regio nal) m od el life tables. T he b asicdata in these tables cam e m osdy from Chin a' s p op ula tio n ce nsu s in 1982. A few of th etab les c ame from ea rlier su rv ey d ata. s uc h as v ita l re gis tra tio n data and th e ca nce r su rv eyfrom 1973 to 1975. By m athem atical processing and analysis of a large part of th e originald a t a , th e p attern s o f m ortality fo r th e C hin ese p op ulatio n cou ld be classified into five

- 2 -


5/46

familie s. T his c la ss ific atio n is q uite c on sis te nt w ith r eg io na l s oc ia l a nd e co nom ic c ha ra cte r-is tic s. a s w ell a s n atio na lity a nd o th er c on ditio ns , a n d a ls o p ro vid es a re lia ble d emogr ap hicb asis fo r fu rth er research w ork inva rious f ie lds .

3 60 mod el life tab les in f iv e " fam il ie s" ( i.e . c at ego ri es ) a re l is te d in this b oo k, o fw hich 1 80 tab les each are fo r m ales and fem ales w ith life ex pectan cies rank ed from age 40to 7 5. T he fiv e fam ilies inclu de: 1 . S ou th-w est C hina, II. M idd le and E ast C hin a, ill.North C hina. IV . N orth -east C hin a, and V . X in jiang au ton om ou s reg ion . An appendixreg ard ing the use o f th e m od el life tab les is attach ed for the users' co nv en ience. S oftw areap plicatio ns h av e b ee n d ev elo ped . as w ell. F rom a life tab le fo r a c ertain ar e a, the softwaremay be u se d. t o id en tify th e mo rta lity p atte rn fo r th at a re a a nd to c la ss if y i t into a cer ta infam ily. G iven the pattern o f m ortality an d on e of th e p aram eters of the life table fo r a cer-ta in area (su ch as life ex pectan cy at certain ag e. in fan t m on ality rate. etc.), a co rresp on din glife ta ble may be produced.

The developmen t and c omp ila tio n o f th e Ch in a c la ss ifie d (r eg io na l) mode l life ta ble shas taken fou r years. T he prelim in ary versio n o f the m odel life tab les w as d ev elop ed in1987 a nd th e c omp ila tio n c omm itte e in vite d e xp erts f rom r ele va nt d ep artm e nts to e va lu atet he ou tpu t inB eijing on M arch 17 of that year. E x pen s a nd p ro fe ss ors from th e T ec hn ic alE co nomy S ocial D ev elo pm en t C en ter u nd er th e S tate C ou ncil, the 710 I ns titu te o f Av ia tio nMinistry, t he App li ed Ma thema ti cs Institute a nd th e S ystem In stitu te o f th e C hin ese A cad -em y o f S cien ces. th e S tate S tatistical B ureau . th e Me dica l R esearch In stitu tes, an d o th erre la ted ag en cies g av e an a ffin nativ e ev alu atio n o n th e p relim in ary v ersio n o f th e C hin aclassified (reg io nal) m od el life tab les an d m ad e sev eral su gg estio ns. A fter th e m eetin g.im pro vemen ts w ere m ad e acco rd in g to th ese su gg estion s. O n D ecem ber 26 , 19 87 , th ec omp ila tio n c omm itte e o f th e mode l life ta ble s in vite d e xp erts f rom re le va nt u nits to m a k e afo rma l a pp ra isa l o f Ch in a c la ssif ie d (r eg io na l) mode l life ta ble s inBeijin g. E xp erts an dp ro fesso rs from th e A pp lied Math em atics In stitu te o f th e C hin ese A cad emy o f S cien ces, th eBeijin g Medic al Un iv ers ity . th e 7 10 I ns titu te o f A e ro na utic al a nd Space Ministry an d th e

- 3 -


6/46

S ta te E co nom ic In fo nn atio n Cen te r and o th er a ge nc ie s. 1 2 units altogether . participated inth e meetin g. S eve n exp erts and profe sso rs w ere in vite d to a ct a s a n a pp ra is al c ommitte eheaded by P rofessor S houren W ang. M embers of the appraisal comm ittee and relevantexperts directed questions w hich w ere answ ered by the w orking group T he appraisal com -m itte e th en met an d reac hed a un an imous con clu sion tha t the reliab ility of the original datas ele cte d f or e sta blish in g th e Chin a c la ss ifie d ( re gio na l) mode l life ta ble s is q uite h ig h. th et echnique appl ied in th e re se ar ch is r ig oro usly sc ie ntif ic . a nd is original compared to meth-ods used b y fo reig n c ou ntries. T he ap praisal cen ificate g iv en b y th e ap praisal comm ittee isincluded in the first part of the book.

In the p ro cess of d eve lop ing a nd comp ilin g o f th e C hin a classified (reg ion al) mod ell ife tab les , we enjoyed ene rg etic suppo rt a nd a ss is ta nc e from popula tio n c en su s o ff ic es .electronic computer c en te rs ( sta tio ns ) and vital r eg is tr atio n agenc ie s o f v ar io us p rovin ce s.a uto nomou s r eg ions and munic ip alitie s. th e Publ ic Hea lth Ministry an d T u m o r ResearchIn stitute . etc. T he model life ta bles comp ilatio n c omm itte e an d de velo pmen t w ork ing gro upw ish to co nvey to these officials ou r sin cere th ank s.

Inthe sh ort h istory o f co llectio n o f mortality d ata in Chi na. there h a s been r ap idc han ge in th e soc io-econ om ic co nd itio ns of our c ou ntry . Chin a c la ss ifie d ( re gio na l) mode llif e ta ble s may. th er efor e. be somewha t inadequat e t o r ef lect some prosperties of themOrtality o f th e Chin ese p op ula tio n. Use rs are welcome to offer criticisms. comments ands ug ge stio ns . A s mo re mo rta lity data b ec ome a va ila ble . w e in te nd to fwth er r ev is e andimp ro ve th es e mode l lif e ta ble s.

Li, ChengruiDirector of C h i n a c la s s if ie d (regional)model l ife t ab le compi la ti on committee

A ugust 15. 1988

- 4 -


7/46

The List of Numbers of Compilation Committee ofChina Classified (Regional) Model Life Tables

Chairman: Li,ChengruiVice Chairmen; Sun Jingxin, Ma An, Wang Yingluo, Yu Wang, Zhou Qifeng, Ding DijiMembers: Jiang Zhenghua, Xiao Zhenyu, Wang Weizhi, Fan Murong, Zha Ruichuan,

Dong Qing, Sun Yiren, Shen Yiming, Mei Xiangfu, Wang Shouxiang

Compilation and Working Group of China Classified(Regional) Model Life Tables

Group Leader: Jiang Zhenghua,Vice Leaders: Wang Shouxiang, Luo JiyuMembers: LiNan. Xing Ying, Su Xiankang, Luo Changrong, Shan Ming, Li Xiru, Qi

Xing,LuiZonghua

- 5 -


8/46

Certificate of Appraisal for theClassified (Regional) Model Life Tables

Research and Established by:The Leading Group Office of the Third Population Census under the State Council

of China.The Population Research Institute of Xi' an Jiaotong University.The Computer Center of Sichuan Provincial Planning Commission and Economy

Commission.

The Appraisal Committee was organized by:The State Statistical Bureau of China. Date of Appraisal; December 26,1987.

A. Briefintroduction to the research on the classified (regional) model life tables of China.a. The classified (regional) life tables of China include five families of sex-specific

life tables basically classified by regions. The corresponding regions for these familieswere given as follows:

The first family: South West China, the second family: Middle and East China, thethird family: North China, the fourth family: Nonh East China, the fifth family: Xinjiangautonomous region.

The outputs of the research provided to the appraisal committee were: printedtables, data on magnetic disk, technical a nd users' manual, related applications software.

b. The raw materials used for developing the classified (regional) model life tablesare 3136 sex-specific life tables(l) compiled from the 1982 population census by censusoffices and computer centers of different provinces. autonomous regions and municipal-ities. Raw materials also include the sample survey data collected since 1958 and 50 life

- 6 -


9/46

tables(2) compiled from data of the nation-wide cancer survey conducted inthe period from1973 to 1975.

c. The research that produced the model life tables was divided into three stages:cluster analysis; model building and parameter identification; establishment of the model lifetables. The cluster analysis was used to determine the geographical classification of thepattern of mortality in China by identifying the mortality patterns of the origina1life tablesand to give the changes by age-specific rates corresponding to the different levels of mortal-ity. The corresponding technique for the formal problem is cluster and model life tables --to form the centers for different mortality families. The well-known Legit transform wasapplied primarily for clustering. The parameters a. and J 3 in Logit transform represent thelevel and pattern ofmonality separately. Taking L(1J as the Logit transform oflife table

> I < > I I


10/46

* 1\where C J " and are the param eters of the standard life table a nd that of t he id en tif ie d li feqltab le. T he mod el h as a n ex cellen t c ha ra cter o f e qu al firs t three momen ts f or th e id en tif ie dan d original life ta ble s. Mo re ov er, th e p ro ce du re to c alc ula te p ar ame te rs C o , CI, and C2. iss imp ly to s olv e lin ea r e qu atio ns .

F ittin g smoo th in g c urv es to id en tif ie d p aramete rs C o . C}. a nd C2 . th e p ar ame te rsc orre sp on din g to in teg er liv e ex pe ctan cy a t b irth o f 40 to 75 years. A pplying the modelmen tio ned ab ov e a gain . th e mod el life ta bles c ou ld be easily estab li shed .

d. T he research project began in June 1984 a nd ended in S eptem ber 1986. T heproject w as supported and ' fm anced by the O ffice of the Third Pop ula tio n Cen su s L e ad in gG roup under the S tate C ouncil of C hina. T he P opulation R esearch Institute of Xi' anJiao tong Univers ity a nd the C omputer C enter of the P lanning and E conomy C omm issiono f S ich ua n P ro vin ce h av e d on e th e re sea rch w ork jo in tly .

e. T he R esearch of C hina C lassified (R egional) M odel Life T ables w as a large-sc ale ap plic atio n o f th e 1 98 2 C hin a p op ulatio n c en su s d a t a . N one of th e fiv e mo del lifeta ble s pub lis hed p revious ly made u se o f C hin ese data in th e re sea rc h p ro ce ss. H ow ev er,China is a h ug e c ou ntr y w ith g re at d if fe re nc es in s ub -re gio na l g eo gr ap hic al, s oc io -economic cond it ions. It is essen ti al t o es tab li sh China c la ss if ie d ( regiona l) model lif e ta ble sus ing Ch inese data. The mod el co uld be ' applied indemograph ica l analys is , popu la ti onp ro je ctio n and p lann ing in d if fe re nt s ub -r eg io ns , T he r es ea rc h a ls o e na ble d d if fe re nt pop-u la tio n s ur ve y d ata to serve soc io -e conomic con st ru ctio n o f Ch in a inw id er a nd deeperways.

N ote: (1) C ollected by the O ffice of the T hird P opulation C ensus under the S tate C ouncilof C hina and the C om puter C enter of the Planning and E conom y C omm ission of S ichuanProvince.

(2) Compiled by th e C an cer R es ea rc h In stitu te o f th e Me dic al A cad emy o f China.- 8 -


11/46

B. The Minutes of the Appraisal Committee of "China Classified (Regional) Model LifeTables"

Referring to the compiling Committee of China Classified (Regional) Model LifeTables, the Appraisal Committee had tested the research output of the project "ChinaClassified (Regional) Model Life Tables" in Beijing on 26 December 1987 and accepted thesummary report of the working group. The members of the Appraisal Committee andrelated specialists raised points for argument and discussion, and reached the followingconclusions:

The research on the relationship between the levels and patterns of mortality isclosely related to work in many fields. Although research in this aspect has been done inother countries several times, none of these made use of original Chinese data. This leadsto a relatively high error in China's population projection and data analysis by applyingindirect methods to determine the effects of the change inmottality level on the populationage and sex structure and other related socio-economic problems. This is the firstopportunity to establish the China Classified (Regional) Model Life Tables successfully.The research filled an important gap in China population studies. The outputs of theresearch h a s supplied a reliable basis for the research in the fields of demography,education and health care. and will attract the attention of population scholars everywhere.

The original data used in this research are of high reliability due to the strictselection procedure. Thus. the compiled life tables can be usefully applied. The censusdata of one billion population a nd other historical materials were tested. As a result. 963original life tables were accepted for each sex. A lot of machine time was spent andresearchers from different departments and academic fields cooperated to analyze the dataand compute the outputs. The large scale of the research was rare in the world. Themethods applied incompiling model life tables were strictly scientific. Comparing it tosimilar work. abroad. the research has a unique character of its own and is of excellentquality.

- 9 -


12/46

T he research m ain ly u sed th e 1 98 2 p op ulatio n cen su s data. The cormni tteesu gg ests th at a rev isio n b e ma d e when more new data become available inth e future so thatth e re su lts c ou ld be applicable to new s itua tions. It is also nec es sa ry to pub li sh relatedcomp uter so ftw are as so on a s possib le s o that ap plicatio n o f the m eth od s can b ecomewidespread.

The Appraisal Committee o f "Ch ina C la ss if ie d(R eg io na l) Mode l L ife T ab le s"

26 D ecem ber 1987

c. T he co nclu sio n o f th e S tate S tatistical B ureau o f C hin a ag rees with th e co nclu sio n o f"Ch in a C la ss ifie d (R eg io nal) Mode l L ife T ab le s" m a d e by t he Appr ais al Committ ee .

20 D ecem ber 1987

- 10 -


13/46

D . T he L ist and S ignatures of the M em bers of the A ppraisal C ommittee to "ChinaC la ss if ie d (Reg io na l) Model L if e Tab le s"

Name Working Unit Title, PostChairman Wang Shouren Applied Mathematics Research Fellow, Research Institute (Professor)o f Sc ienceVice Cha irman Yu Jingyuan 710 Institute of the Research Fellowt Aviation M inistry (Professor)Members LiTIan1in B eijing M edical UD iversity P rofessor,

Xu Shaoyu Population Research Assoc ia te Resea rch, Institute of B eijing FellowEco nomy Sch oo lShenQiuhua Popula tion Sta tist ica l D eputy H ead and, Dep artmen t o f th e S ta te Assoc ia te Profes so rSta tist ica l BureauLiHuijing Economy Rese ar ch In sti tu te Associat e Research, o f th e S ta te P la nn in g FellowCommissionZhang Dayang Sta te Economy Informat ion Senior Eng ineer, Center

- 11 -


14/46

E. The List of Participants of the Appraisal Meeting of "China Classified (Regional) ModelLife Tables"Name Working Unit Title, Post

State Statistical Bureau Fonnal Directorand ProfessorLi Chengrui,Jiang Zhenghua

J

Shen Yiming,Population Research Institute of Director and ProfessorXi' an Jiaotong UniversityPopulation Statistical Department Directorof the State Statistical BureauApplied Mathematics Research Research FellowInstituteWang Shouren,

YuJingyuan, 710 Institute of the Aviation Research FellowMinistryUTlanlin, Beijing Medical University Professor

Population Research Institute Associate Researchof Beijing Economy School FellowState Economy Information Center Associate ResearchFellow

Xu Shaoyu,Wang Sulin,Xiao Zhengyu,WangWeizhi,

China Population Infonnation Center Senior StatisticianPopulation.Research Institute ofThe Social Science Academyof China

Associate ResearchFellow

Shen,Qiuhua

Population Research Institute of Professorthe Chinese People' s UniversityPopulation Statistical Department Deputy Director andof the State Statistical Bureau Associate Professor

Zha Ruichuan,

LiHuijing, . Economy Institute of the State Associate ResearchPlanning Commission FellowEconomy Information Center Associate Researchof Sichuan Province FellowXingYing,

SunDaqi, State Economy Information Center Deputy Director of aDivisionWang Guoyi

)Population Research Institute of LecturerBeijing University

- 12 -


15/46

Name Working Unit Title, PostYao Keqin Planning a nd S ta tis tic al Dep arttn en t C hie f Do cto r, o f th e H ealth M in istry o f C hin aWang Renan Beijing M edical University Assoc ia te Profes so r,LiNan Population Research Institute o f Ph.D . S tu de nt, Xi 'a n J ia ot ong Un iv er si tyW ang Shouxiang Population Statistical Department Deputy Di rec to r, o f t he S ta te S ta ti st ic al Bu re au o f a D iv isio nUBohua China Population Information Center,YaoFeng S ta te Economy Inf ormat io n Cent er,

- 13 -


16/46

The Tec hn ic al D es crip tio n o f th e E sta blis hment o f th eC la ss ifie d (R eg io na l) Mode l L ife T ab le s fo r Chin a

I. C luster Analysis on the O riginal L ife TablesI. T he P rinc ip les fo r C lu ste r A naly sis o n th e O rig in al L ife T ablesIt is g en er ally a cc ep te d th at th e le ve l a nd p atte rn o f mo rta lity is bas ic ally d et ermin ed by

th e b io lo gic al c on ditio n o f th e p op ula tio n liv in g in a certain area . H ow ev er, mortality is a ls os tr on gly a ff ec te d b y e xte rn al fa cto rs lik e s oc io -e co nom ic c on ditio n, h ea lth a nd medic al c are ,c ultu ra l b ac kg ro un d, e nv iro nment, e tc . A lo ng w ith in cr ea sin g s oc ia l d ev elo pment. a ge -specific death probabilities for all ages in an area w ill decline. F igure 1 gives the curves ofa ge -s pe cific d ea th p ro ba bility r ate s c orr es po nd in g to d if fe re nt s ta ge s o f d ev elo pment in a.population. T hey are a set of uncrossed curves shifting gradually from a J shape to a Usha pe . Ob vio us ly , th e c ur ve s whic h re pr es en t h ig he r lif e e xp ec ta nc y and low er mo rta lity a recloser to th e h or iz on ta l a xis a nd c or re sp on d to a h ig her le vel o f socia l d ev elo pmen t. T hecu rve s far ab ov e th e h oriz ontal ax is rep re sen t a low er life ex pec tan cy a nd h ig he r morta lity ,and c orresp on d to a low er stan dard of so cial de velo pmen t p ro ce ss. Ifa f am ily o f mo rta lityprobability curves given in figure 1 has sim ilar shapes and do not intersect one another, w esay that the fam ily of curves has the sam e pattern b ut d iff er en t le ve ls o f mo rta lity . D iff er en tpo pu la tio ns w ith g re at d iffe re nc es in so cio-ec onomic an d demo grap hic c on dition s may differ alo t in a ge -s pe cific d ea th p ro ba bilitie s. e ve n th ou gh th ey h av e th e s ame life e xp ec ta nc ie s a tb irth , In th is ca se, th e mortalities of the p op ulatio ns are sa id to be at the sam e level but toha ve d ifferen t pa ttern s. T he mortality p attern an d lev el c ou ld also be presented interms ofs ur viv al p ro ba bility . A f am ily o f s ur viv al p ro ba bility c urv es c or re sp on din g to th e deathp rob ability c urv es o f fig ure 1 is illu strate d in fig ure 2.

C hina is a populous country w ith many rac es. T he re are b ig d ifferen ce s inc ultu re a ndsocial cu stoms amon g d ifferen t race s an d re gio ns. T he mortality c on dition s may a lso d ifferf rom reg ion to r eg io n d ue to th e d iff er en ce s ingeographic al e nv ironmen t and economic

- 14 -


17/46

d ev elo pment. T o p op ula tio ns liv in g c lo se to ge th er, th e fa cto rs th at d ete rm in e mo rta lity a rem ore o r less the sam e, so th at the pattern s of m ortality in th ese region s shou ld be ab out th esame . T o p op ula tio ns liv in g in re gio ns far a pa rt, th e fa cto rs d ete rm in in g morta lity m ay d iffera lo t, so th at the differences in the p attern of m ortality m ay also b e great T his suggests that itsh ou ld b e p ossib le to d iv id e C hin a's p op ula tio n in to se ve ra l g ro up s a cc ord in g to th e p atte rn o fm orta lity in g eo gra ph ica l a rea s. Sm all p op ula tio ns w ith in a la rg er a re a m ay h av e a p atte rn o fmo rta lity wh ic h d iff er s from th at o f th e ma jo rity o f th e s urro un din g p op ula tio n. In t hi s c as e,th ese sm all g ro up s w ill be c lassifie d a s b elo ng in g to o th er familie s. T he p urp ose o f d oin gc lu ste r a na ly sis o n th e o rig in al life ta ble s wa s to i dent if y s ev er al d if fe rent f am i li es o f mo rt al it yp atte rn from th e m ix ed o rig in al ta ble s. E ac h family o f life ta ble s re pre se nte d a p atte rn o fm orta lity ty pica l to a c erta in re gio n o f C hin a. T he re su lts o f th e c lu ster a na ly sis c ou ld b ea pp lie d to d ec isio n a na ly sis a nd re se arc h in so cio -e co nom ic a nd o th er fie ld s.

2. The C luster Analysis M ethodBo th th e s urv iv al p ro ba bility ra te s a nd age -s pe ci fi c d ea th p robabi li ty r at es cou ld be

u se d in c lu ste r an aly sis o f th e o rig in al life ta ble s. T he cla ssific atio n p ro ce du re w as dividedin to tw o sta ge s a nd te ste d b y g ra ph ic al a nd c omp uta tio na l m eth od s. T he L og it tra nsfo nn a-tio n w as a pp lie d fo r p re lim in ary c la ssific atio n. A cc ord in g to W . B ra ss, if two life ta ble sa re e xp re sse d b y ag e-sp ec ific su rv iv al p ro ba bility ra te s (l ], 12. 1 3, .... In ) an d ( l ; . 1;,

. . . . . .13, ... 1 n )' th e fo llow in g re la tio nsh ip c an be used:. . .Logi t (Ii) =a+ ~. Logit (1. )1 (1)i-I, 2, 3, ... n

where L og it is a fun ctio n defm ed b yL ogit (x) = 0.5 In IX .-x

Here a and ~ are constants w hich can be fo un d b y th e fo llow in g p ro ce du re :

- 15 -


18/46

According to formula (1), w e h av e ' "ogit (11) =a+ ~.Logit (11)*ogit ( 1 v = a+ ~.Logit (12)

' "egit (In) = e x + ~.Logit (1 )n

B y the minimum least e qu al erro r m eth od , w e h av e

n *L [Logit (1. ) - A] [Logit (Ij) - B]i=I 1~ = - - - - - - - - - - - - - - - - - - - - -n *L [Logit (1. ) - A]2i=1 1

and

where

n ,..L [Logit (1.ni=1 1A=-----nnL [Legit (li)]i=l, B = n .

, . .Given n points [logit (I, ), logit ( I i > ] (i - I, 2, 3... n), ~ is the slope of th e linefound by one-dimensional linear regression, i.e. tan 9 =~and a is the intersection of the lineon th e vertical axis (see Figure 3),

- 16 -


19/46

Figure 1 Age specific death probability curves

1x1

o Age_ ~~~~-- xFigure 2 Age specific survival ratio curves

- 17 -

Agex


20/46

.:/./~':. . .j".,. " ./ Q . ; . . :o

. .)/.

F ig ure 3 . G ra ph ic al illu stra tio n o f th e p aramete r id en tific atio n

Inthis calculat ion, a reflects the difference in m ortality ~ of two life tables and ~th e differe nc e in p attern o f th e life ta ble s. Inp articu lar, if tw o life tab les are e xa ctly the same,then w e w ill have a= 0 and ~ :::::1. T aking the legit transform of survival probabilities of aset o f life tab les w ith th e same le vel o f life ex pec tan cy . as and ~s can be id en tif ie d w ith th oseof a standard life table. T he differences in the pattern of m ortality am ong these life tables canbe recognized by com paring the values of~. A life table w ith higher infant m ortality andlow er old age m ortality w ill produce a sm aller ~ than a life table w ith low er infant m ortalityand higher old age m ortality. S im ilar patterns of m ortality for tw o life tables w ill result in ~sthat are close together. T herefore, the param eter ~ can be used as a crude classification on thepattern of m ortality for a set of life tables w ith the sam e level of m ortality. S ince only tw op arameters are u sed to e xpress w hat is in re ality a n on -lin ea r mortality cu rv e, th e classificationm entioned above is only a prelim inary one.

M ore accurate classification w as done by applying the K -m ean m ethod w ith thein fo rma tio n f rom th e p re lim in ary c la ss if ic atio n. The tra ditio na l K -me an me th od was modif ie d

- 18 -


21/46

by the authors to minimize the uncertainty of the classification. This modified K-meanmethod was named the Qprimized K-mean method.

Let two vectors of n dimensions beA = (aj, a2... an)

andB = (b], b2..... bn) .

Define the distance between the two vectors as

It is obvious that the smaller this distance between two vectors is. the closer the two points Aa nd B are. In mathematical language, the original life tables are points disttibuted in an n-dimension space. The optimized K-mean method was applied to find several curves in thespace which could collect all the points around them with the minimum distance. Thesecurves form typical model life tables of several families. Ifa set of a-dimensional vectors canbe clustered into K classes, they form K demographic groups. Based on the preliminaryclassification, dividing the sum of elements with the same level of mortality in a group by thenumber of the vectors in the corresponding group, new vectors could be formed for eachgroup of original life tables. These vectors should be the centers of different groups.Comparing the distances from each vector to different center vectors. the grouping of thevectors is then adjusted by classifying them within the class with minimum distance betweenthe center and observed vectors. The new classification could serve as a reformed originalclass in the next classification procedure. The procedure mentioned above is repeated untileach vector is placed in the class with minimum distance between that vector and its owncenter. A stable K families of original life tables were considered to be satisfied. This thendefines a stable K-family of original life tables.

- 19 -


22/46

3. Data Processing and Cluster AnalysisA fte r r elia bility te stin g, 1 93 6 o rig in al lif e ta ble s fo r ma le s a nd f ema le s w er e a cc ep te d.

Among these life tables, the low est and highest life expectancy at birth are 4 3 and 73 years form ales and 4 3 and 78 y ears for fem ales. S ince the m ortality pattern of X injiang populationw as very different from the others, the original data from this region w ere taken as a separatefamily to d eal w ith . T he re st o f th e life tab le s rep res en te d th e mortality p atte rn s o f a ll o th erp ro vin ce s, a uto nomou s r eg io ns a nd munic ip alitie s e xc ep t X in jia ng and Tibet,

It is q uite p oss ib le th at life tab les w ith d iffe ren t p attern s b ut th e s ame le vels m ay beclu ste re d in a g ro up ac co rd in g to lev el rath er th an a cc ord in g to pattern. T o rem ove this prob-lem , th e sp ac e o f life ta ble s w as d iv id ed in to se gmen ts b y in teg ra l d ig its o f life ex pe ctan cie s,i.e., the original life tables in each segm ent have the sam e rounded value of life expectancy atb in h. T hu s c lu ster a naly sis co uld be d on e amo ng life ta bles w ith th e same lev el o f mortality .A ccording to th e population census of the year 1982, the life expectancy at birth w as 6 6.4years for m ales, and 6 9.4 years for fem ales in the year 1981. A s expected.. the origin al lifetab les from th e th ird c en su s d ata w ere mo re co nc en tra te d in five segm ents of 6 S years to 69years for m ales and 68 years to 72 years for fem ales. E ach segm ent contains original lifeta ble s c om in g from almost all p ro vin ce s a nd au to nomo us reg io ns. A ll o f th e d iffere nt p atte rn so f morta lity in C hin a c ou ld b e id en tifie d from e ach segment.

A p ply in g th e Lo git tr an sfo rm me th od mentio ne d a bo ve . p re lim in ar y c la ss if ic atio n o nthe pattern of m ortality of the original life tables could be do ne by identifying the values of ~o f d iffe ren t life ta ble s w ith a s tan dard life tab le. T he d eta ils fo llow.

Cons id er in g age -spe ci fi c surviv al p robabilit ie s o f e ach original life ta ble as a v ecto r,sum u p th e c orre sp on din g e leme nts o f a ll o rig in al life ta bles ina segm ent and divide by thenum ber of life tables in this segm ent. A n abridged life table could be established in this wayas a standard. A ll the original life tables could be represented by the standard life table withtw o p arame te rs o f a an d B , It w as found that am ong life tables w ith life expectancy of 6 5 to69 years for m ales and 68 to 72 years for fem ales, four groups could be identified inevery

- 20 -


23/46

segment according to the values of~. The great majority of the original life tables in a groupcame from the same region inChina. Different groups of life tables correspond to differentregions. This means that the mortality patterns of China could be classified into four familiesexcept for Xinjiang. The geographical classification for the mortality patterns of males andfemales was perfectly consistent.

The preliminary classification is based on the values of ~s only. Obviously, it is notaccurate to describe the characteristic difference of whole life tables with one parameter. Theabove procedure gave a preliminary classification and determined the number of families asK=4. The optimizing K-mean method was applied to the preliminary classification results,after adjusting and testing the validity of regional classification, to produce the finalclassification of original life tables. The vast majority of the original life tables belong to fourseparate regions. The four regions are: South-West China. Middle and East China. Northand North-West China, and the North-East China.

After identifying the classification of mortality pattern a nd corresponding regions, thecenter vector of each family of life tables serves as a model of the mortality pattern of thefamily. This procedure was applied to both males and females life tables. Because few of theoriginal life tables had very low or very high life expectancies. whole families could not befully covered. Thus, the characteristics of the original life tables had to be tested before theycould be classified into a certain family.

The socio-economic and geographical conditions of different sub-regions in a provinceor autonomous region may differ so that life tables may differ in many ways even amongthose with the same life expectancy and some of these differences may be large. It wasnecessary to remove the life tables with clearly different characters from its own family bydirect observation, graphical comparison and minimizing the distance between the life tables tothe center of the family. All the removed life tables were treated separately. The rest of thelife tables of each family in the segment were classified again by K-mean method. Th e

- 21 -


24/46

procedure of removing and classifying was repeated until all the life tables in each family wereconsistent with the geographical locality.

4. ResultsAfter identifying the center vectors of different families in each segment, the center

vectors of the same region in different segments formed a family. Ranking the life tables bylife expectancy, four families of model life tables were found Each family of model lifetables represented a typical mortality pattern in China After removing life tables with largevariations. the original life tables which were accepted in order to create the regional model lifetables of China included 963 for males and females separately. Among them, 358 original lifetables were used to create the first family of model life tables, 244 for the second family, 207for the third family, and 154 original life tables for the fourth family of model life tables.Xinjiang autonomous region has a special pattern of mortality. A special family of model lifetables was established separately on the basis of original life tables in this region. Thus, fivefamilies of model life tables were found for the Chinese population by cluster analysis.

- 22 -


25/46

n . T he M athem atical M odel and G eneration of M odel L ife T ables1 . Establishm ent of the model and identification of param eters:

W e develop here a m ore accurate and easily calculated m ortality m odel and applythe m odel to construct the classified (regional) m odel life tables of C hina.

B y cluster analysis. the m ortality patterns in C hina w ere classified into five fam iliesaccording to the resem blance of m ortality patterns across geographical areas. T hegeographical areas corresponding to each category are given below :

The first pattern: South-west ChinaThe second pattern: M iddle and east ChinaThe third pattern: North ChinaThe fourth pattern: North-east ChinaThe fifth pattern: X injiang autonomous region

Each of the five patterns contains a set of m odel life tables for life expectancy atbirth betw een 40 years and 75 years w ith one-year increm ents.

L et tl. t2 ..... tm be m observed life tables w ith the same pattern. and TI 2. "',Tm be m prelim inary standard life tables. each Ti (i = 1,2. '" m) is th e a rithmetica l a ve ra geof the observed life tables whose differences in life expectancies is less than one year. .C hoose a representative life table w ith good inheritence Tk (1 S k S m ) as the standard lifetable, the m athem atical m odel of m onality pattern w e construct is

n


26/46

The c ha rac ter o f th e mo del w ill b e illu strated in th e s ec tio n o n e stim atio n o fparameters.

O bv io usly , w he n a c erta in s eries o f p arame te rs n o , aI' ..... an are given. a m odelta ble w ith s pe cif ie d lif e e xp ec ta nc y e o can be u niq ue ly o bta in ed f rom equ atio n (2 .1 ). If th ev ariation trend o f param eter ai (i = 0, ... N ) is know n. a m odel table w ith arbitrary lifeexpectancy e o can easily b e o btain ed . T hen a fam ily of m od el life tab les can be generated.So the first step is to identify the p aram eters ai (i = 0, ... n) of each Ti (i = 1 .m ) a cc or din gto e qu atio n (2 .1 ).

T he values of a(t), a(t).a(t) w ith resp ect to Tl can be o bta in ed b y so lv in g th eo 1 n

fo ll owing equa tion .r min L( q(t). q)2 / ( q(k) t N)x n x n x n x x (2.2)

T ak e th e p ar tia l d if fe re ntia l w ith r es pe ct to a .( t) in ob je ctiv e fun ct io n ( 2.2 ).1

Note: g - ~ t (i+j-I)-N q(k),ij - ~ x n x i=O-N, j=I-N+lb. = L ti-I.N q(l)1 x x n xG = (gij)(N+I) x (N + 1)B = (bi)(N + I) x 1A - t) (t) (tT- a ,a , .... a (N + I) x 1o 1 N

- 24 -


27/46

Then

A=G-l. BObviously. the parameters aCt) can be derived from this set oflinear equations.1Therefore, this model is simply to calculate.

Inaddition, we can direct!y get:L ( q(t~ (l)h= L (nq . (l)h,x Ox tx' X x (xl i= 0, I, 2..... N

Where,n__= (a(t) + a(t)t + ... + aCt) tN) q(k) is the estimated value of a~l).n-ix a 1 x N x n x n '"

It means that the model has equal Nth order moments of n~t) and o'lx'Now we discuss the selection of standard life table Tk and the weighted coefficient

tx _(1) For reducing error in parameter identification, the standard life table Tk must

represent the general pattern of mortality in its category.(2) As the weighting coefficients, a reasonable value of t x can make the estimation

errors quite uniform for all ages. Increasing the value of txo could make the estimation ofn~'L) more accurate.

The explanation mentioned above is only a brief description of the generalcharacters of this model. As an example, in equation (2.1), choose N=2 as our model ofgenerating model life tables of China in five year age group. Thus, it is a three-parametermodel:

- 25 -


28/46

Wheret o = tl = 30t2 = t3 = 10tk = 1 (k = 4, ... 17)

T he p aram eters can be obta in ed b y s olv in g th e fo llow in g p ro blem:

R eg ard in g th e accu racy o f o ur m od el. th e fo llowin g ex am ple g iv es a comp ariso nbetw een the C oale and D em eny m odel and this one. Table 2.1 and Table 2.2 present errorsof fitting tw o m odels to the observed data, w here C -D m ethod denotes the C oale andD em eny m odel and J-L m od el (Jiang-L i) is the one presented in th is b oo k.

T ab le 2 .1 F ittin g E rro rs o f F ittin g fo r P attern 4 , m alese = max (qj _~(l))/~(l)

The p r el im inary T he m ax im um o pp osite The prelim inary The maximum opposites tandard l ife s:rm[St; standard life erroa ET ab le T i CoD model I J-L model Table T i CoD model I J-L m od el

Tl -0.5911 0.5698 T6 0.3514 0.0976T 2 0.4707 -0.2152 1'7 0.3953 0.0000T3 0.2838 0.1746 T8 0.2139 0.2189T4 -0.3441 0.2528 T9 0.1879 0.1799T5 0.1891 -0.2135

- 26 -


29/46

Table 2.2 .Fitting Errors of qo for Pattern 4, males

The pre liminary E o The p re lim ina ry E os tanda rd l if e s tanda rd l if eta ble T i CoD model I J-L model ta ble T i CoD Model I J-L M od elT1 ~.2989 0.3342 T6 0.1952 ~.00021'2 0.1456 0.0621 1 '7 0.2056 0.000013 0.0837 0.0514 T8 0.2139 0.0434T4 -0.3441 0.2528 T9 0.0854 -0.0544T5 0.1891 0.1144

Among the center life tables which were available at the beginning, the fitness.of the fourthfamily of males was the worst among the five families. Various types of data were used tocompare the identification accuracy of our model with that of other models. The results for thecomparison between the worst fourth family of our model for males and the Coale-Demeny modelwas taken as an example The maximum relative error is given in table 2.1 and is the maximum ofthe relative errors in fitting age-specific death probabilities to the original life tables. For instance,the maximum relative error is 0.284 by using the Coale-Demeny model for the third original lifetable of the fourth family for males, and 0.175 by using our new model. Among the age-specificdeath probabilities, the infant mortality rate is very important, and higher identification accuracy isrequired in estimating IMR. Thus the absolute value of relative error between infant mortality ratesfor the original fife table and the estimated life table should be smaller. The relative errors of fittingthe life tables in the fourth family are given in table 2.2. Taking the third original life table as anexample, the relative fitting error of the infant mortality rate is 0.0837 by using the Coale-Demenymodel and 0.0514 by using our new model. Tables 2.1 and 2.2 show that among nine original lifetables of the fourth family for males, the maximum relative fitting errors of two tables is inferior byu sin g o ur mod el th an by using Coale-Demeny, and fo r ano th er ta ble , our model is a little in fe rio r tothe Coale-Demeny model. Only for one table. the fitness of infant mortality rate by using our new

- 27


30/46

model is inferior to that by using the Coale-Demeny model. So that we may conclude that ourm odel has quite high accuracy . T his new m athem atical m odel has the characteristics of clearm eaning . h igh accu racy and easy calculation .

2. The Establishm ent of the M odel L ife TablesU sing original center life tab les T 1. T 2 ... T m- a standard tab le T k could be chosen to

identify the param eters of m group . The trend of the changes in three param eters, a o . al . and a2could be found . F rom equation sqx = ( a o + altx + a2tx2) s~k). once the standard table Tk andw eighting coefficient tx w ere determ ined, the death probabilities w ere fu lly determ ined by a o , a} ,an d a2 ' H igher values of a o . a 1. an d a2 will lead to a low er life expectancy of the correspondinglife table. W e call this the property of m onotonicity. The process of establishing life tab les isreally the p rocess of searching p aram eters a o . al. and a2' The changes of a o . al . and a2 reflect thecharacters of different fam ily center life tab les. The effect of changing param eters could be foundby fitting a curve to the param eters. But t he i de nt if ie d p ar ame te rs do not necessarily have am onotonic relationsh ip w ith the change of life expectancy due to the lim itation of the num ber ofcenter life tables and the consistency of the patterns. In particu lar, w e have very lim ited num bersof life tables w ith very h igh and low life expectancies. Therefore, special m ethods a re needed todeterm ine the trace of the param eter curve for life tab les w ith high and low life expectancy.

In light of th e characters of param eters a o . a 1. and a2 for the first fam ily m odel which wasestab lished on a large am ount of original data, the logistic curve w as found to be g oo d fo r fittin gthe observed relationsh ip betw een param eter and life expectancy . T he logist curve is described asfollows: kY = 1 + ea+ bx + e K>O (2.3)

- 28 -


31/46

I t h as th e follow ing 2 cha ra ct er is tic s:(1) The u pp er lim it is k + a t the low er lim it is a .(2) It will be mono to nic ally in cre as in g w he n b > 0 a nd mono to nic ally d ec re as in g wh en

b 0 a re ad ju s tab le pa ramete rs.

Obviously:K>ao-9

T ak e lo ga rithm s o f (2.4) ka + ben = In ( -1 ) = f (ac).aO - 9 (2.5)so that for the m orig inal center tables, w e have

mmin F (a.b) = min I(aa + beoi - f (aai2i=l ( 2 . 6 )

- 29 -


32/46

T ake partial differentials of F (a.b) for a and b and set them equal to z er o to o bta in :

m -1 m( ~ ) = N :Leoi L !(aoi)i=l i=l (2.7)Leo mi=l Ie2 . m01 L !(aoi) eoii=1 '=1T he d eterm in atio n o f p aram eters a an d b w ill lead to a cen ain fo nn o f relatio nship

between a o and e~ . T his form is d ecided by th e ch aracteristics o f th e clu sters o f o rig in al lifetables.

S im ilarly . th e relatio n cu rves o f param eters al an d a2 to e~ can b e fitted.F or d ev elo pin g m o del life tab le s w ith low life e xp ec tan cy at b irth . th e rela tio ns

am on g in fan t. ch ild an d y ou th m ortality m ay n eed to be adjusted to keep the c on si st en t andev en ch an ge o f d eath p ro babilities in o ne fam ily of m odel. T his m ay be r ea liz ed b y a dju st-i ng we igh ti ng coe ff ic ie nt s t o . t1 0t2 . t3 . an d 1 4 ap pro priately , amo ng which th e ra tio q ()"q i>for the life table of e~ = 40 . i s c al cu la te d thr ough r eg re ss ion ext ra po la ti on .

N ot o ny th e ap pro priate adju stm en t o f th e m odel fo r e~ = 40 is needed but also th ew eig hted coefficien ts to , .... 1 4o f th e m od el tab les w ith e~ u nder that o f th e stan da rd ta blem ust b e ad ju sted ac co rd in g to th e p rin cip le o f ev en c ha ng e. T he w eig htin g c oe fficien ts ofthe stan dard tab le an d tab les w ith e~ g reater th an that o f th e stan dard tab le are kept asco nstan ts. T he a dju stm en t fo rm ula fo r th e w eig hted . co effic ien t is as fo llows:

- 30 -


33/46

i= 0, ... 4 (2.8)

where ~ is the adjusted weighting coefficient, t~40)is the weighting coefficient for e~ = 40.ti is the unadjusted weighting coefficient, e~) and e~ are the life expectancies of the standard

life table a nd rnodellife tables to be generated respectively.Inthis way. a1136 mofel life tables with integral digit life expectancies e~ from 40 to

75 years and life tables can be developed. Any life table with non-integral digit life expect-ancy from 40 to 75 years could also be generated as needed. The range of life expectanciescan be extended if necessary.

Attachment: Symbol explanation of the model life tablesQx - the death probability of age x.Ix - number of persons alive among 10.000 live binhs at exact age x.Dx - number of deaths in age group x.

L x - number of person years in age group x.Tx - cumulative number of person years of age x and above.Ex -life expectancy at age x.

- 31 -


34/46

Appendix I. The Application of the China Classified (Regional) Model Life Tables

Since they combine levels and patterns of mortality, model life tables simplify manyaspects of population studies, and are widely used in population analysis. projection andother studies. They are used in a variety of ways and the details can best be illustrated inconcrete applications. some of which we now present All calculations illustrated beloware for quick estimation by hand. More accurate calculation can be done using theapplication software developed by the authors.I. Calculation of a corresponding life table with desired life expectancy at birth

Ifthe pattern of mortality and life expectancy at birth are known. the correspondinglife table can be calculated by using interpolation.

Example: Assume that the monality pattern of some region confonns to pattern Ifor females in model life tables of China. and the life expectancy at birth, e o . is 65.74years. Calculate the corresponding life table.

Step 1. Find two model tables in pattern I for females whose life expectancies atbirth are e~5= 65 and e~ = 66 years respectively. as shown in Tables 1 and 2. Denote theprobabilities of dying during (x, x+S) as 5q~5and 5q~. respectively. Let e : = 65.74.

- 32 -


35/46

Table 1

x Q x lx Ox Lx 'IX E 'x0 .04119 100000 4119 97240 6500000 65.001- .03332 95881 3195 377133 . 6402760 66.785- .00791 92686 734 461594 6025627 65.0110- .00449 91952 413 458728 5564033 60.5115- .00865 91539 791 455717 5105305 55.7720- .00983 90748 , 892 451509 4649588 51.2425- .01120 89856 1006 446764 4198079 46.7230 - .01289 88850 1145 441386 3751315 42.2235- .01669 87705 1464 434864 3309929 37.7440- .01981 86241 1708 426935 2875065 33.3445- .02847 84533 2406 416649 2448131 28.9650- .04225 82127 3470 401985 2031482 24.7455- .06253 78657 4919 380987 1629524 2.0.7260- .10657 73738 7858 349046 1248536 16.9365- .16206 65880 10676 302710' 899491 13.6570- .23102 55204 12753 244136 596781 10.8175- .33211 42451 14099 177007 352644 8.3l80- .44808 26352 12704 110001 175637 6.1985+ 1.00000 15648 15648 65636 65636 4.19.. .Table 2x Q x lx O x Lx 'I'x E 'x

0 .03749 100000 3749 97488 6600000 66.001- .03026 96251 .2914 379174 6502512 67.565- .00710 93337 663 465025 6123338 65.6010- '.00408 9']1574 37 9 462422 5658312 61.0615- .00777 92295 717 459685 5195890 56.3020- .00943 91578 863 455733 4736205 51.7225- .01075 90715 976 451135 4280472 47.1930- .01237 89739 1110 445921 3829337 42.6735- .01602 68629 1420 439596 3383416 38.1740- .01901 67209 1656 431901 2943819 33.7645- .02733 65551 . 2338 421912 2511918 29.3650- .04056 63213 3375 407630 2090007 25.1255- .05975 79638 4770 387267 1682377 21.0760- .10186. 75066 7646 356227 1295110 17.2565- .15521 67422 1 0 4 6 4 310950 938883 13.9370- .22209 56958 12650 2 5 3 1 6 4 627933 11.~275- .32194 44306 14264 185678 374769 8.4680- .43711 30044 13133 117387 188691 6.2985+ 1.00000 16911 16911 71504 71504 4.2.3

- 33 -


36/46

.. .Step 2. Calculate sq x using the following formulas


37/46

Table 4.

x nqx x 5qx x 5Qx0 " 0.010478 25"': 0.002939 55- 0.0392161- 0.004042 30- 0.003517 60- 0.0684825- 0.002518 35- 0.005678 65- 0.11372510- 0.001426 40- 0.008174 70- 0.18040615- 0.002640 45- 0.011824 75- 0.29581720- 0.002836 5 0- 0.021314 80-84 0.436340

Step 1. Select the representative pattern from the model life tables of China. Com-pare the mortality pattern in the given population with each of those in the model tables formales of China. and determine which one of the five patterns provides the best representa-tion of the mortality pattern of the given population. This can be done as follows:

(a) Find model tables whose life expectancies at birth are all 72.15 years in the fivepatterns of model tables. The age-specific probabilities of dying for these are shown inTable 5.

- 35 -


38/46

Table 5.\ pattern

pattern III pattern IV Pattern Vge \ pattern I pattern IIgroup\ males males males males males0 0.01906 0.01546 0.01300 0.00988 0.033961- 0.01066 0.00785 0.00401 0.00374 0.014765- 0.00442 0.00406 0.00304 0.00258 0.0047010- 0.00258 0.00193 0.00195 0.00206 0.00146IS- 0.00308 0.00269 0.00336 0.00371 0.0027220- 0.00610 0.00391 0.00437 0.00482 0.0066925- 0.00605 0.00401 0.00465 0.00478 0.0059330- 0.00713 0.00501 0.00528 0.00582 0.0062235- 0.00963 0.00727 0.00739 0.00807 0.0073940- 0.01335 0.01144 0.01106 0.01213 0.0104745- 0.01879 0.01828 0.01822 0.02003 0.023225 0- 0.02746 0.03060 0.03083 0.03842 0.034005 5- 0.04525 0.05665 0.05295 0.06336 0.0464760- 0.07598 0.09211 0.06850 0.09741 0.0748965- 0.12376 0.13572 0.13372 0.13887 0.1091970- 0.16456 0.20046 0.21939 0.20546 0.1469775- 0.27149 0.29326 0.33166 0.305 50 0.1921580-84 0.38147 0.42093 0.51165 0.42420 0.23956

(b) Calculate the relative variation, D, between probabilities of dying (5


39/46

T able 6 .

x nqx x sqx x sqx

0 0.03372 25- 0.00630 5 5- 0.088111- 0.01673 30- 0.00787 6 0- 0.141445- 0.00747 35 - 0.01142 6 5 - 0.2040510- 0.00351 40- 0.01798 70- 0.2855715 - 0.00489 45 - 0.02873 75 - 0.3908520- 0.00615 50- 0.04809 80-84 0.52473

m . Popu la ti on Pro jec ti onMode l lif e ta ble s a re v er y impor ta nt in p op ula tio n p ro je ctio n. B ec au se th ey r ef le ct

c ha ng es in mo rta lity , s uita ble mode l ta ble s may make p op ula tio n p ro je ctio n mo re a cc ur ate .E xam ple: In T able 7 , the ag e structure of a m ale pop ulation in a certain regio n on

July 1, 1984 . is p resen ted. the probabilities of dy ing are the sam e as in T ab le 4 a nd t he l if ee xp ec tan cy a t b irth . e o. is 7 2.1 5 y ea rs. S up po se eo is in cre asin g inhalf -y ea r s te psannu ally . C alculate the age structure of th e given p opu lation o n July 1. 1989 . F orc on ve nie nc e, th e in flu en ce o f b irth a nd m ig ra tio n a re n eg le cte d h ere .

- 37 -


40/46

Table 7.

Age group p Age group PAge group P Age group Px x x xx x x x0-4 334241 25-29 471969 50-54 208555 75-79 407435-9 269848 30-34 397317 55-59 175050 80-84 1819610-14 331553 35-39 276621+ 60-r:>4 13038615-19 357281 40-44 196562 65-69 95594

20-24 447889 45-49 208190 70-74 73246

Step 1. Select the appropriate pattern in model life tables of China. According toinformation on death of the given population, a representative mortality pattern can be selected.It was shown in section II that the second pattern for males is the best representation.

Step 2. Find the model table with eo = 72.15 in the family of pattern II for males.Table S shows its age-specific probabilities of dying and Table 8 presents L x . the number ofperson-years lived in each age group.

Table 8.

Age group L Age group L Age group L Age group Ln x n x n x n xx x x x0-4 491235 25-29 481321 50-54 453427 75-79 2254415-9 487414 30-34 479152 55-59 433739 80-84 14741010-14 485954 35-39 476213 60-64 401702 85-89 9373515-19 484834 40-44 471762 65-69 35636420-24 483235 45-49 464760 70-74 291300

- 38 -


41/46

From the formula: Sex) = Lx+s/Lxthe survivorship ratios from age interval (x, x+5) to the next in 1984 can be obtained. Thesevalues are reported as s i o o in Table 9.

Table 9.

Age group S1(x) Age group Sl(x) Age group S1ex) Age group S 1 ex)x x x x

0-4 0.99222 25-29 0.99544 50-54 0.95658 15-19 0.653875-9 0.99700 30-34 0.99392 55-59 0.92614 80-84 0.63,58810-14 0.99770 35-39 0.99065 60-64 0.88714

15-19 0.99670 40-44 0.98516 65-69 0.8342620-24 0.99604 45-49 0.97562 70-74 0.75830

Step 3. The life expectancy at birth of the given population on July 1, 1989, would be72.15 + 2.5 = 74.65 in pattern II for males. The corresponding survivorship ratios can beobtained similarly. In Table 10 they are reported as S2(x).

Table 10.Age group S2(x) Age group S2(x) Age group S2(x) Age group S2(x)

x x x x0--4 0.99399 25-29 0.99651 50-54 0.96609 75-79 0.691335-9 0.99784 30-34 0.99525 55-59 0.94169 80-84 0.6799810-14 0.99834 35-39 0.99277 60-64 0.9097115-19 0.99752 40-44 0.98851 65-59 0.8634920-24 0.99693 45-49 0.96111 70-74 0.79276

- 39 -


42/46

Step 4 . Take S(x) = - i -Sl (x ) + S 2(x )) a s th e su rv iv orsh ip ra tio o f th e g iv enpopulation from (x, x+ 5) to the next age group in the period 1984 -1989. Then, the aged istrib utio n of the g iv en p op ulatio n on July 1 , 1 98 9, can be ca lcu lated by u sin g th efol lowing equat ion :

sP89 = sp84 S(x-5)x x x = 5, 10, 15, .... 85The results are show n in T able 11.

T able 1 1.

Age group P 89 Age group p 8 9 Age group P 8 9 Age group p 8 9x x x xx x x x'25-29 446315 50-54 203686 75-79 568055-9 331936 30-34 470069 55-59 200491 80-84 27404

10-14 269152 35-39 395166 60-64 16348215-19 330697 40-44 275140 65-69 11714220-24 356249 45-49 193974 70-74 61147IV . I nd ir ec t E st ima ti on

Many meth od s o f in dire ct e stim a tio n c an be b ase d o n mo de l life ta ble s. T he fo llowin gis a re ve rse -su rv iv al me th od and illu stra te s th e a pp lic atio n o f mode l life ta ble s in indirectestimation

- 40 -


43/46

Example: Suppose that: (1) in a certain region in 1984 there are N~ = 4038988 malesand N~ = 3916202 females. and their distributions by age group under 10years are given asfollows:

Age group OA 5-9334241311519

269848255814

(2) In 1974 there were 3522888 males and 3401807 females in the total population,thus the average annual growth rate for males is

rro In 403898810 In 3522888 = 0.01367

and for females

rw In 391620\01 n 3401807 = 0.01408.

(3) The mortality patterns of both sexes correspond to pattern II in model life tables ofChina, and the life expectancies at birth for males and females are 72.15 and 74.47 years,respectivel y.

Estimate crude birth rates and sex ratios at birth for the periods 1974-1979 and 1979-1984. The computational procedure is as follows:Step 1. Find a model table with life expectancy at birth of 72.15 years for males and anotherwith e o of 74.47 years for females both in pattern II of model life tables for China. Then thefollowing results can be obtained:

- 41 -


44/46

Age group o 1-4 5-998964 392271 487414

99205 394255 490977

Here n~ and nL; are the number of person-years lived in age interval (x, x+n) for males and

females in the life table.

m m mThen: 5Lo = = Lo + 4Ll = = 98964 + 392271 = = 491235w w wSLo = = Lo + 4Ll = 99203 + 394255 = 493458

Step 2. Calculate mid-period populations by sexes for 1974-1979 and 1979-1984. N r = = N ~ . erm (76.S-S4) = = 4038988 eO.01367(-7.5) = 3645411

N~ =N~. erm (S1.5-84) = 4038988 eO.01367(-2.5) = 3903288N 7 =N~. erw(76.5-84) = 3916202 eO.01408(-7.5) = = 3523738N~ = N~ erw(Sl .S-84) = 3916202 eO.0140S(-2.5) = 3780750

- 42 -


45/46

Step 3. Estimate average annual births by sex in the mid-points of 1974-1979and 1979-1984.

mm SNS 269848Bl = ----;; - 10000 = (487414) -100000 = 55363SLS

ww SN o 311519B2 = -;;. 10000 = (493458) -100000 = 63130SLo

- 43 -


46/46

Step 4. Estimate crude birth rates and sex ratios at birth in the period 1974-1979and 1979-1984. The results are respectively:for 1974-1979: sm + BW

1 1 55363+52103CBRI =-N-m-+-N-w- 1000% =3645411 + 3523738 -1000% = 14.99%1 1

sm -SRI = _1 100 = 106.26

BW1for 1979-1984:

Bm + BW2 2 68041 + 63130CBR2 = -N-m-+-N-w - 1000% = 3903288 + 3780750 1000% = 17.07%2 2

Bm2SR2 = - . 100 = 107.78.BW2

- 44 -

model life tables for china

Documents

th e data w

th e cen su s

th eq uality o f th

s io n

tables cam e

director o f th e popu

th e ch airm

s c hina