manual x chapter 4

8/6/2019 Manual X Chapter 4

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ESTIMATION OF ADULT SURVIVORSHIP PROBABILITIESFROM INFORMATION ON ORPHANHOOD AND WIDOWHOOD

A. BACKGROUNDF METHODS1. Nahm and ure ofindiwct informationon d t ortality

Chapter I11 describes how information on the survivalof close relatives-information from mothers about thesurvival of their childrek-can be used to make indirectestimates of child' mortality. The principle can beextended to information about the survival of otherclose relatives. This chapter describes procedures forobtaining estimates of adult mortality fkom informationconcerning the survival of parents and the survival ofspouses. In both cases, a particular target person isknown to have been alive at the time of some past event(the birth of the respondent in the case of mothers, theconception of the respondent in the case of fathers andmarriage in the case of.spouscs); and some informationis available about both the length of exposure to the riskof dying and about the age at which the exposure began.Sample surveys are the usual source of the necessarydata, though on occasion, the necessary questions havebeen included in censuses. The great advantage of thesemethods of mortality estimation is that they rely uponinformation gathered by questions that are simple andeasy to answer. The normal forms of the questions onorphanhood are "Is your father alive?" and "Is yourmother alive?", the possible answers being yes, no or notknown. For widowhood, all ever-married respondentsarc asked whether their first spouse is still alive, the pos-sible answers again being yes, no or not known. Itshould be mentioned that the term "orphanhood" isused in a rather unusual sense in this chapter, since it isspecific to a parent's sex. Thus, when describingmethods based on proportions with a surviving mother,an orphaned respondent is one whose mother has died,regardless of the father's survival.

Preston' points out that data on child survival col-lected from all women provide information about theoverall proportion of persons in a given population witha surviving mother. In a closed population, the numberof surviving children reported by all women, regardlessof age, should be equal to the number of people in thepopulation with a surviving mother. If no direct infor-mation on orphanhood is available, an estimate of thelevel of adult mortality can be made iteratively byfinding the level of mortality that, when combined with' Samuel H. &ston. "Estimating adult female m o d t y fromrepor&on number of c h i i n n surviving", University of Pennsylvania,Population StudiesCenter, 1980 (mimeographed).

the observed age distribution and age-specific fertilityrates, produces the same overall proportion orphaned asthe proportion calculated from child survival data. Ifdata on the survival of mother are available, the con-sistency of the reported number of respondents with asurviving mother and the total reported number of sur-viving children can be checked; a tendency to omit sur-viving children will reduce the latter figure, and atendency to report incorrectly a surviving mother 'willincrease the former, so typical errors in the two types ofdata should show up clearly.Information on the survival of parents or spouses is anindicator only of adult mortality, since the exposure torisk of the target person begins in adulthood, at thebirth, conception or marriage of the respondent. Thus,strictly speaking, such data should be used only to esti-mate survivorship probabilities from one adult age toanother; the first group of techniques presented in thischapter permits the estimation of such conditional sur-vivorship probabilities, and the methods in this groupare described as "conditional". However, if an estimateof the level of child mortality is available, in the form ofan estimate of 1(2), and some assumption can be madeabout the form of the relationship between child'andadult mortality in the population under study, the infor-

mation on child mortality can be combined with anindicator of adult mortality to estimate an unconditionalsurvivorship probability, that is, the probability of sur-viving from birth to some adult age. The second groupof methods presented in this chapter, described as"direct" methods, allows the estimation of such uncon-ditional survivorship probabilities. In choosing betweenthese two approaches, it should be remembered that theestimates of conditional survivorship are less dependentupon assumed models, but they are also more difiicult toincorporate into a full life table, than are the direct esti-mates, which, in turn, are determined, to a considerableextent, by the estimates of child mortality used in theircalculation.Estimates of adult mortality derived from informationon the survival of close relatives represent averages ofthe mortality experienced over the period during whichthe relatives were exposed to the risk of dying. If mor-tality has been changing in a regular way, each estimatehas a time reference; that is, there is a time before thesurvey with a period life table having the survivorshipprobability that has been estimated. The number ofyears, t , before the survey that define the period towhich this life table refers will depend mainly upon theaverage exposure to risk of the target persons and upon


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the average age range of their exposure; in general, thisnumber of years t will be somewhat less than half theavenge e x k u n to risk. It is possible to estimate thetime period to which the survivorship probabilitiesderived ftom a single survey refer if certain assumptionsabout the regularity of the prevalent mortality trend aremade. Brass and ~amgboye'propose such a method ofestimation. In addition , when data are available fromtwo surveys held five or 10 years apart, it is possible to' i m Bnss and E. A. Bamgboye, "The time location of reof nuvivolrbip: estimates for maternal and OOESthe .ever-widow+", W e g h p e r No. g ; - ? m t b % ! & h l ofHygcne a d r o p d Med~cme, en- for Population Studies 1981.

derive estimates refemng to the intersurvey period byreconstructing the experience of a hypothetical intersur-vey cohort.2. Oganizationo this chqterAll the estimation methods described in this chaptershare two characteristics in common: they allow theestimation of probabilities of survivorship to ages inadulthood (beyond age 20); and they use data on the

survival of close relatives (parents and spouses). Themethods presented can 'be classified into severalcategories according to whether they permit the estima-tion of direct o r conditional probabilities of survivorship

TABU85. H HE MA TIC GUIDETOCONTENTSOF CHMIER IV

B. E d ml io n of conditional B.2(b) Maternal orphanhoodd u l t avvivonbip data: Brass method

B.2(b) Paternal orpha nhoo d&ta: Brass method

B . ~ c ) Maternal orphanhooddata: regressionmethod

B.%d) Maternal orpha nhoo ddata from two surveys

sk.r/r*uhNumber of respondents withmother alive (or dead) classifiedby five-year age grou pTotal number of respondentsclasified by five-year age grou pNumber of respondents who didnot know or did not state thesurvivorship status of theirmothersNumber of births occurring in agiven year classified by five-yearage group of motherNumb er of respondents with fatheralive (or dead) classified byfive-year age gro upTotal number of respondentsclassified by five-year age grou pNumber of respondents who didnot know or did not declare thesurvivorship status of theirfathersNumber of births occurring in ayear classified by five-year age

group of father (or of mother. ifnothing elre is availabk)Number of mpondents withmother alive (or dea d) classifiedby five-year age grou pTotal number of respondentsclassified by Bve-year age groupNumber of respondents who didnot know or did not declare thesurvivonhip status of theirmotherNumber of births occurring in agiven year classified by five-yearage group of motherNumber of respondents withmother alive (or dea d) classifiedby five-year age group. for twosurveys five or 10 years apartTot.1 number d espondents byfive-year age group for the sametwo WNe)5Number of respondents who didnot declare the survival status oftheir mother for the same twoSUNCYSNumber of births occurring in agiven year (preferably duringthe intersurvey period) classifiedby five-year age gro up of mo ther

Probabilities of surviving from age25 to ages 35, 40, 45, ..., 85 forfemales

Probabilities of surviving from age32.5 to ages 45, 50. 55. ....90; orthe probabilities of survivingfrom age 37.5 to ages 50.55.60,.... 95 (depending upon thevalue of M, he mean age offathers at the birth of theirchildre n) for males

Probabilities of surviving from age25 to ages 35. 40,45. .... 75 forfemalesThe time period to which each es-timated probability refers whenmort.lity is changing

Probabilities of surviving from age25 to ages 35, 40, 45. ....75 forfemales during the intersurveyperiod


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Scum ~ I b n r J m l WB. Estimation of conditional B.Z(d) Paternal orphanh oodadult survivorship data from two surveys

B.3(b) Widow hood data by age

B.3(c) Widowhood data byduration o f first mar-riage

B.3(d) Widow hood data fromtwo surveys

Number o f respondents with fatheralive (or dead) classified byfiye-year age group for two sur-veys five or I 0 years apartTotal number of respondents byfive-year age group for the sametwo surveysNumber of respondents who didnot declare the survivorshipstatus o f their fathe r, classifiedby five-year age group for thesame two surveysNumber of births occurring in agiven year (preferably duringthe intersurvey period) classifiedby five-year age group of father(or of mother, i f nothing else isavailable)Number o f ever-married male (fe-male) respondents with firstspouse alive (or dead) classifiedby five-year age groupTotal number of ever-marriedmale (female) respondents byfive-year age grou pNumber o f ever-married male (fe-male) respondents who did notdeclare or di d not kno w the sur-vivo rship status o f first spouse.by five-year age groupSingulate mean ages at marriagefor both males and females (seeannex I)Number o f ever-married male (fe-male) respondents with firstspouse alive (or dead) classi-fied by five-year duration-of-marriage group'rota1 number of ever-marriedmale (female) respondents

classified by five-year duration-of-marriage groupNumb er o f ever-married male (fe-male) respondents who did notdeclare or did not know the sur-vivorship status of first spouse.by five-year duration groupSingulate mean age at marriage forfemales (males)Num ber o f ever-married male (fe-male) respondents with spousealive (or dead) classified byfive-year age (duration) groupfor two surveys five or 10 yearsapartTotal number of ever-marriedmale (female) respondentsclassified by five-year age (dura-tion) group from the same twosurveysNumb er o f ever-married male (fe-male) respondents who did notstate the survivorship status oftheir first spouse classified byfive-year age (duration) groupfor the same tw o surveysSingulate mean ages at marriagefor. males and females for thetwo surveys

The prob abilities-oi surviving fromages 32.5 to ages 45.50. ...,90: orthe probabilities of survivingfrom age 37.5 to ages 50, 55, ....95 (depending upon the value ofM, he mean age of fathers atthe birth of their children) formales during the intersurveyperiod

The probabilities of surviving fromage 20 to ages 25. 30. 35, ..., 55(60) for females (males)The time period to which each es-timate refers when mortality ischanging

Probabilities o f surviving from age20 to ages 25, 30. .... 50 for fe-males (males)The time period to which each es-timate refers when mortality ischanging

Probability of surviving from age20 to ages 25. 30. .... 50 for fe-males (males) during the inter-survey period


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7 TABLE5 (continued)-- - -- - - - - --- . - -- -- - - - - . ---C. Estimation of survivorship C.2 Maternal orphanhoo d data Num ber of respondents with l(4 5) . (5 0) . ... l(7 5) for female\to adulthood from birth mother alive (or dead) classifiedby five-year age grou pTotal number of respondents byfive-year age grou pNumber of respondents who didnot declare or did not know thesurvivorship status of theirmother classified by five-yearage groupNumber of births in a yearclassified by five-year age gro upof motherAn estimate of l(2 ) for females

C.3(b) Widow hood data by age Nu mbe r of ever-married male (fe- I(25). l(30) .... (55) (or l(60)) formale) respondents with first females (male s)spouse alive (or dead) classifiedby five-year age gro upTotal number of ever-marriedmale (female) respondents byfive-year age grou pNumber of ever-married male (fe-male) respondents who did notdeclare the surv ivonhip status offirst spou se, by five-y ear agegroupSingulate mean age at marriage forboth males and femalesAn estimate of I(2) for females(males)C.3(c) Widowhood data by Number of ever-married male (fe- I(2O). l(25) .... (40) for femalesduration of first mar- male) responden ts with 61x1 (male s)riage spouse alive (or dead) classi-fied by five-year duration-of-marriage groupTotal number of ever-marriedmale (female) respondentsclassified by fiv e-ye ar dura tiongroupNumber of ever-married male (fe-male) respondents who did notdeclare the survivorship status offirst spouse. by five-year dura-tion groupSingulate mean age at marriage forfemales (males)An estimate of I(2) for females(males)

and according to the type of data each method requires:information on the orphanhood status; or data on thewidowhood status of respondents. The organization ofthe following sections closely follows this classification.In order to aid the user in selecting the method bestsuited for a particular application, brief descriptions ofeach section follow (see also table 85):Section B. Estimation o conditional adult survivorship.This section presents methods that allow the estimationof conditional probabilities of survivorship, that is,probabilities of surviving to age x given that the targetpopulation has already survived to agey , wherey is lessthan x . This section is divided into two main subsec-tions. In the first, methods using data on the orphan-hood status of respondents are described; in the second,methods using data on the widowhood status of ever-married respondents are presented;

Section C. Estimation o survivorship to adulthoodfrombirth. This section presents methods that permit the esti-mation of the probabilities of surviving from birth tocertain ages x , all greater than 20. All the methodsdescribed require as input an estimate of 1(2) ,the proba-bility of surviving from birth to exact age 2. The twomain parts of this section present methods based onorphanhood and widowhood data, respectively.

B. ESTIMATIONF CONDITIONAL ADULTSURVIVORSHIP

1. General characteristicsThe methods described in this section deal with theestimation of adult mortality from information about the


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orphanhood or widowhood status of respondents. Thesemethods may be called "conditional" because theestimated parameters are conditional probabilities ofsurvival that do not, by themselves, define a completelife table. To derive life-table values from them moreinformation is needed about child mortality; methodsfor combining information about child and adult mor-tality are described in chapter VI.2. Estimation o orkc11 survivorship bared onproprriom nor orphaned

(a) h i s methodand its rationaleData on the proportions of respondents whose mother(father) is still living can yield plausible estimates ofadult mortality. The first method of estimation based onthis type of data was proposed by ~ rass?who estab-lished an equation relating the female probability of sur-viving from age 2'5 to age 25 + n to the proportions ofrespondents in two contiguous five-year age groupswhose mother was still alive at the time of the interview.This equation has the form

where S(n) is the proportion of respondents aged fromn to n +4 with mother alive, and W(n) is a weightingfactor employed to make allowance for typical age pat-terns of fertility and mortality. The set of W(n) valuesthat Brass proposed was estimated on the basis of datasimulated by using a single mortality pattern (the Afri-can standard) and model fertility schedules of fixedshape but variable age locations. Each age location ofthe fertility schedule is associated with a particular valueof M, the mean age of women (or men) at the birth oftheir children. Thus, the weights, W(n ), depend bothupon n, the central point of the age groups being con-sidered, and upon M.

More recently, Hill and ~russel l~roposed anotherestimation procedure based on the following equation;

where a (n), b (n ) and c(n) are coefficients estimated byusing linear regression to fit equation (B.2) to data from900 simulated cases. These cases were derived by usingseveral fertility schedules generated by the Coale-~russell' model and a variety of mortality schedules

W. Brass and K. H. Hill. "Estimatin adult mortality from orphan-h m ~ , d n g s o t k ~ nt er na ti otu k~ olation Conference, fi1 e,,973 (Libge, International Union for the &ientific Study of po pu f-tion. 1973). vol. 3, pp. 1 1 1-123.K . Hill and J. Trussell. "Further develo ments in indirect mortal-it estimation". Population Studies, "01. X X ~ .o. 2 (July, 1977). pp.3i3-333.

' ~ n s l e ~. Coale and T. James Trussell. "Model fertility schedules:vanatlons In the a ge structure of childbearing in human populations",Popukrtion Index , vol. 40, No. 2 (April 1974). pp. 185-258.

generated by the logit system6 with the four different~oale- erne en^^ mortality patterns as standards (seechapter I, subsections B.2 and B.4).When both methods of estimation were applied tonearly 1,000 other cases simulated by using different fer-tility and mortality schedules, the method proposed byBrass was found to perform as well as or better than theregression method for n not exceeding 30 years, whilethe regression method produced substantially better

results for higher values of n.Even though this comparison was camed out only onsimulated cases that are not subject to the type of errorscommonly found in real data, it does reveal the prob-able strengths and weaknesses of the different methods.Since these methods seem to complement each other inthe realm of theory, both of them are described here inthe hope that they may also complement each other inpractice.Before proceeding with their description, some gen-eral features of methods that attempt the estimation ofadult mortality based on the reported proportions oforphaned respondents by age and an indicator of theage pattern of fertility should be pointed out.First, the estimated probabilities of survivorship donot refer to the entire population, since they reflect onlythe mortality experience of parents with surviving chil-dren. Furthermore, if questions on orphanhood areasked of the entire population, parents with several sur-viving children will tend to be overrepresented.Theoretically, this problem can be avoided by using anadditional filter question that seeks to identify only onechild per parent, such as the oldest surviving child or thefirst-born child. However, in practice, errors in thereporting of family-order status have been found to beso large that methods of analysis using data where theresponses have been limited to one child per parent have

not yielded better estimates than the original methods*Secondly, survivorship estimates based on reports byyoung respondents, and thus corresponding to smallvalues of n (under 20), tend to be affected by misreport- .ing of orphanhood status: young orphaned children areoften adopted by relatives who report them as their ownchildren. This phenomenon artificially inflates the pro-portion of young respondents having a surviving parentand biases upward the estimated survivorship probabili-ties of younger adults.Lastly, the estimated probabilities of survival do not,strictly speaking, refer to specific time periods, since theyrepresent average measures over the somewhat ill-defined intervals of exposure to the risk of dying of thetarget population. In cases where mortality hasremained essentially constant, problems in the interpre-

William Brass, Methods for Erritnutin Fertility and Mortalityf iLimited and Lkj2ctive O a Chapl ~ i lorth Carolina. CarolinaPopulation Center, Laboratories for Population Sta tistics 1975).Ansley J. Coale and Paul Demeny, kgionol Modrl U/r T&s andStable Popukztiom (Princeton, New Jersey. Princeton Un~versityRtq

1%6).Kenneth Hill. Hugo Behm and Augusto Soliz. h i twib n dc 1mortalidad en Bolivia (La Paz, lnstituto Nacional de Estadlstica, 1976).


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tation of the estimates obtained d o not arise since, in the index i to denote the different five-year age group s inabsence of data errors. they should all imply the same, the reproductive life span of a woman and lets B(i) beunchanging mortality level. However, because mortality the number of births during a particular period tohas not remained constant during the recent past in most women in age group i , thencountries, the interpretation of survivorship probab ilities 7derived from orphanho od data is not always straightfor- 7ward. Interpretation and assessment are easier when the M = a ( i ) ~ ( i ) / x ( i ) (B.3)r = l 1 = Iestimates obtained can be related to specific timeper iods. I f one can ~ s u m ehat mortality has been where a(i) is the mid-point of age group i. (The userchanging regularly (for instance, by assuming that in the should be reminded that, just a s stated in chap ters I1 andlogit system the parameter indicating mortality level, a, 111, i = 1 represents age grou p 15-19, i = 2 is 20-24 andhas been changing linearly with respect to time) and that so on.) Note that if the births used to calculate M ar ethe adult mortality pattern of the target population is those reported as occurring during the year precedingsimilar to that represented by the general standard (see the survey and are tabulated by age of mother at thechapter I, subsection B.4), it is possible to estimate the time of the survey, the true age g roup of the m other willtime t (as number of years before the survey) to which be, on average, six months younger than stated, so thateach survivorship probability refers. Th e estimation six months should be subtracted from the mid-point ofmethod used for this purpose has only recently been each age grou p when calculatingM.proposed Brass and B ~ ~ ~ ~ Y ~ ; ~nd it ha s Th e estimation of M for males is one of the additionalnot yet been widely tested, its general rde va nce w an an ts problems associated with the estimation of male adultits inclusion in this chapter. mortality from the proportions of respondents with a(b) Brass method surviving fathe r. Fertility ques tions are generally not

(i) m a equiredasked of males, so the information from which thefemale M is estimated is usually not available forThe following data a re required for this method: fathers. Births during the year preceding a survey are(a) The propodon with a surviving sometimes tabulated by age of husband, but this tabula-mother (father) in each fiv e- ~e ar ge g roup from r~ to tion is generally limited to those cases in which then +4. This propodo n is S(n) - The set mother and her husband are enumerated in the sameproportions S(n) can be when any two th e household. Calculating the male M from such a tabula-following items arc available: (a ) the num ber of tion often biases its value upw ard because young fathersrespondents with mother (father) alive; ( b ) the number are more likely to be temporarily absent.of ~ s p o n d e n a ith mother (father) dead: (c) the total A more mbust proceduR for estimating Br malesnumber of r c s ~ n d e n u mother's (father's) sur- of adjusting the female M by using informationviva1 status is known . It is important to exclud e from on status. The median age of the cunently mar-the calculations all respondents who did not know o r did ried population can k calculated by w x, an d thenot declare their mother's (father's) survival status. In difference between the male an d female medians ca n beevery data must be by five-year age added to the previously calculated female M to obtaingroup; an estimate of the male M . Medians are used to reduce(b ) The number of births in a given Year classified by the influence of older and, for the purpose in hand,five-year age gro up of mother (father)- This largely irrelevant couples. In principle, the assum ptionis needed to estimate M * he mean age of mothers of constant fertility can be relaxed by employing(fathers) at the birth of their children in the population different values of M for different age group s of respon-being studied- The M to be estimated is the mean dents, but in practice the data required to follow thisage of the fertility schedule (also known as "mean age of approach are often lacking.childbearing"); it is, rather, a mean age of the fertility Step 2: calculation of wightingfactors. For th e valueschedule weighted by the age distribution of the female of calculated above an d for each value of n , thepopulation. It may be regarded as an estimate of weighting factors, W(n ), are calculated by interpolatingth e average age mother (father) and linearly (see ann ex 1") in table 86 (table 87 s used whenchild in the population. thus being an indicator of the fathers are beingaverage age at which the target persons (parents) begin

their exposure to the risk o f dying.Step 3: calculation o survivorship probabilities. If the

survivorship of mothers is being considered, the proba-(ii) Computational procedure bilities of surviving from age 25 to age 25 + n are cal-The steps of the computational procedure are culated by using the equatio ndescribed below.Step 1: Cdc1(larionofmean age at maternity (paternity). I f (25 +n )/Ij (25)= W(n )S(n -5) +( I .0-W(n ))S(n )The mean age of mothers (fathers) at the birth of agroup of children (normally those born in the year (B.4)the survey) is denoted by If One uses the wherp S( n ) is the proportion of respondents aged fromn to n +4 who declared that their mothers were alive at

W. Brass and E.A. Bamgboyc,op. cir. the time of the interview.102


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TMLE86 WUOHTINO FACTORS, W(n ). FOR CONVERSION OF PROPORTIONS OF RESPO NDENTS WITH MOTHER ALIVEINTOs u R v ~ v o ~ n ~ Po w l L m e s FOR FEMALES

Estimation equation:lf(25+n)/lf(25)= W(n)S(n -5) + 1. (n ) )S ( n )TMLE 7. WE 10H n N 0 FACTORS , W(n ). FOR CONVERSION OF PROFQRTIONS O F RESFOND ENTS WITH FATHER ALIVE

INTO SURVIVORSHIP IRO BA IIW IE S FOR MALES

(a )Bum 32.5-0.388 0.4550.429 0.5220.406 0.5230.335 0.4740.1 62' 0.319-0.047 0.109-0.379 -0.203-0.65 1 -0.495-0.776 -0.651-0.758 -0.667

Estimation equation:lm(35+n)/Im(32.5)= W( n)S (n 5) + (1.0- W(n))S ( n )(b)From 37.5yruvs

39 400.613 0.6870.690 0.7900.708 0.8330.6 13 0.7590.450 0.6140.152 0.32 1-0.157 0.003-0.372 -0.237-0.47 1 -0.377-0.425 -0.366

Estimation quat ion :Im(40+n)/lm(37.5)= W (n )S(n -5) + (1.0- W(n))S ( n )

In the case of fathers. the value 25 is replaced by the if the weighting factors W(n) are obtained from tablevalues 32.5 or 37.5 to allow for the fact that men are 87. pan (a); andd l y lder than women at the birth of their children.The survivorship probabilities are estimated in this case 1 (40+n)/l, (37.5)= W(n )S(n -5) +(1.0 -W(n ))S(n )by tke following equations:lm(35+n)/l,(32.5)= W(n)S(n -5)+(1.0- W(n))S(n) (B.6)

(B.5) if the weighting factors are obtained from table 87. part103


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TABLE8. VALU ES OF THE STA NDAR D FUNCTION FOR CALCULATION OF THE TIME REFERENCEFOR INDIRECT ESllMATES OF ADULT SURVIVORSHIP

(b). In both cases, S( n ) stands for the proportion ofrespondents in the age group from n to n +4 whosefather was alive at the time of the interview. A decisionhas to be mad e whether to use the weights given in table87, part (a), and hence equation (B.5); or those in table87, part (b), and hence equatio n (B.6). In general, thechoice depends upon the estimated value of M: if it isless than or equal to 36, part (a) should be used; if it isgreater than 36, part (b) should be used.Step 4: calculation of number of years before the surveyto which the survivorship estimates refer. It can be shownthat if the level of mortality is declining linearly on thelogit scale (see chap ter I, subsection B.4), the survivor-ship estimates obtained from orphan hood da ta are e qualto those prevalent at specific time periods prior to thesurvey and that the time location of these periods islargely independent of the rate of the mortalitychange.'' In addition, when female mortality in adu lt-hood has a pattern similar to that embodied by the gen-eral standard (see chapte r I, subsection B.4), the numberof years before the survey to which each estimatederived from maternal orphan hood data refers, denotedby t (n ), can be estimated as

where

and

where Z(x) is still obtained from table 88; loS,,-s owrepresents the proportion of respondents aged fromn -5 to n + 4 with father alive; M is the mean age offathers at the time of their children's birth; n is the mid-point of the age group considered; and 0.75 or threequarters of a year have been added to make allowancefor the fact that a fathe r must have been alive at the timeof conception, but not necessarily at the time of the birthof his offspring.(iii) A &toiled e-leThe d ata shown in tables 89 and 90 were collected bythe National Dem ographic Survey cam ed out in Bolivia

durin g 1975." They are used here to illustrate the way inwhich the estimation of female adult mortality is carriedout using data on the maternal orphanhood status ofrespondents.Table 89 shows both the raw data g athered by the sur-vey and the proportions of respondents whose motherwas alive at the time of the interview. Note that in calcu-u(n )=0.3333 In (loS,,-5)+Z(M +n )+0.0037(27 -M), TABLE9. DATA N MATERNAL OW.AN HOOD STATUS AND PROPORTIONS

IMnanllmmd ol%r- P - * ,k,-'%"-

sn)and the value of Z (M + n ) is obtained by interpolating (2) (3) (5)linearlv in table 88. 15-19....... 5540 448 6 0.925220-24 ....... 3 995Note that in this case loS,5 represents the proportion 25-29 886 541 10 0.8807........ 769 8 0.78%of respondents in the age group from n -5 to n +4 with 30-34........ 910 852 9 0.69 15mother alive, and that n is the mid-poin t of the 10-year 35.39 ........ I 661 I 234 1 1 0.5737........ge group being considered. 40-44 I 027 I 272 7 0.44674549 ....... 855 I 556 6 0.3546When data on paternal orphanhood are being used, 5654 ........ 369 1 243 6 0.2289equations (B.7) and (B.8) become

I ' K.Hill.H. ehm and A.Soliz,op. cit.104


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TABLE90. CHILDRENORN DURING THE 12 MONTHS PRECEDINGTHE SURVEY. BY AGE OF MOTHER AT TIME OFTHE SURVEY. BOLIVIA. 1975

W-dn,sixmo4uh)

#O(3)172227

lating the latter figure, the numbers of respondents whowere classified in the category of "unknown maternalorphanhood status" were ignored. In the case of the1975 survey in Bolivia, the numbers in the unknowncategory are very small and their inclusion in thedenominators of the proportions with surviving motherwould not have affected the final results. However,because greater levels of non-response may occur, it isimportant to exclude the non-responses when calculat-ing the proportions with surviving mother. As an exam-ple, S(50) is computed explicitly below:

The computational procedure is described below.Step I: calculation o mean age at materniry. Accordingto equation (B.3):

where a (i) is the mid-point of age group i; and B(i) isthe number of births to women in age group i .However,to obtain a correct estimate of the mean age at mater-nity, M, the births used ,in equation (B.3) should beclassified by age of mother at the time the birthoccurred. In the present case, the age reported by awoman is that at the time of the interview and not that atthe time of the birth; some allowance must be made forthis fact. The simplest strategy is to assume that birthsare uniformly distributed in time and with respect tomother's age. Hence, on average, women were sixmonths younger at the time of the birth they reported

than at the time of the interview, so that, for example,the current age group spanning exact ages 15-20 may beconsidered to span exact ages from 14.5 to 19.5 at thetime the births occurred. All subsequent age groups aresimilarly affected. Therefore, the central a(i ) values tobe used in applying equation (B.3) are 17 instead of17.5, 22 instead of 22.5, and so on, as shown in column(3) of table 90.Applying equation (B.3), M is calculated as follows:

72 a(i)B( i)= (17)(136)+(22)(409)+. +i = l

and

Step 2: calculation o neighring factors. Since data onmaternal orphanhood are being considered, theappropriate weights, W(n), are obtained by interpolat-ing linearly (see annex IV) between the values given intable 86. The value of M is 28.8 years, so the weights inthe columns labelled "28" and "29" are used asextremes in the interpolation. For all values of n, theinterpolation factor8 s

Therefore, to obtain W(35), for example, the values forn = 35 appearing under the columns for M qua1 to 28and 29 of table 86 are weighted by 8 and (1-8).respectively, yielding

The complete set of W(n ) values corresponding to M= 28.8 is shown in column (2) of table 91.Step 3: calculation o survivorship probabilities. Oncethe weighting factors corresponding to the observed Mhave been calculated, the values of 1/ (25 +n ) / l f (25) are

TABLE1. ESTIMATIONF FEMALE ADULT SURVIVORSHIP FROM PROPORTIONS OF RESPONDENTSWITH SURVlVlNO MOTHER. USING THE BRASSMETHOD. BOLIVIA, 1975

W Y twarAf nMoUlyMI1) 17)


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computed as indicated by equation (B.4). Table 91 sum-marizes these computations. This table also shows themortality level in the Coale-Demeny West family ofmodel life tables associated with each survivorship ratio.These levels were calculated by interpolating linearlybetween the values given in table 210 (see annex VI).They provide a useful index for examining the con-sistency of the survivorship ratios themselves.As an example, the calculation of the level consistentwith the estimated 1 (65)/1/(25) is camed out below indetail. The value o l l (65)/1/(25) according to table 91is 0,604. The column abelled "65" of table 2 10 in annexVI is then used to locate the two values that bracket theestimated value. They are 0.58183, corresponding tolevel 13; and 0.60583, comesponding to level 14. Hence,the interpolation factor8 is

t (n ). In step 1,M was found to be equal to 28.8 years.Therefore, the values of x =M +n range from 48.8 to78.8, increasing in steps of five years. The standard func-tion, Z(x), for calculation of the reference period, isobtained for these values by linear interpolation withintable 88. Note that because the table does not cover therange 78-79, the last Z value calculated is Z(73.8). Thus,Z(53.8) is obtained as

All Z(x) values are given in column (4) of table 92.Using these values and substituting those of toS,-s, Mand n in equation (B.8), one obtains the set of values forthe correction function, u(n), given in column (5)of thesame table. To illustrate the procedure followed, u (30) iscalculated below:and the estimated level is

The complete set of mortality levels is shown in column(7) of table 91.Step 4: calculation o number o years before the surveyto which the survivorship estimates refer. The referenceperiod of each of the estimates obtained in the previousstep, denoted by t(n), is calculated from loSn5, the pro-portion of respondents aged from n -5 to n +4 whosemother was alive at the time of the interview. These pro-portions are calculated by dividing the number ofrespondents with surviving mother in the two adjacentage groups from n -5 to n- and from n to n +4 bythe sum of the number of respondents in both agegroups who provided information on mother's survival.For example,

The complete set of loSn-s values is shown in column(2) of table 92.Because the data at hand refer to maternal orphan-hood, equations (B.7) and (B.8) are used in calculating

Then, using equation ;B.7), t (n ) is calculated for eachcase. Continuing with the illustration of the case n =30,

The survey in Bolivia was camed out between I5June and 31 October 1975, so the mid-point of thatperiod may be considered its reference date. The sim-plest way of obtaining the decimal equivalent of themid-period date is by transforming the end-points intodecimal equivalents and finding their mean. Thus, 15June 1975, is equivalent to 1975.45 (the number of daysfrom 1 January to 15 June, both included, being 166,and 166/365 = 0.45) and 31 October 1975 is equivalentto 1975.83, so that the desired mean is 1975.64. There-fore, the lf(55)/1/(25) estimate listed in table 91 refersapproximately to 11.8 years before 1975.6, that is, to1963.8. The complete set of reference dates and the Westmortality levels associated with them are shown incolumns (7) and (8), respectively, of table 92.

TABLE 92 . ESTIMATION OF TIME REFERENCE PERIODS FOR SURVIVORSHIP ESTIMATES DERIVED FROMMATERNAL ORPHANHOOD DATA, BOLIVIA,1975

-a N u m b e r o f y e a r s p r i o r to t h e s u r v e y t o w h i c h s u r v i v o r s h i p e s t i m a t e s r e f e r.

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It is worth mentioning that the estimates obtained TABLE3. COEFFICIENTSOR ESTIMATION OF FEMALE SURVIVORSHIPfrom maternal orphanhood data refer, when adult mor- PROBABILITIES FROM AGE 25 FROM PROPORTIONS WITH SURVIVINGtality has been changing steadily, to periods between 8 MOTHERand 15 years prior to the survey. When judged with *f CQ&~~~HId n ) t i n ) c ln )respect to these estimated dates, the mortality levels (1 ) (2 ) 13) (4 )obtained in the previous step seem less satisfactory than 20............,. -0.1798 0.004 76 1.0505at first sight. The estimates imply that female adult mor- 25.............. -0.2267 0.00737 1.029Itality in Bolivia fell by an average of one mortality level 30.............. -0.3 108 0.01072 1.0287per an num durin g the period 1960-1965. It is difficult to 35........s..*9. -0.4259 0.01473 1.047340... -0.55 66 0.0 1903 1.0818believe that such rapid gains in life expectancy could 45,.,.,,.,....., -0.6676 0.02256 1.1228actually have taken place. Further, the similarity of the so. ............ -0.698 I 0.02344 1.1454levels corresponding to the two most recent periods issuspect. It suggests that even up to age groups 20-24 and Estimation equation:25-29 the adoption effect may be biasing the survivor- 1 , ( 2 5 +n ) / I f ( 2 5 ) = a ( n ) +b ( n )M + c ( n ) S ( n -5 )ship estimates upward. T o what extent such an effect isalso oper ating at ages 30-34 and 35-39 is difficu lt to in the age group from n -5 to n - whose mother wasascertain, but the possibility that the estimates for 1962- alive at the time oft he interview.1964 may also be subject to an undesirable positive bias No!e should be taken that this method themust be in mind' these brief it estimation of female survivorship only. No regressiontranspires that the data for Bolivia may not be as con- method has been developed to estimate male survivor-sistent as one would desire, and that further analysis and ship, u, if pateml orphanhood data are available, theadditional, independent evidence may be necessary to original Brass method has to be used.establish with a greater degree of certainty the mortalitylevels to which the female population of Bolivia has It should also be pointed out that the conditionalbeen subject. probabilities of survivorship obtained by using thismethod do not, by themselves, define a complete life(c) Method based on an equa tionfirted by wing regression table, unless one can assume that the mortality pattern(i) Lhata required of the population studied is identical to that of somemodel set of life tables. When inform ation on child mor-The required lo lhiSmethod in lhe case of tality is available, the techniques described in chapter VIare the same as lhoSe ca n be used to construct a life table that does not dep endneeded for the Brass method (see subsection 8.2 (b) (i); upon a restrictiveassumption.however, no regression method has been developed forthe estimation of male mortality from data on paternal 3: cdculation ofnrunberofyears bejbre theorphanhood. The data required are: to which the survivorship estimates refer. In cases whereadult mortality has been changing regularly during the(a) The proportion 3frespondents 'nother was I5 or 20 years preceding the survey (that is. the changealive at the time of the interview, classified by five-year has been linear on the logit estimates of the timeage group. The P ~ P ~ ~ ~ ~ ~n lhe age group to which each of the su~ivonhip robabilities obtainedfrom n to n +4 is denoted by S(n); in the prev ious step refers, den oted by t (n ), can be cal-(b ) The number o f children born during a given year* culated by m eans of the following equations:classified by five-year age grou p of mother. These dataallow the estimation of M , the mean age of m others at u( n) = 0.3333 In S(n -5 )+Z (M +n -2.5)+the birth of a particular group of children.(ii) Compututionalprocedure 0.0037(27-M) (8 .12)The following steps are required in the com putational an dprocedure.Step 1: calculation of mean age at maternity. Th is step t( n )= (n -2.5)(1.0-u(n))/2.0 (B.13)is identical to that described in subsection B.2 (b) (ii). Itis not described again here. where S (n -5) is the proportion of respondents agedStep 2: calculation o survivorship probabilities. Using from M -5 to M - whose mothe r was alive at the timethe value s of the coefficients a (n ), b (n ) and c (n ) shown of the survey; M is the mean age of mothers at the timein table 93, female survivorship probabilities are of the birth of their child ren; n -2.5 is a rough indicatorobtained by substituting these values in the following of the mean age of respondents; an d Z ( x ) s a standardequation: function of age whose values in any given case can beobtained by interpolating between the values listed inIf (25+n)/lf (25)= a ( n ) + b ( n ) M + c ( n ) S (n - 5 ) table 88. Strictly speaking, the estimates yielded byequations (8.12) and (B.13) when the Z ( x ) values of(B . l l ) table 88 are used as input are based on the assumptionthat female adult mortality in the population being stud-where M is the mean age of mothers at the birth of their ied has a similar pattern to that em bodied by the generalchildren; and S (n -5) is the proportion of respondents standard (see chapter I, subsection 8.4). In practice.

107


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however, the impact of deviations from this assumption group, S(n -5). They are shown in column (2) of tableis unlikely to be large, at least in comparison with the 95. The values of Z(x) for x ranging from 46.3 to 76.3effects on the final estimates of deviations from the in steps of five years are obtained by interpolatingassumed linear trend followed by adult mortality linearly between the values listed in table 88. For exam-changes over time. ple, Z(66.3) is obtained as(iii) A &tailed exampleThe application of the regression method is illustrated 2(66.3)= 0.7Z(56)+0.3Z(67)=below by again using the data for Bolivia presented insubsection B.2 (b) (iii) (see table 89). 0.7(0.452) +0.3(0.473) =0.458.Step I: calculation o mean age at maternity. Refer tostep 1 of subsection B.2 (b) (iii), where the mean age ofmothers at maternity,M , as estimated as 28.8 years forBolivian women.Step 2: calculation o survivorship probabilities. Table94 summarizes the calculations needed to estimate theseprobabilities. It shows that the application of equation(B.ll) is straightforward and that the estimation processis very simple. Once again, the mortality level consistentwith each survivorship ratio in the West family of modellife tables has been computed by interpolating in table210 in annex Vl (see step 3 of subsection B.2 (b) (iii)).Note that for a value of n under 40, the survivorshipestimates obtained by this method are very similar tothose obtained by the Brass procedure (see table 91).For values of n of 40 and over, however, the differencesincrease with n ; or these ages, the regression-based esti-mates would probably be preferable.Step 3: calculation o number o yervs kfow the surveyto which the swviwrship estimates refer. As stated byequations (B.12) and (B.13), the values of this referenceperiod, denoted by t (n), depend in this case upon theobserved proportions not orphaned in each five-year age

Since the values listed in table 88 do not cover the agerange ffom 76 to 77, linear extrapolation was used tocalculate Z(76.3). The complete set of Z(x) values isshown in column (4) of table 95. Using equation (B.12)and these values as input, the set of u(n) values can becalculated. For example,

so that

The complete set of t(n) values, the number of yearsprior to the survey to which each survivorship estimaterefers, is shown in column (6) of table 95. Since the sur-vey reference date is 1975.6, these t(n) estimates can betransformed to dates by subtraction, the results beingshown in column (7) of table 95. The West mortality lev-els implied by each survivorship estimate are also shownTABLE4. CALCULATION OF FEMALE SURVIVORSHIP PROBABILITIES FROM ~ 0 ~ 2 5SINO

PROPORTIONSW I TH SURVIVINO MOTHERS AND THE REOR eSSlON METHOD. BOLIVIA.975MF-

F a d e d u bwnld b - w4-3 R"""I, S+m)lI ,05)(4) (5)0.9252 0.9290.8807 0.8920.78% 0.8 100.69 15 0.7220.5737 0.6120.4467 0.4840.3546 0.383

TABLE 95. ESTIMATION OF TIME REFERENCE PERIODS FOR SURV IVORSHIPPROBABILITIES DERIVED PROMMATERNAL ORPHANHOOD DATA USlNO THE REORESSION METHOD. B OWVIA, 1975

umber of prior to the survey to which survivorship estimates refer.108


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in table 95 to facilitate comparisons and assessment.Note that, as already pointed out in subsection 8.2 (b)(iii), the pace of the mortality change implied by theseestimates between 1960 and 1966 seems implausiblyrapid. It is also clear that the relatively low mortalityassociated with values of n of 20,25 and 50 are suspect,that for 50 because it does not conform with the trend ofthe rest, and those for 20 and 25 because they imply afairly important decline in mortality from mid-1964 to1966, compared with a very moderate decline in pre-vious years. It is very likely that the survivorship esti-mates for n values of 20 and 25 are biased upward bythe adoption effect and that the survivorship ratio for nof 50 is distorted both by overreporting of age and bymisreporting of maternal survival.(d) Use o dorafrom nvo surveysOne of the problems faced when estimating adult sur-vivorship probabilities from data on the survival ofparents is that if mortality has been changing, the esti-mates refer to some fairly distant point in the past. If,however, information of orphanhood has been collectedby two censuses or surveys and the period between themis a multiple of five years, cohorts from the first censuscan be identified at the second, and survivorship proba-bilities applicable to the intersurvey period can beestimated from the constructed proportions notorphaned for a hypothetical intersurvey cohort ofrespondents. It is assumed, of course, that migrationdoes not affect either set of proportions not orphanedand that no relationship exists between the mortality ofthe respondents and that of their mothers. If the periodbetween the two surveys is not a multiple of five years, aset of proportions not orphaned for a suitable point intime can, in some circumstances, be estimated by inter-polation between or extrapolation beyond the observedsets.

(i) lh ta requiredThe following data are required for this method:(a) The proportion of respondents with survivingmother in each five-year age group from n to n +4 fortwo points five or 10 years apart, denoted by S(n, 1) forthe first survey and S(n, 2) for the second;

TABLE6 . CALCULATIONF PROPORTIINTERSURVEY COHORT.

( 6 )The number of births in a year classified by five-year age group of mother from one of the surveys (andpreferably from both) or for some year in the intersurveyperiod.(ii) Computational pruceahreThe steps of the computational procedure are givenbelow.Step 1: calculation o mean age at maternity. See thedescription of step 1 given in subsection B.2 (b) (ii). Ifthe data required for the calculation of the mean age,M, are available from both surveys, M can be calcu-lated for both, and the average can be used in theanalysis. If the required data are available for only oneof the two surveys, or for some intermediate point, thenthe value of M obtained for that point should be used.Step 2: calculation ofproport~~nsot orphaned for ahypothetical intersurvey cohort o respondents. The con-cept of a hypothetical cohort has already been discussed,and other uses of it appear in chapters I1 and 111. Theprocedure followed in calculating the proportions notorphaned among a hypothetical intersurvey cohort ofrespondents depends upon the length of the interval

between the surveys, which is denoted by T. The valueof T should be a multiple of five, so that the survivors ofa standard five-year age group at the earlier survey canbe identified as belonging to a standard five-year agegroup at the second survey. If one denotes by S(n, 1)the proportion of persons in the age group from n ton +4 whose mother was alive at the time of the first sur-vey and by S(n + T, 2) the equivalent proportion at thesecond survey, then S(n + T, 2) is the proportion notorphaned at the second survey among the survivors ofthose whose proportion not orphaned at the first surveywas S(n, 1). Using this notation, and assuming that T isa multiple of five, the proportions not orphaned for ahypothetical intersurvey cohort, S(n, s), are obtained asfollows:

S(n, s)=S(n -T, s ) S(n, 2)/S(n -T, 1) for n )T(B. 15)

IONS NOT ORPHANED FOR A HYPOTHETICALCONSTRUCTED EXAMPLE

Proportions taken directly from second survey.109


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Step 3: txlculation of survivorship probabilities. Oncethe proportions not orphaned for the hypothetical coho rthave been calculated, survivorship probabilities areobtained from them and from the mean age at mater-nity, M , stimated in step 1 by using either the Brass orthe regression method (see subsections B.2 (b) (ii) andB.2 (c) (ii)). The survivorship probabilities obtainedfrom hypothetical-cohort data should reflect adu lt mor-tality levels during the intersurvey period, so there is noneed in this case to estimate the reference period, r ( n ) ,to which each estimate refers.(iii) A &tailed exampleThe only difference in the app lications just describedand the intersurvey m ethod lies in the ca lculation of theproportions with surviving mother in a hypotheticalcohort. Thus, a constructed example of the way in whichto calculate these proportions (step 2) is given in table96. Column (3) shows the proportions not orphaned atthe first survey, classified by five-year age group; andcolumn (4) shows the corresponding proportions at thesecond survey.Column (5) shows the hypothetical-cohort propo rtionsnot orphaned for an intersurvey period of five years. Toillustrate, the value for age group 15-19, 0.835, isobtained as follows (from equations (B.14) and (B.15)):

It may be mentioned that this value may also beobtained by

Column (6) shows the hypothetical-cohort proportionsnot orpha ned for an intersurvey period of 10 years. Thevalue for age group 20-24, 0.739, is obtained as follows(from equations (B. 14) and (B. 15)):

Again this value may be obtained by

Intersurvey adult survivorship probabilities, allowingfor both mortality and fertility change (if M can be cal-culated from data at the beginning and at the end of theintersurvey period) can now be obtained by applyingeither the Brass or the regression method to thehypothetical-cohort proportions not orphaned (see sub-sections 8.2 (b) (ii) and B.2 (c) (ii)).

3. Ertimation of odttlr survivorshiph e d on proportions widowd(a ) Basis of method and its rationaleThe propo rtions of ever-married persons classified byage whose first spouse is still alive can be used to esti-mate adult survivorship probabilities in much the sameway as the proportions not orphaned can be so used. Inthe case of w idowhood, an additional possibility existsbecause the ever-married respondents can be classifiedby duration of mamage. In order to ensure that thereshall be only one person at risk of dying per respondentand to minimize the problems arising because of remar-riage, the da ta collected should refer only to the survivalof the first spouse of each respondent. Hence, the meas-ure of duration required is the time elapsed since therespondent's first union. Classification of the data byduration of first man iage has the methodological advan-tage of establishing directly the length o f expo sure to therisk of dying of the spouse; and it may have a very realpractical advantage in some parts of the world, whereduration of marriage appears to be more accuratelyreported than age. When da ta a re classified by age, thelength of exposure to the risk of dying has to beestimated from the current age of the respondent and ameasure of the average age at marriage of all respon-dents.The estimation of adult mortality based on in-formation about widowhood status has several advan-tages over that based on the orphanhood status ofrespondents. Since only first marriages are being con-sidered, there is in all cases only one respondent fo r eachtarget person. Further, no type of adoption effect islikely to affect this type of data, so proportions notwidowed for the shorter average exposure periods mayproduce acceptable survivorship estimates referring tomore recent periods which are generally those of greaterinterest. Lastly, experience suggests that because themost reliable information about survival of first spouseis provided by women, these methods provide a fairlygood means of estimating male adu lt mortality, the esti-mation of which from data on paternal orphanhood isthe weakest both from a methodological point of viewand for reasons related to data quality.However, this estimation .method share s several of thedisadvantages associated with those based on orphan-hood. Once more, the universe to which the estimatedsurvival probabilities refer is not the entire population:in this case, they refer only to the ever-married portionof it. Yet, in cou ntries where almost universal marriageis the rule, the biases that may be introd uced by assum-ing that mortality risks among the never-married aresimilar to those estimated for the ever-married are likelyto be sm all. A second disadvantage of this method is thatit assumes that mortality and nuptiality have remainedconstant in the recent past and that the survival of therespondent is independent of that of his or her spouse,assumptions that are not likely to hold strictly in prac-tice. In addition, a disadvantage specific to widowhooddata classified by age is that the length of exposure tothe risk of dy ing of a respondent's first spouse has to be


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estimated from the respondent's age and from a sum-mary measure of age at first mamage, thus introducingyet another level of approximation. Lastly, in terms ofdata quality, it is likely that in cases where the respon-dent has remarried and shares a household with the newspouse, the interviewer may neglect to ask explicitly thequestion about survivorship of the first spouse.There is little that the analyst can do to minimize thenegative effects of some of these disadvantages on theh a1 estimates of survivorship. With respect to data

quality, it is obviously important to make sure that theinterviewer understands the concepts being used andavoids mistakes such as that cited above. The assump-tion of constant mortality has been relaxed by the workof Brass and ~ a m ~ b o ~ e , "ho propose a method toestimate the time reference of the survivorship probabil-ities derived from data on widowhood. Of course, thechange of mortality assumed in deriving referenceperiod estimates does not represent all possible patternsof change, but it is general enough to be adequate inmost circumstances.The assumption of constant nuptiality in the past canbe relaxed if the mean ages of both respondents and

their spouses at first marriage are estimated for each ageor marriage-duration group of respondents. In such acase, the SMAM, and SMAM values appearing inequations (B.17HB.26) should C eplaced by age orduration-specific values (according to the respondents'age group being used). This approach is possible ifreliable data on the retrospective mamage history of thepopulation under study are available.This introductory section may be concluded by point-ing out that the widowhood methods presented in thischapter represent recent revisions of the procedures firstproposed by ill" and later revised by Hill and~russell.'~

(b) Wdowhooddata cl ~ ~~ i f i dy age(i) DotarequiredThis method seeks to estimate male (female) proba-bilities of surviving from age 20 to age n from the fol-lowing data:(a) The observed singulate'mean ages at marriage formen (SMAM,) and for women (SMAMf ). For adescription of the way in which to estimate these agesfrom the proportions single claksified by age, see annexI. To calculate the proportions single in each age group,the basic data reauired are: the total male (female)population; and h e male (female) population single;both classified by five-year age group;(b) The proportion of ever-married women (men) in

each five-year age group whose first husband (wife) wasalive at the time of the interview. The proportion ofrespondents aged from n to n +4 with first spouse aliveis denoted by NW(n). Note that the set of proportions

non-widowed, NW(n), can be obtained by having asinput any two of the following: (a) the number ofrespondents whose first spouse was alive at the time ofthe interview classified by five-year age group; (b ) thenumber of respondents whose first spouse was deadclassified by five-year age group; and (c) the totalnumber of ever-married respondents who declared thesurvivorship status of their first spouse, also classified byfive-year age group. Since category (c) is supposed to bethe sum of the first two, it is important to make sure thatit excludes all respondents who did not know or did notdeclare the survivorship status of their first spouse.

(ii) Computational procehreThe steps of the computational procedure aredescribed below.Step 1: calculation o proportions not widowd The pro-portion of female (male) respondents not widowed inthe age group from n to n +4, denoted by NW(n), is'equal to the ratio of female (male) respondents agedfrom n to n +4 whose first husband (wife) was alive atthe time of the interview and the total number of ever-married female (male) respondents aged from n to n +4who declared the survivorship status of their first hus-

band (wife). Since the latter figure equals the sum of thefemale (male) respondents aged from n to n +4 withfirst husband (wife) alive and those of the same agegroup with first husband (wife) dead, knowledge of onlythese two categories of respondents is sufficient to calcu-late the desired proportions.In order to estimate survivorship probabilities formales, the method requires that the proportions offemales not widowed be known, while knowledge of themale proportions not widowed is necessary to estimatefemale survivorship probabilities.Step2: calculation o singulatemean age at mammamageorboth males and d e s . For a description of the pro-cedure required to calculate the singulate mean age at

marriage, SMAM, see annex I. Both values are neces-sary when estimating either male or female survivorshipprobabilities from widowhood data classified by age.Subindicesf and m are used to distinguish the valuesreferring to females from'those referring to males.Step 3:' dculation o survivorship probabilities. Therelationship between the variables calculated in the pre-vious steps and certain probabilities of survivorship wasestablished by using least-squares regression to fit thefollowing equations:

l 2 W. Brass and E. A. Bamgboyc, op. cit.13 K. nmWi, levelr f,,,,,, infomation on to data generated by using model mortality and nuptial-widowhood". p o p d m a d , 01. X ~X I , O. I (March l m ) , pp. ity schedules (see chapter I). The estimated regression

75-84. coefficients ah ) , b(n), c(n) and d(n) are shown, forl4 K . Hill and J. Trussell, loc.cit. each'value of ;,-in tables 97 ind 98. Also shown in each


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T M L E 9 7 . COEFFICIENT S FOR ESTIMATION OF CONDITIONAL MALE SURVIVORSHIP PROBABILITIESFROM DATA ON THE WIDOWHOOD STATUS OF FEMALE RESPONDENTS

C a ~ r n

/I ) 73 %) 7:

Estimation equation:Im(n)/Im(2O)=a(n)+b(n)SMAMfc(n)SMAMm +d(n)NWf n -5)

TMLE98. COEFFICIENTSOR ESTIMATION OF CO NDITIONAL FEMALE SURVIVORSHIP PROBABILITIESFROM DATA ON THE WIDOWHOOD STATUS OF MALE RESPONDENTS

25 ............. -0.0208 0.00052 - .00 137 1.045130.. ........... -0.2135 0.00104 -0.00329 1.279135 ............. -0.1896 0.00 62 - .00492 1.288440............. -0.1290 0.00236 - .00624 1.248345............. -0.07 13 0.00340 - .00742 1.200550............. -0.0327 0.00502 -0.00860 1.159055 ............. -0.0139 0.00749 -0.01019 1.1297

Estimation equation:If(n)/If 2O)= (n )+b(n)SMAMf +c(n )S MAMm d(n )N Wm(n

table is the estimation equation to be used in each case. Equations (B.17) and (B.18) are to be used when theNote that according to table 97 male survivorship prob- estimated survivorship probabilities refer to males, whileabilities are estimated by using as input the proportions equations (B.19) and (B.20) are to be used when theyof non-widowed females, while according to table 98 refer to females. Note that the form of these sets of twofemale survivorship probabilities are estimated from the equations is essentially the same, with the subindicesobserved proportions of non-widowed males. indicating the sex categcry reversed. In both casesStep 4: calmlotion ofnumber ofyears before the (males and females), the quantity n +2.5 - MM isto n~ li ~hivorship estimaes refer. cases wheremar- used as an indicator of the mean duration of first mar-

tality has been changing regularly (that is, when the riage of respondents aged from n to n +%and thechange can be assumed to be linear on the logit scale) values of the standard function Z(x) are obtained bydunng the 15 or 20 years preceding the survey, estimates i~te'polatinginearly between those listed in table 88. ASof the time, denoted by t(n), to which each of the sur- the detailed example illustrates, the use of these equa-vivorship probabilities obtained in the previous step tions is simple; and the timing estimates they yield,refers can be calculated by using the following equa- though not exact, are adequate indicators of the refer-tions: ence periods to which survivorship probabilities apply, ifthe change in mortality has been fairly steady andu,,,(n)=0.3333 In NW,-(n - 5)+ smooth.(iii) A &ailed exampleConsidered here is the case of ever-mamed women

+0.0037(27.0 - SMAM ) who were questioned about the survival status of their(B'17) first husband durine the National Demoera~hic urvevt,(n)= (n -2.5-SMAM')(l.O-um(n))/2.0 (B.18)

of Bolivia, camed Gut between 15 June ind.31 ~ctobe;1975.15 Data gathered by this survey have already beenor

uf ( n ) = 0.3333 In NW, (n)+used in llus'irating thd applicati&n of the estimationmethod based on the orphanhood status of respondents.

The steps of the computational procedure are givenZ(SMAM, +n +2.5-SUAM,) below.Step 1: calculation o proportions notwidowed Table 99

+0.0037(27.0 - A-4M' ) (B-19) shows the raw data gathered by the survey and the pro-


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T me 99. Even-MARRIEDE~ULE~OPULATION NDNUMBEROFWOWENWHOS@FIRST H U M A N D WAS DEAD AT TIM E OF INTERVIEW, BY AOEonour. B~IVIA975

portions of ever-married women whose husbands werestill alive at the time of the interview. NW/ (n). In com-puting these proportions, the denominator used is thenumber of ever-married women rather than all womenand there were no cases of non-response. The propor-tions NW ( n ) shown in column (4) are obtained bydividing e num ber of women with Rnt husband dead(appearing in column (3) of table 99) by the number ofever-married women reporting first husband's survivor-ship status (listed in column (2)) and subtracting theresult from 1.0. Thus,NWf (25) is calculated as

Step 2: oal&ion of singulate mean age at marriagefortznhmoltsa d emales. For a detailed explanation of the

way in which to calculate the necessary SMAM values,see annex I. In the case of Bolivia in 1975, these valuesare: S MM , = 25.3 years; and SMAMr = 23.2 years.Step 3: calculation ./survivorship P&ilitiesfoi males.Using the coefficients shown in table 97 and applyingthe version of equation (B.16) appearing in this tablewith SMM, = 23.2 years, SMAM, = 25.3 years, andthe observed proportions not widowed, NWf(n), thesurvivorship probabilities listed in column (6) of table

100 are obtain ed. As an exam ple, 1 (50)/1, (20) is calcu-lated below:

Also shown in table 100 (column (7)) are the mortal-ity levels to which each estimated survivorship probab il-ity corresponds in the Coale-Demeny West family ofmodel life tables. These levels are obtained by interpo-lating linearly between the values listed in table 226 (seeannex VII). See step 3 of subsection B.2 (b) iii) for adetailed example of the interpolation procedure used.Step 4: calculation of number of years *re the surveyto which the male survivorship probabu'lities refer. Sincemale survivolship is being estimated, equations (B.17)and (B.18) are used to calculate this reference period,denoted by t (n). Table 101 shows the necessary input

TMLE100. ESTIMATION F MALE SURVlVORSHlP PROBABlLlT ES FROM NFOR MATIONON SURVIVAL O F FIRST HUSBAND. BOLIVIA 1975

warTzv(7)

14.316.316.615.715.415.313.314.7

TABLE01. ESTIMATION OF T IME REFERENCE PERIODS FOR MAL E SURVIVORSHIP PROBABILITIES DERIVE D FROMINFORMATION O N SURVIVAL OF FIRST HUSBAND. BOLIVIA. 1975

warYZvm16.316.615.715.415.313.314.7

a Number f years prior to the survey to which survivorship estimates refer.


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information and some of the intermediate results. The 1963, is also disregarded, the improvement in adult mor-observed proportions of females who were not widowed, tality is more modest (about one level) and perhapsNWf(n - ), are listed in column (2); and the values of somewhat more plausible. Unfortunately, the mortality(SMAM, + n -2.5-SMAM,) for each n are denoted levels that refer to periods comparable to those alreadyby x, ( n ) and are listed in column (3). The function Z is estimated' for Bolivia from maternal orphanhood datacalculated for each of these x (n) values by interpolat- (see tables 92 and 95) do not quite match the latter; anding in table 88. Note that the data for the first value of n , although they refer to different sexes (the orphanhood25, have been excluded from table 101 because the tabu- estimates refer to female mortality, while those derivedlated values of Z do not cover the value of x(25) = here correspond to males), the lack of consistency25.6; and although extrapolation is possible, it leads to observed suggests that at least one of these sets of esti-an estimate of t (25) that is negative, a clearly unaccept- mates may be seriously deficient. Although at this stageable outcome. However, this result indicates that in a i t seems more likely, given the trend they imply, that thepopulation where the female singulate mean age at mar- estimates derived from orphanhood information are theriage is 23 and that of males is 25, an estimate of the deficient set, further evidence is required to make amale probability of surviving from age 20 to age 25 definite decision.derived from reports of ever-mamed females aged 20-24is likely to be fairly poor, since a sizeable proportion of (c) Mdowhooddata classified by duration offirst marriage.the women in this age group are not yet married and the (i) ma equiredmean exposure to the risk of dying of their husbands is Male (or probabilities of surviving from agerelatively short. Such an indication is corroborated, in 20 to age can also be estimated from data on survivalthis case, by the fact that the mortality level associated of first spouse by duration of marriage (mess-L(25)/1m(20) is quite Out '' ine with the ured as the time elapsed since first union). The specificsubsequent estimates (see table 100). data required for the application of this variant of the

Returning to the calculation of Z(x), the procedure method are listed below:followed in the case of n = 45 is illustrated. First: (a) The observed singulate mean age at marriage,X ( 4 5 ) ~5.3+ 45.0- 2.5- 2 3 . 2 ~4.6 S UAM , for males (females). This age is estimated fromthe proportion of single males (females) in each five-

then year age group. For a detailed description of the pro-cedure used in estimating the singulate mean age atZm(44.6)=0.4(0.130)+ 0.6(0.139)= 0.135. marriage, see annex I. Observe that when widowhooddata are classified by duration of first marriage, only oneTherefore, using equation (B. 17), value of S U M is needed in each case: if male mortal-ity is being estimated from data from female respon-um (45)= 0.3333 In NWf(40)+ 2(44.6)+ 0.0037(1.7) dents, only the value of SMAM for males is required;and vice versa;= 0.3333 In (0.8735)+ 0.135+ 0.0063 (b) The proportion of ever-married women (men)= 0.0962; whose first husbands (wives) were alive at the time of theinterview, classified by five-year duration-of-marriage

and according to equation (B.18), group. These proportions are again denoted by N W(k ),where k now represents the lower limit of each durationtm 45)= (45.0- 2.5- 23.2) (1 O- 0.0962)/2.0 group, that is, NW(k) is the proportion not widowedamong respondents first married from k to k + 4 years= (9.65) (0.9038)= 8.7. before the survey. As in the case of data classified byage, the proportions not widowed can be calculated

That is, the estimate of 1 (45)/1, (20), which from any two of the following: ( 0 ) the number ofcorresponds to a West mortality level of 15.4, refers respondents whose first spouse was alive at the time ofapproximately to 8.7 years before the survey; and since the interview classified by five-~ear uration group; (b)the latter has as reference date 1975.6 (see step 4 of sub- the number of respondents Whose first Spouse was deadsection B.2 (b) (iii), the reference date for 1 (45)/1,, (20) a t Ihe time Ih e interview classified by five-year dura-is end of ,966, as shown in column (7) of table 101. tion group; (i) he total mmber of ever-married respon-

The mortality levels with the survivorship dents who declared the survivorship status of their firstprobabilities obtained in the previous step are listed in 'larsified f i v e - ~ e a r uration group. Sincecolumn (8) of table 101 to allow a quick assessment of category (c) is to be th e sum th e first two- itthe rrsulfs obtained. It is worth note that, disre- is important that it exclude all respondents who did notgarding the ntimate (/,,, (60)/L (20)). the adult mar- know Or did no t Ihe s u r v i v o n h i ~ tatus of theirtality trend followed by the other estimates seems rapid SPOUY.though not impossible, implying a gain of some three (ii) ComPutationalPr~e~remortality levels (or approximately 7.5 years in expecta- The steps of the computational procedure aretion of life) during 10 years or so. If the estimate associ- described below.ated with lm(55)/Im(20), referring to approximately Step I : calculation o proportions not widowed among

114


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female (male) respondents. The proportion of female(male) respondents not widowed in the duration groupfrom k to k + 4, denoted by NW(k), is equal to theratio of the number of female (male) respondents firstmarried from k to k + 4 years earlier whose first hus-band (wife) was alive at the time of the interviewdivided by the total number of ever-mamed female(male) respondents in the same duration group whodeclared the survivorship status of their first husband(wife).Note that male survivorship probabilities areestimated from the proportions of non-widowedfemales, whereas female survivorship probabilities areestimated from male proportions not widowed.

Step2: calculation o singulate mean age at marriage ormales (jemales). The necessary value of SMAM is calcu-lated according to the procedure described in annex I.Note that the value of SMAM required is thatcorresponding to the sex category for which survivorshipprobabilities are being estimated.

where the male proportion not widowed and the femaleSMAM replace the female proportion not widowed andthe male SUAM of equation (B.21). Note, however, thatthe estimated regression coefficients to be used alsochange; those listed in table 102 a n to be employed inestimating male survivorship, while those given in table103 should be employed in estimating the survivorshipprobabilities for females.TABLE 103. COEFFICIENTSOR ESTIMATION OF CONDITIONAL FEMALESURVIVORSHIP PROBABILITIES FROM THE WIDOWHOOD STATUS OF MALE

ltEM)NDENt$ CLASSIFIED BY DURATION OF MARRIAGE

Step 3: calculation o swviwrshipprobabilities for males Estimation equation:(jemales). When male mortality is being estimated, the l f (n ) / l f ( 2O)= o (n ) + b ( n ) S M A M f + c ( n )N W m ( n - 2 5 )~robabilitvof survival from age 20 is related to the;ingulate hean age at marriag;for males and the pro- Step o/ndr brfire theportion of women in a given duration-of-marriage group to de Miwrship poMilitier In caseswith a surviving first husband. The following equation is where has been changing regularly during theused: 15 or 20 years preceding the survey (that is, when thechange in mortality has been roughly linear on the logit1m(n)/1m(20)=a(n)+b(n)SMAMm c(n) NW/(n -25) scale), estimates of the time to which each of the sur-vivokhip probabilities obtained in the previous step(B.21) refers can be calculated using the following equations:where a(n ), b (n) and c (n) are the coefficients shown intable 102; and NWf (n -25) is the proportion of womenin the duration group from n -25 to n -21 having a sur-viving first husband. As in the case of the method basedon data classified by age, the coefficients of equation(B.2 1) were estimated by using least-squares regressionto fit this equation to data generated from model mortal-ity and nuptiality schedules (see chapter I).

An equivalent relation is used to estimate female sur-vival probabilities, namely,

TABLE02. COEFFICIENTSOR ESTIMATION OF CONDITIONAL MALESURVIVORSHIP PROBABILITIES FROM THE WIDOWHOOD STATUS OF FE-MALE RESPONDENTS.CLASSIFIED BY DURATION OF MARRIAGE

Estimation equation:lm(n) l lm(20)= (n )+b(n )SMAMm +c(n)NWf (n -25)

u,(n)= 0.3333 In NWf (n - 25)+Z(n -22.5 + SMAM,)

uj(n)= 0.3333 In NW,(n -25)+Z(n -22.5 + SMAM')

Equations (B.23) and (B.24) are used when theestimated survivorship probabilities refer to males, whileequations (8.25) and (8.26) are used when they refer tofemales. Note that the form of these sets of eqhations isessentially the same, with the subindices indicating sexcategory reversed. In both cases (males and females), thequantity n -22.5 is the mid-point of the duration groupto which the respondents belong, and the values of thefunction Z(x ) are obtained by interpolating linearlybetween those listed in table 88.

(iii) A detailed exampleThe use of data on the survivorship of first spouseclassified by duration of marriage to estimate adult sur-vivorship probabilities is illustrated by using data gath-


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ered by the National Demographic Survey camed out inPanama between August and October 1976. Table 104shows ever-mamed males classified by time since firstmamage and survival of first spouse. The survivorshipprobabilities to be estimated refer to females. Note thatin table 104 the number of men who said they had beenmamed, but who did not know the duration of theirmamage, constituted a sizeable proportion of the total(slightly over 10 per cent). This level of non-responsesuggests that the data may not be reliable: in a countrywhere some 10 per cent of the mamed male populationdoes not declare or does not know the duration of theirmarriage, .i t is unlikely both that duration will becorrectly reported by the rest and that those of unknownduration have the same characteristics as those whodeclared a duration. One possible treatment for the casesof non-response is to divide them among the differentdurations according to the proportions implied by theknown cases. This procedure would be based on theassumption that, for a given survival of first spouse(dead or alive), all persons are equally likely to fail tostate the duration of their union. However, it seems rea-sonable to suppose that people belonging to unions thathave lasted longer are more likely to have forgotten thedate when the union began, so a more complicatedredistribution procedure should probably be used. Yet,such a procedure would require that an assumption bemade about how the probability of not stating a date offirst union changes according to the time elapsed sincethe union began. Because no information is available onthis matter, the likelihood of introducing substantialbiases in the final results by making incorrect assump-tions is high. In view of this possibility, it seems safer toignore completely the cases of unknown duration whencarrying out this analysis, while bearing their existencein mind at the point of interpreting the final results.

The computational procedure for this example isgiven below.Step I : calculation of proportions not wido wd amongmale respondents. Table 104 shows all the data categoriesthat can be derived from a simple question on the sur-vivorship status of the first wife of each ever-marriedmale respondent. For some duration groups, the numberof respondents who did not declare the survivorshipstatus of first wife constitutes a sizeable proportion of

those who declared that their first wives were dead.Again, this characteristic of the data renders their qual-ity suspect, since if one were to assume that all cases ofnon-response were, in fact, cases of dead first wives, thesurvivorship estimates obtained would be substantiallydifferent, especially at lower mamage durations. Strictlyspeaking, however, there is no basis for making such anassumption; and in the rest of this analysis, the respon-dents who did not declare the survivorship status of theirfirst wife are ignored. Therefore, to calculate the pro-portion of respondents not widowed, NW,(k) in dura-tion group 10-14, for example, the number with first wifealive is divided by the sum of those with first wife aliveand those with first wife dead, as shown below:NW (lo)= 1,096/(1,096 +23)= 0.9794.

Note that NW,(lO) can also be obtained by dividingthe number with first wife alive by the differencebetween all possible respondents and those whose wives*survival status is not known. Thus,

The full set of NW (k) values is shown in column (7)of table 104.Step 2: calculation of singulate mean age at marriage forf d e s . For a detailed description of the procedure to beused in calculating the values of SMAM for females, seeannex I. In this case, SMAMf was found to equal 22.01years.Step 3: calculation of survivorship probabilities forfemales. Equation (B.22) is used to calculate the femaleprobabilities of surviving from age 20 to age n . Theseprobabilities are calculated by using as input the propor-tions not widowed among ever-mamed respondents, thefemale singulate mean age at mamage, and thecoefficients listed in table 103. Table 105 shows thedetails of the calculations. Note that the final estimatesof survivorship probabilities are not, in this case, verydifferent from the original proportions of respondentswith a surviving first wife. The West mortality level asso-ciated with each survivorship probability has beenestimated by interpolating linearly between the valuesshown in table 222 in annex VII (see step 3 of subsection

TABLE04 . DATAN SURVIVORSHIP STATUS OF FIRST WIFE.MALERESPONDENTS.PANAMA. 9 7 6

Unknown


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TABLE05. ESTIMATIONF ADULT FEMALE SURVIVORSHIP PROBABILITIESFROM MALE DATAON SURVIVAL OF FIRST SPOUSE. BY DURATION OF MARRIAGE. PANAMA, 1976

B.2 (b) (iii) for a detailed example of the procedure forcalculating them). These levels imply, in general, thatmortality among females in Panama was very low dur-ing the early 1970s.Step 4: calculatiq o monbr of ymrs before the surveyto which the male -wrshipprobabilities nfer. The Westmortality levels shown in column (6) of table 105 suggestthat mortality has not remained constant in Panama,since the reports of respondents who have been marriedfor a longer period of time are associated with lower lev-els (that is, higher mortality) than are those associatedwith the reports of respondents married for a shorter andmore recent period. It is important, therefore, to obtainsome indication about the time periods, denoted by t(n),

to which the estimated survivorship probabilities refer.Equations (B.25) and (B.26) are therefore used to esti-mate r / (n) . Table 106 shows some of the intermediatevalues required for their calculation and the final t/(n)estimates. The values of Z'(x) are obtained by interpo-lating linearly between the values listed in table 88, andx (n) is used to denote the quantity n -22.5 +SMAM,.dnce the sumy took place between August andOctober of 1976, the mid-point of this period,corresponding in decimal form to 1976.7, is used as itsreference date (see step4 of subsection B.2 (b) (iii) for anexample of how to calculate the decimal equivalent ofthis mid-point). Hence, by subtracting the estimatedr/(n) values from 1976.7, the date to which each sur-vivorship probability refers can be obtained. Thesereference dates are shown in column (7) of table 106;and to facilitate comparison, the West mortality levelsobtained in the previous step are listed in column (8) ofthe same table. Without any other evidence, the trend of

the dated survivorship estimates implying an improve-ment in adult female mortality of about a quarter of alevel per annum seems plausible, although the estimatedlevels are relatively high. Because of the deficienciesnoted earlier in this data set, the possibility that thereports on widowhood status of male respondents mayunderstate true widowhood levels must be exploredfurther before the estimates shown in table 105 can befully accepted.(d) Ukeof wi&nvhoid&a fmtw surveysMortality estimates from widowhood share with thosefrom orphanhood the problem of referring to periodslocated a substantial length of time before the survey,and the time reference of the estimates can only beidentified when mortality has been changing regularly.A method for obtaining estimates or mortality that referto a well-defined period of time by using orphanhooddata from two surveys has been suggested earlier.Strictly speaking, a parallel approach cannot be usedwith widowhood data classified by age of respondentbecause age group do not define marriage cohorts; andthe change in proportions not widowed from one agegroup to the next is generally deflated by the influx ofnewly married respondents whose exposure to the risk ofdying while married is below the average of the group.However, the same phenomena affect the proportionsnot widowed by age group at each survey; thus, as longas age patterns of marriage remain more or less con-stant, the proportions not widowed for a hypotheticalintersurvey cohort will be representative of the intersur-vey mortality experience. These proportions can be cal-culated following the method described in the case of

TABLE06. ESTIMATIONF TIME REFERENCE PERIODS FOR FEMALE SURVIVORSHIP PROBABILITIESDERIVED FROM MALE WIDOWHOOD DATA, PANU 1976

' umber of years prior to the survey towhich survivorship estimates rrfer.


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orphanhood data (see subsection B.2 (d)) and can thenbe analysed by using the method described in subsectionB.3 (b), with average male and female singulate meanages at mamage during the intersurvey period as addi-tional inputs (these average ages can be calculated as themean of those corresponding to the beginning and endof the intersurvey period).On the other hand, when widowhood data areclassified by time elapsed since first marriage, five-yearduration groups do represent mamage cohorts; and pro-portions not widowed for a hypothetical mamage cohortrepresenting intersurvey experience may be calculatedfrom two sets of observed proportions not widowedclassified by duration and obtained from surveys or cen-suses five or 10 years apart. The calculation of the pro-portions not widowed representing the experience of ahypothetical intersurvey cohort is carried out exactly asdescribed in step 2 of subsection B.2 (d) (ii) for the caseof proportions not orphaned, with duration groups sub-stituted throughout for age groups. Because the com-plete computational procedure to be followed in thiscase is very similar to that followed in the case of dataon orphanhood and because no data sets on widowhoodby duration from two surveys are as yet available, adetailed description of the application of this method isomitted.

C. ESTIMATIONF SURVIVO RSHIP TOADUL THO OD FROM BIRTH

1. General chamcteristics o methodrIn this section, an extension of the conditionalmethods discussed in section B is presented. Themethods that are described allow the estimation of prob-abilities of survival from birth, that is, of probabilitiesthat are no longer conditional. These probabilities areestimated by means of equations that include 1(2), theprobability of surviving from birth to exact age 2, as anindependent variable, to take account of mortality inchildhood. In practice, the value of l(2) used is almostinvariably obtained by applying the child mortality esti-mation techniques described in chapter 111.Since the methods to be presented here are extensionsof the conditional methods of adult survivorship estima-tion described above, they are based on very similarassumptions. In fact, all the comments made in sectionB about the validity of the estimates of conditional sur-vivorship probabilities apply also to the validity of esti-mates of direct probabilities; and because the calcula-tion of the latter values demands extra information, for

example, 1 2), certain other assumptions are necessary.The methods presented here are based on equationsfitted by least-squares regression to simulated data, sothe main assumption made is that the models used insimulating data represent reality adequately. An addi-tional assumption necessary in this case is that the rela-tionship between adult and child mortality embodied in

the experience of the population studied falls outside therange covered by the models used. The estimation ofadult survivorship from birth is particularly sensitive tothe relationship between child and adult mortality, andto the pattern of the mortality schedule between ages 5and 25. This sensitivity can be reduced by including inthe estimation equation an independent variable allow-ing for known variations in mortality between ages 2and 20 or 25. The model cases to which the estimationequations were fitted were generated by using mortalityschedules based on the logit system, using as standardsthe four life tables for females at level 16 of the Coale-Demeny models. Therefore, the probability of survivingfrom age 2 to age 20 (or 25 in the case of orphanhood) inthe appropriate standard life table is included as anindependent variable in the estimation equation in orderto allow for the age pattern of mortality between child-hood and early adulthood. It remains true, however, thatestimates of survivorship from birth based on orphan-hood or widowhood data from younger respondents areoverwhelmingly determined by the estimation of 1(2),while the proportions not orphaned or not widowedhave a relatively low predictive value,'(' a fact that mayaffect the consistency of the estimates obtained fromdata corresponding to younger respondents.It has

manual x chapter 4

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