ST/SOA/Series A/42

Manuals on methods of estimating

MANUALIV Methods ofEstimating Basic Demographic Measures from Incomplete

UNITED NATIONS Department of Economic and Social Affairs POPULATION STUDIES, No. 42

Manuals on methods of estimatingpopulation

MANUAL IV Methods ofEstimating Basic Demographic Measures from Incomplete Data

UNITED NATIONS New York, 1967 NOTE

Symbols of United Nations documents are composed of capital letters combined with figures. Mention of such a symbol indicates a reference to a United Nations document.

ST/SOA/Series A/42

UNITED NATIONS PUBLICATION

Sales No.: 67. XIII. 2

Price: $U. S. 2.00 (or equivalent in other currencies) FOREWORD

The United Nations manuals on demographic methodology, as a of disseminating international experience, have a history extending over more than a decade.Asa firststage,the Population Divisionofthe United Nations prepared a series of manuals on methods of derivingpopulation estimates.Manual I dealt with the meth­ ods of estimating total population for current dates. Manual II described the proce­ dures for appraising the quality of basic demographic data. Manual III presented methods of calculating future population estimates by sex and age. While those manuals have found widespread use, the Population Commission expressed the need for additional technical manuals dealing with methods of esti­ mating fundamental demographic variables, particularly in countries where the necessary are incomplete. A technical study was prepared entitled The Concept of Stable Population-Application to the Study of of Countries with Incomplete .t As a further step in assisting technicians in developingcountries, the Population Commission recommended at its twelfth session that "at the earliest possible date a manual should be prepared on methods of esti­ mating fundamental demographic measures from incomplete data 2 ". Owing to the long-standing interest and international experience of the Office of Population Research, Princeton University, in the development and application of methods of estimating fertility and mortality from defectivedata, that office was asked to under­ take the preparation of the Manual for the United Nations. The present study, which is the fourth in this series of manuals, is the result. This Manual was written by Professor Ansley J. Coale, Director, Office of Popu­ lation Research, Princeton University, and Professor Paul Demeny of the same office, with valuable assistance from Mr. R. D. Esten, Mrs. Erna Harm, and Mr. S. B. Mukherjee. In recognizingthe important contribution of the two authors, the United Nations also wishes to mention the help rendered by Princeton University, whose facilities, including those of the electronic computer, were utilized in the work. The tables in annexes I and II are taken from Regional Model Life Tables and Stable Populations, (Princeton University Press, copyright 1966). The Office of Population Research has also provided the graphs included in the Manual. ".

1 Population Studies, No. 39, United Nations publication, Sales No.: 6S.XIII.3. 2 Official Records of the Economic and Social Council, thirty-fifth Session, Supplement No.2, para. 48.

iii

CONTENTS

Chapter Pages

INTRODUCTION . 1

Part One. Methods ofestimation I. METHODS OF ESTIMATION BASED ON RECORDS OF AND DISTRIBUTION BY AGE ...... • • . . •. 7 A. Estimation of mortality from survival rates and the consequent estimation of birth and death rates ...... 7 1. Model life tables...... 7 2. Selection of a model life table consistent with census survival rates 8 B. Estimation of fertility and mortality by stable population analysis when fertility and mortality have been constant ...... 12 1. Model stable populations ...... 14 2. Selecting a model stable population on the basis of an accurately recorded age distribution ...... 15 3. Characteristic forms of age-mis-statement. .... 17 (a) Female age distributions with large distortions . 19 (b) Female age distributions with smaller distortions 21 (c) Distortions in male and female age distributions 21 4. Selecting a model stable population on the basis of a distorted age distribution ...... 22 5. Assigning the characteristics of a model stable population . . . . 23 C. Adjustment of estimates based on model stable populations for the effectsof recent decreases in mortality ...... 25 D. Concluding remarks on estimates adjusted for the effects of recent declines in mortality...... 28 E. The estimation of fertility from the age distribution recorded in one census...... 29

II. METHODS OF ESTIMATION BASEDON RESPONSESTO QUESTIONS ABOUT FERTILITY AND MORTALITY • ...... • ...... 31 A. Estimation of fertility from reports on childbearing in the past . . . . 31 B. Estimates of mortality based on proportions surviving among chil­ dren ever born ...... 34

III. ESTIMATES OF FERTILITY AND MORTALITY BASED ON REPORTED AGE DISTRI­ BUTIONS AND REPORTED CHILD SURVIVAL. ..•..•...... 37 A. Estimation of birth and death rates from childhood survival rates and a single enumeration by reverse projection ...... 37 B. Estimation of birth and death rates from child survival rates and a single enumeration by model stable populations...... 38 C. Estimation of birth and death rates from child survival rates and age distribution in a population enumerated several times ...... 39 1. Estimation of birth and death rates in a non-stable population . . 39 2. Estimation of birth and death rates in a stable population enumer­ ated more than once ...... 39 D. Adjustment of estimates of fertility derived from child survival rates and the age distribution when mortality has been declining 40

IV. ACCURACY OF ESTIMATION • . . . • • . . . . • . . . . . • . . . • 41 A. Differencesbetween assumed and actual conditions ...... 41 1. Errors arising from differences between the actual age pattern of mortality and that embodied in the model life tables ...... 41 v ,t

Chapter Pages (a) Effects ofassumed age patterns ofmortality on estimates derived from census survival rates ...... 41 (b) Effects ofassumed age patterns ofmortality on estimates derived from stable populations chosen on the basis ofC (x) and r. .. 42 (c) Effects ofassumed age patterns ofmortality on estimates derived from reported child survival in combination with records ofthe age distribution originating from one census, or from two or more ...... 44 2. Errors caused by non-stability ofa population assumed to be stable 46 B. Errors caused by faulty data ...... 48 1. Differential rates of omission in consecutive censuses ...... 48 2. Age-misreporting in censuses or surveys...... 49 (a) Age-mis-statement and mortality estimation by census survival 49 (b) Age-mis-statement and stable population analysis ...... 50 (c) Age-mis-statement and the estimation of fertility and mortality from special questions on past experience .. 50 C. Suggestions for best estimation ...... 51 V. DATA USEFUL FOR ESTIMATES OFFERTILITY AND MORTALITY 53 A. Data on age ...... 53 B. Data on children ever born 53 C. Data on the age structure of fertility 54

Part Two. Examples ofEstimation VI. EXAMPLES OF ESTIMATES BASED ON RECORDS OF POPULATION GROWTH AND DISTRIBUTION l3Y AGE ...... 57 A. Estimation ofmortality and ofthe birth rate from census survival rates 57 B. Estimation of fertility and mortality by stable population analysis 61 1. England and Wales, 1871 ...... 61 2. India, 1911 ...... 65 3. Brazil, 1950...... 67 C. Estimation offertility and mortality by stable population analysis when the population is quasi-stable. 68 1. India, 1911 ...... 68 2. Mexico, 1960 ...... 70 VII. EXAMPLES OF ESTIMATES BASED ON QUESTIONS ABOUT FERTILITY AND MOR- TALITY ...... 73 A. Estimation of fertility from reports on childbearing in censuses or surveys ...... 73 B. Estimation of mortality from reported numbers of children ever born, and children surviving ...... 74 VIII. EXAMPLES OF ESTIMATION BASED ON CHILD SURVIVAL AND AGE DISTRIBUTIONS 76 A. Estimation of fertility and mortality from data in a single census that records the age composition of the population, the number ofchildren ever born and the number surviving...... 76 B. Estimation of fertility and mortality from data on age distribution, the intercensal rate ofincrease, and survival rates ofchildren ever born 77

.Annex Part Three. A1UIexes I. Model life tables 81 II. Model stable populations . 95 III. Tables for adjusting stable estimates for the effects of declining mortality 119 IV. Tables for estimating cumulated fertility from age-specific fertility rates 124 V. Tables for estimating mortality from child survivorship rates 125 VI. Anote on interpolation .. .. 126 vi INTRODUCTION

There is a serious gap between the quantitative infor­ the basis of reasonably complete and accurate current mation about populations that is essential for many statistics. purposes and the amount and quality of data actually The only satisfactory method of recording the demo­ available. Among the kinds of fundamental information graphic data needed in the administration of a modern needed about populations are age and sex composition country is a sequence of carefully designed and accurate and fertility and mortality rates. Data of this sort are censuses, supplemented by frequent sample surveys to needed in assessing problems and formulating plans for collect social and economic data required at short inter­ health, education, employment, social services and many vals, and by the prompt and complete registration of other critical functions of government and private organi­ births, deaths, marriages, and dissolutions of marriage. zations. In countries with fully developed statistical Individual records are needed for many purposes-by systems data on the state of the population are obtained the courts in adjudicating inheritance, by administrators of from censuses at decennial intervals or less, often supple­ health programmes, by individuals in establishing proof mented by large-scale sample surveys; and information of citizenship, age or nativity etc. But the establishment about births, deaths, marriages and divorces is usually of complete registration of vital statistics requires an obtained from the continuous registration of these events extensive administrative apparatus and a thorough re­ as they occur. When a national system of education of the public, and, even if given high priority, has been operating for many years in a country where cannot be achieved in less than one or two decades in almost all persons of school age and beyond are literate, the less developed countries. In the meantime, it is the demographic information needed for administrative essential to make use of data already available to deter­ purposes and for social and economic planning is usually mine at least approximately the principal demographic adequate; but in less developed countries the registration characteristics. of vital events is often nearly non-existent or at best only fractional in its coverage, and information about the In some countries there has been an effort to establish population contained in censusesor demographic surveys registration of births and deaths in a sample list of villages is often unavoidably deficient particularly with regard and urban areas, and in others attempts to collect vital to the reported ages of the population. statistics by periodic surveys.Such methods of continuing The disadvantages of not possessing adequate demo­ measurement are a promising interim substitute for graphic data in developing countries are becoming ever complete registration while the latter is being designedand more acute. Governments in these countries are expand­ instituted. But there is still a third source of estimates of ing their efforts to promote social and economic develop­ population parameters: data on age and on growth ment, but they cannot formulate practical plans and usually contained in or derivable from population censuses programmes without data on the current size and compo­ or broad-purpose surveys, plus information on recent sition of the population, and on the rate of increase of fertility and mortality that is sometimes included. The the total population and of various important subgroups, subject of this Manual is estimation from this last kind such as children of school age and persons in the ages of of information. principal economic activity. Specifically, planning agencies The original plans for this Manual, when work on the first are finding some form of population projections an draft began, were for a non-technical, do-it-yourself set almost indispensable tool for their work. Moreover of instructions for methods of estimation that have already the importance of population data is accentuated because been widelyemployed and discussed,but are not accessible of the rapid changes in population that are taking place. to the majority of demographers and because The precipitous fall in the death rate that many less detailed instructions for their use have not been published. developed countries have experienced, combined with a The original purpose was to show in a simple non­ continued high level of fertility, has produced a rapid and technical manner how to use stable population analysis, accelerating increase in numbers. One consequence has and how to modify this analysis when mortality had been that in many countries each census has tended to recently been declining rather than remaining constant. provide a considerable surprise in revealing the unexpec­ There was already a substantial body of published material tedly great increase in the population since the preceding on stable populations, a United Nations manual expound­ census. In other words, the assumption that population ing stable population theory and a recent book tabulating remains more or less as it was in the most recent census model stable populations; but this material contains is no 'Ionger serviceable. Another result of the rapid rather complicated mathematical discussion, and is not acceleration of population growth in some developing designed to guide the estimator to the most efficient countries has been the formulation of policies intended methods of extracting information from data of various to affect the rate of increase itself. Obviously such policies kinds. There was therefore an apparent need for a can be designed and implemented intelligently only on manual that would make it possible for a demographer- 1 with only a moderate levelof training, perhaps choose a model life table, the analysis of typical patterns working in isolation in a provincial capital of a less of age misreporting in different populations in under­ developed country, to derive the maximum of reliable developed countries, the estimation of total fertility from information from data in a census or demographic survey. the average parity of women 20 to 24 and 25 to 29, and the discussion of the susceptibility of the various methods The process of writing the manual, however, disclosed to different sorts of error-that it is hoped will be worth­ the advisability of extending its scope to include a while reading for professional demographers. number of different methods of analysis and estimation, and since some of these were developed after the work There is one respect, at least, in which the examples had started, and are therefore not described anywhere fall short of the procedure that we would recommend else,it was essentialto provide a fuller and more technical for the construction of optimum estimates. A thorough­ exposition at some points than was originally intended. going examination of primary sources, of the nature of It is still hoped that all the methods can be employed by the individual censusesand surveys (wording of questions, persons with no more than a one-year special course in evidence of completeness of coverage, and the like) . To minimize the difficulty of actual esti­ should be part of the effort to arrive at the best possible mation, there are assembled in part two a set of examples approximation of population parameters. Since the in which actual calculations of the most important forms purpose of the examples is to illustrate the mechanics of of estimation are fully worked out. These examples estimation, data have been taken from the Demographic contain references to the discussion in part one. A person Yearbook of the United Nations, except where otherwise interested in obtaining figures for a specific population noted, without examination of census procedures, inter­ can follow the procedures in the appropriate sections of view schedules, and the like. part two, and read perhaps only a few pertinent pages The following table is provided to help the estimator to from part one. But there are some wholly or partly new find the examples and the discussion relevant to his points in part one-the use of census survival ratios to purposes.

GUIDE TO USE OF MANUAL: METHODS OF ESTIMATION SUITABLE FOil VAIlIOUS KINDS OF DATA

MetluHl01est/_tlolt Paramete,. Descriptlolt Discussiolt Examples est/_ed olmetluHl 0/ pm:islolt

I Two or more censuses, with Selection ofmodel life table 0eo, 0es, b, d Chapter IA Chapters Chapter VIA age distributions duplicating census survival by IVA,IVB population projection

2 Two or more censuses with Selection of model stable popu- °eo, °es, b, d, and Chapter IB Chapters Chapter VIB age distributions, plus evidence lation from ogive of age adjusted age IVA,IVB ofconstant fertility and distribution and intercensal distribution mortality rate of increase

3 Proportions married by age, Use of "standard" age schedule m-themean Chapter IB Chapter VIB evidence of small proportion of of marital fertility age of the ferti­ births to non-married women lity schedule and little use of birth control

4 Average number ofchildren Use of regression equation Chapter IB ever born to women 20-24 and 25-29 (P2 and Pa)

5 Data listed in (2), plus those Determination of ORR in model Gross reproduc- Chapter IB Chapter VIB listed (3) or (4) stable population tion and total fertility

6 Two or more censuses with age Selection of model stable popu- b, d, °eo, GRR, Chapter IC Chapter Chapter VIC distributions, and evidence of lation, estimation ofrate and TF IVB constant fertility and declining duration ofmortality decline, mortality (accelerating popu- adjustment ofparameters in lation growth, or changing age model stable population composition of deaths)

7 Singlecensus with age distribu- Selection ofmodel stable popu- b, GRR, TF, Chapter ID Chapter don, rough guess ofr or 0eo lation ID

8 Children ever born and births Adjustment of reported fertility GRR, TF, b Chapter IIA Chapter Chapter VIlA last year, by age ofwoman rates to match reported parity IVB ofyounger women

2 GUIDE TO USE OF MANUAL (continued)

NaturtJ 01data Method 01estll1UJtlon PIR_"" Description Dlscussloll Exomples estlmatM ofmethod 01prtlClslon

9 Children ever born tabulated Regression ofTF/Pa on Pa/P2 TF Chapter IIA Chapter by age ofwoman (Pl, P2, ..., IIA P7)

10 Children ever born, and Selection of adjustment factors lqO, 2qO, aqo, liqo, Chapter liB Chapters, Chapter VIIB surviving children, by age of converting proportion dead loqO, 15qO, 20QO IIB,IVB woman among children ever born to nQO

11 Age distribution, children ever Reverse projection based on b, d, r, °eo ChapterIHA Chapter born and children surviving, model life table selected from IVA by age ofwoman estimated /2

12 Same as (11), plus evidence of Selection of model stable popu- b, d, r, 0eo Chapter nIB Chapters Chapter constant fertility and mortality lation from ogive of age adjusted age IVA,IVB VIllA distribution and estimated /2 distribution

13 Two censuses with age distribu- Life table up to 5 from child b, d, r, Deli, 0eo Chapter IIIC Chapter tion, children ever born and survival, above 5 by IVA surviving children, by age of census survival woman

14 Same as (13), plus evidence of Fertility estimated from model b, d, r, GRR TF Chapter IllC Chapters Chapter constant fertility and mortality stable population matching IVA,IVB VIllB C(x) and /2, death rate as b-r

15 Same as (14) Mortality under age 5 estimated 0eo, 0eli Chapter IllC Chapters Chapter from child survival, above IVA,IVB VIllB age 5 from C(x) and r

16 Two or more censuses with age Same as (14), adjusted for effect b, d, GRR, TF Chapter IllD distributions, children ever born ofmortality decline and surviving children, by age of woman, plus evidence of censtant fertility and declining mortality

3

Part One METHODS OF ESTIMATION

Chapter I

METHODS OF ESTIMATION BASED ON RECORDS OF POPULATION GROWm AND DISTRIBUTION BY AGE

There are many populations that have no usable direct There is a substantial literature on the construction of records of births or deaths, but have been enumerated life tables from census survival rates, and the method has in one or more censuses in which the age and sex of each been applied to data from India, Egypt, Brazil and other person was recorded. In this chapter methods of esti­ countries in Latin America. 1 The demographers and mating fertility and mortality from such census data are actuaries using census survival rates have been forced described. The underlying rationale of such methods is to adjust the original census age distributions before that the growth and age structure of a population are calculating survival, or to adjust the rates after calcu­ determined by the mortality, fertility, and external lation, because of the usual effects of age-misreporting migration to which it has been subject, and consequently combined with differential omission by age. Unadjusted the possibility exists of estimating those forces from survival rates that are over one or that are absurdly records of the evolving size and structure of the popu­ low are common. The methods of adjusting the age lation. distributions (or the raw survival rates) to remove the The description of estimation in this chapter proceeds effects of age-mis-reporting are essentially arbitrary, from methods applicable to any closed population (or and when the reported age distributions are seriously to any for which records of gains and losses by migration distorted, the age pattern of mortality embodied in the exist) to methods applicable to closed populations with estimated life table contains a strong component of the special histories-of essentially unchanging fertility and smoothing procedure used as well as of the actual age mortality, and of unchanging fertility combined with schedule ofmortality. declining mortality. Once a life table-however approximate it may be in form above age five, and in level ofchildhood mortality­ has been constructed from census survival rates, it can A. ESTIMATION OF MORTALITY FROM CENSUS SURVIVAL be combined with additional data from the two censuses RATES AND THE CONSEQUENT ESTIMATION OF BIRTH AND to provide estimates of birth and death rates during the DEATH RATES intercensal interval. A good approximation to the death rate (on the assumption that the censuses are accurate Suppose that a closed population is enumerated in two and the life table valid) can be obtained by calculating censuses at an interval of exactly ten years, and that each the average of the two age distributions and applying census contains tabulations of males and females by age, the me values from the life table. The birth rate can then in five-year intervals. Each cohort enumerated in the be estimated by adding the average annual rate ofincrease first census is counted again ten years later, and it is a to the estimated death rate. simple matter to calculate the apparent fraction of each cohort surviving the decade. Thus the ratio of persons 20 to 24 in the later census to those 10 to 14 in the earlier 1. Model life tables is equivalent to SL20!sLIO in a life table representing The two problematic aspects of constructing a life the mortality risks of the intercensal decade. A sequence table from survival rates are the estimation of infant and of life table values can be based on the sequence of child mortality, and the determination of the age pattern calculated census survival ratios, and by well-tested actuarial procedures, a life table can be constructed for ages above five-provided that the two censuses achieved 1 For general methodological discussion, practical application and for further references to earlier writings, see Clyde V. Kiser, accurate coverage of the population, and that ages were "The Demographic Position of Egypt", The Milbank Memorial accurately recorded. However, this procedure does not Fund Quarteriy, vol. XXII, No.4 (October 1944), pp. 383 to 408; yield estimates of survival rates in infancy and childhood Giorgio Mortara, Methods of Using Census Statistics for the Calculation of Life Tables and Other Demographic Measures (with unless the number of births during the intercensal decade Application to the PopulationofBrazil), see United Nations publica­ has been recorded. Therefore, when, as is usual, adequate tion, Sales No.: 50.XIII.3; Kingsley Davis, The Population of records of births are lacking, a complete life table can be India and Pakistan (princeton, Princeton University Press, 1951) based on census survival rates only by estimating infant pages. 238 to 242; Hugh H. Wolfenden, Population Statistics and Their Compilation (University of Chicago Press, 1954), pp. 115 and child mortality indirectly-for example, by assuming to 117, and Jorge Somoza, "Trends of Mortality and Expectation a typical relationship between mortality rates under age of Life in Latin America", The Milbank Memorial Fund Quarterly, five and rates for persons over five. vol. XLIII, No.4 (October 1965), part. 2, pp. 219 to 233. 7 of mortality above childhood from data distorted by life tables rather than the earlier United Nations tables. age-misreporting. A convenient solution to these diffi­ In most instances estimates based on these model life culties is provided by model life tables once suitable tables are little different from those that would be obtained models have been calculated and published. An abridged from the earlier United Nations life tables; however, set of such tables is reproduced in annex 1. The tables are the absence of associated model stable populations means brieflydescribed in this section, and their use with census that estimation from the earlier United Nations life tables survival rates outlined in the following section. would be much more laborious. It has often been observed that the mortality risks This is not the place for an extended description of experienced by different age-and-sex-defined segments alternative possible forms of model life tables, nor even of of a population are interrelated: i.e., if death rates are the four families of which one was chosen for use in this relatively high among (for example) middle-aged women Manual. Three of the sets summarize mortality patterns in a given population, the normal expectation is that characteristic of regions of Europe, and the fourth­ infant mortality is also relatively high. There is a great the one partially reproduced in annex I, the so-called deal of statistical evidencein support of this commonsense " West" family-expresses an age pattern of mortality re1ationship-a relationship expressing the fact that when common to twenty-one countries (Australia, Canada, health conditions are especially good or especially poor Israel, Japan, New Zealand, South Africa, Taiwan, the for one group in the population, conditions tend to be United States and thirteen in western Europe). The age good or poor for other groups as well. specific mortality rates in this set of model tables are The result of a tendency for death rates experienced by matched quite closely at the appropriate mortality level different groups to be uniquely related would be that by the published life tables of these twenty-one rather death rates for all age-sex groups but one could be diverse populations. The 125 life tables for each sex precisely estimated from knowledge of the mortality of from which these model tables were calculated were that one group. It would then be possible to construct selected because they showed no systematic tendency to a set of life tables that stated the proportions surviving deviate from a preliminary set of model tables designed from birth to each age under mortality.conditions ranging to express recorded world experience. In contrast, from the highest to the lowest death rates observed in each of the other families of model tables was based on human populations, and a life table appropriate to a regional patterns of consistent and persistent deviation givenpopulation could then be chosen (with interpolation, from average world age patterns of mortality. For if necessary) from this set of model tables. example, one regional set is based on life tables from contiguous populations in central Europe with a persistent Of course the mortality experienced by different popu­ tendency towards unusually high infant mortality at lations is not in fact so perfectly uniform. Although there each level of adult mortality. is a strong general tendency for relatively high rates to occur among all segments of a population if they occur The modellifetables in annex I can be logicallyemployed in any, there are populations with especially high (or to construct an approximate schedule of mortality for low) rates at certain ages, for one or both sexes. Various a population with an unknown age pattern of mortality approaches have been tried in efforts to express in analy­ provided there is no specific evidenceof an unusual pattern. tical or tabular form the variety of frequently observed Of course the absence of specific evidence of an unusual sex and age patterns of mortality. 2 The most widely used mortality pattern does not imply that the"usual" pattern model tables are those previously published by the United in fact prevails. It is highly probable that, if accurate Nations, and these might have been (but were not) records of mortality existed for all populations, patterns employed in this Manual; instead, a set of model life of deviation would be found different from and more tables rather closelyresemblingthe earlier United Nations extreme than in the four regional families. Nevertheless, tables were employed-a set based on a large body of the existence of high intercorrelations among mortality accurately recorded national mortality experience con­ rates at different ages, and the existence of patterns to forming closely to a single pattern of death rates by age. which many mortality schedules closely conform provide This group of model tables is one of four calculated at the soundest empirical basis of estimation available at the Office of Population Research, Princeton University.3 present. The selection of this set of model tables-extracts from a more extended set appear in annex I-for use in this 2. Selection of a model life table consistent with census Manual was based primarily on convenience. The model survival rates life tables reproduced in annex I are accompanied in annex II by a set of model stable populations that include If two consecutive censuses of a closed population were a wide of useful parameters, and it is the existence perfectlyaccurate, and if the mortality scheduleexpressing of these auxiliary tables, already calculated on an elec­ average experience during the interval conformed exactly tronic computer, that dictated the use of this set of model to the model life tables in annex I, the appropriate table could readily be located by comparing the values of + 101 2 see Age and Sex Patterns of Mortality, Model Life Tables sLx sLx in the model tables with the corresponding for .Uuder-Developed Countries (United Nations publication, survival rates calculated from the two censuses. Under Sales No.: 55.XIII.9); Methods for Population Projections by these hypothetical circumstances all of the survival rates Sex and Age, Manual III United Nations publication, Sales No.: would fall between the values for the same pair of model 56.XIII.3) ;and other tabulations cited in the work listedin foot-note 3. tables, and an interveningmodel table could be constructed 3 A.J. Coale and Paul Demeny, Regional Model Life Tables and Stable Populations (Princeton, Princeton University Press, 1966). by interpolation. These circumstances are, for instance, 8 rather closely approximated by the censuses of Korea of estimating the level of mortality from a sequence of before the Second World War as indicated by the com­ erratic rates based on erroneous age distribution would parison of female survival rates from 1925 to 1935 in be to determine the level implied by the survival rate of that country with survival rates found in model life each cohort, and then to take some sort of average of the tables shown in figure I. levels so indicated. Specifically, the mortality levels in In most censuses of populations without vital statistics the model tables indicated by the census survival rates (so that mortality must be estimated) the recorded survival of persons 0-4, 5-9, .. " 40-44 in the earlier census could rates are highly erratic in the mortality level they indicate be determined, and ranked from highest to lowest indi­ because of the effect ofage-misreporting. As an illustration cated level, and the median selected as the final estimate. of such situations figure I also shows female survival rates (It is prudent to avoid survival values for persons above calculated from Indian and Turkish censuses 4. One way sixty or seventy because of the prevalence of systematic age-misreporting among older persons, and the possibility of special age patterns of mortality.) In fact, as figure I 4 Note the striking similarity in pattern in the sequence ofapparent survival rates for Turkey and India, doubtless reflecting a basic indicates, the individual survival values are often so similarity in the pattern ofage-mis-statement. See section B.3 below, affected by age-misreporting that many of the survival

PROPOR1"ION SURVIViNG ·.:0.. 1.2 · . ; \rURKEV, 1935-1945

; I. 1.1 \ :.. . :., \ : / :. .\ :/ . ;' \ I

.9 • WES-r,' LEVEL"1 j ~("45.01 Il ,\ \ ~....-.,~. c . : ..' \", / i ! \ .8 i \./ \.... ! ~\ \.,.f .7

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Figure I. Census survivorship rates of females from age x to x+5 at time t to age x+10 to x+15 at time t +10, according to censuses of India, Korea and Turkey, and according to selected ..West" model life tables 9 rates are outside the limits of the model tables (which influenced not only by mortality but also by the age extend from an expectation of life at twenty to one of distribution of the population in question. The compu­ seventy-five years), and little confidence could be attached tational procedure that permits the translation of these to the median of such a wildly erratic sequence. It is better rates into mortality levels requires that the initial popu­ to take advantage of the dampening effect that cumulation lation be projected to the later date by applying the has on age-misreporting, and to try to determine the level survival rates of model life tables at different levels of of mortality from the proportions surviving from the mortality, e.g., levels 3, 5, 7 and 9. Each projection yields entire earlier population to age ten and over in the later a ratio of the surviving population over ten to the initial census, the proportion five and over that survives to population, of the surviving population over fifteen to age fifteen and over, and so on. Survival rates ofthe latter the initial population over age five, etc. By comparing type (calculated from the same census data from which the recorded survival ratios with those obtained by projec­ the cohort survival rates plotted in figure I were obtained) tions with alternative model tables, one obtains a series are shown in figure II. Naturally, unlike simple cohort of estimates of the level of mortality--a series consistent survival rates, such rates cannot be directly expressed with the recorded survival of the whole population, the in terms of mortality levels since their value is significantly population five and over, ten and over, etc., rather than

_PORTION SURVIVING

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.8 ...... ,KOREA.1925-1935 ...... ~ .7

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.1

o 5 10 15 20 25 30 35 40 45 50 55 60 65 AGE

Figure II. Census survivorship rates of females from age x and over at time t to age x +10 and over at time t+10 according to censuses of India, Korea and Turkey 10 with the recorded survival of individual cohorts. 5 The Near East, plus India, Indonesia and Pakistan) have sequence of mortality levels obtained in this manner is census age distributions by five-year intervals that are much less affected by age-misreporting than the series quite substantially distorted by age-misreporting, in a based on the recorded survival of individual cohorts. pattern that has many common features. In contrast, The reduced effectof age-misreporting is seenin a com­ censusesin the Philippines and Latin America have five­ parison of figure III and figure I. Determining the level year age distributions that are much less distorted, and of mortality by examining the survival rates of large censuses in parts of Asia, including China (Taiwan), segments of the age distribution minimizes the distorting the Republic of Korea and Thailand have five-year age effect of age-misreporting because the survival rates distributions that appear distorted only to a minor are not affected by age misreporting within the groups extent by age-misreporting. whose survival is calculated. For example, the survival The large distortions in the African-Indian-Indonesian­ ratio of the population over twenty in the later census Pakistani censuses that the application of the to the population over ten in the earlier is distorted by method described above of determiningthe levelof mortal­ age-mis-statements that transfer persons across twenty ity from census survival rates produces a series of esti­ in the later census and across age ten in the earlier, but is mates with a rather wide range, and with a characteristic unaffected by all other forms of age-mis-statement.6 sequence of ups and downs. The sequence is consistent This method of estimating the level of mortality pro­ with the characteristics of age-misreporting (overestima­ duces a series of alternative figures that are confined to a tion of the age of late adolescent girls and young women, narrow range when age-misreporting is mild, but that for example) discerned by stable population analysis. vary extensively when age-misreporting is extreme. In This analysis suggests that certain survival rates are section B of this chapter stable populations methods are overstated and others understated for these populations, used to show that certain populations (including many and that the level of mortality estimated from such rates in tropical Africa, some in northern Africa and the is biased. Even when ratios with predictable biases are discarded, the remaining censussurvivalrates may indicate levelsof mortality that would lead to estimates of a death a A worked-out example of this method is given in chapter VI. rate differing by several points. The range of mortality 8 Age-misreporting can, however, affect the number of survivors levels consistent with census survival rates in Latin projected with a given life table, and hence influences the level of mortality that matches the reported number surviving. Specifically, America is typically much smaller and in censuses little overstatement of age for persons past middle age reduces the disturbed by age-misreporting-such as in Korea from projected number of survivors, and leads to the selection of a model 1925 to 1940-the mortality levels indicated by different life table with too low mortality rates, or too high an expectation survival ratios are confined within a narrow range. The of life at birth. This effect is important only in the projection of the population that is older than forty and over in the earlier most satisfactory rule of thumb appears to be the selection census. of the median level of mortality indicated by the propor-

MORTALlnr------:------... LEVEL ~. 21 70

65

II 45

9 40

Figure III. Mortality levelsin "West" model life tables implied by female survival rates from age x and over at time 1 to age x+l0 and over at time 1+10 according to censuses of India, Korea and Turkey 11 77"

tion surviving among the first nine groups-i.e., all the birth rate by adding the intercensal rate of increase persons, persons five and over, ten and over, ... , forty to the estimated death rate, the uncertainty surrounding and over. the level of infant and child mortality affects the estimated This procedure permits the selection of a model life birth rate precisely as it affects the estimated death rate. table consistent with the proportions recorded as surviving from one census to the next. It is possible to use the same B. ESTIMATION OF FERTILITY AND MORTALITY BY STABLE procedure to select a table from other families of model POPULATION ANALYSIS WHEN FERTILITY AND MOR­ life tables embodying other age patterns of mortality. TALITY HAVE BEEN CONSTANT It is interesting to note that experimental comparisons of life tables chosen in this way show that the expectation Populations subject to approximately constant mortality of life at ages five, ten and fifteen in model tables selected and fertility schedules come to have the age composition from the four families differs very little but that the characteristic ofAlfred J. Lotka's stable populations. expectation of life at birth varies widely among the life This Manual is not the place for a summary of the tables selected. Also, the death rates estimated by applying extensive literature on stable populations;" instead it the selected model table to an estimated mid-decade tries only to show when and how useful estimates of population differ widely from one family of model tables fertility and mortality can be based on stable population to another, while the estimated death rates for the theory. The question of when estimates can be based on population over age five are very nearly the same. The stable analysis is easily answered in principle: whenever reason for these similarities and differences is that the fertility has been subject to no more than low-amplitude census survival method establishes essentially the and short duration variations during the previous five or mortality of the non-child population-the population six decades, and mortality has changed only slightly and over age five-and that therefore the life table selected and gradually during the past generation. The approxi­ from any family of model tables must have the over-all mate constancy of fertility is a very common, if not non-child mortality indicated by census survival. universal, feature of populations that are mainly agricul­ Consider the mortality indicated by a comparison tural, and low in .literacy and income, except when of the population over ten with the whole population ten fertility has been affected by wars, revolutions, major years earlier. The difference between these populations epidemics or other such episodes. The absence of major is precisely the deaths that occurred during the decade trends in mortality has also been a common characteristic to the persons alive at the time of the earlier census. of less developed areas until the past few decades when At the midpoint of the decade, this population" at risk" very rapid declines in death rates have been frequent. is the population five and over, so that the survival rate In this section the use of stable population techniques is for the whole population is very closely linked to the discussed in those instances where it is clearly appropriate, average death rate for the population past age five. All namely, when there have been no major trends or fluc­ model life tables that give a projected population ten and tuations in fertility, and no sustained important changes over equal to that recorded at the end ofthe decade must in mortality. connote about the same death rate for the population over Stable population analysis has also been applied by five. Since the expectation oflife at age five is the recipro­ demographers to populations whose mortality has been cal of the death rate over age five in the stationary popu­ declining, although it has been demonstrated that the lation, it is not surprising that the different model life resultant estimates are biased. In section C ofthis chapter tables have nearly equal °ess. a method of adjustment is described for altering the In other words, the use ofcensus survival rates to choose estimates derived from stable analysis to compensate a model life table at a fitting level of mortality permits the for the effect of a history of recent decreases in mortality. estimation ofthe death rate over age five with some confi­ A stable population is generated by the continuation of dence. However, the death rate for the whole population is a fixed schedule of fertility and a fixed schedule of mor­ strongly affected in populations of high fertility and tality; it is characterized by an unchanging proportionate moderate to high mortality by death rates among age distribution, and a constant annual rate of increase. infants and young children. The death rate obtained by In populations essentially closed to migration where applying the age specific mortality rates in the selected there is no evidence of spreading use of deliberate birth model table to the estimated mid-period population is control or of changing patterns of nuptiality, and no valid only if the relation of infant and child mortality reason to believe that mortality is changing rapidly, to mortality above age five in the family of model tables confirmation ofthe conjecture that the population may be matches the relation in the population in question. And stable can be sought in comparisons of the recorded this relationship is far from the same in the four families age distributions in successive censuses, and of successive of model life tables." Thus census survival rates (if derived intercensal rates ofincrease. Ifthe census age distributions from accurate enumerations not excessively distorted by show marked differences (e.g., such as are seen in the age-misreporting) establish the level of mortality for censuses of Turkey from 1935 to 1960), it is probable persons five and over, but leave uncertain (unless the age pattern of mortality is known) the level ofchild mortality, 8 The fundamental work on the subject is Alfred J. Lotka's and thus the expectation of life at birth and the over-all Theorte des associations biologiques, deuxleme partie (Paris, Her­ death rate. Since this procedure yields an estimate of mann et Cie., 1939). For a comprehensive treatment see the United Nations Study entitled The Concept ofa Stable Population. Applica­ tion to the Study ofPopulations of Countries with Incomplete Demo­ 7 For a detailed discussion of this point see chapter IV. graphic Statistics (Sales No.: 65.XIII.3). 12 that fertility has not been constant, in the cited example mortality conditions and the average annual rate of because the series of wars and other disturbances expe­ increase. The birth rate b is determined by the fact that rienced by the Turkish population undoubtedly caused the sum of the proportions at all ages must be equal to major fluctuations in fertility, and in mortality as well, one. atleast for males. When the rate ofincrease rises markedly, In England and Wales in 1881 the age composition of it is likely that mortality is falling. But essentially constant the female population was much the same as it had been age distributions and rates of increase observed in a series in the two preceding censuses, and the estimated inter­ of censuses can be considered justification for considering censal rate of natural increase had been nearly constant. the population stable. To be sure, the population was not closed, primarily The age distribution of a stable population is described because of a flow of emigrants to America, but between by a well-known formula of Lotka's: 1871 and 1881 the rate of average annual loss for females was only .00070, so that the effect on the age distribution c(a) = be-,a p(a) (1) was minor. The theory of stable populations leads us where c(a) is the proportion of the population at age a, b to expect, then, that the age composition of the female is the birth rate of the stable population, e is the base of population of England and ~ Wales in 1881 is closely natural or Naperian logarithms, r is the annual rate of approximated by a synthetic age distribution calculated increase, and p (a) is the proportion surviving from birth according to formula (1), using the life table for 1871-1881 to age a according to the prevalent mortality risks. to provide p(a), and equating r to the average annual The proportion p(a) is an alternative expression for the intercensal rate of natural increase in that decade. survivor function (lz/lo) in the life table, and all tbat is Figure IV provides a comparison of this calculated age needed to fix the proportion at every age in a stable distribution with that recorded in the censuses of 1881. population is the life table expressing the constant and gives a demonstration by example that theoretically

cI!)'------.

.14

.13

.12.

.11

.10

•••••~NGLAND AND WALES, FEMALES,I881 .09 .... .08 ...

.07

••• STABLE POPULATION WITH r •.01414, .06 • BASEDON OFFICIAL LIFE TABLE FOR 1871-1881

.05

.04

.03

.02

.01

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 AGE

Figure IV. Female age distribution in England and Wales, by five-year intervals, as recorded in the census of 1881 and as approximated by a stable population constructed on the basis of the intercensal (1871-1881) rate of natural increase (r = .01414) and the official English life table for the same period, both for females 13 derived stable age distributions fit actual populations levels. A set of such model tables embodying what can very closely. be considered "normal" or " typical" world mortality This example does not show the potential usefulness of experience is reproduced in abridged form in annex I. stable populations as a way of estimating demographic Corresponding to each model life table is a set of possible variables, because when data existto calculate an accurate stable populations at rates of increase corresponding life table, as in England and Wales in the 1870s, indirect to the highest and lowest levels of fertility that might estimation is not required. Suppose, therefore, that death accompany the life table. Annex II presents such a set of registration had been non-existent or very incomplete. stable populations for each model life table, with rates of Couldthe theory of stablepopulations and the near stability increase ranging from - .010 to .050. of the population of England and Walesbeusedto estimate This set of model stable populations includes a range of mortality and fertility? A positive answer is provided age distributions bracketing all those likelyto be found in through the device of model stable populations based on actual populations that (a) are themselves approximately model lifetables. stable (i.e., have had approximately constant fertility and mortality), and (b) experience mortality where the age 1. Model stable populations pattern conforms more or less closely to that embodied in the set of model life tables. If an actual population As is indicated by equation (1) above, the age distri­ appears, in the light of observed characteristics, to bution of a stable population is jointly determined by the belong to the family of model stable populations, it is mortality schedule (or life table) and the annual rate of possible to locate the model stable population matching increase. In the preceding section there is a description the actual one, and to estimate various demographic of model life tables which can be constructed to embody parameters of the actual population by attributing to it typical age patterns of mortality at different mortality the parameters of the model stable population.

c(~)

.14

.13

.12

.11

.10 ••••..ENGLAND AND WALES, FEMALES,.188\ .09 ...... 08 . .07 ·WEST" STABLE POPULAT ION, C(20l =.4514 .06 '.'"

.05

.04

.03

.02

.01

o 5 \0 15 20 25 :30 :35 40 45 50 55 60 65 70 75 80 85 AGE

Figure V. Female age distribution in England and Wales, 1881, by five-year intervals, as recorded in census and as approximated by "West" female stable population constructed with the same proportion under age twenty as recorded in 1881 and with the intercensal (1871-1881) rate of natural increase of the female population (r = .01414) 14 In the comparison of a calculated stable age distri­ most closely resembling the recorded population in age 0 bution and a recorded age distribution represented in composition.! figure IV, the life table and the rate of increase were How can one judge whether a stable age distribution known, and the stable age distribution was calculated. "closely resembles" a recorded age distribution? If When stable population analysis is used as a method of the recorded age distribution is generated by genuinely estimation, the process is partially reversed. The age constant fertility and mortality, if the coverage of the distribution is known, as is some other parameter such censuses is complete, ifthe population is not substantially as the rate of increase, and this knowledge is used to affected by migration, and if ages are accurately reported, locate the appropriate stable population among the various alternative features of the recorded age distri­ family of stable populations, and the birth rate, death bution would serve to locate essentially the same model rate, expectation of life at birth and other characteristics stable population. The proportion under age ten, or tabulated for the model population are ascribed to the five to fourteen years of age, or over age fifty would lead actual. (The details of the method are illustrated in the to the selection of much the same stable distribution. For examples given in part two, especially in the example example, the population of England and Wales in 1881 where a model stable population is fitted to the female fits the model stable population remarkably well. But population of England and Wales enumerated in the 1871 age-misreporting in the 1881 census was limited, and the census.) Figure V shows a comparison of the 1881 age age pattern of English mortality conforms quite closely distribution with a model stable with a growth rate equal to the so-called "West" family of model life tables. to the estimated average yearly rate of natural increase In many of the less developed countries the age distri­ between 1871 and 1881 and the same proportion under bution reported in censuses or demographic surveys is age twenty as recorded in the census. It is essential to note affected by gross misreporting of age. The distribution by that this is not the same stable population shown in figure IV. Both incorporate the same rate of increase, single years of age is very often conspicuously distorted but one incorporates the recorded female life table of the by "age heaping" - by a tendency for many persons to decade and does not involve the 1881 age distribution report a preferred nearby number (one ending in zero or in any direct way, so that its virtual identity with this 5, for example) rather than the correct age. Indeed it distribution can be taken as confirming the stable nature appears likely that often many more ages are misreported of the actual population. The other calculated stable than given correctly. A reported number of persons at population, in contrast, is not based at all on the recorded age sixty greater than ages sixty-one to sixty-nine com­ English life table, but on the recorded rate of increase, bined is not unusual. the presumed stability of the population, and the assump­ On a priori grounds, it is clear that the effects ofage­ tion that the age pattern of mortality conforms to this misreporting on the cumulative age distribution (the family of model life tables. The model stable population so-called ogive) are less than on the proportion in a par­ selected in this manner has an expectation of life at birth ticular five-year interval. In fact the proportion under within .3 of a year of the recorded value, and a birth rate age x (which will be designated C (x» is affected only by apparently closer to the actual figure than the registered those age-mis-statements that cause a net transfer of birth rate, because of a slight under-registration of births persons across age x. The proportion reported as under in England and Wales during the decade in question. age thirty is not altered if children under age five are reported as five, six or seven, or if persons in their late fifties or early sixties are reported as sixty years old. 2. Selecting a model stable population on the basis of an One of the advantages of the cumulative age distri­ accurately recorded age distribution bution is the simple general relation that exists among the ogives of model stable populations with the same rate of increase: stable populations with higher fertility Characteristics ofan actual population can be estimated (and hence higher mortality at the same rate of increase) by locating the model stable population that best fits have greater cumulative proportions to every age than certain recorded or calculated features of the population lower fertility stable populations (see figure VI). In other in question, and then assigning the characteristics of words, the ogives of stable populations with the same rate the model stable to the actual population. Relevant ofincrease do not cross, and each model stable population features ofthe actual population that are usually recorded is thus completely determined by knowledge of the rate include its age distribution and the average rate of natural of increase and C (x) for any value of x. Consequently, increase between two censuses estimated by adjusting one stable population is identified by the intercensal rate the average annual intercensal rate of increase for any of increase and C (5), another by the rate of increase and known difference in completeness of coverage, or for any 9 C (10) etc. If the reported population in fact has a stable known minor rates of net gain or loss through migration • form, and if age reporting is accurate, the series of stable The problem of selecting the appropriate model stable populations identified with C(5), C(lO), ..., C(65), will population usually reduces, then, to finding a stable exhibit little variation, as would be evident in a very population with a given annual rate of increase and tight cluster of ogives of the stable populations thus

10 If the proportion of children surviving to age two is known (from the methods described in chapter II) the problem is one of 9 If gains and losses because of migration are substantial, the selecting a model stable population with the given 12 and most use of stable population analysis becomes questionable. closely resembling the recorded population in age composition. 15 selected. Deviations from constant fertility (and to a less rate of increase. 11 There will be a highest and a lowest extent from constant mortality) in the recent past would cumulative stable distribution, setting upper and lower cause the stable populations identified with cumulative limits to the choice of a model stable population. If the proportions under some ages to differ from those iden­ highest stable ogive is accepted as approximating the true tified with the proportions under other ages. The set of age distribution, it follows that C (x) at all other ages model stable ogivesconsistent with C(5), C(lO), ... , C(65) from zero to forty is too low, and that therefore there would then be spread out rather than tightly clustered. must have been a net transfer of persons by age overstate­ The same effect of diverse rather than consistent esti­ ment at all ages except that where the ogive agrees with mation of the appropriate model stable population would the highest stable. Conversely, acceptance of the lowest also result from age-misreporting of a sort that caused ogive implies net understatement of age across all ages large net transfers of persons across ages divisible by 5 divisible by five except one. (cf. figureVII). Suppose a population has been subject to approximately 11 It is wise to avoid comparisons at the older ages because stable conditions and that its cumulative age distribution differences in age-patterns of mortality have an increasing effect on the cumulative age distribution above age forty or fifty. and iscompared with the ogivesof stable populations that are because of the prevalence of systematic age-misreporting at older consistent with C(5), C(lO), ... , C(40) and the intercensal ages.

.",.....;:::-;.~.;. ... »:»:...... :.~--:...... ".,.,'",...... - ,,',' .., /'/ .... .9 /',' .... £•.060/',' ./ '/. , .. ' , " ..' ".,',,',...... ' .8 . i' ..' /','", .....' -b'~OJ.·/, ..,/ .7 . , .' . IIII..'.' ./ .' .,I, .:' I, .: " . .6 I, ... il .... " . I, ... /, ., :... .5 .,I'.... f· .I, ... I' ... .tI,...... 4 .,I,.:. .It,.... It .: .,:I': .3 III t ./:i / ./:IF .2 .,.".: ','It; .1 l

o 5 \0 15 20 25 30 35 40 45 50 55 60 65 70 75 AGE

Figure VI. Ogives of age distributions (proportions up to age x) in stable populations ("West" females) with a growth rate of .02 and with birth rates (b) as indicated 16 These relationships are sometimes useful in detecting were the "West" model tables of annex II, with "patterns" of age-mis-statement, and are so employed 0eo = 40 years. 12 in the next section. 2. The calculation of c(0-4) ...,-----,c(40-44) 3. Characteristic forms ofage-mis-statement £:,(0-4) £:,(40-44) We have compared a large number of recorded age when c(0-4) is the proportion aged 0-4 in the given distributions with stable ogives as a way of uncovering population, and C8 (0-4) is the proportion aged 0 to 4 typical patterns of deviation from stable populations in in the median stable population defined above. certain categories of censuses or surveys. The comparison If the given age distribution conformed exactly to the was twofold:

1. The calculation of C (x) - Co (x), where C (x) is the 12 The reader may naturally suspect that the employment of cumulative age distribution of the given (male or ogives of alternative model stable populations with a given life female) population, and C8(X) is the middle (median) table is very different from using ogives with a given rate of increase In fact, the comparisons obtained by holding 0eo constant at forty stable population of those with ogives that agree with years are virtually indistinguishable from those that would be C(lO), C(15), ..., C(4O). The set of stables employed obtained from a fixed value of r (see figure IX.)

, DIA, FEMA LES,I911 ,, .7 .. ,, ..... ,, .... ,.I ... ,, .6 STABLE. ·WEST" FEMALES,",)' r"·00725, CuO)' .2~."" ~

.5

.4 .,.:" /STABLE:WEST" FEMALES r •.00725. C(20)' .455 I:?"'"I I .3 I / I ,,I .2 I r,I

.1

o 5 10 15 20 25 30 35 40 AGE

Figure VII. Ogive of the age distribution ofthe female population ofIndia as reported by the census of 1911 and ogives of the age distributions in "West" female stable populations with a growth rate same as the female intercensal (1901-1911) rate ofincrease in India (r = .00725) and with the highest and lowest ogives consistent with values of C(5), C(lO), ..., C(40) in the census population 17 ctll,IN01A C( ,INDIA c (!)~STAILE C ! ,STABLE 1.3 1.3

1.2 1.2

\, 1.\ 1.1 \ 1 \ 1 1.0 1.0 \

.9 .9

.8

5 10 15 20 25 30 35 40 5 \0 15 20 25 30 35 40 AGE AGE

CtIC),IN01A-C tIC ),STAILE C(~ ),INOIA -CC !),STABLE .04 - - .04

.02 .02 ...... -.. '..,. , ••• ... / <, ... . ',',••MEDIAN•••,.··' ""- ···,.MEDIAN..•••··' \'. , " , -...... " . " " \ .. ' , -.02 ..' \ ,~ HIGH \,..../

-.04

I , I II o 5 10 15 20 25 30 35 40 o 5 \0 15 20 25 30 35 40 AGE AGE

Figure VIII. Comparisons of the reported female age distribution Figure IX. Comparisons of the reported female age distribution by five-year age groups-<:(x)-and its ogiv~C(x)-as reported by five-year age groups-<:(x)-and its ogive -C(x)-as reported in the 1911 census of India with corresponding values in three in the 1911 census of India with corresponding values in three model stable populations defined by the Indian female growth model stable populations defined by an expectation of life at birth rate for 1901·1911 and by agreement with C(lO). C(20) and C(15) of forty years and by agreement with C(35). C(20) and C(15) in in the census population resulting in the highest. the lowest and the census population resulting in the highest. the lowest and the the median ogive respectively among those corresponding to C(5). median ogive respectively among those corresponding to C(5). C(lO)..... C(40) C(lO)..... C(40) 18 """1'1"&'" n T 7! n

model stable, C(x)-C,(x) would be zero at each age, and in one set of censuses, and the more modest deviations, usually with a saw-tooth quality (positive deviations c(x-y) followed by negative ones) in the other set of censuses. c.(x-y) This contrast is manifested in a U-shaped sequence of large differences between the ogive of the reported distri­ would be one in each age interval. A positive value of bution and the model stable in one group of censuses, C(x)-C,(x) implies age understatement that shifted and a more or less alternating pattern of small differences persons across age x, and a in the other group. The implication of these contrasting c(x-y) patterns is obvious: in the first group of censuses, there c.(x- y) is a systematic form of unidirectional age-misreporting over a broad range that distorts the reported age distri­ greater than one implies that age-misreporting has butions even as an ogive; in the second group, although inflated the reported number of persons. in the given age age-misreporting causes pronounced age heaping by single interval-both implications following if C,(x) is accepted years, the distribution by five-year intervals tends to alter­ as a valid estimate of the true age distribution. But nate excessesand deficits, and the cumulative distribution figure VIII shows that the sequential pattern of ups is not much distorted. and downs of C(x)-C,(x) and c(x-y) (a) Female age distributions with large distortions c.(x-y) An examination of twenty-nine female age distributions is maintained whether the comparison stable is the of the sort affected by large-scale misreporting shows the highest, the lowest or an intermediate ogive. It is evident following commo~ characteristics in the pattern of the in figure IX that the apparently arbitrary choice of stable cumulative age distribution, as revealed by C(x)-C,(x): ogives with °eo = 40 years has no important influence 1. The cumulative age distribution rises (relative to the on the sequence. stable) from age 5 to 10; 13 13 The purpose of the comparisons is to bring to light 2. It falls from age 10to 15 and from 15to 20; 14 common patterns of deviation from the stable form when 3. It rises from 25 to 30,13and from 30 to 35.15 approximate conformity to a model stable population The proportion in five-yearintervals shows the following might be expected. Over 150 censuses or surveys of characteristics, relative to the stable: populations of each sex thought to have a history of approximately constant fertility were analyzed in this way. 1. The proportion 5-9is above the stable; 13 2. The proportions 10_14 13 and 15-19 14 are below the The analysis reveals the existence of certain general stable; patterns of deviation from approximately appropriate 13 15 model stable populations-patterns in each instance 3. The proportions 25-29 and 30-34 are above the shared by censuses and surveys of several different popu­ stable. lations. One pattern is clearly the product primarily In other words, the age distributions have a surplus at of age-misreporting, and others appear to result in large 5-9, and a deficit in the adolescent age intervals (10-14 part from past variations in fertility and mortality-the and 15-19) followed by a surplus in the central ages of result of departures from the prerequisites of a genuinely child-bearing (25-34). Censuses and surveys in all of the stable population, in other words. countries of tropical Africa, in India, Indonesia, Morocco, Censuses and surveyswith extreme age heaping evident and Pakistan show this pattern. It is repeated in all of the in the single-year age distributions could be expected censuses of India before partition (except the census of to have ogives and distributions by five-yearintervals that 1931, where the published age distribution was smoothed) do not conform closely to model stable populations. But as well as in both Pakistan and India in 1961. The under­ in fact some censuses and surveys obviously subject to lying pattern of age-misreporting can be detected in the poor age-reporting (by single years) deviate only slightly, censuses of most Near Eastern and North African and some quite strongly from the expected form of countries. It is not evident in Latin American countries, ogives and five-year age distributions. Figure X (upper Ceylon, Taiwan (China), Malaya, the Philippines, the panel shows Republic of Korea, or Thailand. Why should African censuses and surveys show the c(x-y) same kinds of misreporting of age as censuses of India, c.(x-y) Indonesia and Pakistan ? A plausible explanation of the similar patterns is that in these enumerations the age for females in the Philippines (1960), Colombia (1951), entered on the interview schedule was often an estimate Venezuela (1950) and Ecuador (1950) on the one hand, made by the interviewer, rather than the transcription of a and in India (1911), Morocco (1960), Ghana (1960) and number supplied by the respondent. In other words the Indonesia (1961) on the other. All are populations in common form of distorted age distributions is caused by which age heaping is extensive. The lower panel of the the common biases in the estimation of women's ages figure compares C(x)-C,(x) for the same censuses. The most conspicuous contrast in the upper panel of figure X 13 No exception in twenty-nine censuses or surveys. is between the large deviations from the stable, with two or 1. Only one exception in twenty-nine cases. more consecutive age intervals deviating in the same sense 15 Two exceptions in twenty-nine cases. 19 by another person. Unfortunately, this explanation of the basis for selecting the appropriate stable population, and similar patterns of distorted age distributions does not by by considering these it is possible to obtain an insight into itself indicate what the true age distribution is. It suggests the typical age-mis-statements prevailing in these that the age distributions affectedare distorted in a similar instances-and presumably in other censuses and surveys way, but to determine where there is net overestimation with the same pattern of distortion in the age distribution. of age and where net underestimation it is necessary to In the sample census of the Congo of 1955-1957, the ages find by some other means which of the alternative model of all but a minority of young children were verified by stable populations does in fact resemble the actual age the interviewer through the examination of birth certi­ distributions. ficates or of entries in the mother's identity booklets, There are a few instances where there is an independent and the stable population consistent with the proportion

~ 1.3 ~\ll

1.2 .-.~ 1.2

1.1

1.0 ; ; .9 ,i ; I• .8 .8 _ COLOMBIA,I951 _ INDIA,1911 ___ ECUA DOR,I950 '" '. ~l ! 1: ., ___ INDONESIA,1961 ...... PHILIPPINES,I960 ,I , ....h. GHANA,I960 ._••• VENEZUELA,I950 .f ~'I,. ._._. MOROCCO,1960 .7 \ i .6 \J .6

'( I I II I I II '( I I I I I I I 0 5 10 15 eo 25 30 3!l 40 0 5 10 15 20 25 30 35 40 AGE AGE Cll!l-C.ll!l C~l-C.(!l .06 .06 1\ i \ .04 i \ .04 ,; ,1\ \\ .02 l·"\\ \ .02 ~\ \ I"' ••.~.\ \ 0 0

-D2 _ COLOMBIA,I951 ___ ECUADOR"I950 ...... PHILIPPINES,I960 '.'_' VENEZUELA,1950 -.04

II I II , 'C I I I I I I I I 'C II 0 5 10 15 20 25 30 35 40 0 5 10· 15 20 25 30 3!l 40 , AGE AGE -~

Figure X. Comparisons of the female age distribution by five-year age groups-c(x)-and its ogive-C(x)-as reported in various censuses. with corresponding values in stable populations median among those defined by an 0eoof forty years and by agreement with C(5). C(IO)..... C(40) in the census populations. The comparisons illustrate two typical patterns of age-misreporting; the"African-South Asian" pattern (left-hand diagrams) and the"Latin American" pattern (right- hand diagrams) 20 of children can be accepted as a valid fit. In several other expected from respondent's errors. For example, there African territories fertility and mortality could be esti­ may be a reluctance to pass certain milestones, such as mated on the basis of retrospective data on children ever age thirty or age forty. The relatively inflated age groups born, children surviving, and births and deaths in the of 25-29 and 35-39 found in several of the censuses of year before the census, and a stable population chosen Latin America and the Philippines perhaps include some to conform to these fertility and mortality estimates. women who have really passed thirty and forty, respec­ The stable populations selected in this way in some tively. This possibility is reinforced by the relative deficit instances agreed with the reported cumulated distribution found at ages 30-34 and 40-44. up to age ten, and in other instances had a lower propor­ tion under age ten but a larger proportion under age 15 (c) Distortions in male and female age distributions than the census or survey. If these examples are represen­ tative, the distorted age distributions of Africa, India, Estimations of population parameters can be based on Indonesia and Pakistan are the result of the following the age distribution of either sex. The stable population typical errors in estimating the ages of females: methods of estimation described in this section can be 1. A tendencyto overestimatethe age of young children, applied either to males or females, as can the census contributing to the typical excess proportion at ages 5-9, survival techniques outlined earlier. The ratio of male and the relative deficitat 0-4. births to female births is confined to the limits of 1.04 to 1 to 1.07 to 1 in almost all populations where birth 2. A tendency to overestimate the age of girls 10-14 registration is essentially complete, the consistent excep­ who have passed puberty, especially if they are married, tions being in populations of African origin, where the combined sometimes,but not universally, with a tendency ratio varies from 1.02to 1.04.Becauseof the approximate to underestimate the ages of girls 10-14 who have not constancy of the sex ratio at birth, it is possible to check .reached puberty, causing a net transfer downwards across the consistency of estimates derived from male age age ten, and contributing to the peak at ages 5-9. distributions on the one hand, and female on the other: 3. A tendency toward overestimation like that affecting the estimated male births should exceed the female by some of the 10 to 14 year olds, for females 15-19, 20-24, about 5 or 6 per cent in non-African populations, and and 25-29, causing net upward transfers across ages 15, by 2 or 3 per cent in African populations. 20, 25, and 30, and causing deficits at 10-14 and 15-19, The approximate constancy of the sexratio at birth also and excessive proportions at 25-29 and 30-34. This makes it possible to base the estimates for both sexes on overestimation of the age of young women may be caused the analysis of the age distributions of only one. For by an unconscious upward bias associated with marriage example, if female births are estimated by stable popu­ and child-bearing, or from a mechanical assumption that lation techniques, male births can be estimated as 6 per women were married at some alleged conventional age at marriage and have then experienced an allegedly cent more than the female. This estimate divided by the recorded male population gives an estimated male birth typical passage of time between marriage and first birth, rate. The male birth rate less the intercensal rate of and in each subsequent interbirth interval. increase of the male population gives the male death rate. (b) Female age distributions with smaller distortions The results of analysing age-misreporting in female When, on the other hand, the recorded age is supplied populations with approximately stable age distributions by the respondent, the age distribution is naturally less by comparisons with model stable populations have distorted- first of all because when most respondents already been described. The corresponding male popula­ supplya plausiblefigureto the enumerators, the maximum tions have been examined in precisely the same manner. error is generally below the level of the ridiculous, and In general, age distortions among male populations show the averageerror is thus diminished.That is, when respon­ similarities analogous to those found among females: dents are usually prepared to supply an age on request, one pattern of major distortions in both ogives and five­ and then the figure given is acceptable in the sense of year distributions is found to characterize surveys in being only rarely absurd (e.g, seven for a grown man most of Africa and southern Asia while another pattern with a beard), it is plausible that members of the popu­ of substantial age heaping, but relatively minor distor­ lation know their approximate age, and that the broad tions in ogivesand five-year distributions, is characteristic outline of the age distribution is approximately correct. of Latin America and the Philippines.However,among the Rough knowledge of age on the part of most persons male populations in Africa-South Asia, the similarities does not requirea high levelof literacybut merelya culture in distortion from country to country are less for males in which numerical age has importance. Apparently ages than for females, and the distortions themselves appear ... were accurately known in Sweden in the middle of the larger on the average. In the Latin American populations, eighteenth century, for example. Distortions in the cumu­ the distortions of the female age distributions are almost lative age distributions are relatively minor in the Philip­ always larger than in the male; moreover there is usually pines and Latin America---even in Honduras, where the a slight systematic bias found among the female age . proportion of persons over fifteen recorded as illiterate distributions (ogives that yield continuously increasing is nearly as high as in Indonesia. estimates of fertility as age increases from ten to forty) Some of the features of the less pronounced age-mis­ not found in the males. reporting in the populations where apparently ages were A summary comparison of the effectof distortions in age supplied by the respondents are just what would be distribution on estimation is shown in figure XI. Birth 21 MALES bma -b

.020

.018 • • .016 • " " .014 "

.012 " " .010 " .008 "

..

" AFRICAN-SOUTH ASIAN PATTERN • LATIN AMERICAN PATTERN

.018 FEMALES bmax-bmin

Figure XI. Ranges ofstable population estimates of the female birth rate (b) versus those of the male birth rate in various censuses. Ranges are the difference between the highest and the lowest birth rate in stable populations having an 0eo of forty years and agreeing with C(5), C(lO), ..., C(40) in the census populations. Estimates shown were obtained from two groups ofcensuses, each character- ized by a typical pattern of age-misreporting rates were estimated (assuming °eo = 40) on the basis exist, or are lessevident, for males. The smaller distortions of ogives to age 5, 10, 15, ... ,40 for males and females, affecting male age distributions in Latin America are employing "West" model stable populations. These also consistent with the surmise that age in these popula­ calculationswerelimitedto age distributions not obviously tions is usually supplied by the respondent. The more affected by rapid changes in fertility or mortality. The extensive education and greater worldliness of the male range (highest estimate minus lowest estimate) for males would lead to a better average knowledge of age. and females in the African - southern Asian and Latin American censuses so selected were then plotted. Note 4. Selecting a model stable population on the basis of a that the range is much less for both sexes in the Latin distorted age distribution American group; that in a large proportion of Latin American censuses the male age distribution leads to a The observations on age-misreporting in the preceding smaller range than the female; but that in the African­ section imply that the pattern of distortion must be taken southern Asian populations, the opposite is seen-a larger into account when determining what model stable popu­ range of estimates when based on male age distributions. lation best fitsthe population in question. As a preliminary ... The smaller gross displacements in female than in step, the female age distribution may be compared with male age distributions in the African - southern Asian a model stable population with the same proportion under censuses is consistent with the hypothesis that in these age thirty five(and 0eo = 40) to see if it has the character­ surveys age is often estimated by someone other than istics (African-southern Asian)listedunder sub-section (a) the person in question. The estimation of female ages is above. If the age distribution does have these character­ assistedbyclues-bodilychangesassociatedwithmenarche, istics it is likely that the female age distribution is a better number and age of children, and a fairly well-defined source of estimated population characteristics than is upper limit to the ages of childbearing-that do not the male. The choice of a female stable population is 22 made and the consequent estimate of the birth rate is toward mildly increasing estimated birth rates from age obtained as follows: ten to forty or forty five. 16 (1) The female model stable populations with a rate of (3) Select the model stable population for males that increase equal to that of the female population and produces the median value of the birth rate among those with the same C(5), C(10), ..., C(40) are selected and agreeing with C(5), C(10), ... , C(45). The birth rate of the birth rate of each is recorded. the female stable population can then be taken as (2) The sequence of estimated birth rates will typically male birth rate male population have a peak at age ten, fall at ages fifteen and twenty, sex ratio at birth x female population and rise to a second peak at thirty-five or forty, if the This estimate can be checked by comparing it with the age distribution is of the African - southern Asian type. average of the birth rates in the female model stable On the basis of the analysis of this pattern given earlier, populations that agree with the recorded female popu­ the birth rate estimate based on C(lO) may be about right, lations in growth rate and C (20), and in growth rate and or may be too large, because there is sometimes, but not C(25). always, understatement of the ages of enough 10-14year­ olds to inflate the proportion under ten. The birth rate estimated from C(15), C(20), C(25) and C(30) would 5. Assigningthe characteristics ofa modelstablepopulation be too low because of the almost universal overstatement Once a model stable population has been selected as a of age among young women in these populations. The close approximation of the actual population, the charac­ estimate based on C (35) is expected to be at about the teristics of the stable population can be ascribed to the right level, except for the possible effects of recent mor­ population in question. At a later point, in chapter IV, tality or fertilitytrends. we shall comment on the accuracy of correspondence (3) A minimum estimate of the birth rate in the female between the model stable population and the actual with population is obtained from rand C(15). regard to mortality and fertility. At this place we wish only to indicate what characteristics of the population (4) If evidence in favour of stability is convincing (i.e., can usually be attributed to the actual. little change in intercensal growth rate or in age distri­ bution for two or three decades), the estimate based on When circumstanceswarrant the assumption of approx­ C (35) can be accepted, and with more confidence if imate stability of the population (i.e., when there are confirmed by the figure derived from C(lO). If growth indications of a past history of approximately constant has been accelerating, the estimate based on C (35) can be fertility and mortality), the age distribution of the stable adjusted by methods describedin section C of this chapter. population can be attributed to the actual population. This attribution is useful in those populations with large (5) The birth rate in the male population is estimated as systematic errors in age reporting. For example, an sexratio at birth x female birth rate x fe~ale POP;I~tion appropriately chosen stable population undoubtedly has ma e popu ation an age distribution closer to reality than the reported age In the absence of reliable direct information on the sex distribution for the population of India in 1911. In fact ratio at birth a value of 1.05 should be taken in popu­ the stable population age distribution can be used to lations other than those of tropical Africa. In African provide a base population for population projections populations a multiplier of 1.03should be employed. that will provide a more valid basis for estimating the Now suppose that the comparison of the female age future evolution of the population of schoolgoing age distribution with the model stable population with the or of the ages of labour force participation or the like same C (35) shows that the recorded population has a than would be obtained by basing a projection on the substantial deficit under age five relative to the stable, often highly distorted age distribution recorded in a and a tendency for the proportions under age ten, fifteen, census or survey. The principal purpose of the stable twenty and twenty-five also to fall somewhat short of population analysis described in this Manual is, however, this stable, but to a minor and decreasingextent, and the to provide estimates of fertility and mortality. proportion in successive five-year age groups does not As will be seen later, the estimates of fertility derived depart by more than perhaps 5 to 10 per cent from the from stable analysis are ordinarily more trustworthy than stable from age ten to age forty. It may then be assumed estimates of mortality. Strictly speaking, information on that the age distribution is of the Latin American pattern age composition and growth which permit the choice of a and the choice of a stable population is made as follows: matching model stable population provide an estimate of the crude birth rate as a measure of fertility and do not (1) Find the male model stable populations with a rate enable us to estimate total fertility (the number of children of increase equal to that of the male population, and with the same proportions under age 5, 10, 15, ... ,40 as in the recorded male population, and note the birth rate in 16 Frequent exceptions are found in the Latin American censuses each. Perform the analogous calculations with the female taken since 1950 because of the effect on the age distributions of population; declining mortality, combined in some instances with a slight rise in fertility. The effect is to produce estimates ofthe birth rate among (2) The sequence of male birth rates will typically rise males that fall from age ten to forty or forty-five, and among from age five to ten, and then fluctuate mildly until age females that are about constant, or declining less sharply than the male estimates. The normal sequence of approximate constancy forty or forty-five. The female birth rate will follow a for the males and slightly rising estimates for the females is restored similar sequence except that there will be a tendency by the adjustments described in section C of this chapter. 23 born per woman passing through the fertile part of life) The rates shown in table I for ages 20-49 are based on or the gross reproduction rate (the number of daughters average experience of a number of populations in which born per woman in the same span). The reason is that two little or no birth control is practised. 18 The value for age populations with the same fertility in the sense of the same 15-19 is a rough approximation according to which if, birth rate and withthe same agecomposition can have quite in the absence of birth control, marital fertility for age different numbers of children per woman passing through 20-24, i.e., 1 (20-24), and the proportion of married the childbearing span, depending upon whether, on the females at age 15-19, i.e., m(15-l9), are known,/(15-l9) average, births occur relatively early or relatively late may be estimated as 1.2/(20-24)-.7 [(20-24) m(15-l9). in the span. When fertility is as high as it is in almost all In a population in which it is known that the fertility of of the populations for which stable analysis is appropriate, non-married women is a negligible factor in the over-all a low mean age of child bearing permits women with a birth rate, and where the census includes a tabulation of smaller total fertility to produce the same birth I ate as marital status by age, the age pattern of fertility can be achieved by a population in which women of higher total approximated by multiplying this standard marital fertility produce their children somewhat later in life. fertility schedule by the proportion married among the This relationship-i.e., that low fertility and early child­ women in each age group and from this approximated bearing produces the same birth rate as higher fertility schedule the mean age of the fertility schedule calculated. and late child-bearing-is represented in our tabulation The method described in the previous paragraph is not of model stable populations by the presentation of four applicable in populations with high proportions of births gross reproduction rates with each population; each gross outside of marriage, or in which there is a variety of reproduction rate being associated with a particular mean 1 sanctioned sexual unions, including for example wide­ age of the fertility schedule. 7 This multiple tabulation of spread consensual unions. gross reproduction rates with each stable population of course implies that the identification of a model stable When women have been asked about the number of population as essentially identical with a given actual children ever born, and the responses have been tabulated population is not a sufficientbasis for estimation of total by age of woman, it is possible to estimate the mean age fertility or the gross reproduction rate. In addition, the of the fertility schedule from the relation of average analyst must have some knowledge of the mean age of parity at age 20-24 to average parity at ages 25-29. If the the fertilityschedulecharacterizing the women in the popu­ earlier average parity is called P2 and the later Pa, the lation in question. ratio P3/P2 (in a population not practising birth control) depends primarily on the ages at which women begin The mean age of the fertility schedule can be calculated their child-bearing, a high value of P3/P2 indicates a late from tabulated responses to a question about births occurr­ start, and a low value an early start. It may be assumed ing in the preceding year (with due allowance for the fact that the decline of fertility with age in populations not that women who report a birth during the preceding year practising birth control follows a fairly common pattern, would on the average have been six months younger at so that the mean age of the fertility schedule is determined the time of birth than at the time of the survey). In many primarily by the rising portions. The relationship between populations no such direct evidence on the age pattern the mean age of the fertility schedule and the value of of fertility is available. In such populations the mean age P3/P2 has been calculated on the basis of the recorded of the schedule can sometimes be estimated by various schedules of a number of populations in which there is indirect approaches. One possibility is to estimate the little practice of birth control. The mean age of each age pattern of fertility by assuming that marital fertility schedule was calculated, and the values of P2 and P3 follows a standard pattern. Table 1 shows a pattern of implied by the schedules computed. The relation of iii marital fertility-fertility rates expressed in terms of the to P3/P2 is very close, as is shown in figure XII, and is rate for age 20 to 24-that can serve this purpose. well expressed by the following linear expression:

iii = 2.25 P3 + 23.95 TABLE 1. STANDARD AGE PA'ITIlRN OF FEMALE MARITAL FERTILITY P2 RATES When, due to the lack of required data, neither of the Age Index ofmarital fertility rate approximations described above is applicable, the best alternative is to assign to the population in question the mean age of the fertility schedule of another population 15-19 . 1.2-.7m(15-19)Q presumed to have similar factors affecting the age of the 20-24 . 1.000 25-29 . .935 fertility schedule. Thus in a Latin American country a 30-34 . .853 possible expedient is to utilize the mean age of child­ 35-39 . .685 bearing from a nearby neighbour, if the two countries in 40-44 . .349 question are both known to have a roughly similar 45-49 . .051 prevalence of consensual unions and of marriages per­ formed by the State or the church. The reader's attention is directed to the expedient used in examples worked out Q m(15-19) = proportion of married females at ages 15-19. in part two of this Manual.

17 The arithmetical mean of the schedule itself, unweighted by 18 They are adopted from Louis Henry, "Some Data on Natural the age distribution. Fertility", Eugenics Quarterly, vol. 8, No.2 (June 1961),pages 81-91. 24 34

33

32

3\

30

29

28

27

2.0 2.2 2.4 2.6 2.8 3.0 3.2 v.4 3.6 3.8 4.0 42. 4.4 Pa/P2

Figure XII. Mean age of the fertility schedule in populations not practising birth control versus ratio of Ps (average parity at age 25-29) to P2 (average parity at age 20-24)

C. ADJUSTMENT OF ESTIMATES BASED ON MODEL STABLE epidemics such as the world-wide infleunza epidemic of POPULATIONS FOR THE EFFECTS OF RECENT DECREASES 1918 to 1920 cause a temporary reduction in fertility IN MORTALITY and an excess of infant and child mortality that again produce a small cohort evident in subsequent age distri­ It goes without saying that many populations have age butions. distributions that do not conform at all closely to any of On the other hand, in developing countries where the the model stable populations. Major fluctuations in fertility create unusually large or unusually small birth population is little affected by international migration, cohorts that stand above or below the corresponding age and in the absence of major catastrophes such as wars or group in any stable population during their lifetime. great epidemics, fertility tends to remain at a fairly level Persistent trends in fertility can create an age distribution plateau. The only apparent exception is in areas where far different from any stable population. Because the relatively late marriage is the custom, such as in western populations of all highly industrialized countries of the Europe before the systematic decline in marital fertility began. In these populations, for example, in Tuscany world have experienced sustained decreases in fertility amounting in almost every instance to a 50 per cent during the nineteenth century, changes in total fertility reduction of total fertility, and since most of these amounting to 20 or 25 per cent can occur caused by long populations have experienced substantial fluctuations in intervals in which the average age of marriage was fertility in addition to the downward trend, age distri­ increased and other intervals in which it was reduced. It butions of industrialized populations cannot in general appears possible that in Latin America changes in age at be matched by those of model stable populations. Events marriage, and perhaps differential recourse at different in the history of some of the less developed countries have times to marriage on the one hand and to less stable also caused irregularities in the age distribution that concensual unions on the other have raised and lowered the average level of fertility. would find no match in one of the model stable popu­ lations. Some populations have age compositions strongly The conclusion that emerges from these observations is influenced by age-and-sex-selective migration. Such that there are populations in less developed countries that migration is typical of the urban populations ofdeveloping cannot be analysed by stable population techniques, and countries and a non-stable form of age distribution is others in which the precision of the estimates may be therefore to be found at certain census dates in such degraded, even though the general approach remains populations as that of Singapore' and Hong Kong. The useful. Among the contemporary populations for which prolonged mobilization of a large fraction of the young the method does not appear useful are those strongly adult male population into military service can have a affected by migration and those that have suffered a pronounced effect on fertility and create a lasting notch sequence of wars or other major disturbances. There in the age distribution. Military casualties of course affect remains the majority of populations in the less developed the age distribution of the male population. Invasions and countries, where the assumption of a history of more revolutions can leave similar traces. Finally, major or less constant fertility is warranted, but where mortality 25 has followed a strong and sustained downward trend in levels of the initial expectation of life at birth. These recent years. populations were then projected for forty years with The prevalence of rapidly falling death rates in the less steadily rising expectation oflife at birth (i.e., with steadily developed countries is well known and need not be falling mortality) at a pace which in each year was equi­ described in detail in this Manual. Falling mortality has yalent to a one per cent increase in fertility in the principal followed as many different courses (if these are considered influence on changing age distribution. These projections were performed on an electronic calculator. The pro­ in detail) as there are identifiable populations in the developing countries. It is of course not possible to grammed computations included the calculation of the describe how every imaginable decline in mortality would average rate of increase during each five-year period and affect the age composition of a population. Many demo­ also during each ten-year "intercensal" interval, birth graphers have noticed that different mortality schedules rates, death rates, age distributions in five year groups produce only slightly different stable populations, and at the end of each five-year time interval, and ogives of that populations experiencing approximately constant these distributions. Birth rates and total fertility rates fertility and changing mortality show only restricted were then "estimated" at five-year intervals by choosing alterations in age composition. Finally, it has been a model stable population with the same rate of increase noted that the age composition produced at each as the. "intercensal" rate of increase during either the of time during a prolonged period of declining mortality preceding five years or ten years in the population an~ value~ bears a closer relationship to the stable implied by the prOjectIOn, the of C(5), C(lO) etc. in the pro­ current fertility and mortality conditions than to the age jected population. The difference between the estimated distribution of an earlier period when mortality was rate calculated in this fashion and the average value of high~r. ~uch observations have been used to justify the the "true" birth rate or of the "true" total fertility application of the methods of stable population analysis during .the intercensal period (calculated as part of the to populations experiencing approximately constant projection programme) was then obtained. These differ­ fertility and steadily declining mortality. Indeed a ences, expressed as a proportion of the true value of the special designation of quasi-stable has been invented for birth rate or of total fertility, turned out to be very nearly these populations. Coale and Demeny have analysed the same whether the population projection was for a the .w.ays in which populations that have experienced f~rtility o.f5, 6 or 7 births per woman. Thus these propor­ declining rather than constant mortality differ from the tionate differences can be used as the basis for adjustment stable population that would have resulted had current factors to reconstruct the true birth rate from the birth mortality and fertility conditions prevailed throughout rate ~stimated from the model stable population- if the dur~tIOn the past. 19 They noted that in spite of the close visual of the decline in mortality is known and if the resemblance between the age composition of a stable falling death. rates ar~ at a. J?ace equivalent to a one per population and of a so-called quasi-stable population an cent annual increase m fertility so far as age composition estimate of total fertility or the birth rate based on'the effectsare concerned. That is so since the calculations also ogive of the quasi-stable population and the current rate showed that. the adJustments needed, e.g., at fifteen years of increase can be in error by as much as 10to 15 per cent. ~fter mortality decbn~ ?~gan, were for practical purposes, In this section a method is described by which the demo­ independent ?f the initial level of mortality, i.e., they graphic analyst can adjust estimates of population para­ wer~ subst~ntIally the same whether the initial expectation meters extracted from model stable populations to com­ of life at birth before the onset of declining mortality was, pensate for the distorting effects of a history of recently for example, twenty years or thirty years. Finally sets of declining mortality. population projections were programmed that'differed ~It~ resI?ect t~ the speed of mortality decline, e.g., pro­ On the basis of previous analytical work it is known jections m whI~h the decline in mortality was only half that th~ principal effect of declining mortality on age as fast as specified above; in short, in which mortality, C?~Posltl.o.n closely reseJ?ble~ the influence of steadily c~nges were equivalent in their principal age composi­ nsmg fe~Ility. Over a specifiedinterval of falling mortality, tional effects to an annual increase in fertility of one-half It IS possible to find a proportionate change in fertility­ of one per cent. These projections showed that the differ­ say a 7-per cent increase-that is equivalent to the ences in the estimated and actual values of birth rates and recorded decline in mortality so far as the effect on age tota~ fert~lity :were proportionate to the speed of mortality composition is concerned. It is possible, by an extension decltne, i.e., m the example just given they were almost of this idea, to determine what sequence of annually exactly half as great as with the more rapid decline in rising expectation of life in the "West" model life tables mortality. given in annex I would be equivalent to an annual increase This Manual includes as a result of these calculations a i~ fertility of one per cent. A series of population projec­ tabl~ of adj~stments (see annex III, table I1I.1) to be tions were constructed along these lines in which the initial applied to birth rates and to gross reproduction rates population was a model stable population with a total (or total fertility rates) derived from the model stable fertility of about 5, 6 and 7 respectively, and with various populations- adjustments that are appropriate for ogives up to age 5, 10, ... , 40 and which assume different values 18 Ansley J. Coale, "Estimates of Various Demographic Mea­ at t = 5, 10, ... , 40, where t is the time after the decline sures Through the Quasi-Stable Age Distribution", in Emerging in mortality begins. These adjustments should be applied Techniques in Population Research, Milbank Memorial Fund,1963, ~h~n pp. 175-193,and Paul Demeny, "Estimating Vital Rates for Popula­ t?e value of k equals .01, where k is a parameter tions in the Process of Destabilization", Demography (Chicago), indicating the rate of mortality change expressed in vol. 2, 1965, pp. 516-530. terms of the equivalent proportionate annual increase in 26 fertility in so far as age:distribution effects are concerned. chapter II a method of estimating child mortality from For values of k other than .00-i.e., in the general case special census data is described, and in chapter III the when the age distribution effects of the changing mortality combination of estimated child mortality and the age are equivalent to an average annual fertility change that distribution to selecta model stable population is outlined. is greater or smaller than one per cent per year-the At the end of chapter III it is noted that changes in the tabulated adjustments are to be scaled up or down in estimated level of child mortality from census to census the same proportion as the actual value of k differs from can be used to determine approximate values of k and t 21. .01. For instance if the value of k is .012 the appropriate Another basis for estimating k and t is the changing age adjustment factors are to be increased by 20 per cent. composition of deaths. The proportion of deaths by age Annex table III.1 thus enables the analyst who has made changes when mortality declines and fertility remains a preliminary set of stable population estimates of the constant. It is possible to obtain a very rough estimate of k birth rate and of the gross reproduction rate (from the age and t from even an inexact record of the changing age distribution cumulated to age 5, 10 etc. in conjunction composition of deaths. The reader should note that the with the intercensal rate of increase) to adjust these method outlined in.thenextparagraphs is clearlyimprecise, preliminary estimates in order to correct the bias present but that it is used only to indicate the magnitude of an in the stable estimates due to the fact that contrary to adjustment for the effects of declining mortality, and the assumptions underlying the stable estimates mortality that an approximate adjustment usually produces an in fact has been declining. A quasi-stable estimate of the estimate superior to the unadjusted stable value. death rates is obtained by subtracting the intercensal growth rate from the adjusted birth rate. A quasi-stable The estimation procedure rests on the following estimate of the expectation of life at birth is finallyderived supporting facts and relationships: by findingthe °eo of the stablepopulation characterized by (1) When initially stable populations are projected this death rate plus any of the other parameters (ORR, b with declining mortality but constant fertility, certain or r) for which the quasi-stable estimates have previously changes in the age composition of deaths are very closely been calculated. correlated with the associated change in expectation of The application of the method described in the pre­ life at birth. The most usable index of the changingcompo­ ceding paragraph assumes that estimates of the duration sition of deaths is perhaps the ratio of deaths to persons and average pace of mortality decline-i.e., estimates of over sixty-five to deaths to persons over age five. This the parameters t and k-have already been obtained. index is as closelyrelated as any other to lioeo, and has the Preferably such estimates should be based on information merit of omitting infant and child mortality, which could concerning the rate of population growth in the decades have an overidingand possiblymisleadingeffect, especially preceding the census that is being analysed- information when completeness of registration changes. that is sufficient to locate approximately the time when the (2) The change in the proportion of deaths over sixty­ departure from the stable state has occurred and which five may be faithfully represented in registered deaths, indicates the tempo at which destabilization has taken even when these are so incomplete in coverage that place. Specifically, on the basis of the same calculations neither the level nor the trend of the crude death rate that were outlined above, it was found that k can be esti­ can be derived directly. mated as 17.8x lir/lit where fir is the absolute change in the rate of growth as compared to the original stable (3) An approximate value of Ii°eo can be obtained from rate, and lit is the number of years that have elapsed annex table 111.2 as the difference between two stable while that change took place.20 population estimates of °eo referring to different periods of time, each based on a value of the index deaths The acceleration of population growth is not the only 65+/deaths 5+ and on a rough measure of fertility evidence from which the parameters k and t can be esti­ mated. What is needed is any approximate indication of births of a given sex )22 the duration and pace of recent mortality declines. In ( persons 15-44of the same sex (4) The value of k. t associated with a given lioeo 20 This formula assumes that the acceleration of growth is attributable in its entirety to a change in mortality, i.e, that fertility depends rather strongly on the base level ofoeo, and an has remained constant. It may be noted at this juncture that in a approximate value of the latter is needed before k. t formal sense the procedure adjusting stable estimates for the effects can be estimated from table III.3. A crude but usable of changing mortality may be extended to the case where destabili­ indication of the terminal level (which lessIi°eo gives the zation has been brought about by changing fertility, or a mixture of changing fertility and mortality, rather than by changing mor­ base level) of °eo can again be obtained by the provisional tality alone. Assume, for example, that fertility has been changing assumption of stability. It is recommended that the for t years (following an original stable situation) while mortality provisional terminal level be taken as the average of the has remained constant. The adjustment factors tabulated in table 111.1 °eo's in the stable populations associatedwithr and C (10), would still be applicable-with proper attention to sign, i.e., multiplied by -1 in case of declining fertility-the value of k to and rand C (15). The same procedure is to be used in be used being simply the average annual change in fertility. Note obtaining the rough measure of fertility that is needed that the value of the multiplier connecting k and ti.rlti.t would in estimating Ii°eo, as explained in point (3) above. then be about t.wice as large as in the case of mortality change; approximately 36 instead of 17.8. In view of the fact that sustained changes in fertility are less regular and, under the conditions 21 See chapter II, section B, chapter III, section C.2 and annex necessitating the application of quasi-stable techniques for estima­ table I1I.4. ting vital rates, are less common, no systematic discussion of this 22 This ratio is one of the stable population parameters tabulated topic is offered in this Manual. in annex II. 27 It must be noted that the recommended procedure for 40 years. in figure XIV the plot of C(x)-Cs(x) of several estimating kt from the changing age composition of Latin American populations with censuses in the early deaths (i.e., approximating aOeo and the levelof °eo from 1960s is shown in comparison with the projected popu­ the provisional assumption of stability before using lation where mortality has been declining for twenty table 111.3) generates provisional data on mortality­ years. The imprint of the effect of declining mortality namely the levelofoeo-that are useful only in determining on the age composition of these populations is clearly a correction factor, and that must not be confused with evident. a final estimate of that parameter.

D. CoNCLUDING REMARKS ON ESTIMATESADJUSTED FOR THE EFFECTS OF RECENT DECLINES IN MORTALITY C (x) - C lx) .04- $- Whether adequate direct or indirect information on past changes in mortality are available or not it is to be expected that with reasonably good reporting of ages such a deviation from stability will be discernable from the age .02 distribution itself. The nature of the effect of a history of declining mortality on the stable age distribution is indicated in figure XIII which shows C(x)-Cs(x) where 01---..lj,L----~1IO:""""----- , ...- --.-.-. C(x)-Clx) - s- 04

Cl!)' _ MODEL QUASI STABLE, ~".Ol, 1-20 02 ___ VENEZUELA,I961 •••••• HONDURAS, 1961 •__• PERU,I961 o o 5 10 15 20 25 30 35 40 AGE

-.02 Figure XIV. Comparisons of the ogive-C(x)-of a model quasi-stable population and of male populations as recorded in recent Latin American censuses, with that of stable populations Cl~): MODELQUASI STABLE (h.Il.O!) -Cs(x)-having the same proportion under age twenty and an 0eo of forty years t - 15 == i- 20 ..... i· 25 The age distributional effects of declining mortality are __._. t~ 30 also present in recent censuses of such populations as ··_··1=35 that of India and Pakistan, but because of the nature of the distortions in the reported age distribution the effects are not so easy to discover as in the Latin American cen­ o 5 10 15 20 25 30 35 40 suses. However it is interesting to note that when stable AGE population analysis is applied to the Indian age distri­ bution (females) in 1911 there emerges from the inter­ censal rate of increase and the proportionunder ageten and Figure XIII. Comparisons of the ogive-C(x)-of model quasi­ stable populations (populations which were originally stable but under age thirty-five estimates of total fertility in approx­ which have experienced a decline of mortality for t years). with imate agreement-estimates of 6.5 children per woman that of stable populations-Cs(x)-having the same proportion in the former case and 6.3 in the latter case. When stable under age twenty and an 0eo of forty years population analysis is applied to the Indian census of 1961 the total fertility estimated from the proportion under age ten and the intercensal rate of increase is 6.6 C (x) is the ogive of the projected population with mortal­ and from the intercensal rate of increase and the propor­ ity declining for IS, 20, ... , 35 years and Cs(x) is that of tion under thirty-five is only 5.8. However, when the the stable populations having the same C (20) and °eo of appropriate adjustments are made to allow for the influence 28 of declining mortality in the preceding forty years at a enumerated population in fact departs from the stable age pace estimated from the average rate of acceleration of distribution. growththe resultant estimatesare 6.6and 6.4, respectively. In the hypothetical example just described, the impli­ cation that the demographer would know that the proportion under age 10 is .3 is unrealistic, since if a E. THE ESTIMATION OF FERTILITY FROM THE AGE DISTRI­ population has had its age distribution recorded only once, BUTION RECORDED IN ONE CENSUS it is likely that the recorded distribution is seriously distorted by age-misreporting. A more realistic example What is the minimum information that permits demo­ is provided by considering the problem of estimating vital graphers to form approximate estimates of fertility or rates for Indonesia on the basis of the age distribution mortality? In the earlier sections of this chapter methods information provided by the 1961 census of that country. are described for basing estimates on a series of two or As shown in figure XV combination of various indices more population censuses, and in the next chapter there of this age distribution (proportions up to age 5, 10, ...etc.) is an outline of techniques of estimation to extract the with specified hypothetical growth rates or levels of maximum of reliable inference from special questions mortality imply widely varying birth rates. Unless some consciously inserted in a census or survey to measure particular indices of the age distribution can be accepted fertility and mortality. What can be learned from a as more reliably reported than others, thus leading to a singlecensus(so that no intercensal rate of increase can be narrower range of uncertainty, the information provided calculated) that provides no special fertility or mortality in figure XV would be of little value. Inspection of this tabulations? figure as well as comparisons of the Indonesian age Age composition is more strongly affected by fertility distribution in cumulative form with the ogives of model than by mortality, so that with minimal information more stablepopulations revealsthe pattern of age-mis-statement reliable estimates can be made of the birth rate than of characteristic of censuses in India, Pakistan and many the death rate. Forexample,if a closedpopulation contains African populations. It is therefore appropriate to use a very high proportion of young children, it must be a the rules of estimation devised for such age distributions. population that has recently experienced high fertility­ Table 2 highlights the figuresthat are relevant in applying an inferencethat is valid whether the expectation of life at these rules. birth is high or low. However, because high infant and child mortality diminish the fraction under age five or ten, TABLE 2. STABLE ESTIMATES OF THE FEMALE BIRTH RATE OF INDONESIA a population with a large proportion of children would DERIVED FROM THE 1961 FEMALE AGE DISTRIBUTION be a moderately high fertility population if mortality were low, and a very high fertility population if mortality Assumption about Birth rates based on assumed leoels mortality or rate ofmortality or rate ofIncrease, werehigh. ofIncrease and proportion rmder age: Consider a specific example: suppose the proportion under ten years of age in a population was 30 per cent. 10 IS 3S If mortality were assumed to be that of the model life table with 0eo = 30 years, births in the precedingten years °eo = 30 ...... 0511 .0492 .0502 could be estimated as 30per cent of the current population = 35 •• 0 ••••••••••••••••• .0534 .0460 .0472 times a factor derived from the model life table expressing = 40 '0 •••••••••••••••••• .0505 .0434 .0450 the reciprocal of the proportion surviving, and the guess may be hazarded that with such a low expectation of life, r = .010 ...... 0811° .0559 .0565 = .015 ...... 0615 .0491 .0503 the population grew moderately-at perhaps 1.5 per cent = .020 ...... 0594 .0435 .0462 per year so that it was about 7.5 per cent larger than it was at the midpoint of the preceding ten years. On these assumptions the average birth rate during the decade OJ Extrapolated figure. preceding the census would be 50 per thousand. On the other hand, if mortality were assumed to match that of the model life table with °eo = 40 years, and if a more The estimates based on C(l5) are a series of minimum rapid rate of population growth-say, 2 per cent annually estimates, those based on C(1O) appear in this instance -were viewed as a sensible guess, the resultant estimate to be a series of maximum estimates, and those based of the birth rate would be 44.3. Often the conditions in on C(35), to be the best obtainable. While the interpre­ which the population lives are known well enough, and tation of the Indonesian age distribution in the light of the mortality of other populations in similar circumstances outside experience does drastically reduce the width of are well enough recorded to give some confidence that a the interval within which the birth rate is likely to be broad estimated range of the level of mortality probably located, lacking further information it would be unwar­ encompasses the actual figure. The use of model stable ranted to go beyond the cautious assertion that the birth populations can be used to translate a recorded proportion rate in Indonesia is probably not lessthan 45 per thousand under age five, ten or fifteen and an assumed level of and not more than 56 per thousand. mortality into an estimated birth rate without elaborate The Indonesian population was enumerated in 1930in calculations. If the basis of estimation is confined to ages a census that did not record the distribution by chrono­ under fifteen, the procedure is closely equivalent to logical age. Strictly speaking, then, Indonesia is not a reverse projection and the estimate is valid even if the proper example of a single census. For example, over- 29 BIRTH. BIRTH RATE RATE

,07 .07 4 . ,.\ 1::\ .06 ,:":\ .06 ,.'i :1.\ ! h . :\ : :1 /\ .05 u .05 / ••••\ :\ ..' \\ : \ I ••···r=.02 \\ :\ I. - :\ I·' '.' r:, '. \_01 : ... '-'.... -. : .04 •••••••••• .04 -. : ...... :

,03 .03

,02 .02

,01 .01

o'----''---~_--'-_--L._ _'__-L-- _ 0'-----''-----'---1.---'---'---'------10 20 30 40 50 60 AGE AGE

Figure XV. Stable population estimates of the female birth rate in Indonesia derived from C(x) as reported by the census of 1961and from hypothetical rates of population growth (left panel) or hypothetical levels of mortality (right panel) looking differences in geographical coverage and com­ ration in the growth rate of about 5 per thousand each pleteness of enumeration, one can calculate the intercensal decade, the estimated female birth rate based on C(35) rate of increase of the female population (for thirty-one and r = .02 should be increased by 6 to 10 per cent-to years) as 1.63per cent per annum. This information would 50 per thousand, or a little higher. The male birth rate is lead us to guess that the growth rate just before 1961 was probably about 8 per cent higher than the female, because 2 per cent (or more), and that the estimate of 46.2 per the number of males is about 2 per cent less, and the thousand for the birth rate was therefore to be preferred number ofmale births is normally about 5 per cent greater. to the 56.5 and 50.3associated with lower rates of increase. As a result of the above arguments then, it is possible to But if a rate of increase above the intercensal average is conclude that the birth rate in Indonesia was probably accepted for the period just before the census, population at least 2 or 3 points above 50 per thousand in the years growth must have been accelerating, primarily, one before the census. A corresponding minimum estimate assumes, because of falling mortality. If mortality had for total fertility, assuming that the mean age of the been falling for fifteen years at a rate causing an accele- fertility schedule is twenty-nine years, is 6.5.

30 Chapter II

METHODS OF ESTIMATION BASED ON RESPONSES TO QUESTIONS ABOUT FERTILITY AND MORTALITY

A. EsTIMATION OF FERTILITY FROM REPORTS ON CHILD­ ever born is very frequently a downward-biased estimate BEARING IN THE PAST l of the cumulative fertility experience of women over thirty or thirty-five, and the average parity of women In many censuses and surveys there appear data on the past age forty-five or fifty, being typically understated, number of children women have ever borne, tabulated by would usually provide an underestimate of total fertility. age of woman. If fertility rates have been approximately On the other hand, younger women presumably report constant in the recent history of the population in question, the number of children ever born to them with much if the reported fertilityhistories have not been substantially better accuracy. Such women are not asked to recall affected by migration, and if differential mortality accord­ events from the remote past or to count accurately to a ing to the prolificacy of the woman has not had an impor­ large number; a higher proportion of the children ever tant effect on the survival of mothers, the average number born have survived to the time of the interview and few, of children ever born to women past age forty-five or fifty if any, have left the household. If age-misreporting does equals total fertility. MOle precisely, the average number not cause excessive distortion, the sequence of average of children ever born per woman aged forty-five to forty­ numbers of children ever born by age of woman duplicates nine equals the total fertility of this cohort of women, closelythe curve of cumulative age specific fertility rates, which in turn is about the same as the total fertility of the until an age is reached where the proportion of children population at the time of the census or survey, provided ever born that are omitted by the respondents becomes fertility rates in the population have been approximately significant. In other words, the early part of a curve constant. If the assumptions stated above are valid, it is showing the rise in the average parity of women with age possible to estimate the age specific fertility rate for each should resemble closely the cumulation by age of fertility five-year age interval within the child-bearing span by rates. On the other hand, towards the end of the child­ (for example)fitting a polynomial to the reported number bearing interval, as the average number of children ever of children ever born by age of mother. born approaches total fertility, the tendency towards The uncritical use of the reported average number of omission or understatement causes reported parity to children ever born as a means of estimating the fertility fall short of cumulated fertility rates. To provide a more of a population is risky, however, because of a widespread valid estimate, then, it would be useful to splice to the tendency for the number of children ever born to be rising curve of children ever born with age at the younger under-reported, especially for older women. In many ages a curve which continued to rise with age in a way censuses or demographic surveys the average number of reflecting more accurately the actual fertility rates above children ever born increases too gradually with age, age thirty or so. If we could be sure of the approximate especially at ages above thirty or thirty-five, and indeed relationship between fertility rates at different ages in the a common feature of many censuses is reported average given population, we could determine what fertility rates numbers of children ever born that declinewith age above at the younger ages would account for the average number age forty-five 01 fifty. One can speculate about the causes of children ever born reported by the younger women, of the apparent under-reporting of the average number and then ascribe to the older women fertility rates of children on the part of older women. Perhaps the most consistent with these rates established at the younger ages important factor is that some women tend to omit children of child-bearing. For example, if all populations in less who have grown up, or who have left home. A second developed countries had approximately the same age possibility is the inability of some illiterate respondents pattern of fertility, and differed only according to the size to report large numbers accurately. A third hypothesis of a factor expressing the level at which this age pattern is that older women tend to omit offspring who died, of fertility operated, the level could be ascertained by especially many years earlier, although there are many looking at the average number of children ever born by instances in which the proportion reported as dead rises younger women-say, at ages twenty to twenty-four and with the age of woman in a consistent fashion. For twenty-five to twenty-nine. It would then be possible to whatever reason, the reported average number of children prepare a rough estimate of the remainder of the fertility schedule and thereby to estimate total fertility. However, no such universal age pattern of fertility exists, although 1 W. Brass. A.J. Coale, P. Demeny, D. Heisel, F. Lorimer, A. Romaniuk, and E. van de Walle, The Demography of Tropical it is probably true that many populations share a similar Africa (Princeton, Princeton University Press), (in press), chapter III. age pattern of fecundability, that is a similar age pattern 31 of the probability of conception among women living demographic survey. The number of births reported in regularly in sexual union, without practising birth control. response to such questions has not proven accurate. The age pattern of fecundability is not identical in all Experience seems to indicate that the source of the inaccu­ populations primarily because of differences in the racy is not a systematic tendency for women to fail to incidence of secondary sterility, or in the average age at report births that have occurred or to exaggerate the which childbearing ceases. But even if differences in the number of these births but rather the difficulty that age pattern of fecundability could be ignored, it would still respondents have in identifying properly the length of the not be possible to ignore differences by age in exposure to interval for which births should be reported. In some the risk of pregnancy. The major source of such differences surveys there is a net tendency for women to report births is in the age pattern ofthe establishment ofregular sexual that occurred in a period that is less than a year and in unions-through marriage or other socially sanctioned other surveys to report on the average events that occurred institutions. The average age at first marriage for females during more than the preceding year. The factors causing in populations not commonly practising contraception this reference period error seem likely to depend on gene­ varies from less than fifteen in India before the second ral cultural conditions, the circumstances of the particular World War to over twenty-five in many parts of Europe survey, including the wording of questions, the instruc­ during the nineteenth and early twentieth centuries. In tions to enumerators and the like. There seems no reason societies where consensual unions and formal marriages to expect an association between errors in reference period are both common, the age of entry into the former is and the age of respondents. Moreover, the population typically much younger than into the latter, and con.. covered by questions on births during the preceding censual unions do not ordinarily involve as regular expo­ interval is the same as that covered by questions about sure to intercourse as marriage. There are similar though children ever born so that inconsistencies caused by quantitatively less important differences in the dissolution differences in coverage or in forms ofage-misreporting are of sexual unions. The incidence of widowhood depends avoided-differences that may exist between a population upon the level of mortality, and the of divorce included in a register and a population covered by a upon law and custom. Societies also differ with regard survey. to remarriage of the widowed and divorced. The necessary questions on births during the preceding To adjust the reported numbers of children ever born year on the one hand and on children ever born on the for a characteristic omission on the part of older women other, both tabulated by age of woman, have been included it is therefore necessary to have reliable evidence of the in a number of demographic surveys in Africa. William particular age pattern of fertility in the population in Brass has designed for use primarily in Africa a method of question. There are two potential sources of direct fertility estimation that accepts as essentially correct the information on child-bearing rates by age of mother. The pattern of fertility rates by age indicated by the births first is birth registration, and the second responses to a reported as occurring during the preceding year, and that surveyor census question on births during the year accepts as an essentially correct indication of the level preceding the survey, either tabulated by age of woman. of fertility the average number of children reported as Of course if either of these sources were known to be ever born by younger women. The method requires the accurate in the coverage of births the resultant data could estimation of the average value of cumulative fertility by be used directly to construct fertility measures. But even age over the same age intervals (usually 15 to 19, 20 to when the fertility rates derived from these sources are 24, 25 to 29 etc.) for which the average number of children not accurate, they may be approximately correct in form. ever born is reported. It is then assumed that the source of In other words there is the possibility that even if the the difference between the estimated average value of number of births registered or reported understates or cumulated fertility at the younger ages (such as 20 to 24 overstates the true number, the degree ofunderstatement or 25 to 29) and the average number of children ever born or overstatement is not age selective. This possibility is a at these ages is an erroneous perception of the reference plausible one with regard to the events reported by period by the respondents. The multiplier that would be respondents for the preceding year, but is much less so for needed to bring cumulative fertility at the younger ages registered births. Incomplete registration cannot ordi­ in line with the reported average number of children ever narily be used to indicate the age pattern of fertility born is determined, and the reported numbers of births at because there is no reason for supposing that the popu­ all ages are multiplied by this factor. lation covered by registration is representative of the The basic principle underlying the Brass method of whole population with respect to the fertility schedule. fertility estimation is simple enough, and the computations If birth registration is associated with literacy, for example, are complicated only by the difficulty of estimating the there may well be a tendency for the births occurring to average value of cumulated fertility for the same age younger women to be more completely registered than intervals for which average children ever born are given", those occurring to older women. Other likely sources of Ifapproximate age specificfertility rates based on reported bias include differential completeness of registration in births by age of mother are tabulated only in five-year age regions that are not uniform with respect to the age intervals, cumulative fertility can be calculated directly pattern of fertility-in urban areas as compared to rural, only at the boundaries of these age intervals. Thus five for example. times the age specific fertility rate ofwomen 15 to 19 gives The other potential source ofinformation about the age the cumulative fertility to age 20; this value plus five pattern offertility is survey data on births occurring during some preceding period, typically a year, before a census or 2 W. Brass, et al. loco cit. 32 times the fertility rate of women 20 to 24 gives the cumu­ governed more by decliningfecundability than by customs lativefertilityto age 25,and so on. Itis necessarytherefore and institutions. The decline of fecundability with age in to estimate from cumulativefertility to ages 20, 25,30etc., different populations is likely to follow a roughly similar what would be the average value of cumulative fertility pattern while no such similarity is found with respect to in the five-year age groups for which average parity (or the age of entry into cohabitation. These considerations average number of children ever born) is reported. If age suggestthe hypothesis that the ratio of the average parity specific fertility rates were constant within each five-year of women at the end of child-bearing to the average age interval the average value of cumulative fertility parity of a younger group (say women 25-29) is closely would be closely approximated by simple linear inter­ related to the relative parity of women early and late in polation to the midpoint of each age interval. In fact, their twenties. The reasoning behind the hypothesis is as the typical pattern of fertility rates (increasing as they do follows: (a) if the average parity of women 25-29 is an from the earliest age of child-bearing to a peak usually in unusually large multiple of the average parity of women the late twenties) creates a curve of cumulative fertility 20-24, child-bearingdoes not begin early in this population that is not linear and that therefore requires a more com­ since the high ratio implies that fertility is unusually high plicated form of estimation than simple linear interpo­ at ages 22.S to 27.S compared to before 22.S. It follows lation. Brasshas calculated variable interpolation factors that in this population an unusually large fraction of total to be applied to the readily calculated values of cumulated fertility occurs in the later years of child-bearing, and the fertilityto the boundaries of the ageintervals.The selection ratio of final average parity to the average at ages 2Sto 29 of appropriate interpolating factors is determined by how is therefore unusually large; (b) on the other hand, an rapidly the reported fertility rates increase from the first unusually low ratio of parity at 2S-29 to parity at 20-24 to the second age group of women, sinceit is the steepness indicates that high rates of child-bearing began early, of the rise in fertility that determines the curvature in that an unusually small fraction of total fertility occurs cumulative fertility with age over the early portion of the in the later years of child-bearing, and that the ratio of cumulative fertility function. Tables in annex IV present final averageparity to the average at 2S-29 is unusually low. the interpolation factors-table IV.1 to be used when age Suppose that the average number of children ever born specific fertility rates are available for the five-year age (average parity) to women lS-19 is designated PI, to groups bounded by exact ages l4.S and 19.5, 19.5 and women 20-24P2, and so on, until P7designates the average 24.S etc.; 3 and table IV.2 to be used when such rates are parity of women 4S-49, Suppose the average parity of available for the conventional five-year age groups. In women reaching age S0-assumed to be the upper limit of chapter VII there is a fully worked out example of the child-bearing-is designated TF (for total fertility). Our estimation of fertility by this method. hypothesis is, then, that TF/P3 is closelyrelated to P3/PS. There are many censuses and surveys that have included The usefulness of this hypothetical relationship, should it a question on children ever born, with the responses prove valid, is that it provides a possible method of tabulated by age of woman, but that have not included a estimating total fertility when older women under-report question on births in the precedingyear, so that there is no the number of children they have borne, and younger possibility of using the method just described. As stated women report parity accurately. earlier, it is a mistake to assume that differentpopulations The hypothesis obviously cannot be tested by examining have the same age pattern of fertility-even populations the relationships among average numbers of children not employing contraception or abortion. However, the ever born reported by women over SO, 2S-29 and 20-24in form of the fertility schedulesin populations that employ a number of populations that do not practise birth control, little birth control differs primarily in the way in which because it is the very inaccuracy of these reports that fertility rises from the first ages of childbearing to the leads us to consider the expected relationship. The test ageswherefertilityis a maximum, and relatively much less employed is based on the cumulation of age specific in the way fertility declines after the peak is reached. fertility rates in a number of apparently reliable fertility This greater relativevariability in the early part of fertility schedules (based on virtually complete birth registration) schedules results from the fact that the rise of fertility to construct a set of necessarilyconsistent average parity with age is strongly affected by customs and institutions schedules.Thus the average number of children ever born governing the establishment of sexual unions-strongly was calculated at ages 20-24,2S-29 and on reaching age SO affected, that is to say, by the age pattern of nuptiality in in a group of women subject to each of the given fertility societies where formal marriage is a principal determinant schedules. By constructing the average parties in this way, of cohabitation. In other words, when birth control is not we insured that there would be no distortion from faulty widelypractised, the shape of the early part of the fertility reporting, or changing fertility.The ratios ofTF/P3 and of schedule is dominated by the age pattern of entry into P3/P2 are plotted in figure XVI. The relationship is regular sexual relations rather than by the age schedule gratifyingly close, and is well represented by simple of fecundability; and in contrast the decline of fertility equality of the two fractions, or by the formula TF with age-when birth control is not practised-is generally = (P 3)2/P2· This formula provides an estimate of total fertility 8 When age specific ferti1ity rates are based on births reported in the preceding year, the mothers were approximately one half-year under the followingconditions: younger when the births occurred than at the time of the survey. (1) Fertility at ages lS-29 has been constant in the Therefore births reported by women between exact ages 15 and recent past; 20 serve as a basis for estimating fertility rates for women 14.5 to 19.5.This slight displacement of age must be allowed for in calcula­ (2) The age pattern of fertility conforms to the typical ting cumulative fertility. age relationships found in populations practising little 33 • 4.4

4.2

4.0

3.8 • 3.6

3.4

3.2

3.0

2.8 • 2.6

2.4

2.2

2.0 • 1.8

o 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 P3/p'

Figure XVI. Ratio ofTF (total fertility) to Pa (average parity at age 25-29) versus ratio of Ps to Pa (average parity at age 20-24) calculated from fertility schedules of populations not practising birth control birth control, implying (a) that the age pattern of declining B. ESTIMATES OF MORTALITY BASED ON PROPORTIONS fecundability is typical; and (b) that widowhood, divorce, SURVIVING AMONG CHILDREN EVER BORN and other forms of dissolution of sexual unions do not have an unusual age incidence from age thirty to forty-five In some of the censuses and demographic surveys in in the population in question. which women are asked to report the total number of The value of (P3)2IP2 can be compared with the average children ever born to them, there is an additional question parity reported by women over 50, and 45-49. Lower asking the number of surviving children. It has long been parity reported by women over 50 than at 45-49 is an realized that the proportion surviving depends on the indication of likely omission of children ever born by level of infant and child mortality, and census reports in older women. If under these circumstances (P3)2IP2 which data on surviving children are included often con­ exceeds P7, the estimate gains in credibility and is to be tain comments that variations within the enumerated preferred to the numbers supplied by the older women. population in proportions surviving can be considered Approximate equality of P7' the average parity reported an index of differential mortality. Recently William by women over 50, and (P 3)2IP2 indicates that any of the Brass has greatly increased the usefulness of data of this three figures is an acceptable estimate of total fertility. sort by developing a method of translating proportions But if (P3)2IP2 is substantially lessthan P7' or if the average surviving and proportions dead among the children ever parity reported by women over 50 is much greater than born to women in different age groups into conventional P7' the wisest course is not to attempt an estimate of measures of mortality. His technique makes it possible current fertility by manupulation of the data on children under certain circumstances to estimate the proportion ever born. ofchildren born who survive to age 1, 2, 3, 5, 10, 15, ... ,35 34 from the proportion reported as surviving among children the proportion dead among the children ever born to ever born to women 15 to 19, 20 to 24, 25 to 29, ..., 60 older women, the index of early or late fertility is the to 64. Because a full account of this technique appears mean age (iii) or the median age (iii') of the fertility in another publication4 there is no attempt in this schedule. These adjustment factors are given in annex V; Manual to explain the somewhat surprising correspon­ in table V.l (to be used when children ever born and dence between Ix values and data on child survival in surviving are tabulated by the conventional five-year age women's reports of children ever born, nor to justify groups) and in table V.2 (to be used when the census in detail the adjustments described below. Our discussion tabulations are by ten year groups of women: 15-24, is restricted to a brief outline of the conditions under 25-34 etc.). Examples of estimating mortality by this which the technique may be applied, and of the steps method are worked out in chapter VII. involved in constructing estimates. Of course the conditions specified as ideal are seldom if The Brass method of estimating mortality enables the ever completely fulfilled. Because of the sensitivity of the analyst to construct the of a life table estimate of the proportion dying before age one to up to early adult ages. The conditions that would make peculiarities or defects in the data the estimate of infant this computation accurate are: mortality directly derived by this technique does not (1) The age specific fertility schedule has been approxi­ justify much confidence. On the other hand, estimates of mately constant in the recent past (at least for the younger child mortality up to older childhood ages based on the women), and the approximate form of the schedule is reports by older women of still-living children and of known; children who have died are especially subject to reporting (2) Infant and child mortality rates have been approxi­ errors and also to the effects of the possibly different mately constant in recent years; mortality levels in the more distant past. The estimates that appear to reflect the best chance of minimizing error (3) There is no powerful association between age of from various sources are those for the proportion dying mother and infant mortality or between death rates of before age two and age three. Because of the prevalence mothers and of their children; of falling death rates in many of the less developed coun­ (4) Omission rates of dead children and of surviving tries in recent years, it should be borne in mind that the children are about the same in the reported numbers life table value estimated by these procedures represents ever born; the average mortality experience over the preceding four (5) The age pattern of mortality among infants and or five years in the determination of the proportion dead children conforms approximately to the model life tables. before the second birthday, and in the preceding six to Under such ideal conditions it has been shown that the eight years, in the estimate of proportion dead before the proportion of children dying before their first birthday third birthday. is not very different from the proportion dead among those The ideal conditions listed above were not fully realized ever born to women 15 to 19, the proportion dying before in the population of Hungary before the censuses of the second birthday not very different from the proportion 1930, 1949 1960; nor in the population of Canada just dead among the children ever born to women 20 to 24, before the census of 1941. Mortality was changing, the before the third birthday to the proportion dead among age pattern of mortality in Hungary does not conform children ever born to women 25 to 29, before the fifth to the "West" pattern, and fertility was not constant. birthday to the proportion dead among children ever born Questions in the censuses concerning children ever born to women 30 to 34, before the tenth equal to the propor­ and surviving were limited to married women only. tion dead among children ever born to women 35 to 39 etc. Nonetheless, when estimates are made by the Brass These approximations are very close for a population techniques they come remarkably close to estimating the characterized neither by a very early nor by a very late proportions dying before age two in the four or five years start in child-bearing. If fertility begins at a very early preceding each of the censuses, even though the range age the children ever born to women in each age group of mortality estimated is from a high of over 200 per would be exposed to more prolonged risks of mortality, thousand to a low of less than 70 per thousand, as shown and, therefore, the proportion dead would tend to be in table 3. Columns 2 and 3 of table 3 were obtained by higher for each age group of mothers than when fertility calculating the average level of infant mortality in the has a later start. Brass has constructed a set of adjustment given country and time for a four and five year period factors that can be used to modify the estimates of before the given census and then using the relationship proportions dying defore age 1, 2, 3, 5 and so on, in between lqO and 2QO as shown in the nearest official life accordance with whether the starting point of fertility tables to estimate the proportion dying before age two. in the population in question is early or late. The index Column 4 is derived from the reported proportions of early or late fertility is the ratio of the average number surviving to women aged twenty to twenty-four, using the ofchildren ever born in the first two age groups ofwomen multipliers of annex table V.I, selected according to the -ptiP2' In making adjustment for estimates derived from reported PtiP2 ratios. When the potential sources of bias are considered, it is evident that the estimation of child mortality by this 4 See the discussion in W. Brass, et al., op. cit. For an earlier procedure tends almost always to err on the low side, exposition see W. Brass, "The Construction of Life Tables from if the estimate is not accurate. The presumption of bias Child Survivorship Ratios", in Union internationale, International Population Conference, New York, 1961, (London, 1963), vol. I, in this direction is based on a judgement that respondents pages 294-301. are much more likely to omit from a summary of their 35 TABLE 3. PROPORTIONS DYING BBFORE AGB 2 (VALUES OF 2QO) IN experience to date children who have died than those who SPBCIFIED PBRIODS AS DBRIVED FROM VITAL REGISTRATION DATA have survived. FOR HUNGARY AND CANADA, AND AS ESTIMATED FROM PRO­ PORTIONS OF CHILDREN SURVIVING REPORTED IN CBNSUSES OF Estimates of infant mortality have been obtained in a THE SAME COUNTRIES number of populations by first applying the method here described to detmine the proportion dying before the From vital reglstratlolldata second birthday and then assuming that this proportion is related to infant mortality as in the "West" model Average 0/4 Average ofj Estimated/rom census Census years preced/llg years precedlllg survivorship life tables. It is remarkable in view of the expected down­ the census the census rates ward bias that these estimates exceed (often by a major extent) the average level of infant mortality derived from registered births and deaths in the four years before the Hungary 1930 ...... 204 .203 .203 survey in the countries shown in table 4. This comparison 1949 ...... 138 .133 .137 indicates the widespread prevalence of under-registration 1960 ...... 063 .063 .067 of infant mortality. Canada The demonstrated accuracy of the Brass method of 1941 ...... , .. .069 .071 .072 estimating infant and child mortality in Hungary and Canada, in conjunction with the fact that it appears in many ofthe less developed countries to provide an estimate TABLE 4. INFANT MORTALITY RATES (tqo) ASSOCIATED WITH aqo much closer to actuality than can be obtained from VALUES THAT WERE ESTIMATED FROM PROPORTIONS OF CHILDREN registered data suggests that this method will prove a SURVIVING AS REPORTED IN VARIOUS CENSUSES AND AVERAGE powerful and welcome addition to the techniques available INFANT MORTALITY RATES DURING THE FOUR YEARS PRECEDING to demographic analysts. Unfortunately, the method does EACH CENSUS AS REGISTERED IN VITAL STATISTICS not provide information about adult mortality." To Estimates of lqO estimate the expectation of life at birth or the crude Av#rageQ/'lqO III obtained from ,qO death rate from these childhood survival rates, it is thefort year, prece­ according to dlllg the cellSUS, "West" /llfe necessary to make an assumption about the relationship accordlllgto vital table" based all Coulltry Year registrat1011 data censusreports between death rates at different ages. It is a simple mechanical procedure to select a model life table having the same proportions surviving to age two as is indicated Barbados 1946 .161 .162 British Guiana 1946 .119 .131 by this technique of estimation. However, there is little Brunei 1960 .097 .119 basis for confidence thatthe relationship between mortality Cyprus 1960 .030 .050 rates at different ages in the "West" model tables holds Egypt 1947 .152 .212 closely in a population in Africa, Asia or Latin America. Fiji Islands 1946 .067 .115 Fiji Islands 1956 .051 .093 North Borneo 1951 .114 .186 Peru 1940 .1280 .202 Sarawak 1960 .065 .114 Seychelles 1960 .049 .069 Western Samoa 1956 .038 .095 Western Samoa 1961 .023 .080 Windward Islands .. 1946 .112 .129 Yugoslavia 1953 .117 .133 6 Concerning the possibility of obtaining such information from census reports on the proportion of orphans in a population, see however Louis Henry, .. Mesure indirecte de la mortalite des adultes", a 1940 only. Population, 1960, No.3, pages 457-466.

36 T

Chapter III

ESTIMATES OF FERTILITY AND MORTALITY BASED ON REPORTED AGE DISTRIBUTIONS AND REPORTED CHILD SURVIVAL

In chapter I there are described methods of estimating A. ESTIMATION OF BIRTH AND DEATH RATES FROM can.o­ fertility and mortality that make use of the age distribution HOOD SURVIVAL RATES AND A SINGLE ENUMERATION BY of a population recorded in one or more enumerations. A REVERSE PROJECTION secondenumeration providesvaluablesupplementaryinfor­ mation in the form of the rate of increase and survival An accurate census that records the number of persons rates. When there has been only one usable enumeration in each five-year age interval provides the basis for of a population, only very rough estimation is usually reconstructing recent birth and death rates, if migration possible. However, in chapter II a technique of cal­ either is known accurately or is negligible in magnitude, culation is outlined that makes it possible to determine and if survival rates by age are known for recent periods. proportions survivingfrom birth to age, 2, 3, 5 and some­ The method of estimation is simply to reversethe custom­ times to ages up to 30 or 35 from data supplied by wom­ ary procedures of population projection-to reconstruct en about the history of the children they have borne. by reverse survival the births that brought into being the Knowledge of childhood mortality is an extremely children recorded as under age five or ten, and to recon­ useful adjunct to knowing the age composition of a struct the population among which the births occurred population, particularly for the estimation of fertility. by reversesurvival of persons over fiveor ten in the census. In fact, responses about the survival of children ever born The specific steps employed in reverse projection are to are probably more useful in estimating recent fertility divide the population under five by sLo/5 x 10 from a life from the age composition recorded in a census or survey table representing the mortality of the preceding five than is the existence of an earlier enumeration. Speci­ years to obtain an estimate of births, and to divide each fically, if C (x) is known, 12 is a more useful supplementary five-year age group by the appropriate survival factor datum than r, the rate of natural increase, in estimating from the same life table to reconstruct age group by age the birth rate. On the other hand, knowledge of 12 (with group the population five years before. The sum of all or without the age distribution) gives only inferential such estimated age groups is the estimated total population evidence about adult mortality (i.e., mortality above age five years earlier. The estimated average annual number of five). From data on childhood mortality, mortality above births can then be divided by the average of the enumer­ age five can be estimated only on the basis of assumed ated population at the end and the estimated population regular relations between mortality at different ages, and at the beginning of the period to give an average birth in different populations the relation of child mortality rate during the preceding five years. The average annual to adult mortality varies substantially. For example, in rate of increase (l/510ge Pt/Pt-s) can be subtracted from the four families of model life tables expressing average the birth rate to estimate the death rate. Similarly, the mortality patterns in different "regions", expectations of population 5-9 can be projected back to provide the lifewere calculated at age 5 ranging from forty-six years to estimated birth rate in the next earlier five years, etc. over fifty-three years associated with the same proportion The estimated total population becomes subject to increas­ (about 75 per cent) surviving from birth to age two. ing uncertainty as the reverseprojection procedes,however, Two enumerations spaced five or ten years apart, on even if mortality is somehow accurately known: the oldest the other hand, provide a good indication of the level of age group in the past population has no current survivors, adult mortality, but no direct evidence on childhood and this segment of the past population must be estimated mortality. Expectations of life at age five estimated from by some assumption about the nature of the past age model stable populations with a given C (30) and a given distribution. For earlier and earlier dates, the portion rate of increase vary only from 51.7 to 52.4 years when of the population estimated in this way is larger and larger. based on the different "regional" model tables; the esti­ mated level of mortality above age five in this instance is Given an accurate census, the crucial additional essentially independent of variation in age pattern. But element for reverse projection is an appropriate life table. when rand C (x) are known, childhood mortality must The Brass method of estimating child survival provides be approximated by assuming some kind of "normal" approximate values of 12 (during the preceding four or association between child and older age death rates. It five years) and 13 (during the preceding six or eight). will be seen in each of the techniques presented in this A model life table can be selected with the given 12 , and chapter that a solidly based figure for child mortality survival factors from this table employed for the reverse is needed for any precision in an estimate of the birth rate projection. The value of sLo in the model table is very derived from an enumeration of a population. close to the correct one, if the data from which 12 was 37 estimated are accurate. Differences in age patterns of based on age data from one sex than when based on the mortality do not much affect the ratio of 12 to sLo• On other. It is then necessary to make some rough estimate the other hand survival rates above age five are estimated of the 12 or 13 implied for males and females by 12 or 13 by assuming that the age pattern of mortality conforms for the two sexescombined. Often there is indirect evidence to the "West" familyofmodellife tables, and ifone judges indicating the direction and approximate extent of sex by differences among life tables based on accurate data, differences in mortality: the sex ratio of a closed popula­ the actual survival rates above five may diverge from the tion that has not experienced large sex selective military "West"family.However,differences in mortality aboveage deaths indicates the sex incidence of mortality since the five in life tables with a given 12 would rarely produce sex ratio at birth can often be closely estimated; and the estimates of over-all population 2.5 years earlier differing sex ratio of mortality in registration areas, in sample by more than one per cent, so that the error in the esti­ surveys, or even in neighbouring populations can be mated birth rate from this source would rarely exceed taken as relevant evidence. half a point (e.g., an estimate of 50.5 per thousand in­ Reverse projection cannot be recommended as a stead of 50). generally satisfactory basis of estimating birth rates even If the questionson children everborn and survivinghave when calculation of child survival rates is possible because been asked and recorded separately for males and of the frequent unreliability of recorded age distributions. females, separate estimates of child mortality can be made The tendency for the proportion under five to be under­ for each sex and (if the internal consistency of the data is reported in many censuses and surveys has often led acceptable) the model life table selected for each sex can demographers to base estimates of the birth rate on the be based on this separate evidence. However,the questions reverse projection of persons five to nine. However, the on children ever born are typically asked (or tabulated) proportion of the population five to nine is frequently only for both sexes combined. One is tempted to derive, overstated by a wide margin (because of understatement from values of 12 and 13 for the two sexes together;: of the ages of adolescent girls, for example, and because estimates for each sex on the assumption that the relation there is a common tendency to overstate the age of some of female to male child mortality in the given population children under five), so that this procedure cannot be is the same as in the population whose experience under­ endorsed as always valid. In fact, if there is evidence of liesthe "West" model tables. The typical relation between substantial age-misreporting, it is not possible to make male and female mortality in these populations is that good use of reverse projection unless some means is male and female life tables tend to be at about the same available for identifying a valid part of the reported age level; i.e., when 12 for females is .72765 (level7, °eo = 35.0), distribution or of adjusting the reported figures. the typical 12 for males is .69537 (also level7, °eo = 32.48). It is easy to construct a table containing the combined l« values for x = I to 5 for both sexes (assuming a typical sex ratio at birth of 105 males per 100 females) at each B. ESTIMATION OF BIRTH AND DEATH RATES FROM CHILD "level", and then to assume that male and female mor­ SURVIVAL RATES AND A SINGLE ENUMERATION BY MODEL tality is expressed by the life table for each sex at the STABLE POPULATIONS "level" indicated by the value of 12 and 13 for the two sexes together. However, the evidence available on sex The use of model stable populations to estimate charac­ differences in mortality in the less developed countries teristics of a population requires the identification of a does not warrant the assumption that these differences stable population among the tabulated age distributions alwaysconform to the relations found in the experience­ that shares some of the observed or inferred characteristics primarily from Europe, North America, and Oceania­ of the recorded population.' In chapter I, the identifying underlying the model life tables. It is possible to find 1 features used to locate a model population were the many examples of female mortality higher than male intercensal rate of increase (assumed equal to the rate of mortality and this resort to male and female model tables growth of the stable population), and the cumulative at the same level may introduce a mortality differential age distribution or ogive up to some age that depends on opposite to the actual one. the apparent pattern of age-misreporting. Under the If the age distributions of both sexes are about equally conditions considered in this section, a model stable usable as a basis for estimating fertility, the uncertainty of population is identified by the estimate of 12 , which by sex differences in mortality can be ignored, and the values means of table 1.2 in annex I determines the level of mortality, and by the ogive, which is used in a manner of 12 or 13 estimated for the two sexes combined can be employed as if it were a valid estimate for each sex sepa­ wholly analogous to the procedures outlined in section B rately. If in fact the sex differences in child mortality are of chapter I. For example, if the age distribution is of the substantial, estimates based on the common value of 1 Indian-Pakistani-Indonesian-African pattern, a minimum 2 estimate of b can be obtained from C(lS), and a less or 13 will overstate the birth rate and death rate for one sex, and understate the rates for the other, but provide biased estimate from the value associated with C(35). unbiased figures for the whole population. In general, however, the reliability of estimation is greater when a As noted earlier, the identification of a model stable population does not determine unique value of the gross reproduction rate or 1 Pravin M. Visaria, The Sex Ratio of the Population of India of total fertility. To estimate these quantities, the mean age of the (unpublished doctoral dissertation, Princeton University, 1963). fertility schedule must first be estimated. For a discussion of this Available at University Microfilms, Ins., Ann Arbor, Michigan. topic, see section B.5 in chapter I. 38 Two features of estimation by model stable populations of any other population parameters dependent on selected by /2 and C (x) are worth noting: first, if an esti­ mortality in this age range, and to accept the estimates of mate of the birth rate is based on C (5), the result is mortality over age five that can be derived from an essentially identical with the results obtained by reverse analysis of the two censuses-either by , projection, whether or not the population is stable; and or by accepting the mortality above age five consistent second, estimates of the birth rate obtained from 12 with C (x) and r. Of course such general advice is con­ and C (x) are insensitive to differences in age pattern of tingent on the quality of the basic figures. If two censuses mortality, at least the differences found in the four are unequal in completeness of coverage, or if interna­ families of model tables. tional migration is substantial and inadequately recorded, The birth rate in the model stable population with the it would be necessary to rely more on inferences that same /2 and C (5) as the observed population is identical could be based solely on age distribution and estimated to that which would be obtained by applying reverse child survival. Similarly, if internal inconsistencies were projection to the children under five in the stable popu­ apparent in the reports ofchildren ever born, the estimates lation to obtain births, and reverse projection to the of child survival might not be usable. whole stable population to obtain an average number of persons (the denominator of the birth rate). The number 1. Estimation of birth and death rates in a non-stable of births estimated for the actual and the stable popu1a­ population tions is identical, so that the only source of difference between the birth rate estimated by reverse projection Suppose a population is enumerated in censuses at the and that found in the model stable population is in the beginning and end of a decade, and that the second denominator, which in each case is a number obtained census includes data permitting the calculation of 12 by applying reverse projection, with the same life table, to and 13 , The best estimate of the over-all life table is populations with the same number of persons, and the obtained by accepting °es (and the mz and qz values for same number over and under five, but possibly differing age five and over) from the model life table chosen as in the internal age structure above age five. This point best fitting census survival rates, and by taking Is (and loses relevance as the choice of a model stable populationis 51110) from the model life table with the value of 12 or 13 made dependent on C(lO), C(15) and ogives to higher obtained by the Brass methods. The expectation of life ages, because with possibly different internal age compo­ at birth in this hybrid model table is easily calculated. sition in the population (e.g., under fifteen) that is The average death rate for the decade is then calculated. implicitly or explicitly projected back to birth, and with by applying the m, values in this hybrid life table to a differences in the size of the reverse projected denominator rough mid-decade age distribution obtained by averaging becoming more pronounced as the time period of reverse the distributions at the beginning and end. The birth rate projection is extended, the virtual identity of the two can then be estimated as equal to the death rate plus the estimates is lost. This feature does imply, however, that average annual rate of increase. interpolation in stable populations is a convenient way of determining the birth rate implied by reverse projection of children under one, under five, one to five, five to ten 2. Estimation ofbirth and death rates in a stable population or under ten. enumerated more than once

The insensitivity ofbirth rate estimates from 12 and C (x) If the popu1ation enumerated in two or more censuses to differences in age patterns of mortality is an important has closely similar age distributions in each census, the advantage of such estimates. The advantage lies in the methods just described can be supplemented by the fact that the true age pattern of mortality.is usually not following procedures: (a) estimate the birth rate by known, and if estimates based on alternative plausible selecting a model stable population from 12 and C(x); age patterns are widely different, the range of uncertainty and (b) estimate the death rate as the birth rate. less the is great. This point is discussed in greater detail in intercensal rate of increase. A slightly more elaborate chapter IV. procedure may be applied when an estimated life table, an adjusted age distribution, and other detailed parameters are sought. This procedure recognizes the superiority C. ESTIMATION OF BIRTH AND DEATH RATES FROM CHILD of the Brass estimation procedures for determining SURVIVAL RATES AND AGE DISTRIBUTION IN A POPU­ childhood mortality, and of stable population techniques LATION ENUMERATED SEVERAL TIMES using C (x) and r to estimate mortality above age five. The procedure entails: (a) selecting a model stable population When a population has been enumerated more than from 12 and C (x), and accepting the proportion under once at an interval of about five or ten years, the methods five and the child death rate in this population, and (b), of estimation presented in chapter I can be applied, and selecting a model stable population from C (x) and r, at first thought the mortality estimates derived from data and accepting the age specific death rates above five, on survival among children ever born might be considered and the age distribution within the span above five in this merely as a verification of the mortality estimated on the population. The over-all death rate is then estimated as the basis of C (x) and r, or C (x) and a model life table consis­ sum ofthe death rate under five in (a) times the proportion tent with fractions surviving from one census to the next. under five in stable popu1ation (a) plus the death rate However, the best use of such data is to accept the over five in (b) times one minus the" proportion under 5 estimates they provide of mortality under age five, and in stable population (a). 39 D. ADJUSTMENT OF ESTIMATES OF FERTILITY DERIVED FROM obtained previously) can be adjusted to allow for the CIDLD SURVIVAL RATES AND THE AGE DISTRIBUTION effects of declining mortality on the age distribution. The WHEN MORTALITY HAS BEEN DECLINING procedure is the same as that described in section C of chapter I, using the adjustment factors from the appro­ If the population for which childhood survival can be priate part (part (c) or (d)) of table IILl in annex III. estimated appears to have experienced declining mortality If questions on child survival were asked in two consecu­ during the recent past, the stable estimates of the birth tive censuses,it is also possible to estimate the parameter k rate and of the gross reproduction rate (assuming that an by means of table IlIA. estimate of the mean age of the fertility function has been

40 Chapter IV

ACCURACY OF ESTIMATION

The extraction of approximate birth and death rates The four families are described in chapter I. The distinc­ from censuses of varying completeness, in which age, tive age patterns of mortality they embody are examples parity and other relevant data are inaccurately or incom­ of differences found among populations with especially pletely reported, can scarcely be expected to produce accurate data, and undoubtedly do not nearly exhaust the figures of great precision. Moreover, the range of error variety of patterns to be found among all populations. in the estimates about the true figure cannot in general be Nevertheless, it can safely be assumed that forms of esti­ determined at all exactly. The purpose of this chapter is to mation that yield nearly identical figures in all four call attention to the imprecision of estimates based on families are insensitive to age pattern differences, and are data of poor quality, even when they have been made on that account preferable to forms of estimation that with the help of elaborate tables and adjustments; to yield divergent figures. The itself can be taken give some rough impression of the magnitude of error as a minimum index of the uncertainty of estimation that is routinely encountered; and to distinguish the associated with variations in age patterns; it is obvious forms of estimation most subject to large errors from that variations at least this large do occur. those less susceptible. The effectson various estimates ofdifferent age patterns The sources of error discussed are of two principal of mortality will be illustrated by utilizing the four kinds: errors caused by a discrepancy between actual and families of model life tables to make calculations for a assumed conditions, and inaccuracies in the basic data. population that is assumed recorded without error. The Crucial assumptions that may be wrong are that the age "test" population is a "West" female stable population pattern of mortality in a given population conforms to a with an expectation of life at birth of forty years, a gross family of model life tables, and that the age composition reproduction rate of 3.00, and mean age of fertility of of a population has a form stable in the sense of Lotka. twenty-nine years. The various characteristics of this The kinds of inaccuracy that affect almost all forms of population can be obtained from the appropriate table in estimation are omission of persons from censuses and the model stable populations. To illustrate the importance surveys and age-misreporting. The effects of these forms of the assumption that the population has a given age of imprecision are illustrated in this chapter principally pattern of mortality, estimates are made with all four age by synthetic estimates in which each source of error is patterns. The variation among the four figures is the assumed to operate in the absence of others. significant result: the fact that the estimate based on the " West" tables always agrees with the "true" figure is of course a purely automatic consequence of how the A. DIFFERENCES BETWEEN ASSUMED AND ACTUAL CON­ example is formulated. DITIONS (a) Effects of assumed age patterns of mortality on 1. Errors arising from differences between the actual estimates derived from census survival rates age pattern ofmortality and that embodied in the model life tables In section A of chapter I, the reader can find a descrip­ tion ofhow to select a model life table consistent with the There is no conclusive way of delineating the errors numbers recorded in two censuses taken at a ten year that might arise because populations with incomplete interval: project the first population by life tables at records may have age patterns of mortality that differ various levels of mortality and by interpolation find the from those with full records. Even when there are recorded level that matches the recorded total over ten, over, data from which age schedules of mortality can be calcul­ fifteen etc. It is suggested that the median of the first nine lated, it is often uncertain whether an extreme pattern of levels so indicated is a sensible choice. By summing the mortality - e.g., unusually low infant mortality, given the products of the age specific death rates from the life prevalent death rates above age one - is genuine, or the table so determined and of the average intercensal product of unusual inaccuracy in the data rather than an number of persons in the corresponding age groups one unusual pattern of death rates. obtains an estimate of the death rate. This death rate In this section the errors that originate in age patterns added to the average annual rate of growth during the of mortality different from the model life tables ofannex I intercensal period yields an estimate of the birth rate. are illustrated by examples in which estimates based on The procedure at no point makes use of the assumption alternative families of model life tables are compared. of stability. 41 TABLE 5. ExPECTATION OF LIFE AT AGES 0 AND 5IN VARIOUS FAMILIES OF MODEL LIFE TABLES PRODUCING A PROJECTED POPULATION OVER AGE X MATCHING THE TEST POPULATION AT THE END OF A DECADE

Expectation ofII/e at birth Expectation ofII/e at age 5

AgBX '·West" "North" "East" ·'South" "West" "North" «East" ,.South "

10 ...... , .. 40.0 40.5 36.3 37.2 49.7 49.8 50.1 50.6

15 •••••••• 0 ••••••••••• 40.0 38.6 35.5 35.0 49.7 48.6 49.6 49.2 20 ...... 40.0 37.7 35.6 34.5 49.7 47.9 49.6 48.9

25 ••••••••••• 0.0 •••••• 40.0 37.5 35.9 34.3 49.7 47.8 49.8 48.8 30 ...... 40.0 37.6 36.2 34.0 49.7 47.9 50.0 48.6 35 ...... 40.0 37.7 36.4 33.8 49.7 47.9 50.1 48.5 40 ...... 40.0 37.7 36.8 33.8 49.7 47.9 50.4 48.5 45 ...... 40.0 37.5 37.4 34.1 49.7 47.8 50.7 48.7 50 ...... 40.0 37.3 38.1 34.7 49.7 47.7 51.1 49.0 55 ...... 40.0 37.3 39.1 35.6 49.7 47.7 51.7 49.7 60 ...... 40.0 37.6 40.3 37.0 49.7 47.9 52.3 50.5 65 ...... 40.0 38.1 41.6 38.6 49.7 48.2 53.1 51.5 70 ...... , .. 40.0 38.7 42.9 40.3 49.7 48.6 53.8 52.6 75 ...... 40.0 39.1 43.8 41.9 49.7 48.9 54.3 53.6

Median of first 9 ...... 40.0 37.7 36.3 34.3 49.7 47.9 50.1 48.8

TABLE 6. PARAMETERS ASCRIBED TO THE TESTPOPULATION (WEST MODELSTABLE, 0eo = 4O,GRR = 3.00), BY SELECTING THE MEDIAN LEVEL MODEL LIFE TABLE FROM EACH FAMILY FROM AMONG TABLES MATCHING THE PROPORTIONS SURVIVING IN TWO CENSUSES

Est/mated by median model life table

Estimated parameter "West" "North" "East" "South"

°eo . 40.0 37.7 36.3 34.3 °es . 49.7 47.9 50.1 48.8 /2 . .773 .778 .701 .706 /s . .725 .704 .654 .629 Death rate . .0234 .0252 .0276 .0290 Birth rate . .0445 .0463 .0487 .0501 Death rate of population under age 5 . .0720 .0769 .0976 .1019 Death rate of population over age 5 . .0137 .0149 .0137 .0146

Suppose that our hypothetical population (" West" differences in the estimated population death rates, model stable female population, °eo = 40 years, ORR arising primarily from differences in the estimated mor­ = 3.00, iii = 29) were enumerated at the beginning and tality under age five, the estimated birth rates differ by end of a decade, and projections made employing various some 5.6 points or 11 per cent of the largest estimate. levelsof the four sets of model life tables. Note that when there is no evidence of the age pattern of Table 5shows°eo's and °es's in the model life tables that mortality nor separate indications of child mortality, produce the numbers over age 10, 15, 20 etc. in the the assumption of the" West" pattern of mortality pro­ " actual" population at the end of the decade. Table 6 duces low (or conservative) estimates of birth and death shows various parameters that would be ascribed to the rates. test population by assuming that the median life table indicated in the last row in table 5 represented the popu­ (b) Effects of assumed age patterns of mortality on lation's mortality schedule. estimates derived from stable populations chosen on the basis ofC (x) and r The differences in age pattern among the four families produces estimates of over-all mortality (estimated death If the test population conforming exactly to the West rate and expectation of life at birth) and mortality under model stable with °eo= 40 years and ORR = 3.00 were age five (12 , 15' and death rate under five) that are much enumerated twice, the intercensal rate of increase could be more divergent than the estimates of mortality in the calculated, and model stable populations found matching population 5 and over °es and death rate over five). The the given population in the proportion under age x and uniformity diminishes among estimates based primarily in the rate of increase. Table 7 shows the expectation of on survival at the older ages. As a consequence of the life at birth and at age five in model stable populations 42 TABLE 7. EXPECTATION OF LIFE AT AGES OANDS IN MODEL STABLE POPULATIONS BASED ON VARIOUS FAMILIES OF MODEL LIFE TABLESTHAT MATCH THE TEST POPULATION IN PROPORTION UNDER AGE X AND IN THE RATE OF INCIUlASIl

EJq¥ctatlon oflife at birth Expectation oflife at age 5

Age x "West" "North" "East" "South" "West" "North" "East" "South"

5 .0 ..•••••.••...••••• 40.0 42.2 36.8 39.5 49.7 50.9 50.4 52.1

10 .0 •••••••••••••••••• 40.0 41.3 36.1 37.9 49.7 50.3 50.0 51.1 15 ...... 40.0 40.4 35.7 36.9 49.7 49.7 49.7 50.5 20 ...... 40.0 39.6 35.5 36.3 49.7 49.2 49.6 50.1 25 ...... 40.0 39.2 35.4 35.8 49.7 48.9 49.6 49.7 30 '0' ••••••••••••••••• 40.0 38.9 35.5 35.3 49.7 48.7 49.6 49.4 35 .0' ••••••••••••••••• 40.0 38.7 35.5 35.0 49.7 48.6 49.6 49.2 40 ...... 40.0 38.5 35.6 34.7 49.7 48.5 49.7 49.0 45 ...... 40.0 38.4 35.8 34.5 49.7 48.4 49.8 48.9

50 •••••••••••••••••• 0. 40.0 38.2 36.1 34.5 49.7 48.3 50.0 48.9 55 ...... 40.0 38.1 36.6 34.6 49.7 48.2 50.3 49.0 60 ...... 40.0 38.0 37.3 35.1 49.7 48.2 50.7 49.3 65 ...... 40.0 38.0 38.2 35.9 49.7 48.2 51.1 49.8

Median of first 9 ...... 40.0 39.2 35.6 35.8 49.7 48.9 49.7 49.7

IIItTH RATI .070

.060

"SOUTH" .050 ~ "EAST" .=-~------~~---- ",.,...... """.-... -NORTH" ._._._._._._._~ -- ~._.-._.~._._._._.- 0 :.,;,::;;:,;,: W.£5i' 0 ••••••••• ••

.040

.030

.020

.010

10 III 20 211 30 311 40 'Ill !$O 115 60 65 AGE

Figure XVII. Birth rates in the test population derived by the stable population method from G(x) and r assuming various patterns of mortality as the appropriate one 43 TABLE 8. PARAMETERS ASCRIBED TO THE TEST POPULATION BY SELECTING THE MODEL STABLEPOPULA­ TION FROM BACH FAMILY WITH THE MEDIAN BIRTH RATE FROM AMONG THOSE WITH THE SAME r AND C(S), C(10), ... , C(4S)

Estimatedfrom median model stable population

Estimated porQllUlter "West" "North" "East" "South"

°eo ...... 40.0 39.2 35.6 35.8 °e5 ...... 49.7 48.9 49.7 49.7 12 .0 ••. · .•••••...••••••.•••••••••.••••• .773 .790 .694 .719 /5 ...... 725 .720 .646 .646 Death rate ...... 0234 .0244 .0287 .0287

Birth rate ••••••••••••••••••••••• 0 •••••• .0445 .0455 .0498 .0498 ORR (fli = 29) ...... 3.00 3.11 3.36 .342 Death rate of population under age 5 ...... 0720 .0724 .1030 .0992 Death rate of population over age 5 ...... 0137 .0144 .0136 .0138

(based on various families of model life tables) that independent of mortality pattern than when the estimates duplicate the rate of increase and the proportion under are based on C (x) and r. On the other hand parameters age 5, 10, 15 etc. in the test population. Figure XVII shown in table 9 that measure mortality above age five shows the birth rate in these stable populations. Table 8 are directly dependent on the assumed age pattern of shows various parameters that would be ascribed to the mortality, since the only observed quantities relate to test population by assuming that the model stable popu­ child mortality. Thus estimates of °es and the death rate lation with median fertility shown in figure XVII was of persons over five have a much larger range than in representative of the test population. tables 6 and 8. The same features of variation in estimation are seen Now assume that the test population had been enumer­ in these tables as in the preceding two. Again estimates ated in two censuses a decade apart, and that the second of mortality above age five are insensitive to differences in census included the data needed to calculate 12 , It is now age pattern while mortality estimates below age five and apparent that 12 gives estimates of the death rate under other measures strongly affected by infant and early age five that are less sensitive to age patterns of mortality childhood mortality, such as the over-all death rate or the than estimates based on age composition, and that on the expectation of life at birth, are markedly influenced. other hand, estimates of the death rate over age five based on age composition (either census survival or C(x) and r) (c) Effects of assumed age patterns of mortality on are less sensitive to differences in mortality patterns. It is estimates derived from reported child survival in therefore recommended (cf. section C of chapter III) combination with records of the age distribution that parameters related to child mortality be estimated originating from one census, or from two or more from 12 and those related to adult mortality from the age censuses composition; from census survival in the general case or from the intercensal growth rate and the (stable) age Assume that the test population has been enumerated distribution when stability may be assumed. To illustrate in a census that includes questions about the number of the advantages of this procedure in terms of increased children ever born to each woman, and the number still precision, and in particular to illustrate the increased alive at the time of the census, tabulated by sex of the insensitivity of the resulting estimates to differences in child and age of the woman. By methods described in mortality patterns, the following calculations were made: section B of chapter II, proportions surviving to age (a) Death rates estimatedfrom census survival (ages five two and age three (/2 and 13) can be estimated for females, and (as outlined in section B of chapter III) these values and over), and from 12 (ages under five), and b estimatedas can be used to select a model life table, and with C(5), d--r. A model life table was selected on the basis of best C(lO) etc., to select model stable populations at the agreement between the projected population and the population recorded in the second census (the median indicated mortality level. Table 9 shows 15' °eo, and °es level in table 5), and the smx values in this model table were selected on the basis of 12 according to each of the "regional" patterns of mortality. Figure XVIII indicates applied to the recorded (test) population above age five the birth rate in the stable population based on these to obtain the death rate over five. Death rates under age five (tmO and 4ml) from the life table selected on the regional patterns with the given 12 and the proportion under age 5, 10, 15 etc., in the test population. Table 9 basis of 12 (cf. table 9) were applied to the test population also includes various parameters of the "median" model at these ages. The expectation of life at birth was calcu­ stable populations selected on the basis of the fertility lated as sLo+°es x 15110 where sLo and 15 were taken from shown in figure XVIII. Note that the patterns of estimated the life table associated with 12 and °es was taken from the birth rates shown in figure XVIII are much more uniform life table obtained for the population over age five. than in figure XVII, and that, therefore, the estimation (b) Death rates estimated from stable population (ages of the birth rate from C(x) and 12 is much more nearly overfive), andfrom 12 (ages underfive), b estimatedas d +r. 44 I ••,H RATE .0701-

.06(1-

.0501-

~ ---:':. , . ~..~...•....;,;::..._._._._._._.!II'lI'I'.~_._.-...... •.... ._._._.- ·WES. • ·ST• .040f.-

.030 f.-

.020f.-

.010 -

I•• o 10 I5 20 25 30 35 40 45 50 55 60 65 AGE

Figure XVIII. Estimated birth rates in the test population derived by the stable population method from C(x) and la assuming various patterns of mortality as the appropriate one

TABLE 9. PARAMETERS ASCRIBED TO THE TEST POPULATION BY SELECTING THE MODEL STABLE POPULA­ TION FROM EACH FAMILY WITH THE MEDIAN BIRTH RATE FROM AMONG THOSE WITH THE SAME 12 AND C(5), C(10), ... , C(45)

Estimatedfrom median model stable population

Estimated parameter "West" "North" "East" "South"

°eo . 40.0 37.0 44.0 42.6 °e5 . 49.7 47.5 54.4 54.1 15 ...... •...... •...... • .725 .697 .736 .716 Death rate . .0234 .0265 .0207 .0221 Birth rate . .0445 .0465 .0451 .0461 Growth rate . .02II .0201 .0244 .0240 GRR (m = 29) . 3.00 3.18 3.07 3.19 Death rate of population under age 5 . .0720 .0797 .0697 .0730 Death rate of population over age 5 . .0137 .0154 .0109 .0113

45 The death rates for ages over fivewere obtained by select­ fact have the age distribution of a stable population ing the life table underlying the stable population deter­ because of age-selective migration, or past variations in mined from C(x) and r (the median level in table 7); fertility or mortality. otherwise the same procedure was followed as in (0).1 Any extended discussion of the deviations in stability (c) Birth rate estimated from stable population, death that can and do occur is beyond the scope of this Manual. rate as b-r. The birth rate was obtained from C(x) Only a few comments about general principles and fre­ and 12 (table 9), and the death rate was calculated as the quently encountered cases will be attempted. difference between this birth rate and the reported inter­ Ideally, stable estimation should be employed only for censal rate of increase. a closed population with constant mortality during the Table 10showsthe population parameters calculated by preceding 25-30years, and constant fertility for some two these methods. Method (a) is applicable whether the generations. A useful practical test is the absence of population is stable or not, but (b) and (c)can beemployed substantial change in age composition and of intercensal only with stable populations. The most striking feature rate of increase in three consecutive quinquennial or of this table is the small variability in the estimate of such decennial censuses. For example, examination of the age over-all characteristics as the birth and death rates and distributions in Turkey from 1935 on reveals clearly the expectation of life at birth caused by differences in (in spite of conspicuous distortions caused by age-mis­ age pattern of mortality when mortality above age five reporting) that fertility was greatly reduced during certain is derivedfrom the age distribution and intercensal change periods since 1910: the evidence is a low point in the age and mortality under age five from reported child survival. distribution that does not remain at the same age from It is clear how valuable is the supplementary information one census to the next as it would if age-misreporting were provided by data on proportions survivingamong children the cause of the low point, but rather advances by five ever born. years in each subsequent quinquennial census. In contrast, the Indian age distributions from 1891 to 1911, also irreg­ 2. Errors caused by non-stability ofa population assumed ular, are much the same in form, indicating that the to be stable irregularities are caused by age-misreporting, and that the underlying age composition was essentially constant (see If the age distribution of a population conforms closely figure XIX). Stable analysis is appropriate for India in to that of a stable population, estimation of many 1911, but not for Turkey in the years shown. characteristics is greatly simplified, especiallythrough the Few populations for which estimation is necessaryhave use of tabulated model stable populations. The methods been enumerated in an extended series of censuses of of selecting an appropriate model population are given comparable quality, and it is often impossible to apply in chapters I and III; and in the preceding discussion in the suggested criterion of an essentially unchanging age this chapter the errors that may arise because of variations distribution and rate of increase. The assumption of in age patterns of mortality are explained. Another source stability must often be made without much direct evidence of error is that the population in question may not in in its support. In general, stable methods of estimation should be 1 In a less artificial example of the application of the stable attempted only in populations where there is no wide­ population method the life table death rates would be applied not to the reported age distribution but to the stable age distribution spread use of birth control, since where the practice is as explained in section C.2 of chapter III. common there are usually pronounced variations or

TABLE 10. ESTIMATED PARAMETERS OF THE TEST POPULATION CALCULATED BY VARIOUS METHODS BASED ON POPULATION AGE DISTRIBUTIONS FROM TWO CENSUSES AND FROM REPORTS ON CHILD SURVIVAL

Estimatedparameter

Pattern Death rate Death rate of under age over age Methad ofestimation mortality S 5 Oeo °e5 Birth rate Death rate

(a) Death rates from "West" .0720 .0137 40.0 49.7 .0445 .0234 census survival (ages "North" .0791 .0149 37.3 47.9 .0466 .0255 over 5) and from 12 "East" .0697 .0137 40.8 50.1 .0441 .0230 (under 5); b = d-s-r "South" .0734 .0146 38.8 48.8 .0454 .0243

(b) Death rates from "West" .0720 .0137 40.0 49.7 .0445 .0234 C(x) and r (ages over "North" .0791 .0142 38.0 48.9 .0460 .0249 5) and from 12 (under "East" .0697 .0140 40.5 49.7 .0443 .0232 5); b = d-rr "South" .0734 .0139 39.5 49.7 .0448 .0237

(c) Birth rate from C(x) "West" .0445 .0234 and 12;d = br-r "North" .0465 .0254 "East" .0451 .0240 "South" .0461 .0250

46 C(~l It I 8 TURKEY INDIA .16 __ 1945 '16~ _1911 ___ 1901 ___ 1940 ...... 1891 ...... 1935 .14 .14

.12 .12

.10 .10

.08 .08

.06 .06

.04 .04

.02

II o 5 10 15 20 25 30 35 40 45 50 55 60 65 AGE AG E

Figure XIX. Distribution of the female population by age in five-year intervals as recorded in selected censuses in India and in Turkey trends in fertility. Stable analysis should be avoided in past few decades. What may produce poor estimates are populations in which migration has had a pronounced either major swings in fertility producing one or more influence on age composition-as is typical of the popu­ consecutive small five-year cohorts, or a sustained trend lations of many cities in developing countries. One in either fertility or mortality. A marked and continued reason for the emphasis on analysing female age distri­ decline in mortality has occurred in many populations that butions in this Manual is that female distributions are very appear to have essentially constant fertility, and chapter I often less affected by migration than are male. With some includes a section (section C) that shows how estimates conspicuous exceptions (e.g., rural-urban migration in based on stable populations can be adjusted for the Latin America) female migration is usually less than male, effects of a history of falling death rates. But these and because women are usually accompanied by their adjustments can be made only ifthere are clues indicating children, the effect on the age distribution is sometimes approximately how long and how rapidly mortality has negligible, even when migrants form a substantial fraction fallen. When there is no reliable basis for detecting the of the population. It has been shown for example that a downward course of mortality, the adjustments given constant stream of female immigrants, constituting in table I1I.l in annex III can be interpreted as indications annually 4 to 5 per cent of the receiving population, of the errors that occur from stable population calcula­ consisting of young adults and their children, does not tions made in ignorance of a downtrend in mortality. produce an age composition markedly different from what For example, suppose that a stable population is chosen would exist in the absence of immigration.2 with C (30) = .7, and ten-year intercensal r = .020, If mortality and fertility have fluctuated rather than leading to an estimated b of .04912 and GRR29 of 3.312. being constant, but without a regular trend, the stable But if in fact because of falling mortality population population that has the the same ogive up to age fifteen, growth has increased from 0.144 in the next earlier twenty or thirty (and the same rate of increase) has intercensal decade and about .009 in still earlier decades fertility and mortality close to the average during the the approximate value of k would be .01, and the value of t about 20 years. It can be seen, then, in table IlL1 that the true value of the birth rate would be about 8.5 per cent higher, and the true value of the gross reproduc­ I See J.M. Boute, S.J. ,La demographie de la branche indo­ tion rate about 9.6 per cent higher than the estimates paklstanaise d'Afrique, Etudes morales, sociales et juridiques, arrived at on the basis ofthe false assumption of stability. (Louvain, 1965). On this topic see also Leon Tabah and Alberto Cataldi, "Effets d'une immigration dans quelques populations In other words the correct estimates for the birth rate and mod~les", Population, No.4, 1963, pages 683-696. the GRR in this example are .0533 and 3.63, respectively. 47 Table III.I can also be used as an approximate indi­ earlier discussion) that the population is the same "test" cation of the errors caused by the assumption of stability population used before-the" West" femalemodel stable, when fertilityhas recentlyfollowed a rising trend such that with °eo = 40 years, and GRR2 9 = 3.00. Table 11 shows the GRR has risen by one per cent annually-been the levels of mortality required to project the earlier multiplied by (1+k) in each of the last t years, where k = 0.01.The reason that table III.1 indicates the effectof a history of rising fertility is that the age distribution is TABLE 11. EXPECTATION OF LIFE AT BIRTH IN "WEST" MODEL LIFE TABLES PRODUCING A PROJECTED POPULATION OVER AGE X THAT displaced from the stable in an almost identical manner MATCHES THE TEST POPULATION AT THE END OF A DECADE ASSUM­ by a recent trend of falling mortality on the one hand, or ING A 2 PER CENT RELATIVE OVERCOUNT OF THE TEST POPULATION of rising fertility on the other. If the sign of the adjustment AT THE LATTER DATE factors in table 111.1 is reversed, the factors then indicate the errors in stable estimates if in fact fertility has been Age" Age x °00 falling by one per cent annually for t years. It does not seem advisable to attempt to extend the use 10 44.5 50 42.4 of table III.1 or to construct similar or more complicated 15 44.6 55 42.1 tables, to cover estimation for populations in which the 20 44.2 60 41.7 voluntary control of fertility is widespread. The appli­ 25 43.8 65 41.5 cation of stable analysis should probably be confined to 30 43.5 70 41.2 populations that apparently do not practise deliberate 35 43.2 75 .. 41.0 40 43.0 80 40.9 contraception or abortion. But fertility may be subject to 45 .. 42.7 prolonged, if limited, upward and downward movements even when conscious birth control is rare. In tropical Latin America there appears to be little contraception or abortion except in some urban populations, but in many population to the later one at ages over to, 15, 20 etc. countries fertility was reduced in the depressed 1930s, The assumed conditions imply that a mortality level must and rose during the 1940s and 1950s.3 The source of the be chosen producing a proportion surviving that is 2 per variation was changes in marital status reflecting some­ cent above the true numbers. It requires a bigger difference what later marriage in the 1930s. In other populations in mortality level to increase survival of the whole popu­ proportions married increasewhen mortality falls because lation by 2 per cent than of the population over fifty or of a reduced incidence of widowhood, and in still others sixty. Thus the biggest mistake in selecting a level of fecundity is changed by the spread or the conquest of mortality when there is a change in the completeness of pathological sterility. Table III.1 shows the orders of coverage occurs, as is evident in table II, in levels based magnitude of the errors in stable estimates introduced partly on survival of the population at younger ages. by such trends, if not allowed for explicitly. However, as will be seen later, age-mis-statements make the projected number of persons derived from the population over forty especiallyunreliable. B. ERRORS CAUSED BY FAULTY DATA TABLE 12. STABLE POPULATION ESTIMATES OF THE BIRTH RATE IN The discussion of mistaken estimation caused by THE TEST POPULATION DERIVED FROM CORRECTLY REPORTED mistakes in the basic data will be confined to two impor­ PROPORTIONS UP TO AGE X AND FROM THE OBSERVED INTERCENSAL tant forms of defective information: omission of persons GROWTH RATE WHEN THE LATTER IS DISTORTED BY A 2 PER CENT RELATIVE OVERCOUNT IN THE SECOND OF TWO CENSUSES TAKEN who should have been included in a census or survey TEN YEARS APART (or the opposite mistake of erroneous inclusion), and misreporting of age. Distortions in the recorded age Age x Birth rate Age x Birth rate distribution are caused by age selective omissions as well as by age-mis-statement, but it is not generally possible to determine which of these factors has caused a given 5 ...... , ...... 0429 40 ...... 0432 irregularity in age composition, and it will be implicitly 10 ...... 0425 45 ...... 0434 assumed that omissions affect primarily the total number 15 ...... 0425 50 ...... 0436 20 " ...... 0426 55 ..... '" ...... 0438 of persons enumerated, and that distorted age compo­ 25 ...... 0428 60 ...... 0440 sition is caused by misreporting of ages. 30 ...... 0429 65 ..... '" ...... 0442 35 ...... 0431 Test population ...... 0445 1. Differential rates of omission in consecutive censuses Consider a population enumerated in two censuses a decade apart, and suppose the coverage of the second Now suppose that the estimate is made by stable census is 2 per cent more complete. What is the effect on population methods. Ithas been assumed thatthe improve­ estimates of mortality and fertility? ment in coverage has not affected the reported age It is assumed (to maintain comparability with the distribution, but the intercensal rate of increase is over­ estimated by about 2 per thousand. Table 12 shows the estimates of b obtained from this erroneous rand C (5), 3 Cf. O. Andrew Collver, Birth Rates in Latin America, New (to) Estimates of Historical Trends and Fluctuations (Institute of Inter­ C etc. Note that the error in estimating b varies national Studies, University of California, Berkeley, 1965). little from about ages five to forty and that its absolute 48 TABLE 13. EXPECTATION OF LIFE AT BIRTH AND VITAL RATES IN THE TEST POPULATION, AND ESTIMATES OF THESE QUANTITIES BASED ON THE CENSUS SURVIVAL METHOD AND THE STABLE POPULATION METHOD. BIAS OF ESTIMATES REFLECTS AN ASSUMED RELATIVE OVERCOUNT IN THE SECOND OF TWO CENSUSES OF THE TEST POPULATION, TAKEN TEN YEARS APART

oeo Birth rate Death rate Growth rate

Test population ...... 40.0 .0445 .0234 .0211 Census survival method ...... 43.5 .0434 .0203 .0231 Stable population method ...... 44.3 .0429 .0198 .0231

magnitude is larger than when obtained with ogives up stable populations were used to demonstrate the existence to higher ages.Onceagain, however,due to age-mis-state­ of typical distortion in the.ogives of age distribution, and ments at such ages, this observation is of no consequence in distributions by five-year age intervals, caused by in selecting a stable estimate of the birth rate from a characteristic forms of age-misreporting. The most reported age distribution. conspicuous systematic distortions are found in popula­ Table 13 summarizes the values of various parameters tions in which apparently the ages of many persons are based on the census survival method and on the stable estimated by the interviewer rather than the respondent, population method i.e., obtained by accepting the and rules were suggested for selectingages at which ogives median model life table and the median stable population are likely to be most reliable. Of course such rules can among the first nine in tables 11 and 12, respectively. In do no more than minimize errors that are unavoidably the former case the m, values of the selected life table are substantial, and the question remains concerning the combined with an estimated mean age distribution to range of unavoidable variation introduced into estimates of birth and death rates by age-rnis-statements. derive the (crude) death rate; the estimate of the birth rate is obtained by adding to this death rate the observed intercensal growth rate. In the case of the stable popu­ (a) Age-mis-statement and mortality estimation by census lation method, the selection of the median stable popula­ survival tion naturally implies the acceptance of all other stable parameters of that population. Note that with the census The effects of age-mis-statement on the estimation of mortality by finding the model life table that duplicates survival method a more complete enumeration in the second census causes an appearance of higher rates of observed census survival values cannot be simply summa­ rized. There are two principal ways in which age-mis­ survival and hence causes the estimated death rate to be statement affects the level of mortality that gives a too low; of course, the calculated intercensal rate of increase is too high, and the two errors are partially projected population over x +10agreeingwith the enumer­ ated population. First is the effect of age-mis-statement compensating when the birth rate is calculated. In the on the reported numbers over x+ 10 in the later census case of the stable population method, an upward-biased r relative to the effect on those reported as over x in the in combination with C (x) causes b to be underestimated­ earlier census. Both numbers may be inflated or both in the case of the median stable population this error is deflated without causing an error in the estimated level of approximately equal to the error in r. Since its sign is mortality, provided relative inflation or deflation is the opposite, however, the estimate of d is about twice as far same. But if the increase in the proportion over thirty removed from the true value as the estimate of b. in the later census by age overstatement is less than the Variations in completeness of coverage tend to com­ increase in the proportion over twenty in the earlier promise the adjustments for declining mortality. For census, the estimated survival rates will be too low, example, if the middle of three decennial censuses is and estimated mortality too high. Secondly, the estimated especially incomplete, mortality in the earlier decade is level of mortality is affected by age errors that exaggerate overstated and in the later decade understated, creating or understate the proportions at ages of high mortality. the impression of declining mortality when death rates The typical exaggeration of the age of persons past were actually constant, or exaggerating the extent of a fifty or sixty means that too many old persons are reported real decline. in the earlier census. This upward shift of age reduces the expected number of survivors in a projection by any 2. Age-misreporting in censuses or surveys given life table, and therefore requires overstated survival rates (too low mortality) to produce expected survivors A large proportion of the ages recorded in most censuses equal to the actual. show that in the high­ and demographic surveys in less developed countries are fertility populations for which estimates are usually inaccurate. In tabulation of the population by single needed, overstatement of age by the aged tangibly affects years of age, peaks at ages ending in zero and five, and to a only the estimates of mortality based on projections of the less extent at two and eight, are usual, with deficits at the population forty-five and over. Because of the complexity other digits. Numbers reported at forty which are several of the effects of age-mis-statement on census survival times bigger than those at forty-one are not uncommon, ratios, no preferred ages are suggested for estimating the for example. In section B.3 of chapter I comparisons with level of mortality. Selection on the basis of projection 49 of the population at ages above forty-five cannot be ofthe test populationis 44.48 per thousand, the erroneous trusted, and the median of the first nine estimates is a estimates would be 47.53 and 46.19, respectively.) Note neutral choice. that knowledge of 12 yields more "robust" estimates of the birth rate than does knowledge of r-estimates less (b) Age-mis-statement and stable population analysis sensitive to errors in the reported age distribution as well as possible differences in age pattern of mortality. Esti­ The use of model stable populations to estimate various mates of d are in error to the same extent as is b when r parameters involves the selection of a stable age distri­ is known. Because knowledge of 12 implies (within the bution matching the recorded age distribution in some context of a family of model life tables) that the level of way. In this Manual the recommended procedure is to mortality is known, errors in C (x) would generally select a stable population with the same proportion under produce trivial errors in the estimate of d, and errors in the some age-the selection of what age depending on the estimate of r essentially equal to those in estimating b. pattern of apparent distortion of the recorded age distri­ In the earlier discussion of typical pattern of age-mis­ bution. With the extreme distortions characteristic of reporting, the hypothesis was advanced that when age certain Asian and African censuses, it is recommended reporting is not subject to the gross distortions seen in that the stable ogive matching the given female population the censuses of Africa, India, Pakistan and Indonesia, at age thirty-five be used. For age distributions subject populations have an approximate knowledge of age, to less distortion such as found in the Philippines and accounting for the relatively orderly form of the ogive, Latin America use of the male distribution was recom­ and even of the five-year age distribution, even though mended, specifically the selection of the median ogive age heaping is extensive. This hypothesis suggests that among those matching the census at 5, 10, ... ,45. gross overstatement or understatement of age is not the Whatever procedure is followed, it can only avoid norm and that age-mis-statement is usually the result of extreme errors, and cannot insure that the proportion small errors, with a preference for round numbers. In recorded as under the age prescribed by the rule is exact. Latin America and other areas where there appears to be The value of C(x) in the model population selected is knowledge of the approximate age, it is possible to argue thus generally somewhat above or below the true value that ogives to ages divisible by five have a systematic of C(x), because age-mis-statement has caused a net downward bias, because the cumulative age distribution transfer of persons across age x. How much does an error always stops just short of the highly preferred ages ending in C (x) affect the estimated birth rates, death rates, and in zero or five. To make the point in a different way: other parameters ? cumulative proportions under age 11, 16, 21, 26, 31 etc. would indicate higher fertility than ogives to age 10, 15 etc., because the first set would always just include, TABLE 14. VALUES OF Ab/AC(x) FOR THE TEST POPULATION WHEN THE BIRTH RATE IS ESTIMATED BY THE STABLE POPULATION METHOD rather than just exclude, an age containing a greatly GIVEN C(x) AND r, AND C(x) AND /2 exaggerated number. It appears probable that the ogives barely including the ages divisible by five are too large L1b/LlC(x) L1b/LlC(x) because they contain persons really 11 and 12, 16 and 17, glvell C(x) glvell C(x) Q/Idr aIId12 21 and 22 etc., reported as 10, 15, 20 etc. On the other Age x (a) (b) Col. alb hand, ogives ending just short of ages divisible by five are possibly too small, because some persons 8 or 9 are reported as 10, 13 or 14 as 15 etc. To test the extent of the 5 ...... 568 .304 1.868 10 ...... 388 .189 2.053 possible bias caused by this effect of age heaping, a 15 ...... 319 .155 2.058 special analysis was made of the census of Mexico in 20 ...... 293 .146 2.007 1960, when age heaping was extensive. The numbers 25 ...... 283 .145 1.952 recorded in three-year intervals (9-11, 14-16, 19-21, etc.,) 30 ...... , ...... 289 .153 1.889 around each age divisible by five were reapportioned so 35 ...... ,. .305 .171 1.784 as to have the sequence expected in an appropriate stable 40 ...... 336 .192 1.750 population, and the ogive recalculated. The effect was 45 ...... 379 .225 1.684 to increase the proportion below each age divisible by five (except below age fifteen) because some of the persons reported at these preferred ages were reassigned to the younger quinquennium. Despite the considerable age Table 14 shows the ratio of errors in the estimation of b heaping, the differences between the birth rates estimated to errors in the recorded value of C (x) (x = 5 to 45) on the basis of the adjusted C (x) values and those calcu­ when C(x) is combined with r on the one hand and 12 lated from the unadjusted data were at all ages wellwithin on the other. These calculations apply to the same test one per thousand population. population ("West" model female, °eo= 40, GRR2 9 = 3.00) employed before. The meaning of the entries in (c) Age-mis-statement and the estimation of fertility this table is illustrated by this example: Suppose C (35) andmortalityfrom special questions on past experience were recorded as .7627 rather than the correct figure of .7527, or that there were an error of .01 in C(35). If r In chapter II there is a description of methods of were known, the error in estimating b would be (.01) approximate calculation of fertility and mortality from (.305) or .00305, and if 12 were known, the error in the data obtained in asking women about the number of estimating b would be .00171. (Since the "true" birth rate children they have ever borne, and the number of these 50 surviving. It was noted, because of a tendency towards A basic feature of demographic analysis underlying omission in the answers given by older women, that many of the procedures suggested here, but not always reported parity could be accepted as about correct only emphasized, is that there are logically necessary or biolo­ for women under thirty. However, it is a plausible hypo­ gically inevitable interrelations among population para­ thesis that the age pattern of fertility is correctly indicated meters, and these relations should be fully exploited in by responses to questions about births in the preceding examining the consistency of estimates. For example, year. In the method of fertility estimation devised by in populations in which circumstances permit a separate William Brass, a comparison at ages 20-24 or 25-29 of valid estimate of the average number of male and female cumulated fertility indicated by births reported for the births in a given decade, it is important to compare the preceding year with average numbers of children ever estimated numbers to see if they lie within tolerable limits born to these ages provides an appropriate adjustment to of the expected sex ratio at birth. Almost all populations reported births for the last year that can give a good of non-African origin have 105-107 male births for every estimate of the fertility schedule. This method, valid in 100 female births, subject, of course to fluctua­ principle when age is accurately reported, also works tions. Populations of African origin have a sex ratio at with tolerable accuracy in the presence of large and birth of perhaps 102-104. Thus if a male birth rate (male frequent errors in age, provided the errors are not births/male population) and a female birth rate have been systematic. But in the Asian-African populations, where calculated from the age distribution of each sex, it is age is often estimated by another person, it appears that essential to see whether the estimates are consonant or about half the women in some of the child-bearing age inconsistent. If the former, there is reason for added groups are reported in the next higher group, which means confidence, if the latter, the data or the methods are that the cumulated experience of those reported as 15-19 inconsistent (a judgement subject to the crucial reser­ does not in fact correspond to the history ofthose reported vation that sampling must be allowed for). as twenty. It is not possible to analyse the resultant biases here," but merely to warn that the method can be used One of the principles to be borne in mind in making only with the possibility of a wide margin of error in such checks is the importance of noting what aspects populations where massive systematic age-misreporting is of the two estimates being confronted are independent. Thus for example a comparison of the level of mortality apparent. It is much more promising, should the appro­ priate questions be asked in a census or survey, in popu­ indicated by census survival during a decade, and by C(x) lations subject to milder age distortions, such as in the in the second census plus the intercensal rate of increase Philippines or in Latin America. is a comparison of two inferences from essentially the same basic data. Deficiencies in the censuses or special Age-misreporting also affects the estimation of child features in the age pattern of mortality would affect the mortality from proportions dead among children ever two estimates similarly, and perfect agreement would not born to young women. Here again, accurate estimates can be as reassuring as agreement between estimates with be expected only in populations not subject to gross wholly independent bases. An example of a more nearly transfers of women among ages 15-19, 20-24, 25-29 and independent pair of estimates is the male birth rate 30-34. But the sensitivity of the estimated levels of child calculated indirectly from the female age distribution mortality to such transfers is not great, and one can be plus the sex ratio of the population and an assumed confident, for example, if responses about live and dead normal sex ratio at birth, and directly from the male age children are approximately correct, that the adjusted distribution. proportion dead reported by women 20-24 is greater than The comparison that is most acceptable as a confir­ 1qo and less than 3qO even if not equal to 2QO' mation of valid estimation is between figures derived from data of wholly different kinds. For example, in a situation where fertility is constant, data on children ever C. SUGGESTIONS FOR BEST ESTIMATION born by age of woman, if themselves internally consistent, A number of rules of thumb have been offered in this take on added persuasiveness if verified by comparison Manual, representing, however, not any definitive best with the cumulation of current fertility rates based on procedure in each form of estimation, but a preliminary reported births last year. But suppose now that death distillation of the authors' experience with each method. rates over five are calculated from census survival, and In the first application of an unfamiliar method, it may under five from proportions surviving among children be best simply to follow these rules; but in situations ever born, that these rates are combined to form an when many alternative calculations are feasible (surely estimated population death rate, to which the intercensal the usual situation in making estimates for a single rate of increase is added to provide a figure for the birth national population), all of the possibilities here discussed rate. If now the birth rate thus obtained agrees closely should be examined, and a final estimate made only with that derived by the methods outlined earlier in this after a critial examination of these alternatives. The paragraph, the confirmation is a strong one. Of course it is analyst should be sensitiveto patterns ofage-misreporting, always possible that the agreement is fortuitous. An to evidence that the population is not stable, and to the additional principle in judging the importance of agree­ possibility of systematic omission of events or persons ment is its consistency. As a final element in this hypo­ with certain characteristics. thetical example, suppose that the comparison has been made in each of a moderate number-ten or more­ of subdivisions of the same data system, such as regions or 4 See the discussion in Brass et al., op, cit. provinces enumerated in the same national censuses. If 51 the various forms of independent estimation yield mation least sensitive to unknowable uncertainties-to fertilities that agree in the geographical differences they age patterns of mortality when age specific death rates show, the reality of the differences is more or less con­ are not recorded, to systematic age-misreporting etc. clusively established. This principle leads to preferring 12 to r as an adjunct to The last suggestion for making the best of inaccurate C(x) in estimating the birth rate. and incomplete data is always to seek the form of esti-

52 Chapter V

DATA USEFUL FOR ESTIMATES OF FERTILITY AND MORTALITY

The importance of measuring current fertility and Tables should be published showing the distribution mortality has led to a number of experimental attempts to by the standard five-year age intervals for each sex in each obtain current figures by sample registration, by special geographical unit, and by single years for whole popu­ surveys, and by a combination of registration and inter­ lations covered by the survey. The single-yeardistribution views. 1 The design of such sample registration and makes useful analysis ofage-misreporting possible. surveys has been excluded from the discussion in this Manual; the methods outlined here are confined to data that might be obtained in a census, or large-scale sample demographic survey. The purpose of this short final B. DATA ON CHILDREN EVER BORN chapter of part one. is to specify explicitly what kinds of questions should be included in censuses or broad In the absence of vital statistics, data on the number of purpose surveys to make reliable estimation of birth children born in the lifetime of each woman and the and death rates possible. number of these surviving provide a very useful basis for determining fertility and mortality. The following sugges­ The same questions would also be a valuable part of tions are intended to supply the maximum material for the design of intensive repeated surveys instituted for the constructing estimates, and simultaneously to make it specific purpose of estimating vital statitics: for example, possible both to detect and to minimize biases in the Brass-style estimates of infant and child mortality would responses. be a useful check on mortality information obtained in a sample register, or from frequent interviews. 1. Questions on fertility histories should ask for the number of children born alive who are still living in the household, the number born alive who have left the A. DATA ON AGE household, and the number who have died. The question about those who have died is the foundation of estimates Information about age is an essential part of every form of infant and child mortality, and the separate questions of estimation presented in this Manual. Moreover, the about those still at home and those who have left minimize existence of similar patterns of distortion in age distri­ a source of omission, especially for older women with butions in surveys or censuses makes even rough age grown children. distributions the basis of useful estimation. A question on chronological age should be part of every census or 2. Women should be asked the sex of each child demographic survey, and when the respondent is unable reported, and males and females should be tabulated to supply an acceptable figure, the interviewer should separately. This procedure makes possible the estimation be instructed to make an estimate. of male and female child mortality separately. It also provides material for a number of tests ofconsistency-for example, omissions of children ever born are likely to be 1 See notably, C. Chandrasekaran and W.E. Deming, "On a Method of Estimating Birth and Death Rates and the Extent of sex selective, and such a tendency is revealed by a trend Registration", Journal ofthe American Statistical Association, vol. 44, in the sex ratio of the reported children ever born as the March 1949, pages 101·115; Ansley J. Coale, "The Design of an age of woman increases. Another significant form of Experimental Procedure for Obtaining Accurate Vital Statistics", probable omission is to leave out higher proportions of in International Union for the Scientific Study of Publications, International Population Conference, New York, 1961 (London, 1963), dead than of surviving children, especially on the part of Vol. II, pages 372-376; Guanabara Demographic Pilot Survey, older women. This form of omission may also be sex A joint project of the United Nations and the Government of selective, resulting in an implausible contrast in estimated Brazil (United Nations publication, Sales No.: 64.XIII.3). Karol J. child mortality by age for the two sexes. Krotki, "First Report on the Population Growth ", in International Union for the Scientific Study of Population, 3. Questions about children ever born and children International Population Conference, Ottawa, 1963 (Liege, 1964) surviving should be asked of and tabulated preferably pages 159·173; Carmen, Arretx G. and Jorge L. Somoza, "Survey Methods, Based on Periodically Repeated Interviews, Aimed at for all women, not merely married women. If, for some Determining Demographic Rates," Demography (Chicago, 1965), reason, non-married women must be excluded, the vol. II, pages 289-301; cf. also William Brass, "Methods of Obtaining questions should be asked of and tabulated for all Demographic Measures where Census and Vital Statistics Registra­ married women, not merely "mothers". Interviewers tion Systems are Lacking or Defective", background paper presented at the Second Conference (Belgrade, 1965), should be instructed to make an unambiguous entry for No. 409. every respondent, especially to enter a zero for women 53 with no children, rather than leaving a blank, which in the same survey if the technique is to be employed. indicates "no response", rather than "no children". This additional detail creates additional possibilities for 4. The parity distribution in each five-year age interval determining the presumed error in the perception of the should be tabulated-i.e., the number of women with "reference period" (one year) that causes mistakes in no children, with one child, two children etc.-rather reporting recent births. For example, the cumulation of than just the mean parity or number of children ever born. age specific fertility of zero parity women to the age This tabulation makes possible additional tests of consis­ interval 30-34 should equal the proportion of women tency, and additional valuable inferences, such as differen­ at age 30-34 having at least one child, and a correction tial mortality in families with different numbers of can be applied to the births reported for the preceding children. year to insure such equality. If this correction factor is about the same as the one described in chapter II-the correction needed to equate cumulative fertility (all C. DATA ON THE AGE STRUCTURE OF FERTILITY parities) with average parity at age 20-24-the credibility of the adjustment is greatly strengthened. In chapter II a method of estimating fertilityis described Another possibility for ascertaining the age structure of that is based on accepting as accurate the number of fertility is to ask every woman whether she is currently children ever born reported by younger women as an pregnant. Not all pregnancies result in live births; some indication of the level of fertility, and judging the age women may not recognize pregnancy in its early stages; pattern of fertility from births-by-age reported for the and in some populations there may be a tendency for preceding year. This technique is especially promising in women to deny they are pregnant. None of these differ­ populations in which knowledge of approximate age is ences between reported pregnancy and fertility are likely widespread, so that the comparison of cumulated fertility to be age selective, except in ways that can be estimated; with reported average parity is not excessively distorted and this question is a promising supplement (or substitute) by misreporting of age. for a question on births last year. Of course the age of the The question that should be asked to reveal the age woman at the birth of the child would be about four pattern of fertility is whether each woman bore a child months greater than the mean age during a reported in the year before the survey.Answers should be tabulated pregnancy, and an allowance for this difference must be by age, and also by parity-information that is collected made in forming a schedule of fertility.

54 Part Two EXAMPLES OF ESTIMATION

-

Chapter VI

EXAMPLES OF ESTIMATES BASED ON RECORDS OF POPULATION GROWTH AND DISTRIBUTION BY AGE

A. ESTIMATION OF MORTALITY AND OF THE BIRTH RATE intercensal period and the two censuses should refer to FROM CENSUS SURVIVAL RATES the same geographical area. Preliminary adjustment of data. The ideal requirements An application of the method of estimating mortality as stated in the preceding paragraph are seldom perfectly and fertility described in section A.2 of chapter I is satisfied. Deviations from these requirements result in illustrated below using information taken from Turkish biases in the final estimates or cause computational statistics, specifically from the national censuses of 1935 inconveniences. Preliminary adjustments of the basic data and 1945. Estimation could have been based on the can eliminate or reduce either of these effects. The possi­ censuses of 1935 and 1940, but censuses at five-year bility of making such adjustments when they are needed, intervals are rare, and the ten-year interval was chosen and their specific nature may differ from case to case: as more typical. the decision of the analyst concerning the procedures to Required basic data. Distribution of the population by be followed should be influenced in each instance by the five-year age groups as recorded in two successive censuses extent of the deviation from the ideal requirements and taken several-preferably five or ten-years apart. by the amount and quality of the information that is Columns 2 and 3 of table 15 show the female population available for making corrections in the basic data. The in Turkey in 1935 and 1945, classified by age. Ideally the following four types of adjustments may be both com­ population should be closed to migration during the monly needed and feasible:

TABLE 15. FEMALE POPULATION OF TuRKEY BY AGE IN 1935 AND 1945 (THOUSANDS) AND CENSUS SURVIVAL RATES FOR FIVE-YEAR COHORTS

Population reported Adjusted Adjusted by census population. 1935 population, 1945

For migration For "age and boundary For "age unknowns" changes unknown" Ten-year survival Age 10,20, 10.21, (col. 2 (col. 4 (col. 3 rates for Interval 1935 1945 x 1.004.f) x 1.020.f) x 1.00I.f) each cohort" (I) (2) (3) (4) (5) (6) (7)

0-4 ...... J,297 1,185 1,303 1,329 1,187 5-9 ...... 1,128 1,242 1,133 1,156 1,244 10-14 ...... 746.0 1,074 749.2 764.2 1,076 .8096 15-19 ...... 485.9 931.5 488.0 498.0 932.8 .8069 20-24 ...... 640.2 691.7 643.0 656.1 692.7 .9060 25·29 ...... 721.3 619.1 724.4 739.3 620.0 1.2450 30-34 ...... 642.1 699.7 644.9 658.1 700.7 1.0680 35-39 ...... 509.6 578.4 511.8 522.3 579.2 .7834 40-44 ...... 473.9 558.0 476.0 485.7 558.8 .8491 45-49 ...... 314.9 378.5 316.2 322.7 379.0 .7256 50-54 ...... 384.4 434.1 386.J 394.0 434.7 .8950 55-59 ...... 195.2 219.4 196.1 200.1 219.7 .6808 60-64 ...... 297.4 349.2 298.7 304.8 349.7 .8876 65-69 ...... 105.0 J24.6 105.4 107.6 124.8 .6237 70-74 ...... 126.1 133.0 126.6 129.2 133.2 .4370 75 and over .... 118.0 112.3 118.5 120.9 112.5 Unknown ...... 35.81 13.11

TOTAL .••••.•.• 8,221 9,344 8,221 8,388 9,344

a Ratio of number of persons in each age interval in 1945 (from Column 6) to number ten years younger in 1935 (from Column 5). 57 (1) Adjustment for persons not classifiedby age. Unless question involved movements of whole families it is the number of persons not classified by age is very small, unlikely that the age and sex composition of the migrants or their proportion in the total population is very nearly was highly atypical; therefore the remaining bias due to the same in both censuses, "unknowns" with respect to migration after this adjustment is undoubtedly small.1 age should be distributed in a fashion that leaves the Since statistics on migrants by age and sex are seldom distribution of the total population for the given sex with adequate, and since migrants are often concentrated in known ages unchanged. This is performed by multiplying certain age and sex groups it is obvious that a substantial the population classified by age by the ratio: volume of migration may strongly bias the mortality total population estimates obtained from census survival rates. It should total population - population with ages unknown be noted, however, that the availability of certain types of census tabulations may still permit the use of the census Columns 4 and 6 show the femalepopulation of Turkey survival method by identifying population groups that in 1935 and 1945 after such an adjustment has been are more nearly closed to migration than the total performed. Column 4, for example, was obtained by population. An example is the calculation of mortality multiplying the population in each group shown in estimates for well-specified linguistic, racial or religious Column 2 by the factor of 8221/(8221-36) = 1.0044. groups that are not affectedby migration. A possiblymore (2) Adjustment for boundary changes. Other things commonly feasible application may be the use of tabu­ being equal, an increase in the territorial coverage in the lations of the native population classified by age and sex second census would spuriously inflate the census survival in two consecutive censuses in population where immi­ rates; territorial losses in the intercensal period would gration is substantial but out-migration is negligible. introduce an opposite bias. If the two censusesin question Column 5 of table 15 shows the 1935 population of do not refer to the population of the same territory Turkey adjusted for comparability with the 1945 figures comparability must be insured by reckoning the popu­ given in column 6. Column 5 was obtained by multiplying lation in both censuses on an identical territorial basis. through column 4 by 1.0204-the product of 1.0118 If the population involved in the adjustment constitutes (adjustment for territorial coverage) and 1.0085 (adjust­ a substantial portion of the total population, or if its age ment for migration). and sex composition is very atypical, it would be highly (4) Adjustment for the length of the intercensal period. desirable to correct the census figures by individual age Computational convenience and limitations of the data and sex groups: e.g., in case of intercensal territorial gain (e.g. lack of classifications by single-year age groups) to add to the figures of the first census the population of makes it mandatory to base the calculations on two sets the territory in question (as estimated at the time of the of population figures referring to points of time that are first census) as it was actually distributed by sex and age, to a very close approximation five, ten or perhaps fifteen or to remove the population of the affected territory years apart. When this is not the case it is necessary to from the figures of the second census age group by age "move" one of the populations involved over time to group. When the population involved is small, and its establish the desired time distance between the censuses. characteristics are not strongly deviant from those of the No such need arises in the case of the 1935 and 1945 rest of the country, a simpler adjustment is adequate. Turkish censuses that have reference dates of 20 October This statement holds true in the case of Turkey for 1935­ and 21 October, respectively. When an adjustment is 1945. The territory of that country was increased by the needed on this score the simplest procedure to follow is to province of Hatai in 1939. In 1940 the total population assume that both the age distribution and the observed without this province was reported as 17,613,000 and with intercensal growth rate has been, or will remain, un­ Hatai province as 17,821,000. An adjustment for this changed during the period to be removed from, or added territorial change may be performed simply by multiplying to, the actual intercensal time distance in order to make through the 1935 census figures by the factor of (17821/ that distance equal the nearest integer multiple of five 17613) = 1.0ll8. Alternatively the 1945 census figures years. For the sake of an example assume that the two could have been deflated by the factor of .9883. censuses were taken 8.72 years apart and that the average (3) Adjustment for migration. The considerations yearly increase during this time was r = .022. Under governing this adjustment are the same as those under­ these circumstances the population registered at the time lying the adjustment for change in territorial coverage of the second census in each age group is to be multiplied 02 2 outlined in the preceding paragraph. In Turkey it was by the factor of 1.0286-Le. by e· x 1.28; 1.28 being estimated that net immigration during the 1935-1945 the additional number of years that would have elapsed period amounted to some 150,000 persons. Since no beyond the actual intercensal time distance had the second detailed information is available as to the sex-agecompo­ census been taken exactly ten years after the first one." sition of the migrants, and sincethe numbers involved are relatively small, it may be simply assumed that their demographic characteristics (their fertility, mortality, 1 The use of the 1940 population (or the average intercensal population) in the calculation of the correction factor implies that age and sex composition) were the same as those of the the migration over the decade is assumed to have been approximately rest of the population and the adjustment may be per­ evenly distributed. formed by multiplying through the 1935 figures by the 2 Where r (.022) has been calculated as follows: factor of 1.0085, i.e., by the ratio of the mid-period (1940) population plus the net migratory balance to the mid­ 8.72 r = log e P,+8.72 . period population. Since much of the migration in P, 58 Computational procedure. The essence of the compu­ projections that bracket the recorded numbers above age tations is to find a life table (from among the model x in 1945, where x is 10, 15, ..., 50. Levels five to eleven tables) that, employed to project the 1935 population, would have sufficed in this example, but thirteen and produces a 1945 population most consistent with the fifteen have been employed to illustrate the level of recorded one. The computational steps are: mortality implicit in the apparent rate of survival of the older population (over sixty five in 1935). The suggested (1) Apply the ten-year cohort survival rates given in procedure is to begin with a projection using a first guess table I.3 of annex I to the 1935 population distributed by of the mortality level, and then make projections at other age (table 15, column 5) at various "levels" of mortality," An unnecessarily extravagant procedure would be to levels in a spirit of trial and error. Table 16 shows the project with all of the tabulated model tables. In practice, projections at levels five to fifteen; it is sufficient to use a range of mortality levelsto produce (2) The projected populations and the adjusted census populations in 1945 are cumulated (from the "top" down) 3 Ifthe two censuses were fiveyears apart, the survival rates given to obtain figures for the number of females above age x, in column 7 of table 1.1 would be used. If the interval were fifteen x = 10, 15, 20, ... etc. (see table 17); years, it would be necessary to calculate values of SLx+lS/sLx in the model life tables. (3) By interpolation, find what level of mortality

TABLE 16. THE FEMALE POPULATION OF TuRKEY 1945 (THOUSANDS) AS PROJECTED FROM THE ADJUSTED 1935 CENSUS POPULATION WJTH VARIOUS "WEST" MODEL LIPE TABLES produces a projected population x and over exactly 1.1140. The implied annual rate of natural increase is matching the census population in 1945, x = 10, 15, 20 etc. For example, the population over ten in 1945 was r = log e 1.1140 = .0108' 6,914,000, the projected populations based on mortality 10 ' levels seven and nine are 6,891,000 and 7,081,000. The level that would duplicate the 1945 census figure is 7.24. TABLE 19. CALCULATION OF THE AVERAGE FEMALE CRUDE DEATH IN The mortality levels and corresponding values of 0 eo, 0 es TuRKEY IN THE PERIOD 1935-1945 CORRESPONDING TO THE and 12 are given in table 18; MEDIAN LEVEL OF MORTALITY IMPLIED BY CENSUS SURVIVAL RATES (4) Select the median level among the first nine in column 2 of table 18, or level 7.98, as the best single Death rates per Mean population Average annual thousand at age x 1935-1945 (thou- deaths at age x estimate of the level of mortality among Turkish females, in median life sands) (from cols. (thousands) Age x table level 7.98 4 and6 In table 15 (col.2 x col.3) 1935-1945; (I) (2) (3) (4)

TABLE 18. INDICES OF MORTALITY IN "WEST" MODEL LIFE TABLES 0-4 ...... ,. 79.57 1,245 99.06 CORRESPONDING TO CENSUS SURVIVAL RATES FOR TURKEY (1935-1945) FROM AGE X AND OVER TO AGE X+ 10 AND OVER 5-9 ...... 7.68 1,189 9.132 10-14 ...... 5.97 912.6 5.448 15-19 ...... 7.92 710.4 5.626 Age x Level 0/ 0"0 °e5 I. mortallty 20-24 ...... 9.99 667.8 6.671 (1) (2) (3) (4) (5) 25-29 ...... 11.25 672.2 7.562 30-34 ...... 12.75 672.8 8.578 35-39 ...... 14.10 545.5 7.692 0 ...... 7.24 35.62 46.98 .7332 40-44 ...... 15.33 517.4 7.932 5 ...... 8.09 37.73 48.32 .7523 45-49 ...... 16.93 347.6 5.885 10 ...... 10.37 43.41 51.86 .8002 50-54 ...... 22.27 410.4 9.140 15 ...... 10.80 44.50 52.53 .8090 55-59 ...... 29.00 207.9 6.029 20 ...... 7.98 37.44 48.13 .7497 60-64 ...... 43.16 324.2 13.99 5.96 25 ...... 32.39 44.90 .7010 65-69 ...... 59.93 115.1 6.898 30 7.02 35.05 46.62 .7282 ...... 70-74 ...... 89.77 129.9 11.66 35 ...... 7.21 35.53 46.92 .7324 75 and over ...... 171.47 115.5 19.80 40 ...... 9.24 40.61 50.13 .7776 45 ...... 8.42 38.54 48.83 .7595 TOTAL ..•.....•.. 26.31 a 8,783 231.1 50 ...... 10.44 43.61 51.98 .8018 55 ...... 7.31 35.79 47.09 .7348 60 ...... 7.56 36.40 47.47 .7403 a (Sum of col. 4)f(sum of col. 3)• 65 ...... 13.81 54.06 58.17 .8772

(7) An estimate of the average annual female birth rate is obtained as the sum ofthe estimated death rate and (5) An estimate of the average intercensal crude death rate of natural increase already calculated: rate for females can be obtained by calculating the life b = .0263 +.0108 = .0371; table mortality rates corresponding to level 7.98 (shown in (8) An estimate of the male birth rate may be obtained column 2 of table 19)4 and multiplying these rates with the as the product of the female birth rate, the sex ratio at average intercensal population (or the estimated mid­ birth, and the ratio of the average intercensal female period-1940-population) given in column 3 of the population to the male population. Assuming that the same table. The average population is calculated as the sex ratio at birth was 1.05, and estimating the mid-period mean of the reported 1935 and 1945 populations, after male population (analogously to the estimation of the adjustment for ages reported as unknown (columns 4 and female population) as 8,692,000, we have 6 in table 15, respectively). The result of this operation is the average yearly number of deaths by age in the inter­ b(males) = .0371 x 1.05 x 8783 = .0393. censal period, shown in column 4 of table 19. The ratio 8692 of the average yearly number of all deaths and the average intercensal population gives the estimated The average intercensal increase ofthe male population average crude death rate, d. In this example d = (231.1/ was .0153, as estimated from census figures adjusted /8783) = .0263; for migration and boundary changes. This gives a male death rate of .0393-.0153 = .0240. (6) An estimate ofthe increase of the female population in Turkey from 1935 to 1945. is provided by the ratio of Comments. In calculating the crude death rate the age the total populations in these years after adjustments for specific death rates from the estimated life table are migration and changing territorial coverage (columns 5 weighted by an age distribution that is obviously distorted and 6 in table 15), i.e., by the ratio of (9344/8388) = by misreporting of age. Thus the resulting distribution of deaths is also erratic and its detailed features should not be accepted as a valid description of that distlibution. The effect ofage distortions on the calculated total number 4 The death rate for age 0-4 may be obtained from table I.1 as (/0 - 15)f(lLo +4Ll). The death rate for the population aged 75 of deaths can be expected to be much smaller since the and over is calculated as 175fT75. errors to a large extent are compensating ones. Never- 60 theless the analyst should consider the potential bias due 1. England and Wales, 1871 to this source. In particular if the proportion under age five is under-reported (either because of omission of In section B of chapter I, it was shown that a stable young children or overestimation of their ages), the population based on the 1871-1881 English life table and resulting estimate of d (and, given the intercensal r, the on the rate of natural increase during the same period estimate of b) will be downward biased. In the given matches very closely the age distribution as actually example, however, age-misreporting does not appear to recorded in the census of 1881. Conversely, an index of have affected the estimate of d, once a life table has been the recorded age distribution and the rate of growth were shown to define a model stable population the parameters obtained. When weightingof the mx values of that table has been done by an intercensal population adjusted of which provide an excellent approximation of various for age-mis-reporting (by means of a procedure not demographic characteristics of the population, such as discussed in this Manual) the resulting d was .0264, the birth rate or the expectation of life at age zero. instead of .0263 obtained above. However since the values of these parameters were known from direct statistical observations there was little justi­ The estimate of childhood mortality is derived in this fication of applying stable methods of estimation apart method not from the basic data themselves, but is a from proving the power of the technique under conditions simple extrapolation from the estimated adult mortality when age reporting is highly reliable. The mechanics of (or mortality over age five) via the " West" model life the application of this method are illustrated in the tables. If the pattern of mortality characterizing this following paragraphs also by using English data, but under family of life tables is not valid for Turkey, the estimated somewhat less artificial circumstances. Notably stable childhood mortality, hence the derived d and bare estimates of population parameters for the period accordingly biased. This point is discussed in section A.l.a preceding 1871 will be derived from the age distribution as of chapter IV. If, for example, the "South" family of reported in the 1871 census and from the rate of growth model life tables derived from the experience of other between 1861 and 1871. No official life table has been Mediterranean countries more nearly approximates the prepared for the sixteen-year period preceding 1871, and (unknown) true pattern of Turkish mortality, the death the registration of births during that time is known to be and birth rates may be as much as .006 higher than the slightly more defective than in the 1870s-birth statistics estimates given above. Apart from the argument of became virtually complete only after legislation in 1874 geographical, and to some extent cultural, closeness to placed the responsibility for registering births upon the countries known to be characterized by "South" mortality, parents. 5 there exists some evidence from recent surveys that the Conditions for applying the method. Whether alternative age pattern of Turkish mortality isindeed more"southern " methods of estimation are available or not, stable esti­ than "western". mation should be attempted only if a case for the existence The above remarks suggest that the crude birth rate of stability with respect to the relevant demographic just derived (.0382for the population as a whole) is lower conditions can be established. Preferably such a case than the actual level. It should be noted however that should rest on direct evidence, in particular on the during the period in question the actual level of the birth constancy of the age distribution and of the rate of rate itself must have been appreciably lower than its population growth. Examination of the distribution by "normal" level. There are two reasons supporting this age in the decennial censuses from 1841 through 1871 assumption. First, wartime conditions, such as extensive provides a confirmation of approximately stable condi­ mobilization in the early 1940s probably have depressed tions in England and Wales during that period (and fertility. Second, the relative size of the cohorts in the attests to the good quality of age reported) although prime child-bearing ages was much below "normal" masculinity ratios between ages 30-44 in 1871 are notice­ during the period because of depressed fertility and ably smaller than the ones reported in earlier censuses. unusually high mortality due to the Balkan wars and to This appears to reflect the effect of excess male out­ the first World War and its troubled aftermath in Turkey. migration and suggests that the male population in this case is a less satisfactory basis for stable estimates." As to the rate of growth, the slight fluctuations in the inter­ censal rates ofincrease during the three decades preceding B. ESTIMATION OF FERTILITY AND MORTALITY BY STABLE 1871 that show no trends are reassuring but, again, cannot POPULATION ANALYSIS be taken at face value since the population was subject to net outmigration during the period. Explicit consideration The method of deriving estimates of fertility and of the effect ofmigration is clearly necessary. The average mortality from records of the age distribution and from information on the rate of growth under conditions when the population may be considered approximately stable is discussed in section B of chapter I. In the present section applications of this method are illustrated by three 5 cr. D.V. Glass, "A Note on the Under-Registration or Births examples based on data collected in censuses in England in Britain in the Nineteenth century", Population Studies, vol, V, and Wales (1871), India (1911) and Brazil (1950). These No. 1 (July 1951), pp. 70-88. censuses exemplify three different situations with respect 8 No illustration or these points is offered here. For a convenient source see the historical series in General Register Office, Census to the quality of the basic data, in particular with respect 1961, England and Wales, Age, Marital Condition and General Tables to the quality of data concerning age. (London, 1964), pp. 30-32. 61 rates of intercensal growth before and after correction for the female age distribution in England and Wales up to net outmigration are given in table 20.7 age forty-five. It is not suggested to go beyond that age Table 20 shows that there was little change in the for purposes of stable estimation. natural rate of growth over the thirty-year period prior Computational procedure. (I) Obtain values of C(x): to 1871 and the male and female rates were reasonably proportions up to age x(x, 5, 10, ...,45) from table 21. closeto each other. (Perfect stability would imply identical These cumulated proportions after rounding are shown growth rates for the two sexes). The effectof the correction in col. 2 of table 22. for migration on the female growth rate is moderate, (2) From table II in annex II find the parameters of the but is much less so for the male population. On the female stable populations characterized by the C(x) values basis of the preceding observations estimation of fertility will be derived from the female age distribution and on the one hand, and by the rate of natural increase for the decade preceding the census (.0131), on the other hand. growth rate only. This operation may be conveniently executed in the following steps: TABLE 20. AVERAGE ANNUAL RATE OF INCREASE BY DECADES BETWEEN (a) 1841 AND 1871 FOR EACH SEX CALCULATED FROM CENSUS FIGURES Given the value of C(5) as reported, select two AS REPORTED ("INTERCENSAL RATE OF GROWTH") AND AFTER stable populations each having the required growth rate CORRECTION FOR MIGRATION ("NATURAL RATE OF GROWTH"), (i.e., by interpolating between stable populations tabulated ENGLAND AND WALES for r = .010 and r = .015 to get the female growth rate of .0131) and one of the levels of mortality for which Males Females model stable populations are given in table II (i.e., levels I, 3, ... , 23). Specifically the mortality levels should Intercensal Natural Intercensal Natural be so chosen (by means of a rough process of trial and rate of rate of rate of rate of Period growth growth growth growth error) that the C(5) values in the resulting two model populations just bracket the C(5) value in question, i.e., 1841-1851 , , .0124 .0131 .0120 .0121 the ogive at age five in one of them should be just higher, 1851-1861 , . .0108 .0140 .0118 .0128 and in the other just lower, than the reported value. 1861-1871 ., ", .. .0124 .0138 .0125 .0131 The proper levels in this instance are levels nine and eleven. Columns 3.a and 3.b show the values of C(5) in these stable populations and also the parameters (such TABLE 21. FEMALE POPULATION BY AGE, ENGLAND AND WALES, as the birth rate) for which estimates are sought;" 1871" (b) Repeat the above procedure for other values ofC(x), i.e., for x = 10, 15, ... ,45, using additional columns if Population Population C(x)s Age (thousands) (percentage) any of the reported are not bracketed by the ogives of stable populations previously calculated. In the given example no such need arises since none of the C(x) values 0-4 . 1,534.8 13.17 imply a mortality level higher than level eleven, or lower 5-9 , . 1,355.6 11.64 10-14 , ,' , 1,203.5 10.33 than level nine, given the growth rate of .0131; 15-19 . 1,095.7 9.40 (c) For each value of x find the interpolation factors 20-24 ,, 1,052.8 9.04 that would be necessary to obtain the reported C(x)­ 25-29 .....•.... ,. 937.3 8.04 shown in col. 2-from the corresponding values in cols. 30-34 .. ,,,. 813.7 6.98 3.a and 3.b. For example, the reported proportion up to 35-39 " . 700.5 6.01 age 10 is .248 which may be expressed as a weighted 40-44 .. 639.7 5.49 45 and over ". 2,319.7 19.90 average of the figures for C(10) in cols. 3.a and 3.b; TOTAL •••••••• , •••••••• 11,653.3 100.00 specifically as .27 x .256+ .73 x .245 (cf, annex VI). Applying the same interpolation factors to other popu­ lation parameters calculated for the stable populations II Source: See foot-note 1 to the present chapter. in cols. 3.a and 3.b, such as tbe birth rate, one obtains parameters of the stable population defined by the given r Required basic data, Apart from numerical evidence and the observed C(x). For example the birth rate corres­ necessary to establish the case for the applicability of the ponding to r = .0131 and C(IO) = .248 is .27 x .0365+ stable method, and, in the present instance, to make a .73 x .0329 = .0339. The results of these calculations correction for migration, the required basic data are a are given in cols 4.a through 4.e. Note that one of the five-year distribution of the population by sex in one parameters calculated is the population death rate. Once census, and a count of the total population by sex at an the birth rate is calculated the death rate may of course be earlier point in time to provide a rate of growth. The obtained by simply subtracting the specified growth latter information was given in table 20. Table 21 gives rate from the birth rate.

7 The corrections are based on estimates of migration prepared 8 One of the parameters calculated is the gross reproduction rate by Glass (op. cit., pp. 85-86, "method c"). The estimated intercensal associated with iii = 32.1 (the basis for this particular value of iii is net balance in the decade preceding a given census was added to the discussed later). Since table II contains only values of GRR for census population and then the intercensal increase was calculated iii = 27, 29, 31 and 33, it is necessary first to calculate two GRR using this population and the uncorrected population a decade values that bracket the GRR with the correct value of iii, and to earlier. obtain this latter quantity by an additional step of interpolation. 62 ------

TABLE 22. DERIVATION OF STABLE POPULATION ESTIMATES OF FERTILITY AND MORTALITY BASED ON A REPORTED AGE DISTRIBUTION AND THERATE OF GROWTH. ENGLAND AND WALES, FEMALES, 1871

Values ofC (x) and C(x) various parameters in Values of various parameters In (proportion female stable populo- female stable populations with up to age x) tions with r = .0131 C(x) as shown In col. 2 and with r = .0131

Age x Level 9 Level II Birth Death Level 0/ oeo .ORR (m = 32.1) rate rate mortality (1) (2) (3.a) (3.b) (4.a) (4.b) (4.c) (4.d) (4.e)

5 ...... 132 .139 .131 .0334 .0202 10.8 44.4 2.37 10 ...... 248 .256 .245 .0339 .0208 10.5 43.6 2.41 15 ...... 351 .363 .349 .0334 .0203 10.7 44.3 2.38 20 ...... 445 .461 .444 .0331 .0200 10.9 44.7 2.35 25 ...... 536 .548 .530 .0341 .0210 10.3 43.3 2.42 30 ...... 616 .626 .607 .0346 .0215 10.1 42.6 2.46 35 ...... 686 .695 .677 .0347 .0216 10.0 42.5 2.46 40 ...... 746 .756 .739 .0344 .0213 10.2 42.9 2.44 45 ...... 801 .810 .794 .0345 .0214 10.1 42.8 2.45

Birth rate ...... 0365 .0329 Death rate ...... 0234 .0198 °eo ...... 40.0 45.0 GRR(m = 31) ...... 2.52 2.28 GRR(m = 33) ...... 2.65 2.39 GRR (m = 32.1) ..... 2.59 2.34

(3) Ideally each combination of C(x) and r for a given median among those considered provides the best avail­ sex should define the same stable population: the para- able choice. To find this population, rank the estimated meters of this model then could be accepted as valid for nine birth rates according to their absolute values, and the actual population of that sex. In practice however a select the intermediate (the fifth largest) in this series. more or less tightly clustered series of stable populations In the given example the median stable population is the are determined by the various pairs of C(x) and r. The one associated with the reported C(25), giving the procedure of selecting a single best estimate (or selecting following estimates for the female population: birth rate estimates located within a narrower range than the range = .0341, death rate = .0210, expectation of life at birth of all obtained stable estimates) depends on the nature of = 43.3 years, and gross reproduction rate = 2.42. identifiable errors in the data, especially with respect to (4) Estimates for the male population and for the age misreporting. In the given example the consistency of population as a whole may be obtained from the para­ the estimates shown in cols 4.a through 4.e is gratifyingly meters calculated for the females plus the knowledge of high (see figure XX for a graphical representation of the the sex ratio at birth (male births/female births) and the series of birth rates obtained). This finding tends to sex ratio (males/females) in the total population. The confirm the original assumption of stability and the good ratio of registered male births to female births in the five quality of age reporting. Given these circumstances, and years preceding 1871 was 1.041. The number of males lacking information that would single out some reported enumerated in 1871 was 11,058.9 thousands. (The female C(x) values as particularly reliable, or relatively defective, population was givenin table 21.) The male birth rate then the selection of the stable population with an ogive isIcalculated as

femaeI biirthraex t sexratio at birth=. 0341 X--=1.041 ..0374 sex ratio of the population .949 The total birth rate can be obtained as a population- the two sexesor directly from the female birth rate as weighted average of the rates calculated separately for . female population .. female birth rate x x (1+sexratloatbuth) = .0341x.513x2.041 = .0357. total population (5) By subtracting the appropriate rate of natural table II by finding the level of mortality in the male model increase from the estimated male birth rate and total stable population having the estimated male death rate birth rate death rates for males (.0374- .0138 = .0236) (.0236) and the male rate of natural increase (.0138). The and for the total population (.0357- .0134 = .0223) are level is 10.0, implying an °eo of 39.7years. Sincemortality obtained. in England is known to be well described by the model (6) An estimate of the male expectation of life at birth "West" life tables, the level of mortality in this instance (and/or any desired life table parameter) is obtained from may be estimated also by simply assuming that the 63 BIRTH RATE

.050

.045 .".---­ ---..._."..".-­ BRAZIL,I950 MALES .040

.035 ------ENGLAND AND WALES,IB71 FEMALES .030

.025

.020

.015

010

.005

oL-_---L.__...l-_--l__-L__.L.-_--!.__....1-__L-._--'-_---J 5 1510 20 25 30 40 45 AGE

Figure Xx. Stable population estimates of the birth rate derived from reported proportions up to age x-C(x)-in censuses of England and Wales, India and Brazil for the year and sex as indicated and from the average rate of natural increase for the same sex during the ten-year period preceding each census relation of male mortality is the same as in the tables population. Such rates are not available for England and shown in annex I, i.e., that the level of male mortality Wales for the period in question. Section B.5 of chapter I is 10.3,as is for the females,hence thatthemale °eois39.9. describes two methods for indirect estimation of m; (7) Given the sex ratio at birth an estimate of total due to lack of data on children ever born only the first fertility can be obtained from the estimated GRR of these methods-based on the reported proportions married-may be applied here. An illustration of the (m = 32.1): TF = 2.42x 2.041 = 4.94 children per computation is given in table 23. The standard age pattern woman. of marital fertility (applicable for populations whose Estimation of m. The calculation of estimates of the birth control practices are negligible) shown in col. 3 gross reproduction rate as described above presupposes is taken from table 1. Note that the absolute magnitudes the existenceof an estimate of the mean age in the schedule of the hypothetical age specific fertility rates calculated of the age specific fertility rates (m) prevailing in the given in col. 4 have no practical significance. What is relevant 64 Life tablelor Life tablelor for the problem at hand is merely the age pattern ofthese England and Wales, Stable estimate Ellglandand Wales, imputed rates: the mean age of the fertility schedule is 1838-1854 from 1871 census 1871-1880 computed as the average of the central ages in each age Females 41.85 43.3 44.62 interval (col. 5) weighted by the entries in col. 4. Males ..... 39.91 39.9 41.35

TABLE. 23 CALCULATION OF m FROM REPORTED PROPORTIONS According to vital registration for the sixteen-year MARRIED AND FROM THE STANDARD AGE PATTERN OF MARITAL period from 1855 to 1870 the average death rate for the FERTILITY RATES, ENGLAND AND WALES, 1871 total population was .0223. This figure is identical to the stable population estimate of the same quantity derived Standard Hypothetical from the age distribution in the census of 1871. Proportion marital fertility Col. 4 Age ofmarried (ertllity rates Median X tnteroal females rates (col. 2 x col. 3) age col. 5 (I) (2) (3) (4) (5) (6) 2. India, 1911 Another illustration of the technique of obtaining vital 15-19 .... .032 1.178" .0377 17.5 .66 rates by the stable method based on a census record ofthe 20-24 .... .343 1.000 .3430 22.5 7.72 age distribution and an intercensal growth rate is offered 25-29 .... .624 .935 .5834 27.5 16.04 in this section using Indian statistics, notably C(x) from 30-34 .... .735 .853 .6270 32.5 20.38 35-39 .... .766 .685 .5247 37.5 19.68 the census of 1911 and r for the period 1901-1911. In 40-44 .... .758 .349 .2645 42.5 11.24 contrast to the previous example, information on age 45-49 .... .740 .051 .0377 47.5 1.79 distribution and population growth constitutes virtually the only valid basis for establishing vital rates for India 2.4180 77.51 relating to the period in question. The usefulness of that 77.51 Hence, iii = 2,418 = 32.1 information is fortunately greatly enhanced by the applicability of stable population analysis. The argument a ~21-:iJ8~roportion married at age 15-19) = 1.2 -.7 x .032 supporting the assumption of stability rests mainly on two considerations. First, the series of decennial censuses in India up to 1911 shows a remarkable degree ofstability. This point is sufficiently well-illustrated in figure XI. Implicit in the above procedure is the assumption that To be sure, the detailed shape of the reported age distri­ all births occur in marriage. However the proportion of butions is far from what would result from sustained past illegitimate births in England at the time amounted to constancy of vital rates. But the fact that the marked some 6 per cent of all births. It would be possible to peculiarities of the age distribution are reproduced census e~tend the calculation just outlined to obtain a hypothe­ after ce~sus at the same age (as opposedto the same cohort) tical age pattern of fertility that takes into account conclusively proves that the explanation for these peculiar­ illegitimate births as well, e.g., by assuming that illegiti­ ities lies in an essentially unchanged pattern of age-mis­ mate fertility rates by age of mother had the same pattern reporting rather than in violent past deviations from as was recorded in Sweden in the 1860s. Calculations show stable levels of fertility and mortality. that the iii resulting from such assumptions would differ little from the value obtained in table 23-it would be Second, the series of intercensal growth rates preceding less by not more than .2 year. 1911 lack any detectable trend away from the horizontal; rather ups are followed by downs in regular succession.I? Comments. Some of the estimates obtained above may be compared with data from vital registration or other Constancy of age distribution and fluctuating growth estimates. For example, with respect to the birth rate for rates are consistent with indirect or qualitative knowledge the total population we have: on the main features of the demographic situation in pre­ 1911 India. Such features are frequent short-term changes Stable estimate .0357 in mortality conditions but the absence of lasting im­ Vital registration provement or deterioration in the chances of dying; a 1866-1870 0351 sustained high level of fertility explained by the lack of 1861-1865 ...... 0350 contraceptive practices and by quasi-universal and early Vital registration corrected for under-registration of births'' marriage; and, finally, the essential closedness of the 1866-1870 ' .0357 population with respect to external migration. 1861-1865 ' .0358 Stable conditions notwithstanding, no high precision Official life tables are available for the periods 1838­ can be expected from estimates derived by stable analysis 1854and 1871-1880,but not for the sixteen years preceding in the Indian case, primarily because of the defects in the census for which the stable estimates derived above age-reporting mentioned earlier. Nevertheless, identi­ may be considered as relevant. The stable life table fication of systematic deviations from the expected stable estimates do, however, suggest a plausible trend of mor­ distributions, as discussed in detail in section B.c of tality change when compared with the official life tables chapter I, permits an interpretation of even seemingly inconsistent series of stable estimates, thus considerably mentioned. In terms of °eo the following comparison can be made: reducing the apparent range of uncertainty. As was

°Glass, op, cit., p. 85. 10 See Davis, op, cit., pp. 27-28 and 85. 65 TABLE 24. DERIVATION OF STABLE POPULATION ESTIMATES OF FERTILITY AND MORTALITY BASED ONAREPORTED AGE DISTRIBUTION AND THE RATE OF GROWTH. INDIA, 1911, FEMALES

Value. ofC(x) and various parameters in female C(x) stable population. with r = .0073 and level. of Value. ofoarious parameters in female stable population. with C(x) as shoum (proportion mortality a. indicated in col. 2 and with r = .0073 up to age x) India, 1911. Birth Death Level of Age x female. Levell Level 3 Level 5 Level 7 rate rate mortality oeo ORR (m = 28.2) (1) (2) (3.a) (3.b) (3.e) (3.d) (4.a) (4.b) (4.e) (4.d) (4.e)

5 .0 .••••• •• .141 .151 .139 .0421 .0348 4.7 29.2 2.68

10 •• 0 ••••••• .276 .296 .272 .0503 .0430 2.7 24.2 3.19 15 . " ...... 375 .383 .360 .0457 .0384 3.7 26.8 2.91 20 ...... 455 .457 .435 .0403 .0330 5.2 30.4 2.57 25 ...... 548 .573 .546 .0413 .0340 4.9 29.6 2.63 30 ...... 640 .653 .625 .0449 .0376 3.9 27.3 2.85 35 ...... 725 .754 .724 .0487 .0414 2.8 24.8 3.09 40 ...... 782 .784 .759 .0478 .0405 3.2 25.4 3.03 45 ...... 847 .861 .837 .0531 .0458 2.2 22.9 3.38

Birth rate ...... 0597 .0484 .0408 .0353 Death rate ...... 0524 .0411 .0335 .0280 oeo ...... 20.0 25.0 30.0 35.0 GRR (m = 27) ...... 3.67 2.98 2.53 2.21 GRR (m = 29) '0 ••• 3.88 3.13 2.64 2.30 GRR (m = 28.2) ... 3.80 3.07 2.60 2.26

shown in chapter I-see in particular figures VIII, IX The procedure of calculation was the same as that and X-Indian age distributions are characterized by discussedin connexion with table 23 above. what has been described in this Manual as the"African­ South Asian" pattern. Rules for analysing such distri­ Inspection of the sequence of the derived birth rates (column 4.a in table 24), also reproduced in figureXX, butions were set forth in section BA in chapter 1. Accord­ ing to these rules it is preferable to use only the female shows the same characteristic pattern as was found from direct comparisons of the reported population with model population as a basis for estimation as far as the reported age distribution is concerned. Table 24 shows the deriva­ stable populations (cf. figures VIII and IX). This suggests tion of various population parameters from the 1911 that such comparisons are not necessarilyrequired for the female age distribution in India (col. 2) and from the identification of the general character of age-reporting errors. Once a case for applying the stable method has 1901-1911 average female growth rate (r = .0073).11 The computational procedures underlying this table are been established the analysis may proceed directly to the calculation of birth rates and other parameters implied by exactly analogous to those used and explained in connex­ ion with table 22 above. Note that the intermediate step the various pairs of C(x) and r, A judgement on the of calculating parameters for stable populations with the pattern of age-misreporting, hence on the rules of esti­ mation to be applied, then can be based on the results of proper growth rate but with approximate ("bracketing") this calculation, in particular on the sequence of the levels of mortality requires in this instance the use of more birth rates obtained for x = 5, 10, ... , 45. In the present than two mortality levels(see cols. 3.a through 3.d) owing case the rules call for the acceptance of the parameter to the fact the reported age distribution is less consistently close to one single stable distribution than was the case values associated with C(35) as the best single estimates. Given the female rates, parameter values for the male in the previous example. population and for the population as a whole are to be The mean age of the fertility schedule (m = 28.2) used obtained in the same fashion as was shown in the preceding in the computation was estimated from the standard example, i.e., by using the available information on marital fertility schedule shown in table 1 and from the average intercensal growth (in this case .0082 for males proportions of married females in India, 1911. The and .0077for the total population), on the sex ratio of the latter, for five-year age groups, were as follows: 12 population (1.037), and assuming-in the absence of information to the contrary-that the sex ratio at birth 15·19 20-24 25·29 30·34 35·39 40-44 45-49 was 1.05. The male expectation of life and/or other life table indices are determined by finding the level of .818 .902 .882 .814 .742 .597 .522 mortality in the male stable population having the reported male growth rate and the death rate as derived 11 Both the age distribution and the growth rate are for the current (post-partition) territory of India. They were calculated earlier. Some of the stable estimates resulting from these by Mr. S. B. Mukherjee in his"A Study of the Vital Rates in India calculations are summarized in table 25. and West Bengal" (unpublished manuscript, Princeton, 1965) which he kindly made available to the authors. Naturally all figures in table 25 are to be regarded as 12 See the preceding foot-note for the source of these data. rough approximations. Yet, on the basis of knowledge 66 TABLE 25. STABLE POPULATION PARAMETERS FOR INDIA, 1911, of fertility as indicated by 1940 and 1950 census reports DERIVED FROM THE FEMALE AGE DISTRIBUTION, THE FEMALE AND on children ever born; on the relative unimportance of MALE GROWTH RATES, AND THE SEX RATIO OF THE POPULATION AS REPORTED; AND BY ASSUMING THAT THE SEX RATIO AT international migration; and on the little or no change in BIRTH IS 1.05 mortality prior to 1950as evidencedby reports on propor­ tions of children surviving in the 1950and 1940censuses.

Females Males Total population The Latin American character of the pattern of age­ misreporting is revealed by comparing the actual age distributions to model stable distributions, or, more Birth rate . .0487 .0493 .0490 directly, by calculating estimates of the birth rate for males Death rate , .0414 .0411 .0413 and females from C(x) and r, As a result the basic stable Level of mortality . 2.8 4.0 analysis is to be limited to the male population. Table 26 °eo . 24.8 25.3 GRR (m = 28.2) . 3.09 shows the parameters implied by the male age distribution Total fertility . 6.33 (column 2) and the male growth rate (r = .0232). The computations underlying this table were explained above in connexion with table 22 (cf. also table 24). Naturally, the male model stable populations of annex II were of the pattern of age-misreporting it is possible to assert utilized in this case. with a high degree of certainty that the female birth rate was higher than .046-the estimate associated with C(15). TABLE 26. STABLE POPULATION ESTIMATESOF FERTILITY AND MORTA­ Furthermore, the value derived from C(35) is strongly LITY BASED ON THE AGE DISTRIBUTION OF THE MALE POPULATION supported by the only slightly higher estimate (.050) OF BRAZIL AS REPORTED IN THE CENSUS OF 1950 AND ON r = .0232, implied by C(lO) and r, Most likely this latter figure is, THE ANNUAL RATE OF GROWTH OF THAT POPULATION IN THE or is close to, what may be considered a fair upper 1940-1950 INTERCENSAL PERIOD estimate of the birth rate. These statements are qualifiedby C(x) Values of various parameters in male stable the fact that there is no direct evidence confirming the (proportion populations with C (x) as indicated in validity of the "West" pattern of mortality in the Indian up to age x) column 2 and with r = .0232 Brazil. 1950, case. Use of alternative stable population families in the males above calculations would have typically resulted in Age x Birth Death Level of rate rate mortality °eo higher estimates of the birth and death rates, hence in (1) (2) i;(3.a) (3.b) (3.e) (3.d) higher estimates of total fertility and lower expectation of life. As to the relation of male and female mortalities, 5 ...... 164 .0422 .0190 12.1 45.0 the strong masculinity of the population-demonstrated 10 ...... 302 .0430 .0198 11.7 43.9 by the other Indian censuses as well-conclusive1y 15 ...... 424 .0438 .0206 11.3 42.9 indicates that this particular relation incorporated in the 20 ...... 527 .0436 .0204 11.4 43.2 model life tables is not duplicated in India. 25 ...... 619 .0441 .0209 ILl 42.4 30 ...... 698 .0447 .0215 10.9 41.8 35 ...... 760 .0436 .0204 11.4 43.2 3. Brazil, 1950 40 ...... 819 .0447 .0215 10.9 41.8 45 ...... 867 .0456 .0224 10.5 40.9 Using age distribution data from the Brazilian census of 1950 jointly with the rate of growth between 1940and 1950to derive stable estimates exemplifies the application ofthe stable method under conditions when age-reporting The series of male birth rates and other parameters is typically "Latin American" in its characteristics. The given in table 26 are located within rather narrow limits. applicability of the method in this instance is supported The median in the series, which is the best single estimate, by somewhat less satisfactory evidence than in the two is associated with C(lS). Assuming a sex ratio at birth preceding examples because of the lack of an extended of 1.05, and considering that the sex ratio of the popu­ series of previous censuses of reasonably good quality. lation (as reported by the census) was .9933, the female Nevertheless the case for assuming stability is convincing. birth rate is obtained by multiplying the male birth rate It is based on the close similarity of the 1940 and 1950 by the ratio .9933/1.05. The birth rate for the whole age distributions; on high and essentially identical levels population is obtained as male birth rate male population (1 ti t bi th) --_._-- x x +sex ra 10 a If . sex ratio at birth total population Death rates are calculated by subtracting the rates of 15-49 (which in this case yields an estimated fii of 28.8 average intercensal growth (.0238 for females and .0235 years) is open to the objection, serious in the case of for the total population) from the estimated birth rates. Brazil, of ignoring the fertility of women reported as The value of fii necessary to obtain an estimate of the single but living in de facto unions. Thus the estimate (female)gross reproduction rate can be estimated by both obtained from the relation fii = 2.25 P3/P2 +23.95is to be methods suggested in chapter I, section B.5. However preferred. The value of P3/P2-the ratio of children ever the method of applying a standard marital fertility born per woman 25-29 and 20-24- was 2.289 according schedule to the proportions married among females to the 1950census. Hence we have fii = 29.1. The female 67 gross reproduction rate, as well as the female expectation estimates from information on age distribution and of life and other parameters, then can be obtained by growth under conditions of approximate constancy of reading these values in a female stable population deter­ fertility and mortality. When mortality has been declining, mined by any two ofthe parameters previously calculated, but other requisites of stability obtain- a situation often such as the female death rate and the female rate of encountered in the contemporary world- stable analysis growth. Table 27 summarizes the main results. is still frequently attempted, the practice being defended by the argument that the age distribution in such so-called quasi-stable populations is always close to that of a TABLE 27. STABLE POPULATION PARAMETERS FOR BRAZIL, 1950, DERIVED FROM THE MALE AGE DISTRIBUTION, THE MALE AND FEMALE population which is stable in the strict sense, and is GROWTH RATES, AND THE SEX RATIO OF THE POPULATION AS characterized by the current fertility and mortality of the REPORTED; AND BY ASSUMING THAT THE SEX RATIO AT BIRTH population in which mortality has been declining. IS 1.05 However, such estimates contain a bias which, depending on the duration and speed of the change in mortality, Males Females Total population may be substantial. Section C of chapter I describes a method by which such a bias can be eliminated or at Birth rate . .0438 .0414 .0426 least considerably lessened by using information on the Death rate . .0206 .0176 .0191 nature of the mortality decline. The method is illustrated Level of mortality . 11.3 12.0 below by the example of two populations; that of India °eo . 42.9 47.5 in 1961 and of Mexico in 1960. GRR (m = 29.1) .. 2.83 Total fertility . 5.80 1. India, 1961 The assumption of essentially stable demographic conditions, on the basis of which estimates for the India As was the case in the previous example no vital of 1911 were derived above, is less defensible after the statistics are available with which these estimates could be census of 1921. While there are no signs that would confronted. In view of the high consistency of the values indicate a change in fertility, the growth of the population implied by the various C(x)s, the major uncertainty with has been accelerating since the 1920s, undoubtedly respect to the goodness of the estimates originates, once reflecting a more or less steady decline of mortality from again, in the choice of the mortality pattern underlying the high plateau of the 1881-1921 period. Under such the model stable populations utilized:the "West"pattern­ conditions stable estimates should be adjusted to take and in particular the early childhood mortality implied care of the effects of that decline. by a given adult mortality in that pattern- mayor may not be a close approximation of Brazil's actual expe­ Computational procedure. (1) Table 28 shows stable rience.P The application of the census survival method estimates of the birth rate and the gross reproduction rate (cf. the discussion in section A in this chapter and in that are calculated in exactly the same manner explained chapter 1) for the Brazilian male population yields an oeo in connexion with tables 22 and 24 above. The inputs in value of 42.4 that appears to confirm the validity of the this instance are the 1961 female age distribution.!" and mortality estimate shown in table 27 (hence, given the stable age distribution, the validity of the birth rate estimate). But this is not pertinent to the problem stated TABLE 28. STABLE POPULATION ESTIMATES OF THE BIRTH RATE (b) AND OF THE GROSS REPRODUCTION RATE (GRR) BASED ON THE above, since both methods have essentially the same AGE DISTRIBUTION OF THE FEMALE POPULATION OF INDIA AS weakness in estimating childhood mortality. Unlike in REPORTED IN THE CENSUS OF 1961 AND ON THE ANNUAL RATE the case of India, however, census information on child OF NATURAL INCREASE OF THAT POPULATION IN THE 1951-1961 survival rates in Brazil supplies a basis for a direct INTERCENSAL PERIOD (r = .0189) estimation of child mortality thus permitting a check on, and improvement of, the estimates shown in table 27. Values ofb and of GGR in female C(x) stable populations with C(x) as in This topic will be taken up in chapters VII and VIII (proportion col. 2 and u;lth r = .0189 up to age x) below. Age x India. 1961, females Birth rate GRR (flj = 28.8) (1) (2) (3.a) (3.b)

C. EsTIMATION OF FERTILITY AND MORTALITY BY STABLE POPULATION ANALYSIS WHEN THE POPULATION IS 5 ., ...... 154 .0396 2.64 QUASI-STABLE 10 ...... 303 .0476 3.18 15 ...... 412 .0447 2.98 The examples given in the previous section demon­ 20 ...... 493 .0393 2.62 25 ...... 583 .0397 2.64 strated the technical details of extracting population 30 ...... 668 .0415 2.76 35 ...... 738 .0425 2.83 18 An at least qualitatively identifiable source of bias in these calculations also arises from the fact that no allowance was made 40 ...... 794 .0422 2.81 for external migration in reckoning the growth rate. The rate of natural increase may have been perhaps .0002 smaller than the average intercensal rate of growth. If so, the birth rates are under­ estimated by roughly the same amount, while the error introduced into the death rates (also underestimated) is about twice as large. 14 Source: Census of India, 1961 Census, Age Tables, p. 54. 68 the rate of female natural increase for 1951-1961, (r (4) The parameter k can be derived from the average = .0189) that was obtained by adjusting the intercensal rate of acceleration of the growth rate itself using the rate for changed territorial coverage and for net immi­ empirical relation k = 17.8x Sr]11t. The absolute increase gration. The mean age of the fertility schedule was in the growth rate can be obtained by subtracting from estimated from an imputed age specific fertility schedule the 1951-1961 rate of increase (which may be thought of in the same manner as shown in table 23. The proportions as referring to the year 1956), the level of growth that married among females 15-49 that were used in the prevailed, on the average, up to 1921. The latter may be calculation are as follows (1961 census data): estimated from the ratio of 1921 all-India population to the samepopulation in 1881. This ratio is 1.1877, therefore, 15-19 20-24 25-29 30-34 35-39 40-44 45-49 r 19 2 1 = (loge 1.1877)/40 = .00430. The value of !:>.r/I:!.t is then .696 .918 .942 .915 .871 .777 .698 .0189- .0043 = .0146 = .000417 ; The resulting mequals 28.8 years. 1956-1921 35 (2) The preliminary stable estimates of band GRR hence k = .000417 x 17.8 = .0074. (columns 3.a and 3.b in table 28) are to be adjusted for the (5) Column 3 in table 29 shows the adjustments as distorting effects of changing mortality using the adjust­ taken directly from table IIU, which is tabulated for ments listed in table IIU. Since the preliminary estimates k = .01. For other values of k it is necessary to scalethese are based on C(x) and the average rate of growth during fractions up or down in the same proportion that the the ten years preceding the time to which C(x) refers, the actual value of k bears to .Ol-i.e., in this instance by appropriate section of that table is its "Part (a)". To .0074/.01 = .74. This is shown in column 4. extract the correct adjustment factors from the tabulated (6) Column 4 thus contains proportions to be added figures it is first necessaryto estimate values of two indi­ to or subtracted from the preliminary estimates. It is ces; namely t, the approximate length of time (in years) convenient to transform these adjustments into multipliers for which the decline of mortality has been proceeding; by adding 1 to each entry (see column 5). Column 6 gives and k, a parameter that describesthe speedof the decline. the products of these multipliers and the preliminary (3) The value of t may be estimated in this instance as stable estimates, i.e., the adjusted (quasi-stable) estimates 40 years, the time that has elapsed between 1921-the of the female birth rate and the gross reproduction rate. date up to which growth rates for India showed a regular (7) Selectionof a singleestimate for band GRR among sequence of ups and downs, and after which acceleration those associated with the various C(x) values should be of growth was uninterrupted-and 1961, the date of the carried out in the same fashion as was explained and latest census. illustrated for "pure" stable estimates in section B of this

TABLE 29. EsTIMATION OF THE BIRTH RATE AND OF THE GROSS REPRODUCTION RATE FOR THE FEMALE POPULATION OF INDIA, 1961, BY AD1USTMENT OF PRELIMINARY STABLE ESTIMATES OF THESE PARA­ METERS (AS CALCULATED IN TABLE 28) FOR THE EFFECTS OF DECLINING MORTALITY

Stable population Adjustments Adjustments Adjusted estimates derived from table III.I, for t = 40, Adjustment estimates from C(x) and part (a) k = .0074 factors (col. 2 X col. 5) Age x lO-year intercensal r t = 40, k = .01 (col. 3 X .74) o+col. 4) (1) (2) (3) (4) (5) (6)

Birth rate 5 .0 .••••....• 0. .0396 -.043 -.032 .968 .0383 10 ...... 0476 -.032 -.024 .976 .0465 15 ...... 0447 -.004 -.003 .997 .0446 20 ...... 0393 .026 .019 1.019 .0400 25 ...... 0397 .051 .034 1.034 .0410 30 ...... 0415 .073 .054 1.054 .0437 35 ...... 0425 .092 .068 1.068 .0454 40 ...... 0422 .114 .084 1.084 .0457

Gross reproduction rate (fii = 28.8) 5 ...... 2.64 -.010 -.007 .993 2.62 10 ...... 3.18 .001 .001 1.001 3.18 15 ...... 2.98 .031 .023 1.023 3.05 20 ...... 2.62 .062 .046 1.046 2.74 25 ...... 2.64 .088 .065 1.065 2.81 30 ...... 2.76 .111 .082 1.082 2.99 35 ••••••••...• 0. 2.83 .130 .096 1.096 3.10 40 ...... ,., 2.81 .153 .113 1.113 3.13

69 chapter and of chapter I. In the present instance the tion system of long tradition and its statistics on births estimates derived from C(35) are to be preferred to the and deaths for the past two or three decades at least, rest. The male birth rate and the birth rate for the total are considered virtually complete. Also the assumption population are calculated by assuming a sex ratio at of constant fertility underlying both the stable and the birth of 1.05 and by accepting the reported masculinity quasi-stable methods discussed here is apparently valid ratio of the population (1.062). only as a rough approximation. Apart from the violent (8) Death rates (for the two sexes and for the total demographic disturbance caused by the Mexican revolu­ population) are obtained by subtracting the rates of tion in the second decade of the century (the consequences natural increase (adjusted intercensal growth rates) from of which are now less visible than they were in earlier the appropriate birth rate estimates. The expectation of censuses), shifts in the age distribution and trends in life or any other index of mortality is determined by population growth in recent decades reflect the influence reading the level of mortality in the stable populations of a slight but not negligible increase in Mexican fertility (one for the males, one for the females) determined by that subtly reinforces,and is superimposedon, the domi­ the vital rates calculated earlier. No adjustment of such nating effect on changes in those variables exerted by the estimates for quasi-stability is warranted, or indeed very rapid decline of mortality since the mid-1930's. desirable. The principal parameter values derived by There is no practical way to separate such effects in stable following the above steps of calculation are exhibited population analysis. A straightforward application of in table 30. stable methods (including the method of correcting for the presumed effects of declining mortality) for the TABLE 30. ESTIMATES OF VARIOUS POPULATION PARAMETERS FOR analysis of Mexican data then can be expected to reveal INDIA, 1961, OBTAINED BY ADJUSTING STABLE ESTIMATES OF THESE some inconsistencies. The existence of direct information PARAMETERS FOR THE EFFECTS OF DECLINING MORTALITY on vital rates, not used in the stable estimates, offers the advantage of making explicit the nature of such inconsis­ Females Males Total population tencies, and should also reveal other biases involved in stable analysis that may commonly occur in other appli­ Birth rate . .0454 .0449 .0451 cations, yet that are ordinarily not possible to nail down Death rate . .0265 .0254 .0259 in the absence of independent evidence. Level ofmortality . 7.7 9.1 °eo . 36.8 37.5 Not surprisingly the tests proposed earlier for detecting GRR (fii = 28.8) . 3.10 errors in age-reporting when applied to Mexican census Total fertility . 6.36 data reveal the existence of a "Latin American" pattern. Hence the male age distribution is accepted as the main basis for stable estimates. Table 31 sets forth the elements Comments. Application of the stable method to the of the calculation leading to a series of estimates of the population of India in 1961 without adjustment for male birth rate, adjusted for the effects of declining declining mortality results in a seriesof birth rate estimates mortality. The basic data on which these calculations are unlike those typically produced by age distributions built are the male age distribution (column 2) and the subject to the African-South Asian pattern of age­ average male growth rate for the period of 1950-1960 misreporting. When quasi-stability of the underlying (.0316). The stable estimates implied by these variables demographic conditions is allowed for, however, the (column 3) are adjusted for quasi-stability in a manner pattern familiar from the analysis of the 1911 Indian age analogous to the computations shown earlier in table 29. distribution is fully re-established. The apparent range of The parameters k and t used in the adjustment process uncertainty-apart from any error in the observed r, or were obtained by comparing the male growth rate for the in the assumption concerning the pattern of mortality­ 1930-1940 intercensa1 period (.0181) with that for 1950­ as to the actual levelofthe(female) birth rate is remarkably 1960. Specifically, !J.r/at = .0316-.0181/20 = .0135/20 small; what may be tentatively considered maximum and = .000675; and k = !J.r/atx 17.8 = .0120. Some accele­ minimum estimates (those associated with C(lO) and ration of the growth rate increase probably occurred C(5), respectively) differ only by about .002 (.0465 versus before the 1930s, although very little of such acceleration .0446). Comparisons with the estimates derived for 1911 is suggested by the difference between the r's for 1930­ (cf. table 25) show virtually identical gross reproduction 1940 and 1920-1930, or by registered death rates in the rates but appreciably reduced birth rates in 1961. The 1920s and early 1930s. Naturally such comparisons may changed relationship of these two indices is of course be somewhat misleading owing to possible changes in a necessary consequence of destabilization. It may be census coverage and improvement in vital registration. noted that the actual birth rate of the quasi-stable (1961) On the other hand an examination of the shifts in the age population is below the intrinsic rate: maintenance of the distribution of deaths, a not necessarily affected estimated mortality level and of the GRR would result by moderate omission rates, indicates fluctuating mor­ eventually in a higher birth rate than the one shown in tality before the mid-1930s followed by a clear sustained table 30. upturn. In any event it is evident that far the greatest proportion of the improvement in mortality took place 2. Mexico, 1960 in the twenty-five year period preceding the census of Any attempt to estimate Mexican fertility and mortality 1960. Consequently the value of t was taken as 25 years. exclusively from census data is bound to be a highly From column 7 of table 31 the median of the birth rates artificial enterprise since the country has a vital registra- is selected as the single most acceptable estimate among 70 TABLE 31. ESTIMATION OF THE BIRTH RATE FOR THE MALE POPULATION OF MEXICO, 1960, BY ADJUST­ MENT OF PRELIMINARY STABLE ESTIMATES OF THAT PARAMETER (DERIVED FROM REPORTED C (X) FOR 1960 AND THE INTERCENSAL GROWTH RATE-r = .0316-FOR 1950-1960) FOR THE EFFECTS OF DECLINING MORTALITY

Birth rate. In C(x) male stable (proportion populations; Adju.tment. Adiustments Adjusted up to age x), with C(x) a. from table for t ~ 25, Adjustment estimates of Mexico, 1960, In col. 2 and III.l part (a) k = .012 factors the birth rate Age x male. with r = .0316 t = 25; k = .01 (col. 4 X 1.2) (I +col. 5) (col. 3 x col. 6) (1) (2) (3) (4) (5) (6) (7)

5 ...... 169 .0383 -.028 -.034 .966 .0370 10 ...... 325 .0424 -.025 -.030 .970 .0411 15 ...... 454 .0436 -.004 -.005 .995 .0434 20 ...... 554 .0422 .021 .025 1.025 .0433 25 ...... 635 .0405 .053 .064 1.064 .0431 30 ...... 704 .0395 .081 .097 1.097 .0433 35 ...... 762 .0385 .099 .119 1.119 .0431 40 ...... 818 .0394 .106 .127 1.127 .0444 those associated with x = 10, 15, ... ,40. C(20) and C(30) estimates shown in table 32 was based on a value of the are tied for the median position - as a matter of fact all parameter k obtained from the acceleration of the growth birth rates implied by C(l5) through C(35) are virtually rat~. It is inter~sting to check the consistency of that indistinguishable. From the estimated male birth rate the estimate of k with the result of an alternative method male death rate is calculated by subtracting the growth also described in chapter I, section C, that utilizes th~ rate. Mortality indices are then obtainable from the stable population defined by these vital rates. Rates for females and for the total population are calculated in a TABLE 32. ESTIMATES OF VARIOUS POPULATION PARAMETERS FOR MEXICO, 1960, OBTAINED BY ADJUSTING STABLE ESTIMATES OF similar fashion, having first derived the birth rates for THESE PARAMETERS FOR THE EFFECTS OF DECLINING MORTALITY these groups from the male birth rate via the sex ratio at birth (1.05, assumed) and the sex ratio of the population as a whole (.995, reported). Male. Female. Total population A somewhat more roundabout process is to be followed in finding the (female) gross reproduction rate. The Birth rate . .0433 .0410 .0422 following steps are required here: (a) estimate m;15 Death rate . .0117 .0120 .0119 Level of mortality . 16.6 15.4 (b) having selected the median male adjusted birth rate in °eo . 55.5 56.0 column 7, say the one derived from C(20) and r, find (by GRR (m = 28.8) . 2.90 the usual method, i.e., by using the sex ratios at birth Total fertility . 5.95 and in the population) the stable (unadjusted) female birth rate associated with the stable (unadjusted) male birth rate (Column 3) implied by C(20) and r (results: male birth rate, .0422; female birth rate, .0400); (c) this changing composition of deaths to measure the tempo female birth rate plus the reported female growth rate of mortality change. The index of the age distribution (.0290) determine a stable population: read the value of e~p~oyed is the proportion of deaths over sixty-five GRR with the appropriate mfrom this stable population WIthin all deaths over age five. This index for Mexico (GRR (m = 28.8) = 2.75); (d) adjust the GRR thus can be calculated for each year from 1936 on; it shows obtained for quasi-stability using parameter values as clear upward trend with relatively minor yearly fluc­ in the earlier calculation (t = 25, k = .012) and selecting tuations. To minimize the effects of the latter it is better the adjustment factors appropriate for the gross repro­ to calculate the index for periods longer than one year. duction rate and for the proper x, in this instance 20. In the following illustration (which is limited to the male population) the average for 1936-1939 and 1956-1959 (Adjusted GRR (m = 28.8) = 2.75 x 1.054 = 2.90. The corresponding value for total fertility is 5.95. This are used; their values are .220 and .322, respectively. compares with an estimate of total fertility from number This change has occurred in twenty years. If we had an of children ever born, as P~/P2 which gives 5.99). The estimate of the °eo at the base period, and ifwe knew the results of the above calculations are given in table 32. change in the expectation of life at birth during these The correction for quasi-stability that affects all the twenty years it would be possible to read a tabulated value of kt in annex table III.3, hence to estimate k. If fertility is constant, and the age distribution is quasi-stable, it is 15 The method based on standard marital fertility is not applicable possible to obtain just such a base-period value for °eo because of the prevalence of consensual marriages in Mexico.Using and a value for !J.°eo using the indices ofthe age distri­ the reported Pa/P2 ratio (2.141), m is calculated as 28.8 years. bution of deaths given above. The procedure is as follows: This is the value accepted in the following calculation. Note how­ ever, that calculated direct from birth statistics, m is appreciably (1) By means of the tabulation in annex II calculate °eo higher - 29.3 years. in the stable populations defined by the 1950-1960 71 intercensal r, and (a) C(10) in 1960and (b) C(lS) in 1960. estimate of k) is not attributable to mortality decline, (The results are 57.2 and 54.9.) Calculate the average of which is alone measured by the shift in the age distri­ these two figures (56.0). This gives an estimate of the bution of deaths. Fertility increase also biases the terminal (end-period) °eo' latter measure downwards. Naturally if stable estimates (2) Using the same tables calculate the index births/ are corrected only for mortality decline, but the shift in population 15-44 (thisindexoffertility isincludedin tableIn age distribution is reinforced by fertility increase also, definedby the same parameters as givenin point (1) above the resulting estimates will have a downward bias. The (results: .1006 and .1036) and take their average (.1021). actual value of k to be used in this instance actually should be larger than .012 (cf. foot-note 20 to chapter I). ~ (3) From annex table III.2 obtain °eo as the difference Even if the estimates given in table 32 were obtained by betweentwo separately calculated estimates of °eo, each a more adequate correction for quasi-stability, their defined by the average index of births/population 15-49 values would still be affected by a more substantial bias as calculated in point (2), and by the index of the distri­ owingto the inadequate representation of the true pattern bution of death for the two dates as given above (.322 and of mortality in Mexico by the "West" model life tables ~oeo .22). The result is: = 56.6-40.1 = 16.5. which underlie the above calculations. Section A.1.b. (4) Calculate an estimate of the base period °eo as the of chapter I gives a general statement of this problem. difference between the end period °eo-see point (I)-and In the present instance there is ample evidence from the the ~oeo-see point (3)-i.e., as 56.0-16.5 = 39.5. lifetables prepared for Mexico since 1930that the relation of child mortality (e.g., 15) to "adult" mortality (e.g., 0e10) Table III.3 can now be used to get kt the value of which is much closer to the "South" pattern than it is to the in this example is .1686. Hence k = .1686/20 = .0084. "West". For Mexico this factor alone would cause the If the adjustment of the preliminary stable estimates is estimated birth rates to be some .004 lower than their carried out with this value of k, the procedure is exactly actual value, and of course there is a corresponding the same as shown in table 31 with the exception that distortion in the other parameter values as well. This column 4 is multiplied by .84, instead of 1.2, to get example thus shows the basic weakness of stable (or column 5. quasi-stable) estimates derived from C(x) and r: their Comments. There is a substantial difference between the dependence on a well-chosen model life table family. two independent estimates of k obtained above which There is often no information available on the true cannot be attributed to the approximate nature of the pattern, and no basis for a good choice. This difficulty techniquesinvolved,or explainedby biasesin the reporting is however eliminated, and the power of stable techniques of the age of dead persons. A more fundamental cause of greatly increased, when censuses provide data on child this difference is that fertility has been increasing, and survival. Examples of estimation under such circum­ part of the acceleration of growth (reflected in the first stances are discussed in chapter VIII below.

72 Chapter vn

EXAMPLES OF ESTIMATES BASED ON QUESTIONS ABOUT FERTILITY AND MORTALITY

A. ESTIMATION OF FERTILITY FROM REPORTS ON CHILD­ Total fertilility derived from the age specific fertility BEARING IN CENSUSES OR SURVEYS rates reported in the example (column 3) is

7 The estimation of fertility from data on (a) births in 5 L Ii, the year before a surveyor census, and (b) children ever 1=1 born, is illustrated by an example based on hypothetical or 5.24. The adjustment of these fertility rates for a data. A synthetic example is used because the only possible error in the reference period is achieved by instances of surveys containing the requisite information calculating the average value of cumulative fertility in the are in tropical Africa, and, as is noted in chapter IV, the age intervals shown in column 2 (F I in column 7), and validity ofthe method is sometimes seriously impaired by forming the ratio of reported parity (PI) to cumulated the extensive age-misreporting characteristic of surveys fertility (FI)' Reported parity at 20-25 (P2) and 25-30 (P3) and censuses in Africa. The suitability of the method for is assumed to be approximately correct, although the estimating fertility in an African population must be latter may be affected to some slight degree by omissions. decided after a detailed examination of the quality of the The ratio of P2/F2 is a correction factor that makes the data in the survey in question-an examination that would fertility rates consistent with the average number of go beyond the scope ofthis manual. The reader is referred children ever born reported by women 20-25. In column 9 to the detailed studies of different populations in the there are age specific fertility rates that have been multi­ book on the population of Africa recently completed at 1 plied by P2/F2 (1.313). The adjusted estimate of total the Office of Population Research. fertility is five times the sum of the rates in column 9, or The method exemplified in table 33 is of greatest 1.313 times the figure derived from the unadjusted fertility potential value for populations (such as in Latin America, rates. The estimate is 6.88-higher than either the average the Philippines, and especially in the Republic of Korea parity reported by older women (deficient because of and Thailand) for which age-misreporting is less extreme. omissions) or than the cumulation of reported fertility The example shows how data from such areas can be (deficient because of a shortened reference period). employed when the appropriate questions have been Computational procedure. The fertility rates in column 3, included in a census or survey. based on births reported for the year before the survey, pertain to women one-halfyear younger than the intervals 1 Brass, et. al., op. cit. in column 2. Therefore the estimation of average cumu-

TABLE 33. THE ESTIMATION OF TOTAL FERTILITY AS P2/F2 TIMES REPORTED TOTAL FERTILITY

Cumulative Estimated Average fertility average number of at beginning Multiplying cumulative births In Average ofInterval factors for fertility Adju"ted Exact age preceding number of estimating age specific o/woman year per children average value fertility ( I-I ) Interval at time woman ever born fertility Fi= 1:fj+wi/i rates (i) ofsurvey (fj}G Pi (¥Jd) (Wi) , J-O PilFi (J'i =fi x PI/PI) (1) (2) (3) (4) (5) (6) (7) (8) (9)

1 ...... 15-20 .081 .186 0 1.963 .159 1.170 .106 2 ...... 20-25 .242 1.435 .405 2.842 1.093 1.313 .318 3 ...... 25-30 .261 3.109 1.615 3.011 2.401 1.295 .343 4 ...... 30-35 .238 4.176 2.920 3.121 3.663 1.140 .312 5 ...... 35-40 .166 4.710 4.110 3.247 4.650 1.013 .218 6 ...... 40-45 .043 4.761 4.940 3.548 5.093 .935 .056 7 ...... 45-50 .017 4.503 5.155 4.484 5.231 .861 .022

a For age intervals one-half year less than shown in column 2. 73 lative fertility is obtained by the use of multiplyingfactors recorded as non-respondents is ajI +a. The recommended found by interpolation in annex table IV.I. The key to the method of adjusting the non-responses, then, is to make interpolation is fllf2' equal to .335. The multiplying a scatter diagram showingthe proportion of non-responses factors in column 6 were (5/130) x entries where fl1f2 in each age interval on one axis, and the proportion of equals .460plus (125/130) x entries wherefl1f2 equals .330. childless women on the other. If the resultant points are The entries in column 8 are typical of those found in closelyfitted by a straight line, extend this line to the zero African surveys least distorted by age-misreporting. value of the proportion childless. The proportion of non­ responses on the straight line at this point (when the Pl/F 1 is best ignored because of the intrinsic difficulty of proportion childless is hypothetically zero) can be taken estimating Fl' P2/F2 and P3/F3 are approximately equal, and either ratio can be taken as a multiplying factor for as an estimate of the true proportion of non-responses. the correction of fertility rates for bias in the reference The recommended arithmetical adjustment of the data on the average number of children ever born and on the period. The steady decline in PdFI past age thirty is the typical result of progressively greater omission of children average number surviving is the omission from the deno­ ever born by older women. minator of the estimated number oftrue non-respondents, and the inclusion in the denominator of the estimated number of non-respondents who are considered to be in fact childless. In most instances almost all of the non­ B. ESTIMATION OF MORTALITY FROM REPORTED NUMBERS OF CHILDREN EVER BORN, AND CHILDREN SURVIVING respondents fall in the probably childless category, and little or no error is introduced by assuming that all non­ Preliminary adjustment of data. This method of esti­ responses indicate zero children. mation requires responses from a census or survey on the (b) Adjustments when data are tabulated only for married number of children ever born alive to each woman, and women or for mothers. In some censuses and surveys, the number of children surviving, with average numbers questions about children ever born are asked (or at least per woman tabulated for the standard five-year age tabulated) only for ever married women, and in others, intervals. Deficiencies in the data for which adjustments data are shown only for women who have borne at least can sometimesbe made include:(a) a moderate proportion one child. A word about the nature of the resulting biases of women for whom no responsesare obtained; (b) asking is not out of place. First, if illegitimacy is infrequent, the the relevant questions (or tabulating responses) only for proportions surviving among children ever born will be married women, or only for women (" mothers") who adequately representative, and the only effectof obtaining have experienced at least one live birth. information solely from ever married women is that it is (a) Adjustment for non-response. The women for whom necessary to estimate the average parity of all women children ever born are not tabulated are not, ordinarily, in each age group indirectly, usually on the assumption representative in their average parity of the age group to that all births occur to married women. If, in fact, illegiti­ which they belong. There is a widespread tendency to mate births contribute significantly to the average parity leave a blank space instead of a zero for the response to of women 15-19, and not so much to women 20-24, the the question of the number of children ever born for ratio P tiP2 may be underestimated, leading to the selec­ women of zero parity. The evidence of this tendency is a tion of adjustment factors in table V.I that tend tooveresti­ strong positive correlation between the proportion of mate q(2) and q(3). The bias resulting from the higher women reported childless in each age interval with the mortality rates experienced by children born to non­ proportion of non-responses. If all non-responses were married women, on the other hand, tends to cause an of this sort, they should be counted as zeros in calculating underestimate of mortality, although this latter bias is the average number of children ever born in each age minimized by the fact that women under twenty bearing interval. However some of the non-responses represent children while single are often married by the time (at a genuine absence of information, so that the assumption age 20-24 or 25-29) their fertility histories are used to of zero parity would produce a downward-biasedestimate. estimate 2qO and 3QO' 2 EI Badry has proposed a simple but often effective tech­ The adjustment made when information is given only nique for determiningapproximately which non-responses for ever married women is to determine average parity represent zero parity and which the genuine absence of in the two relevant intervals (15-19 and 20-24) by multi­ information (as, for example, when data are supplied by a plying the average parity of ever married women by the neighbour). He suggests fitting a straight line of the form ratio ever married women/total women. This adjustment y = ax-s-b, where the observed values of yare the cannot be accepted as valid in populations where a major proportion of non-respondents in each age interval and x proportion (say more than 10 per cent) of the births to is the proportion reporting zero parity. If the observed women 15-19 occur to the non-married. When data are relationship is very closely fitted by the straight line, it supplied only for "mothers", the adjustment is analogous may plausibly be assumed that the fraction of non­ to that for ever married women, i.e., parity is estimated responses genuinely associated with the absence of infor­ by multiplying the average parity of mothers by the ratio mation is b, and that the proportion of childless women mothers/total women. In both instances of limited data, the reported proportion surviving among children ever born is accepted as representative of the experience of 2 M.A. El Badry, "Failure of enumerators to make entry of all women. In fact the proportion surviving for ever zero: errors in recording childless cases in population censuses". Journal of the American Statistical Association, vol. 56, No. 296, married women is higher than for the non-married, and December 1961, pp. 909-924. this bias of course also holds for "mothers" if the 74 TABLE 34. CALCULATION OF lqO. 2qO. aqo. 5qO. lQqO, 15qo, AND 20QO FOR BRAZIL. BASED ON CHILDREN EVER BORN. AND CHILDREN SURVIVING RECORDED IN THE 1950 CENSUS

Average ,",~r Average number Multipliers Proportion ofchildren ofchildren for colUlllll 5 dead by Interval Ageo! ever born surviving from age x (i) women (P,) (S,) I-S,IP, PIIPa Age x

1 ...... 15-19 .146 .118 .1918 1.058 1 .203 2 ...... 20-24 1.099 .870 .2083 1.050 2 .219 3 ...... 25-29 2.516 1.947 .2262 1.016 3 .230 4 ...... 30-34 3.883 2.935 .2442 1.019 5 .249 5 ...... 35-39 5.065 3.730 .2636 1.029 10 .271 6 ...... 40-44 5.778 4.146 .2825 1.007 15 .284 7 ...... 45-49 6.212 4.353 .2993 1.006 20 .301

>'. Ji ~) Q ;;... ·V"', )< , ~~) ') <:J questions have been asked only of married women, and The level of mortality indicated by the sequence of esti­ tabulated only for those with at least one child. If, in fact, mates of "qo implies the following sequence of °eo's for the questions are asked only of women who have borne both sexes (as calculated from annex tables 1.2 and 1.1) : at least one child, the probable bias is increased because of the likelihood that "mothers" with no surviving child­ ren may have been excluded. 2qO •••... ••...... ••...•••••••• •• 41.2 Computational procedure. The estimation of mortality aqo ...•...... • 42.2 from child survival rates is illustrated in table 34 on the 5QO ••••••••••••••••••••••••••••••• 42.4 basis of data taken from the 1950 census of Brazil. lOqO 42.4 lIiqO •..••••••••.•..•••••••••••••••• 42.6 Columns 3 and 4 show the average number of children 20QO •••••••••••••••••.••••••.•• " .• 42.9 ever born per woman and the average number surviving for the age intervals shown in column 2. The proportion The remarkably consistent sequence of the implied °eo ofnon-survivors is shown in column 5; these proportions values (or of the implied mortality levels in general) are converted into estimates of lqO' 2qO, .... 20qO by suggests that the basic data are of very good quality. multipliers taken from annex table V.l. Pl/P is .146/1.099 2 This is also supported by an examination of the reported = .133. The multipliers in column 6 were obtained by parity distribution itself which shows no obvious signs of interpolating between the columns in table V.I, for which an increasing failure to report children as the age of the P /P is .143 and .090, specifically, by adding (.811) x l 2 reporting women progresses. Naturally these observations (entries where P /P = .143) and (.189) x (entries where l 2 inspire increased confidence in the mortality estimates Pl/P2 = .090). The final estimates are in column 8. derived above. On the other hand, it should be borne in Comments. As is pointed out in the discussion in section mind that survival rates to increasingly higher ages B of chapter II the level of infant mortality derived from reflect the mortality experience of an increasingly longer child survival reported by women 15-19 should not be period prior to 1950. It is possible, therefore, that there regarded seriously, because of the basic weakness of the is a fortuitous element in the high consistency of the method of estimation at this point. A better estimate of mortality levels implied by these rates; namely, the effects lqO is obtained by accepting zqo, and the relationship ofthe actually higher mortality of earlier periods to some between 1 qo and 2qo in the model tables. The resultant extent might have been offset by the probable tendency estimate of infant mortality is .171, if based on the on the part of older women to omit a higher proportion "West" female model tables, and .178 if based on the of their children who are dead than of those who are males, so that an acceptable estimate of lqO is about .175. still alive.

75 Chapter VIII

EXAMPLES OF ESTIMATION BASED ON CHILD SURVIVAL AND AGE DISTRIBUTIONS

A. ESTIMATION OF FERTILITY AND MORTALITY FROM DATA ratio at birth (of, say, 1.05 males/females) and the "West" IN A SINGLE CENSUS THAT RECORDS THE AGE COM­ differences in mortality at the same level. We shall POSITION OF THE POPULATION, THE NUMBER OF CHILDREN assume growth rates of .015 and .025, almost certainly EVER BORN AND THE NUMBER SURVIVING bracketing the Brazilian rate of increase, here provisionally assumed unknown. The sex ratio in a tabulated stable The estimation of child mortality from the reported population is calculated as follows: (a) assume the sex survival of children ever born to women in various age ratio at birth to be 1.000. Note the figure "Pop. size, intervals in the census of Brazil, 1950, (see chapter VII, B(O) = 1" at, for example, mortality level 9, growth rate section B) is combined with the age distribution in the .015. The figure is 26.095for females and 24.701for males, example given in this section, to provide estimates of and the sex ratio of the stable population under these fertility and over-all mortality by methods described in circumstances is 24.701/26.095 or .9466. If the sex ratio chapter III, section B. The example begins with the esti­ at birth is 1.04 or 1.05, the sex ratio of the stable popu­ mates of 2qO in table 34., i.e., 2QO for both sexes equal to lation becomes 1.04x .9466 = .984, or 1.05x .9466 =.994. 0.219. The method of estimation (employing the age Futher calculations of this nature provide the figures in distribution ofthe malepopulation in 1950and the family table 35. of model stable populations) requires determination of The sex ratio among more than 200,000 births registered the level of male mortality. The first step is to ascertain in the Federal District in 1949-1952 was 1.058, so that a whether the relations between male and female mortality sex difference of mortality at least as large as in the in the "West" model life tables can plausibly be assumed "West" model tables is needed to account for the mascu­ to characterize mortality in Brazil. Life tables based on linity of the Brazilian population, and more especially registered deaths in the Federal District have mortality the native population in 1950. Thus it will be assumed differences as large as in the " West" tables; sex differences that 2QO for the male population of Brazil justprior to 1950 in mortality in Latin America are almost without excep­ was 0.232, and that the appropriate life table is level tion unfavourable to males up to age two; and the sex 10.1. Table 36 shows (columns 4.0, 4.b and 4.c) birth rates, ratio of the Brazilian population supports the inference death rates and rates of increase associated with this that sex differences in mortality of about the magnitude mortality level and recorded values of C(5), C(lO), ... , found in the "West" tables have prevailed in Brazil. The C(45) for Brazilian males, 1950. level of mortality identified by a 2QO of 0.219 is 10.1 (both sexes). The question is whether the sex ratio of the Female and total population parameters are estimated Brazilian population (males/females = .993 for the whole as in chapter VI, section C. The resultant values are population, and .988 for the native population in 1950) shown in table 37. is consistent with the sex differences found at level 10.1 Note that these estimates indicate slightly higher ferti­ in the "West" model life tables. The last entries in each lity and mortality than were obtained from the same age model stable population make it possible to calculate distribution and the intercensal rate of increase. The the sex ratio on the assumption of any stipulated sex fertility estimates presented here are based on evidence

TABLE 35. SEX RATIO IN STABLE POPULATIONS WITH THE "WEST" SEX DIFFERENCES IN MORTALITY, VARIOUS MORTALITY LEVELS, RATES OF INCREASE AND SEX RATIO AT BIRTH

Sex ratio at birth 1.04 Sex ratio at birth 1.05

Rate a! increase Level 7 Level 9 Levelll Level 7 Level 9 Levelll

.015 ...... 979 .984 .988 .989 .994 .998 .025 ...... 986 .991 .995 .995 1.001 1.005

76 TABLE 36. CALCULATION OF STABLE POPULATION ESTIMATES OF BIRTH AND DEATH RATES BASED ON A REPORTED AGE DISTRIBUTION AND ON A LEVEL OF MORTALITY DERIVED FROM REPORTED CHILD SURVIVAL RATES, BRAZIL, 1950, MALES.

C(x) (proportioll VailleS ofC(x) IIIIdvarious up to age x) parameters 111 male stable Values ofvarious parameters ill male stable Brazil. /950, populatlolls with a level of populatlolls with C(x) as showlI ill col. 2 IIIId Age x males mortality of/0./ with mortality level of10./ r = .020 r = .025 Birth rate Death rate Growth rate (l) (2) (3.a) (3.b) (4.a) (4.b) (4.c)

5 ...... 164 .160 .177 .0444 .0232 .0212

10 ••••••••••• 0 .302 .293 .321 .0449 .0233 .0216 15 ...... 424 .410 .444 .0453 .0233 .0222 20 ...... 527 .514 .550 .0451 .0233 .0218 25 .0 .••••.•••. .619 .604 .641 .0453 .0233 .0220 30 ...... 698 .682 .717 .0456 .0233 .0223 35 ...... 760 .750 .781 .0449 .0233 .0216 40 ...... 819 .807 .834 .0455 .0233 .0222 45 ...... 867 .855 .878 .0459 .0233 ,0226

Birth rate ...... 0432 .0484 Death rate ...... 0232 .0234

TABLE 37. STABLE POPULATION PARAMETERS FOR BRAZIL, 1950, subtracting the intercensal rate of natural increase from DERIVED FROM THE MALE AGE DISTRIBUTION, LEVEL OF MORTALITY this estimated birth rate. The resulting death rate figures FROM ESTIMATES 2qO FOR BOTH SEXES, THE SEX RATIO OF THE POPULATION AS REPORTED, AND ASSUMED SEX RATIO AT BIRTH are little affectedby an age pattern of mortality that does OF 1.05 not in fact conform to the model life tables (table 38).

Males Females Total populatioll TABLE 38. BIRTH RATES CALCULATED FROM C(x) AND 12 (MALES), AND DEATH RATES FROM b-r, BRAZIL, 1950 Birth rate . .0453 .0428 .0441 Death rate . .0233 .0210 .0222 Males Females Total populatlOll Level of mortality . 10.1 10.1 °eo .. 40.0 42.8 ORR (fii = 29.1) . 2.91 Birth rate ...... 0453 .0428 .0441 Total fertility . 5.97 Intercensal rate of natural increase .0 ..•• .0232 .0238 .0235 Death rate ...... 0221 .0190 .0206 that is in principle preferable, because fertility estimates derived from 12 are less affected by variations in the age pattern of mortality. Moreover, the fertility estima~es It is not difficult to construct a life table approximately obtained in this way are closer to the average panty consistent with these rates, embodying mortality above reported by women 45-49 years of age; which in t~e agefive derived from C(x) and r, and mortality under age Brazilian census of 1950 appears a remarkably valid five from the estimated value of 12, The procedure is as report. On the other hand, estimates of the death rate and follows: (a) find the value of sLo and Is consistent with and of the expectation of life at birth are strongly depen­ the estimated /2 (i.e., at mortality level 10.1 for each sex); dent on how closelythe age pattern of mortality in Brazil (b) find the value of °es in the stable population selected conforms to the "West" model tables. from C(x) and r for males, and estimated for females from the male C(x) and r plus the recorded sex ratio of the population, the estimated sex ratio at birth, and the B. ESTIMATION OF FERTILITY AND MORTALITY FROM DATA femaleintercensalrate of natural increase(i.e., at mortality ON AGE DISTRIBUTION, THE INTERCENSAL RATE OF level 11.3 for males and 12.0 for females); (c) calculate INCREASE, AND SURVIVAL RATES OF CHILDREN EVER BORN °eo as (1/10) (sLo+ls x °es)' If an approximately stable population has been enumer­ The results are as follows: ated twice in about a decade, and if the second census contains information making it possible to calculate Males Females childhood mortality, the birth rate can be estimated (as °es .. 51.3 54.4 in the preceding section) in a way that has relatively little Is . 72700 75200 sensitivity to age-mis-statement and unusual age patterns sLo/lo . 3.898 4.023 of mortality, and the death rate estimated in turn by °eo . 41.2 44.9 77 Comments. The analysis of the male age distributions typical of results found from other censuses) to find in Brazil in 1950, combined with the estimated intercensal reported parity by older women exceeding the estimate rate of natural increase, the recorded sex ratio, and data derived from the age distribution. Moreover, the average on child survival leads to estimates of the birth rate of parity reported by women 45-49 exceeds (p3)2/P2' which about 44 per thousand, a death rate of about 21 per equals 5.76 in this case. But there is enough consistency­ thousand, an expectation of life at birth of about forty­ in the level of mortality indicated by the succession of three years, and a total fertility of about six children per "qo estimates (see chapter VII, section B), in the not very woman passing through the child-bearing years. The only wide divergence between estimates based on C(x) and r, independent information available-the average parity and on C(x) and 12, and in the total fertility as estimated reported by women 45-49-indicates a slightly higher total and reported-tolend substantial authority to the approxi­ fertility of 6.2. It is slightly surprising (and not at all mate values ofthese parameters.

78 Part Three ANNEXES

Annex I

MODEL LIFE TABLES *

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY

LEVEL 1

Age x Ix nmx nqx nLx .Lx+. Tx 0"" .Lx

Females

0 ...... 100,000 .4788 .3652 76,264 .5728" 2,000,000 20.00 1 ...... 63,483 .0790 .2615 210,124 .7886b 1,923,736 30.30 5 ...... 46,883 .0152 .0731 225,847 .9346 1,713,612 36.55 10 ...... 43,456 .0118 .0572 211,069 .9347 1,487,765 34.24

15 ••••• ,0 ••••• 00 40,972 .0154 .0739 197,287 .9175 1,276,696 31.16 20 .0 ..••••..•••• 37,943 .0192 .0918 181,008 .9031 1,079,409 28.45 25 ...... 34,460 .0216 .1025 163,468 .8914 898,402 26.07

30 ••• 0 •••••••••• 30,927 .0245 .1155 145,708 .8797 734,933 23.76 35 ...... 27,356 .0268 .1258 128,178 .8708 589,226 21.54 40 ...... 23,916 .0286 .1332 111,613 .8633 461,048 19.28 45 ...... 20,729 .0303 .1407 96,358 .8418 349,435 16.86 50 ...... 17,814 .0393 .1787 81,111 .8019 253,077 14.21 55 ...... , ...... 14,631 .0499 .2217 65,045 .7382 171,966 11.75

60 •••••••••• 0 ••• 11,387 .0743 .3133 48,017 .6529 106,921 9.39 65 •••••••••••• 0. 7,819 .0988 .3962 31,351 .5540 58,904 7.53 70 ...... 4,721 .1437 .5285 17,368 .4265 27,553 5.84 75 ...... 2,226 .2011 .6691 7,407 .2727c 10,184 4.58

80 •••••• 0 ••••••• 737 .2652 2,778 2,777 3.77

Males

0 ...... 100,000 .5827 .4191 71,922 .5287" 1,803,333 18.03 1 ...... ". 58,093 .0784 .2597 192,420 .7860b 1,731,411 29.80 5 ...... 43,005 .0140 .0675 207,769 .9417 1,538,991 35.79 10 ...... 40,102 .0099 .0484 195,655 .9435 1,331,223 33.20 15 ...... 38,160 .0135 .0651 184,591 .9218 1,135,568 29.76 20 ...... ". 35,676 .0193 .0923 170.153 .9024 950,977 26.66 25 ...... 32,385 .0218 .1035 153,542 .8889 780,824 24.11 30 ••••.••••••. 0. 29,032 .0254 .1196 136,479 .8712 627,282 21.61 35 ...... 25,560 .0299 .1392 118,904 .8485 490,803 19.20 40 ...... 22,002 .0362 .1658 100,889 .8242 371,898 16.90

45 •••••••••••• 0' 18,354 .0415 .1878 83,149 .7929 271,009 14.77 50 ...... 14,906 .0522 .2309 65,926 .7519 187,860 12.60 55 ...... 11,464 .0626 .2705 49,568 .6953 121,933 10.64 60 ...... 8,363 .0853 .3516 34,463 .6136 72,365 8.65 65 ...... 5,422 .1128 .4401 21,146 .5178 37,902 6.99 70 ...... 3,036 .1546 .5575 10,949 .3963 16,757 5.52 75 ...... 1,343 .2193 .7083 4,338 .2530c 5,808 4.32 80 ...... 392 .2667 1,470 1,469 3.75

* These tables are extracted from Coale-Demeny, Regional Model Life Tables and Stable Populations (Princeton University Press, Princeton, 1966). " Pro~ortion surviving from birth to 0-4. b oL50LO. c Tso/T75.

81 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continuecl) LEVEL 3

Age x Ix nmx nqx nLx sLx +s Tx °ex sLx

Females

0 ...... 100,000 .3807 .3052 80,163 .6371a 2,500,000 25.00 I ...... 69,481 .0628 .2155 238,404 .8296b 2,419,837 34.83 5 ...... 54,506 .0125 .0605 264,280 .9459 2,181,433 40.02

10 ...... " 51,206 .0097 .0473 249,975 .9458 1,917,153 37.44 IS ...... 48,783 .0127 .0615 236,422 .9313 1,667,179 34.18 20 ...... 45,786 .0159 .0765 220,171 .9192 1,430,756 31.25 25 ...... 42,283 .0179 .0855 202,371 .9093 1,210,585 28.63 30 ...... 38,666 .0203 .0964 184,008 .8994 1,008,214 26.08

35 '0' .••••.•••.• 34,937 .0222 .1053 165,492 .8915 824,206 23.59 40 ...... 31,259 .0238 .1l21 147,532 .8844 658,715 21.07 45 ...... 27,754 .0254 .1195 130,475 .8649 511,182 18.42 50 ...... 24,436 .0331 .1527 112,853 .8298 380,708 15.58 55 ...... 20,705 .0422 .1910 93,640 .7731 267,855 12.94

60 ••••••••••• II' 16,751 .0628 .2712 72,397 .6964 174,215 10.40

65 •••••••• II •••• 12,208 .0843 .3481 50,416 .6034 101,819 8.34 70 ...... 7,959 .1232 .4710 30,423 .4817 51,403 6.46 75 ...... 4,211 .1746 .6077 14,656 .3014° 20,979 4.98 80 ...... 1,652 .2612 6,324 6,323 3.83

Males

0 ...... 100,000 .4595 .3513 76,462 .5982a 2,285,135 22.85 I ...... 64,868 .0625 .2144 222,637 .8279b 2,208,674 34.05 5 ...... 50,957 .0116 .0562 247,627 .9515 1,986,037 38.97 10 ...... 48,093 .0082 .0404 235,61 I .9527 1,738,410 36.15 IS ...... 46,151 .0112 .0546 224,456 .9343 1,502,799 32.56 20 ...... 43,631 .0161 .0774 209,712 .9182 1,278,343 29.30 25 ...... 40,254 .0181 .0867 192,547 .9070 1,068,632 26.55 30 ...... ,. 36,765 .0211 .1001 174,630 .8921 876,085 23.83

35 •••••••••••• I. 33,087 .0248 .1167 155,785 .8726 701,455 21.20 40 ...... 29,227 .0300 .1395 135,941 .8512 545,670 18.67 45 ...... , 25,149 .0347 .1596 115,714 .8230 409,729 16.29 50 ...... 21,137 .0439 .1979 95,228 .7855 294,015 13.91 55 .0 .••••••..••• 16,955 .0533 .2352 74,803 .7329 198,787 11.73 60 ...... 12,967 .0730 .3088 54,823 .6573 123,983 9.56 65 ...... 8,963 .0974 .3918 36,036 .5659 69,160 7.72 70 ...... , .. 5,452 .1347 .5037 20,393 .4481 33,124 6.08 75 ...... 2,706 .1922 .6491 9,138 .2822° 12,731 4.71 80 ...... 949 .2643 3,593 3,593 3.78

a prohortion surviving from birth to 0-4. b 5L55Lo. c Tso/T75.

82 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 5

Age x Ix nmx nqx nLx 5Lx+5 Tx °ex 5Lx

Females

0 ...... 100,000 .3067 .2557 83,378 .6924a 3,000,000 30.00 1 ...... 74,427 .0503 .1777 262,817 .8618b 2,916,622 39.19 5 ...... 61,205 .0103 .0502 298,352 .9552 2,653,805 43.36 10 ...... 58,135 .0080 .0392 284,979 .9549 2,355,453 40.52 15 ...... 55,856 .0105 .0512 272,136 .9426 2,070,474 37.07 20 ...... 52,998 .0132 .0639 256,525 .9324 1,798,338 33.93 25 ...... 49,612 .0148 .0716 239,185 .9240 1,541,813 31.08 30 ...... 46,062 .0168 .0807 221,017 .9156 1,302,628 28.28 35 ...... 42,345 .0185 .0884 202,364 .9086 1,081,611 25.54 40 '" ...... 38,601 .0199 .0948 183,862 .9017 879,247 22.78 45 ...... 34,944 .0215 .1021 165,796 .8841 695,385 19.90 50 ...... 31,375 .0281 .1313 146,579 .8528 529,589 16.88 55 ...... 27,257 .0361 .1657 124,997 .8021 383,009 14.05 60 ...... 22,742 .0537 .2365 100,263 .7324 258,012 11.35 65 ...... 17,363 .0729 .3083 73,432 .6446 157,750 9.09 70 "0 ••••••••••• 12,010 .1075 .4235 47,332 .5277 84,318 7.02 75 ...... 6,923 .1544 .5570 24,975 .3248e 36,986 5.34 80 ...... 3,067 .2553 12,011 12,011 3.92

Males

0 ...... 100,000 .3684 .2955 80,205 •6580a 2,766,802 27.67 1 ...... 70,454 .0502 .1771 248,774 .8605b 2,686,597 38.13 5 ...... 57,976 .0096 .0469 283,083 .9595 2,437,823 42.05 10 ...... 55,258 .0069 .0337 271,627 .9603 2,154,740 38.99 15 ...... ". 53,393 .0094 .0460 260,829 .9447 1,883,113 35.27 20 ...... 50,938 .0135 .0652 246,394 .9312 1,622,284 31.85 25 ...... 47,619 .0151 .0728 229,434 .9219 1,375,890 28.89 30 ...... 44,154 .0175 .0839 211,509 .9093 1,146,456 25.97 35 ...... 40,449 .0206 .0981 192,329 .8926 934,947 23.11 40 ...... 36,482 .0251 .1179 171,663 .8735 742,618 20.36 45 ...... 32,183 .0292 .1362 149,954 .8478 570,955 17.74 50 ••••••••••• ,0' 27,799 .0373 .1706 127,137 .8133 421,001 15.15 55 .. , ...... 23,056 .0460 .2061 103,400 .7641 293,864 12.75 60 '0' ••••••••••• 18,304 .0634 .2734 79,008 .6935 190,464 10.41 65 ...... 13,299 .0854 .3519 54,795 .6059 111,456 8.38 70 ...... 8,619 .1193 .4593 33,198 .4912 56,660 6.57 75 ...... 4,660 .1716 .6003 16,307 .3050e 23,462 5.04 80 ...... 1,863 .2603 7,155 7,155 3.84

a prohortion surviving from birth to 0-4. b 5L55Lo. e Tso/T75.

83 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 7

Age x Ix nffix nqx nLx 6Lx+6 Tx °ex 6Lx

Females

a 0 .0 ..•••••••••• 100,000 .2484 .2139 86,099 .7407 3,500,000 35.00 b 1 •••..••••••. 0. 78,614 .0403 .1456 284,252 .8881 3,413,901 43.43

5 ••••••••••••• 0 67,169 :0085 .0414 328,896 .9630 3,129,649 46.59 10 .0 .••••...•••• 64,389 .0066 .0323 316,741 .9627 2,800,754 43.50 15 ...... 62,307 .0087 .0425 304,921 .9523 2,484,012 39.87 20 .0 .••••••••••• 59,661 .0109 .0532 290,367 .9436 2,179,092 36.53 25 ...... 56,486 .0123 .0597 273,999 .9366 1,888,724 33.44 30 ...... 53,114 .0140 .0674 256,617 .9294 1,614,726 30.40

35 ••••• 0 •••••••• 49,533 .0154 .0741 238,488 .9230 1,358,109 27.42

40 ••••• 0 •••••••• 45,862 .0167 .0800 220,133 .9164 1,119,620 24.41 45 ...... 42,191 .0183 .0874 201,739 .9003 899,487 21.32 50 '0 •••••••••••• 38,504 .0240 .1131 181,633 .8723 697,748 18.12 55 ...... 34,149 .0311 .1442 158,433 .8268 516,116 15.11 60 ...... 29,224 .0462 .2071 130,989 .7630 357,682 12.24 65 ...... 23,171 .0637 .2747 99,942 .6796 226,694 9.78 70 ...... 16,806 .0949 .3834 67,921 .5667 126,751 7.54

75 '0' ••••••••••• 10,363 .1384 .5142 38,493 .3457" 58,830 5,68 80 ...... 5,035 .2476 20,338 20,338 4.04

Males

0 ...... 100,000 .2977 .2482 83,373 .7103a 3,248,436 32.48 1 .0 .••••••••.••• 75,183 .0403 .1455 271,759 .8868b 3,165,064 42.10 5 .0 •.••••••••••• 64,242 .0080 .0390 314,946 .9663 2,893,305 45.04 10 ...... 61,737 .0057 .02'81 304,343 .9667 2,578,358 41.76 15 ...... 60,000 .0079 .0387 294,202 .9534 2,274,015 37.90 20 ...... 0. 57,680 .0113 .0548 280,501 .9422 1,979,813 34.32 25 ...... 54,520 .0126 .0610 264,287 .9345 1,699,313 31.17 30 ...... 51,195 .0146 .0703 246,981 .9239 1,435,026 28.03 35 ...... 47,598 .0172 .0823 228,190 .9094 1,188,045 24.96 40 .0 ..•••••••••• 43,678 .0209 .0995 207,526 .8925 959,856 21.98 45 ...... 39,332 .0247 .1165 185,206 .8690 752,330 19.13 50 ...... 34,751 .0319 .1476 160,935 .8368 567,124 16.32 55 ...... 29,623 .0399 .1815 134,677 .7906 406,189 13.71 60 ...... 24,248 .0555 .2435 106,477 .7243 271,512 11.20

65 ••••••••• 0 •••• 18,343 .0757 .3182 77,123 .6398 165,036 9.00 70 ...... 12,506 .1069 .4217 49,345 .5280 87,913 7.03 75 ...... 7,232 .1552 .5590 26,052 .3245" 38,567 5.33 80 ...... 3,189 .2548 12,515 12,515 3.92

a Proportion surviving from birth to 0-4. b 5L5/5Lo. " Tso/T75.

84 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 9

Age x nmx nqx nLx liLx+li Tx 0"" liLx

Females

0 ...... 100,000 .2010 .1777 88,447 .7835a 4,000,000 40.00 1 ...... 82,226 .0320 .1179 303,316 .91oob 3,911,553 47.57 5 ...... 72,530 .0069 .0338 356,520 .9698 3,608,237 49.75 10 ...... 70,078 .0054 .0264 345,762 .9694 3,251,718 46.40 15 .0 .••••••••••• 68,227 .0071 .0350 335,172 .9606 2,905,956 42.59 20 ...... 65,842 .0090 .0440 321,964 .9533 2,570,784 39.05 25 ...... 62,944 .0102 .0495 306,933 .9474 2,248,820 35.73 30 ...... 59,829 .0115 .0559 290,781 .9412 1,941,886 32.46 35 ...... 56,483 .0128 .0618 273,690 .9355 1,651,105 29.23 40 ...... 52,993 .0139 .0673 256,043 .9291 1,377,415 25.99 45 ...... 49,424 .0155 .0747 237,894 .9144 1,121,372 22.69 50 ...... 45,733 .0205 .0975 217,525 .8891 883,478 19.32 55 ...... , .. 41,277 .0268 .1257 193,410 .8481 665,953 16.13 60 ...... 36,087 .0400 .1818 164,037 .7895 472,543 13.09 65 ...... 29,527 .0560 .2457 129,500 .7100 308,507 10.45 70 ...... 22,272 .0845 .3488 91,943 .6006 179,007 8.04 75 ...... 14,505 .1253 .4772 55,221 .3657c 87,064 6.00 80 ...... 7,584 .2382 31,843 31,843 4.20

Males

0 ...... 100,000 .2408 .2074 86,106 .7567a 3,730,053 37.30 1 ...... 79,263 .0321 .1183 292,227 .9088b 3,643,947 45.97 5 ...... 69,888 .0065 .0322 343,818 .9722 3,351,720 47.96 10 ...... 67,639 .0047 .0233 334,258 .9722 3,007,902 44.47 15 ...... 66,064 .0066 .0324 324,977 .9610 2,673,644 40.47 20 0" ••••••••••• 63,926 .0094 .0459 312,305 .9517 2,348,667 36.74 25 ...... 60,995 .0104 .0508 297,226 .9454 2,036,363 33.39 30 ...... 57,895 .0121 .0585 281,009 .9365 1,739,137 30.04 35 ...... 54,509 .0142 .0688 263,172 .9240 1,458,128 26.75 40 ...... 50,760 .0175 .0837 243,182 .9088 1,194,955 23.54 45 ...... 46,512 .0209 .0994 220,999 .8872 951,774 20.46 50 ., ...... 41,887 .0273 .1277 196,068 .8572 730,775 17.45 55 ...... 36,540 .0348 .1602 168,066 .8136 534,706 14.63 60 ...... 30,686 .0489 .2177 136,730 .7510 366,640 11.95 65 ...... 24,006 .0676 .2891 102,678 .6693 229,910 9.58 70 ...... 17,066 .0967 .3893 68,718 .5598 127,231 7.46 75 ...... 10,421 .1418 .5234 38,471 .3425c 58,514 5.62 80 ...... 4,967 .2478 20,043 20,043 4.04

a pror:ortion surviving from birth to 0-4. b IIL55Lo. c Tso/T75.

85 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 11

A.g~ x Ix nmx nCb: nLx IiLx+1i Tx 0"" IiLx

Females

0 ...... 100,000 .1615 .1461 90,502 .8219G 4,500,000 45.00 1 .0 .••••••••••. 85,388 .0250 .0937 320,442 .9288b 4,409,498 51.64 5 ...... 77,389 .0055 .0272 381,683 .9758 4,089,056 52.84 10 ...... 75,285 .0043 .0212 372,430 .9752 3,707,373 49.25 15 ·...... 73,687 .0058 .0284 363,207 .9679 3,334,942 45.26 20 ...... 71,596 .0073 .0360 351,543 .9618 2,971,735 41.51 25 ...... 69,022 .0083 .0405 338,115 .9569 2,620,192 37.96 30 ...... 66,224 .0094 .0459 323,525 .9516 2,282,017 34.46 35 .0' ••••••••••• 63,186 .0105 .0510 307,872 .9465 1,958,552 31.00 40 ...... 59,963 .0116 .0562 291,388 .9402 1,650,680 27.53 45 ...... 56,592 .0131 .0636 273,969 .9267 1,359,291 24.02 50 ...... 52,996 .0175 .0837 253,884 .9039 1,085,322 20.48 55 ...... 48,558 .0232 .1095 229,493 .8669 831,439 17.12 60 ...... 43,239 .0347 .1596 198,946 .8127 601,946 13.92 65 ...... 36,339 .0495 .2203 161,682 .7367 403,000 11.09 70 ·...... 28,333 .0758 .3184 119,112 .6304 241,319 8.52 75 ...... 19,311 .1144 .4448 75,083 .3856 122,207 6.33 80 ·...... 10,722 .2275 47,125 47,124 4.40

Males

0 ...... 100,000 .1940 .1717 88,499 .7983G 4,211,576 42.12 1 ...... 82,835 .0252 .0944 310,632 .9274b 4,123,076 49.78 5 ...... 75,015 .0053 .0262 370,157 .9773 3,812,445 50.82 10 .0 .•••••.••••• 73,048 .0038 .0190 361,763 .9771 3,442,288 47.12 15 ...... '" 71,657 .0054 .0268 353,478 .9677 3,080,525 42.99 20 ...... 69,7~4 .0078 .0380 342,042 .9601 2,727,047 39.11 25 .0 ..•••...•••• 67,083 .0086 .0419 328,380 .9550 2,385,005 35.55 30 .0 ..•••..••••. 64,269 .0099 .0482 313,607 .9476 2,056,625 32.00 35 ...... 61,173 .0117 .0569 297,166 .9368 1,743,018 28.49

40 ••••••••••••• 0 57,693 .0145 .0698 278,395 .9231 1,445,852 25.06 45 ...... 53,665 .0177 .0845 256,985 .9032 1,167,457 21.76

50 .0 •••••••••••• 49.129 .0233 .1102 232,105 .8750 910,472 18.53 55 ...... 43.713 .0305 .1416 203,093 .8337 678,366 15.52 60 '0' ••••••••••• 37,524 .0432 .1951 169,318 .7743 475,273 12.67 65 ...... 30,203 .0607 .2636 131,109 .6951 305,955 10.13 70 ...... 22,241 .0881 .3610 91,133 .5879 174,847 7.86 75 ...... 14,213 .1306 .4922 53,575 .3600" 83,714 5.89 80 ...... 7,217 .2395 30,139 30,139 4.18

G pr0ft0rtion surviving from birth to 0-4. b liLs sLo. " Tso/T76.

86 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 13

Age x Ix nmx nqx nLx .Lx+6 Tx OCx 6Lx

Females

0 ...... 100,000 .1282 .1183 92,310 .8566a 5,000,000 50.00 b 1 •••••••• 0 ••••• 88,169 .0188 .0717 335,996 .9453 4,907,690 55.66 5 ...... 81,848 .0043 .0214 404,871 .9810 4,571,694 55.86 10 ...... 80,100 .0034 .0166 397,178 .9804 4,166,823 52.02 15 ...... 78,771 .0046 .0226 389,403 .9743 3,769,645 47.86

20 ••••••••••••• 0 76,990 .0059 .0289 379,397 .9693 3,380,242 43.91 25 ...... 74,769 .0066 .0327 367,738 .9652 3,000,845 40.14 30 ...... 72,326 .0076 .0370 354,934 .9608 2,633,107 36.41 35 ...... 69,647 .0085 .0415 341,009 .9561 2,278,173 32.71 40 ...... 66,756 .0095 .0464 326,031 .9500 1,937,165 29.02 45 ...... 63,656 .0111 .0538 309,724 .9375 1,611,134 25.31 50 ...... 60,234 .0149 .0717 290,374 .9170 1,301,410 21.61 55 ...... 55,916 .0200 .0953 266,259 .8834 1,011,036 18.08 60 ...... 50,587 .0301 .1401 235,225 .8332 744,777 14.72 65 ...... 43,503 .0439 .1980 195,982 .7603 509,552 11.71 70 ...... 34,890 .0683 .2918 149,002 .6566 313,570 8.99 75 ...... 24,711 .1051 .4163 97,836 .4055" 164,568 6.66 80 ...... 14,424 .2162 66,732 66,732 4.63

Males

0 ...... 100,000 .1538 .1394 90,659 .8375a 4,711,432 47.11 1 ...... 86,058 .0186 .0708 328,086 •9449b 4,620,773 53.69 5 ...... 79,961 .0042 .0206 395,689 .9822 4,292,687 53.69 10 ...... 78,315 .0030 .0149 388,655 .9816 3,896,998 49.76

15 ••••••••••••• 0 77,147 .0044 .0219 381,507 .9735 3,508,343 45.48 20 ...... 75,456 .0063 .0311 371,407 .9674 3,126,836 41.44 25 .•••...••... 0. 73,107 .0069 ,0341 359,299 .9634 2,755,429 37.69 30 ...... 70,613 .0080 .0391 346,160 .9573 2,396,130 33.93

35 •••••••••••• 0' 67,851 .0095 .0464 331,379 .9482 2,049,970 30.21 40 .0 •••.•••••••• 64,700 .0119 .0575 314,197 .9358 1,718,590 26.56

45 ••••••••••••• 0 60,979 .0148 .0712 294,038 .9176 1,404,393 23.03 50 '0 •••••••••••• 56,636 .0198 .0944 269,814 .8912 1,110,356 19.61

55 .0 •••••••••••• 51,289 .0266 .1247 240,455 .8519 840,541 16.39 60 .0 •••••••••••• 44,893 .0383 .1748 204,850 .7954 600,086 13.37 65 ...... 37,047 .0547 .2408 162,938 .7183 395,236 10.67 70 ...... 28,128 .0807 .3357 117,033 .6129 232,298 8.26

75 ••••• 0 •••••••• 18,685 .1210 .4644 71,730 .3777" 115,265 6.17 80 ...... 10,007 .2299 43,536 43,535 4.35

a Pr0r.0rtion surviving from birth to 0-4. b 5L5 sLo. " Tso/T7S.

87 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 15

Age x Ix nmx nqx nLx aLx+a Tx 0"" aLx

Females

0 ...... 100,000 .0996 .0934 93,745 .8890a 5,500,000 55.00 b 1 ••••••••••••• 0 90,661 .0129 .0500 350,729 .9613 5,406,255 59.63 5 ...... 86,127 .0032 .0157 427,251 .9860 5,055,527 58.70 10 ...... 84,773 .0025 .0122 421,284 .9852 4,628,276 54.60 15 '0 •••••••••••• 83,740 .0035 .0174 415,061 .9800 4,206,992 50.24 20 ...... 82,284 .0046 .0227 406,751 .9757 3,791,931 46.08 25 ...... 80,416 .0053 .0259 396,874 .9724 3,385,180 42.10 30 .0 ••••••• ,. "0 78,333 .0060 .0294 385,907 .9686 2,988,306 38.15 35 ...... 76,029 .0068 .0334 373,805 .9643 2,602,399 34.23 40 ...... 73,493 .0078 .0382 360,445 .9581 2,228,595 30.32 45 ,0 •••••••••••• 70,686 .0094 .0458 345,343 .9464 1,868,150 26.43 50 ...... 67,452 .0128 .0619 326,821 .9274 1,522,807 22.58 55 ...... 63,276 .0175 .0840 303,101 .8966 1,195,986 18.90 60 .0.0 •••••••••• 57,964 .0266 .1246 271,765 .8492 892,885 15.40 65 ...... 50,742 .0398 .1808 230,771 .7783 621,120 12.24 70 ...... 41,567 .0629 .2716 179,610 .6765 390,349 9.39 c 75 "0 •••••••• "0 30,277 .0984 .3948 121,501 .4235 210,740 6.96 80 ...... 18,323 .2053 89,239 89,238 4.87

Males

0 ...... 100,000 .1203 .1114 92,539 .8720a 5,183,094 51.83 1 ...... 88,864 .0132 .0511 343,443 .9594b 5,090,555 57.29

5 "0 •••••••• "0 84,327 .0032 .0159 418,287 .9862 4,747,112 56.29 10 ...... 82,988 .0024 .0117 412,515 .9854 4,328,825 52.16

15 ••••• 0 •••••••• 82,018 .0036 .0176 406,474 .9787 3,916,310 47.75 20 ...... 80,571 .0051 .0250 397,814 .9739 3,509,837 43.56 25 ...... 78,554 .0055 .0272 387,439 .9709 3,112,023 39.62 30 ...... 76,421 .0063 .0310 376,180 .9659 2,724,584 35.65 35 ...... 74,051 .0076 .0373 363,357 .9579 2,348,404 31.71 40 ...... 71,292 .0096 .0471 348,068 .9464 1,985,047 27.84

45 ••••••••••••• 0 67,936 .0125 .0604 329,422 .9290 1,636,979 24.10 50 ...... 63,833 .0172 .0822 306,046 .9035 1,307,556 20.48 55 ••••.•••••• '0. 58,585 .0237 .1120 276,526 .8657 1,001,510 17.10 60 ., ...... 52,025 .0347 .1595 239,386 .8112 724,984 13.94 65 ...... 43,729 .0504 .2238 194,182 .7355 485,599 11.11 70 ...... 33,944 .0753 .3170 142,825 .6315 291,416 8.59 75 ...... 23,186 .1142 .4440 90,190 .3930c 148,592 6.41 80 ...... 12,890 .2207 58,402 58,401 4.53

a Proportion surviving from birth to 0-4. b sLs/sLo. c Tso/T7s.

88 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 17

m Age x I" n " nq" nLx 5Lx+5 T" °e" 5Lx

Females

0 ...... 100,000 .0745 .0707 94,785 .9171 a 6,000,000 60.00 b 1 ••••••••••••• 0 92,934 .0085 .0332 363,755 .9744 5,905,215 63.54 5 ...... 89,854 .0022 .0110 446,805 .9902 5,541,460 61.67 10 ...... 88,868 .0017 .0085 442,445 .9805 5,094,655 57.33 15 ...... 88,110 .0025 .0125 437,800 .9855 4,652,210 52.80 20 ...... 87,010 .0033 .0165 431,463 .9822 4,214,410 48.44

25 •••••••• 0 ••••• 85,575 .0039 .0191 423,798 .9795 3,782,947 44.21

30 •••• '0 •••••••• 83,944 .0044 .0219 415,125 .9763 3,359,149 40.02 35 ...... 82,106 .0052 .0255 405,299 .9722 2,944,024 35.86 40 ...... 80,014 .0061 .0302 394,022 .9660 2,538,725 31.73 45 .0 ...•.•..•... 77,595 .0077 .0379 380,623 .9550 2,144,704 27.64 50 ...... 74,655 .0108 .0523 363,504 .9378 1,764,080 23.63 55 .... , ...... 70,747 .0151 .0727 340,876 .9097 1,400,576 19.80 60 •••••••••••• ,0 65,603 .0231 .1093 310,088 .8652 1,059,700 16.15 65 ...... 58,432 .0356 .1633 268,300 .7968 749,612 12.83 70 ...... 48,888 .0574 .2508 213,785 .6969 481,312 9.85 75 ...... ,. 36,626 .0917 .3729 148,989 .4431 c 267,528 7.30 80 .0 ...•...•.••• 22,970 .1938 118,539 118,538 5.16

Males

0 ...... 100,000 .0918 .0862 93,882 .9021 a 5,647,390 56.47 b 1 • ••••••• 0 ••••• 91,379 .0089 .0350 357,189 .9717 5,553,508 60.78 5 .0 .•..••...••• 88,184 .0024 .0118 438,323 .9897 5,196,319 58.93 10 ...... 87,145 .0018 .0088 433,805 .9887 4,757,996 54.60

15 , •••••• 0 ••••• 0 86,377 .0028 .0138 428,910 .9834 4,324,192 50.06 20 " ...... 85,187 .0039 .0195 421,781 .9798 3,895,282 45.73 25 ...... 83,526 .0042 .0209 413,271 .9777 3,473,501 41.59

30 • •••••• 0 •••••• 81,782 .0048 .0238 404,055 .9737 3,060,231 37.42 35 ...... 79,839 .0059 .0289 393,429 .9669 2,656,176 33.27 40 ...... 77,532 .0076 .0375 380,399 .9563 2,262,747 29.19 45 • .•....••... 0. 74,627 .0103 .0501 363,783 .9400 1,882,348 25.22 50 ...... 70,886 .0146 .0704 341,950 .9156 1,518,565 21.42 55 .0 ..•••...••.. 65,894 .0209 .0994 313,100 .8794 1,176,615 17.86 60 ·...... 59,346 .0311 .1442 275,333 .8270 863,514 14.55 65 ., ...... 50,788 .0461 .2066 227,706 .7531 588,181 11.58

70 ••• 0 •••••••••• 40,295 .0700 .2978 171,474 .6505 360,475 8.95 75 ...... 28,295 .1073 .4231 111,545 .4098c 189,000 6.68 80 ...... 16,323 .2107 77,456 77,456 4.75

a Pro,;ortion surviving from birth to 0-4. b sLs sLo. c Tso/T7s.

89 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVIlL OF MORTALITY (continued) LEVEL 19

Age x Ix nmx nqx nLx &Lx+& Tx °ex &Lx

Females

a 0 .0 •••••••••••• 100,000 .0520 .0499 96,004 .9428 6,500,000 65.00 b I ...... " 95,006 .0048 .0190 375,407 .9851 6,403,996 67.41 5 ...... ". 93,201 .0014 .0069 464,405 .9939 6,028,589 64.68 10 .0 ...••••.•••• 92,561 .0011 .0054 461,567 .9932 5,564,184 60.11 15 ...... 92,065 .0017 .0082 458,440 .9904 5,102,618 55.42 20 ...... 91,311 .0022 .0111 454,021 .9879 4,644,178 50.86 25 .0 •••••••••••• 90,298 .0026 .0130 448,547 .9859 4,190,157 46.40 30 ...... 89,121 .0031 .0152 442,221 .9833 3,741,610 41.98 35 .0 .•••••••••.• 87,767 .0037 .0183 434,829 .9795 3,299,389 37.59 40 ...... 86,164 .0046 .0228 425,918 .9735 2,864,560 33.25 45 ...... 84,203 .0062 .0303 414,639 .9635 2,438,642 28,96 50 ...... 81,653 .0088 .0430 399,491 .9480 2,024,003 24.79

55 .0 •••••••••••• 78,144 .0127 .0615 378,702 .9228 1,624,511 20.79 60 ...... 73,337 .0197 .0940 349,452 .8815 1,245,809 16.99 65 ...... 66,444 .0314 .1455 308,047 .8158 896,357 13.49 70 ...... 56,775 .0518 .2294 251,317 .7182 588,310 10.36 75 ...... 43,751 .0848 .3499 180,485 .4664c 336,993 7.70 80 ...... 28,442 .1817 156,509 156,509 5.50

Males

a 0 .0 •••••••••••• 100,000 .0661 .0629 95,117 .93oo 6,122,821 61.23 1 ...... 93,713 .0053 .0210 369,858 .9826b 6,027,704 64.32 5 ...... 91,744 .0016 .0081 456,871 .9929 5,657,846 61.67 10 ...... 91,004 .0012 .0062 453,620 .9918 5,200,975 57.15 15 ...... 90,444 .0020 .0102 449,920 .9878 4,747,355 52.49 20 ...... 89,524 .0029 .0143 444,411 .9853 4,297,435 48.00 25 ...... 88,240 .0030 .0151 437,879 .9840 3,853,024 43.67 30 ...... 86,912 .0034 .0171 430,851 .9809 3,415,145 39.29 35 ...... 85,429 .0043 .0211 422,637 .9753 2,984,293 34.93 40 ...... 83,626 .0058 .0284 412,201 .9658 2,561,657 30.63 45 ...... 81,255 .0082 .0402 398,115 .9508 2,149,456 26.45 50 ...... 77,991 .0121 .0587 378,509 .9277 1,751,341 22.46

55 ••••••••••••• 0 73,412 .0181 .0867 351,159 .8933 1,372,832 18.70 60 ...... 67,051 .0275 .1286 313,696 .8433 1,021,673 15.24 65 ...... 58,427 .0417 .1889 264,546 .7713 707,977 12.12 70 '" ...... 47,392 .0645 .2779 204,036 .6705 443,431 9.36 75 ...... 34,223 .1004 .4011 136,796 .4286c 239,394 7.00 80 ...... 20,495 .1998 102,599 102,599 5.01

a ~proh0rtion surviving from birth to 0-4. b liLli liLo. c Tso/T7li.

90 91 TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 23

Age x Ix nmx n'b: nLx 5Lx+5 Tx Oex 5Lx

Females

0 ...... , ..... 100,000 .0154 .0152 98,629 .9840" 7,500,000 75.00 1 ...... 98,484 .0006 .0024 393,346 .9979b 7,401,371 75.15 5 ...... 98,248 .0003 .0013 490,926 .9988 7,008,025 71.33 10 ...... 98,123 .0002 .0011 490,355 .9986 6,517,099 66.42 15 ...... 98,019 .0004 .0018 489,662 .9979 6,026,745 61.49 20 ...... 97,846 .0005 .0025 488,610 .9971 5,537,082 56.59 25 '0' ••••••••••• 97,598 .0006 .0032 487,207 .9963 5,048,472 51.73 30 ...... 97,285 .0008 .0041 485,417 .9950 4,561,265 46.89 35 .0 ...... 96,882 .0012 .0058 483,004 .9927 4,075,848 42.07 40 '0' ••••••••••• 96,319 .0018 .0089 479,464 .9883 3,592,844 37.30 45 ...... 95,466 .0029 .0146 473,851 .9813 3,113,380 32.61 50 ...... '" 94,074 .0046 .0228 465,006 .9707 2,639,529 28.06 55 ...... 91,928 .0073 .0360 451,369 .9531 2,174,523 23.66 60 ...... 88,619 .0120 .0582 430,193 .9211 1,723,154 19.44 65 ...... 83,458 .0212 .1008 396,265 .8652 1,292,961 15.49 70 ...... 75,048 .0378 .1727 342,839 .7759 896,696 11.95 75 ...... 62,087 .0668 .2862 266,015 .5197" 553,857 8.92 80 ...... 44,318 .1540 287,843 287,842 6.50

Males

0 ...... 100,000 .0219 .0214 98,080 .9774" 7,118,759 71.19 1 ...... 97,856 .0009 .0034 390,618 .9967b 7,020,680 71.75 5 ...... 97,521 .0005 .0022 487,062 .9979 6,630,062 67.99 10 ...... 97,303 .0004 .0019 486,055 .9972 6,143,000 63.13 15 ...... 97,119 .0007 .0037 484,692 .9956 5,656,944 58.25 20 ...... 96,758 .0010 .0051 482,547 .9949 5,172,252 53.46 25 ...... 96,261 .0010 .0051 480,084 .9946 4,689,705 48.72 30 ...... 95,773 .0011 .0057 477,500 .9934 4,209,621 43.95 35 ...... 95,227 .0015 .0075 474,348 .9905 3,732,121 39.19 40 ...... 94,512 .0023 .0116 469,820 .9843 3,257,773 34.47 45 ...... 93,416 .0040 .0199 462,430 .9734 2,787,952 29.85 50 ...... 91,556 .0068 .0334 450,149 .9553 2,325,522 25.40 55 ...... 88,503 .0116 .0564 430,029 .9273 1,875,373 21.19 60 ...... 83,508 .0188 .0900 398,759 .8849 1,445,344 17.31 65 ...... 75,995 .0307 .1427 352,874 .8200 1,046,585 13.77 70 ...... 65,154 .0503 .2235 289,364 .7259 693,711 10.65 75 ...... 50,591 .0817 .3394 210,034 .4806" 404,347 7.99 80 ...... 33,422 .1720 194,314 194,313 5.81

" Proh0rtion surviving from birth to 0-4. b liLu liLo. " Tso/T7li.

92 TABLE 1.2. VALUES OF THE FUNCTION Ix (SURVIVORS TO AGE X) FOR X = 1,2, 3 AND 5 IN "WEST" MODEL LIFE TABLES AT VARIOUS LEVELS OF MORTALITY, FOR FEMALES, FOR MALES, AND FOR BOTH SEXES ASSUMING THAT THE SEX RATIO AT BIRTH IS 1.05 (/0 = 100,000)

Level h 12 18 15 Level 11 12 18 15

Females Males (continued)

1 '" .. , ...... , .. 63,483 55,000 51,199 46,883 13 ...... 86,058 82,912 81,534 79,961 3 ...... 69,481 61,829 58,399 54,506 15 ...... 88,864 86,523 85,498 84,327 5 ...... 74,427 67,671 64,643 61,205 17 ...... 91,379 89,790 89,056 88,184 7 ...... 78,614 72,765 70,145 67,169 19 ...... 93,713 92,796 92,338 91,774 9 ...... 82,226 7,1,271 75,051 72,530 21 . " ...... 95,909 95,508 95,285 94,989 11 ...... , .. 85,388 81,300 79,468 77,389 23 ...... 97,856 97,719 97,636 97,521 13 ...... 88,169 84,939 83,492 81,848 15 ...... 90,661 88,364 87,324 86,127 17 ...... 92,934 91,419 90,709 89,854 Both sexes 19 ...... 95,006 94,143 93,724 93,201 21 ...... 96,907 96,559 96,385 96,160 1 ...... 60,722 52,597 48,996 44,897 23 ...... 98,484 98,377 98,321 98,248 3 ...... 67,118 59,709 56,425 52,688 5 ...... 72,392 65,798 62,877 59,551 7 ...... 76,857 71,112 68,567 65,670 Males 9 ...... 80,709 75,813 73,646 71,177 11 ...... 84,080 80,019 78,220 76,173 1 ...... 58,093 50,308 46,898 43,005 13 ...... 87,088 83,901 82,489 80,881 3 ...... 64,868 57,690 54,546 50,957 15 ...... 89,740 87,421 86,389 85,205 5 ...... 70,454 64,015 61,195 57,976 17 ...... 92,137 90,584 89,862 88,999 7 ...... 75,183 69,537 67,064 64,242 19 ...... 94,144 93,453 93,011 92,455 9 ...... 79,263 74,425 72,307 69,888 21 ...... 96,396 96,020 95,822 95,560 11 ...... 82,835 78,800 77,032 75,015 23 ...... 98,162 98,040 97,970 97,876

93 TABLE 1.3. TEN-YEAR SURVIVAL RATES IN "WEST" MODEL LIPB TABLES AT VARIOUS LEVELS OF MORTALITY

Level Level Level Level Level Level Level Level Level Level Level Level Age I 3 5 7 9 II 13 15 17 19 21 23

Females

0-4 to 10-14 ...... 7370 .7847 .8232 .8552 .8826 .9063 .9273 .9478 .9649 .9791 .9906 .9967 5-9 to 15-19 ...... 8736 .8946 .9121 .9271 .9401 .9516 .9618 .9715 .9798 .9872 .9934 .9974 10-14 to 20-24 ...... 8576 .8808 .9001 .9167 .9312 .9439 .9552 .9655 .9752 .9837 .9912 .9964 15-19 to 25-29 ...... 8286 .8560 .8789 .8986 .9158 .9309 .9444 .9562 .9680 .9784 .9878 .9950 20-24 to 30-34 ...... 8050 .8358 .8616 .8838 .9032 .9203 .9355 .9488 .9621 .9740 .9848 .9935 25-29 to 35-39 ...... 7841 .8178 .8461 .8704 .8917 .9106 .9273 .9419 .9563 .9694 .9814 .9914 30-34 to 40-44 ...... 7660 .8018 .8319 .8578 .8805 .9007 .9186 .9340 .9492 .9631 .9762 .9877 35-39 to 45-49 ...... 7518 .7884 .8193 .8459 .8692 .8899 .9083 .9239 .9391 .9536 .9673 .9810 40-44 to 50-54 ...... 7267 .7649 .7972 .8251 .8496 .8713 .8906 .9067 .9225 .9380 .9528 .9698 45-49 to 55-59 ...... 6750 .7177 .7539 .7853 .8130 .8377 .8597 .8777 .8956 .9133 .9308 .9525 50-54 to 60-64 ...... 5920 .6415 .6840 .7216 .7541 .7836 .8101 .8315 .8530 .8747 .8964 .9251 55-59 to 65-69 ...... 4820 .5384 .5875 .6308 .6696 .7045 .7361 .7614 .7871 .8134 .8402 .8779 60-64 to 70-74 ...... 3617 .4202 .4721 .5185 .5605 .5987 .6335 .6609 .6894 .7192 .7501 .7969 65-69 to 75-79 ...... 2362 .2907 .3401 .3852 .4264 .4644 .4992 .5265 .5553 .5859 .6183 .6713 70+ to 80+ .. , ...... 1008 .1230 .1425 .1605 .1789 .1951 .2128 .2286 .2463 .2660 .2879 .3210

Males

0-4 to 10-14...... 7402 .7877 .8257 .8570 .8835 .9064 .9281 .9462 .9617 .9756 .9871 .9946 5-9 to 15-19 ...... 8844 .9064 .9214 .9341 .9452 .9549 .9642 .9718 .9185 .9848 .9904 .9951 10-14 to 20-24 ...... 8697 .8901 .9071 .9217 .9343 .9455 .9556 .9644 .9723 .9197 .9865 .9928 15-19 to 25-29 ...... 8318 .8578 .8796 .8983 .9146 .9290 .9418 .9532 .9635 .9733 .9822 .9905 20-24 to 30-34 ...... 8021 .8327 .8584 .8805 .8998 .9169 .9320 .9456 .9580 .9695 .9800 .9895 25-29 to 35-39 ...... 7744 .8091 .8383 .8634 .8854 .9049 .9223 .9378 .9520 .9652 .9771 .9881 30-34 to 40-44 ...... 7392 .7785 .8116 .8402 .8654 .8877 .9077 .9253 .9415 .9567 .9706 .9839 35-39 to 45-49 ...... 6993 .7428 .7797 .8116 .8397 .8648 .8873 .9066 .9246 .9420 .9579 .9749 40-44 to 50-54 ...... 6535 .7005 .7406 .1755 .8063 .8337 .8587 .8793 .8989 .9183 .9365 .9581 45-49 to 55-59 ...... 5961 .6464 .6895 .7272 .7605 .7903 .8178 .8394 .8607 .8820 .9028 .9299 50-54 to 60-64 ...... 5228 .5757 .6214 .6616 .6974 .7295 .7592 .7822 .8052 .8288 .8523 .8858 55-59 to 65-69 ...... 4266 .4817 .5299 .5727 .6109 .6456 .6776 .7022 .7213 .7533 .7801 .8206 60-64 to 70-74 ...... 3177 .3720 .4202 .4634 .5026 .5382 .5713 .5966 .6228 .6504 .6794 .7257 65-69 to 15-79 ...... 2052 .2536 .2976 .3378 .3747 .4086 .4402 .4645 .4899 .5171 .5463 .5952 70+ to 80+ ...... 0877 .1085 .1263 .1424 .1575 .1724 .1874 .2004 .2149 .2314 .2500 .2801

94 Annex II MODEL STABLE POPULATIONS

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY LEVEL 1 Females (Oeo = 20.00 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0292 .0335 .0381 .0430 .0482 .0535 .0590 .0647 .0704 .0763 .0822 .0882 .0942 1-4 ...... 0824 .0935 .1051 .1171 .1294 .1420 .1547 .1674 .1800 .1926 .2049 .2171 .2290 5-9 ...... 0926 .1027 .1129 .1231 .1330 .1426 .1519 .1607 .1690 .1768 .1840 .1906 .1965 10-14 ...... 0910 .0984 .1055 .1122 .1182 .1237 .1285 .1326 .1360 .1387 .1408 .1422 .1430 15-19 ...... 0894 .0943 .0986 .1022 .1051 .1073 .1087 .1094 .1094 .1088 .1077 .1061 .1041 20-24 ...... 0862 .0887 .0905 .0915 .0917 .0913 .0902 .0885 .0864 .0838 .0809 .0778 .0744 25-29 ...... 0819 .0822 .0817 .0806 .0788 .0765 .0737 .0706 .0672 .0635 .0598 .0561 .0523 30-34 ... , ...... 0767 .0751 .0729 .0701 .0668 .0633 .0594 .0555 .0515 .0475 .0437 .0399 .0363 35-39 ..... " ...... 0709 .0677 .0641 .0601 .0559 .0516 .0473 .0431 .0390 .0351 .0314 .0280 .0249 40-44 ...... 0649 .0605 .0558 .0510 .0463 .0417 .0373 .0331 .0292 .0257 .0224 .0195 .0169 45-49 ...... 0589 .0535 .0482 .0430 .0380 .0334 .0291 .0252 .0217 .0186 .0158 .0134 .0113 50-54 ...... 0522 .0462 .0406 .0353 .0305 .0261 .0222 .0187 .0157 .0131 .0109 .0090 .0074 55-59 . " ...... 0440 .0380 .0325 .0276 .0232 .0194 .0161 .0133 .0109 .0088 .0072 .0058 .0045 60-64 ...... 0341 .0288 .0240 .0199 .0163 .0133 .0108 .0086 .0069 .0055 .0043 .0034 .0027 65-69 ...... 0234 .0193 .0157 .0127 .0101 .0081 .0064 .0050 .0039 .0030 .0023 .0018 .0014 70-74 ...... 0136 .0109 .0087 .0068 .0053 .0041 .0032 .0024 .0018 .0014 .0011 .0008 .0006 I,Q 75-79 ... , ...... , ...... 0061 .0048 .0037 .0028 .0022 .0016 .0012 .0009 .0007 .0005 .0004 .0003 .0002 VI 80+ ...... 0024 .0018 .0014 .0010 .0008 .0006 .0004 .0003 .0002 .0002 .0001 .0001 .0001

Age Proportion under given age 1 ...... 0292 .0335 .0381 .0430 .0482 .0535 .0590 .0647 .0704 .0763 .0822 .0882 .0942 5 ...... 1115 .1269 .1432 .1601 .1776 .1955 .2137 .2320 .2504 .2688 .2871 .3053 .3232 10 ...... 2041 .2297 .2561 .2832 .3106 .3381 .3656 .3928 .4195 .4456 .4711 .4958 .5197 15 ...... 2951 .3281 .3617 .3953 .4288 .4618 .4941 .5253 .5555 .5843 .6119 .6380 .6627 20 ...... 3845 .4225 .4603 .4976 .5339 .5691 .6027 .6347 .6649 .6932 .7196 .7441 .7669 25 ...... 4707 .5112 .5508 .5891 .6257 .6604 .6929 .7232 .7512 .7770 .8005 .8219 .8413 30 ...... 5526 .5934 .6325 .6697 .7045 .7369 .7666 .7938 .8184 .8405 .8603 .8780 .8936 35 ...... 6293 .6685 .7054 .7397 .7713 .8001 .8261 .8493 .8699 .8881 .9040 .9179 .9299 40 ...... 7003 .7362 .7695 .7998 .8272 .8517 .8734 .8924 .9089 .9232 .9355 .9459 .9548 45 ...... 7652 .7967 .8253 .8509 .8736 .8934 .9107 .9255 .9382 .9489 .9579 .9654 .9717 50 ...... 8241 .8502 .8735 .8939 .9116 .9268 .9398 .9507 .9599 .9675 .9737 .9789 .9830 55 ...... 8763 .8964 .9140 .9292 .9420 .9529 .9620 .9695 .9756 .9806 .9847 .9879 .9905 60 ...... 9203 .9344 .9465 .9568 .9653 .9723 .9781 .9827 .9865 .9895 .9918 .9937 .9951 65 ...... 9544 .9632 .9705 .9766 .9816 .9856 .9888 .9914 .9934 .9949 .9962 .9971 .9978 Parameter ofstable populations Birth rate ...... 0380 .0438 .0500 .0566 .0635 .0707 .0782 .0859 .0937 .1018 .1100 .1182 .1266 Death rate ...... 0480 .0488 .0500 .0516 .0535 .0557 .0582 .0609 .0637 .0668 .0700 .0732 .0766 GRR (27) ...... 2.33 2.66 3.03 3.44 3.93 4.46 5.07 5.75 6.52 7.39 8.36 9.45 10.68 GRR (29) ...... 2.39 2.75 3.16 3.64 4.18 4.79 5.49 6.28 7.18 8.20 9.36 10.67 12.16 GRR (31) ...... 2.45 2.85 3.31 3.84 4.45 5.16 5.96 6.89 7.96 9.18 10.57 12.17 13.99 GRR (33) ...... 2.51 2.95 3.47 4.07 4.77 5.58 6.52 7.62 8.90 10.38 12.09 14.07 16.36 Average age ...... , 29.2 27.3 25.5 23.8 22.3 20.8 19.4 18.2 17.0 16.0 15.0 14.1 13.3 Births/population 15-44 ...... 081 .093 .108 .124 .143 .164 .188 .215 .245 .279 .318 .361 .410 TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 3 Females (Oeo = 25.00 years)

A1/IIUiIlrat« ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0241 .0279 .0321 .0365 .0412 .0461 .0511 .0564 .0617 .0672 .0727 .0783 .0840 1-4 ...... 0733 .0840 .0954 .1072 .1194 .1320 .1447 .1575 .1703 .1831 .1957 .2082 .2204 5-9 ...... 0850 .0953 .1057 .1162 .1266 .1367 .1466 .1560 .1650 .1734 .1812 .1885 .1951 10-14 ...... 0846 .0924 .1000 .1072 .1139 .1200 .1254 .1302 .1343 .1377 .1403 .1423 .1437 15-19 ...... 0841 .0896 .0946 .0989 .1024 .1053 .1074 .1087 .1093 .1093 .1087 .1075 .1059 20-24 ...... 0823 .0856 .0881 .0898 .0908 .0910 .0905 .0893 .0876 .0854 .0829 .0799 .0768 25-29 ...... 0795 .0806 .0809 .0805 .0793 .0776 .0752 .0725 .0693 .0659 .0624 .0587 .0550 30-34 ...... 0760 .0752 .0736 .0714 .0686 .0654 .0619 .0581 .0543 .0503 .0464 .0426 .0389 35-39 ...... 0719 .0693 .0662 .0626 .0587 .0546 .0504 .0461 .0420 .0380 .0342 .0306 .0273 40-44 ...... 0674 .0634 .0590 .0544 .0498 .0452 .0406 .0363 .0322 .0284 .0249 .0218 .0189 45-49 ...... 0626 .0575 .0522 .0470 .0419 .0370 .0325 .0283 .0245 .0211 .0181 .0154 .0130 50-54 ...... '" .0570 .0510 .0451 .0396 .0345 .0297 .0254 .0216 .0183 .0153 .0128 .0106 .0088 55-59 ...... 0497 .0434 .0375 .0321 .0272 .0229 .0191 .0158 .0130 .0107 .0087 .0070 .0057 60-64 ...... 0404 .0344 .0290 .0242 .0200 .0164 .0134 .0108 .0087 .0069 .0055 .0043 .0034 65-69 ...... 0296 .0245 .0202 .0164 .0133 .0106 .0084 .0066 .0052 .0041 .0031 .0024 .0019 70-74 ...... 0188 .0152 .0122 .0097 .0076 .0059 .0046 .0035 .0027 .0021 .0015 .0012 .0009 75-79 ...... 0095 .0075 .0059 .0045 .0035 .0027 .0020 .0015 .0011 .0008 .0006 .0004 .0003 \0 80+ ...... 0043 .0033 .0025 .0019 .0014 .0011 .0008 .0006 .0004 .0003 .0002 .0001 .0001 0\ Age Proportion under given age I ...... 0241 .0279 .0321 .0365 .0412 .0461 .0511 .0564 .0617 .0672 .0727 .0783 .0840 5 ...... 0974 .1120 .1274 .1437 .1606 .1780 .1958 .2139 .2320 .2503 .2685 .2865 .3044 10 ...... 1824 .2072 .2331 .2599 .2872 .3147 .3424 .3699 .3970 .4237 .4497 .4750 .4995 15 ...... 2670 .2996 .3331 .3670 .4010 .4347 .4678 .5001 .5313 .5613 .5900 .6173 .6432 20 ...... 3511 .3893 .4277 .4659 .5035 .5400 .5752 .6088 .6406 .6706 .6987 .7248 .7491 25 ...... 4334 .4748 .5158 .5557 .5942 .6310 .6656 .6981 .7283 .7561 .7815 .8048 .8259 30 ...... 5129 .5555 .5967 .6362 .6736 .7085 .7409 .7706 .7976 .8220 .8439 .8635 .8808 35 ...... 5890 .6306 .6703 .7076 .7422 .7739 .8028 .8287 .8518 .8723 .8903 .9061 .9197 40 ...... 6608 .7000 .7365 .7702 .8009 .8285 .8531 .8748 .8938 .9103 .9245 .9367 .9470 45 ...... 7282 .7633 .7955 .8247 .8507 .8737 .8938 .9112 .9261 .9387 .9494 .9584 .9659 50 ...... 7908 .8208 .8477 .8716 .8926 .9107 .9263 .9395 .9506 .9598 .9675 .9738 .9790 55 ...... 8478 .8717 .8929 .9112 .9270 .9405 .9517 .9611 .9688 .9752 .9803 .9844 .9878 60 ...... 8975 .9151 .9303 .9433 .9542 .9633 .9708 .9770 .9819 .9858 .9890 .9915 .9934 65 ...... 9379 .9495 .9593 .9675 .9742 .9798 .9842 .9878 .9906 .9928 .9945 .9958 .9968 Parameter ofstable populations Birth rate ...... 0299 .0347 .0400 .0456 .0516 .0579 .0644 .0712 .0782 .0853 .0926 .0999 .1074 Death rate ...... 0399 .0397 .0400 .0406 .0416 .0429 .0444 .0462 .0482 .0503 .0526 .0549 .0574 GRR (27) ...... 1.88 2.15 2.46 2.80 3.19 3.62 4.12 4.68 5.31 6.01 6.81 7.70 8.71 GRR (29) ...... 1.92 2.21 2.54 2.93 3.36 3.86 4.43 5.07 5.81 6.64 7.58 8.65 9.86 GRR (31) ...... 1.95 2.27 2.64 3.07 3.56 4.13 4.78 5.53 6.39 7.34 8.50 9.79 11.26 GRR (33) ...... 1.99 2.34 2.75 3.22 3.78 4.43 5.18 6.06 7.08 8.26 9.63 11.22 13.06 Average age ...... 31.3 29.2 27.3 25.5 23.8 22.2 20.7 19.3 18.0 16.9 15.8 14.9 14.0 Births/population 15-44 ...... 065 .075 .087 .100 .115 .132 .151 .173 .198 .226 .258 .293 .333 ~...,..",~. TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) <, LEVEL 5 Females (Oco = 30.00 years)

AIUfUDl rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0205 .0240 .0278 .0319 .0362 .0408 .0455 .0505 .0555 .0607 .0659 .0713 .0767 1-4 ...... 0662 .0766 .0876 .0992 .1113 .1238 .1365 .1494 .1623 .1753 .1881 .2008 .2132 5-9 ...... 0787 .0889 .0995 .1101 .1208 .1314 .1416 .1515 .1610 .1700 .1783 .1861 .1933 10-14 ...... 0790 .0871 .0950 .1026 .1098 .1164 .1224 .1277 .1324 .1363 .1395 .1420 .1438 15-19 ...... 0793 .0853 .0907 .0956 .0997 .1031 .1058 .1077 .1088 .1092 .1090 .1083 .1069 20-24 ...... 0786 .0824 .0855 .0879 .0894 .0902 .0902 .0896 .0883 .0864 .0842 .0815 .0785 25-29 ...... 0770 .0788 .0797 .0799 .0793 .0780 .0761 .0737 .0708 .0677 .0642 .0607 .0570 30-34 ...... 0748 .0746 .0737 .0720 .0697 .0669 .0636 .0601 .0563 .0525 .0486 .0448 .0410 35-39 ...... 0720 .0701 .0675 .0643 .0607 .0568 .0527 .0486 .0444 .0403 .0364 .0327 .0293 40-44 ...... 0688 .0653 .0613 .0570 .0525 .0479 .0433 .0389 .0347 .0308 .0271 .0237 .0207 45-49 ...... 0652 .0603 .0553 .0501 .0450 .0401 .0354 .0310 .0269 .0233 .0200 .0171 .0145 50-54 ...... 0606 .0547 .0489 .0432 .0378 .0329 .0283 .0242 .0205 .0173 .0145 .0121 .0100 55-59 ...... 0543 .0478 .0417 .0359 .0307 .0260 .0218 .0182 .0151 .0124 .0101 .0082 .0066 60-64 ...... 0458 .0393 .0334 .0281 .0234 .0193 .0158 .0129 .0104 .0083 .0066 .0053 .0042 65-69 ...... 0353 .0295 .0245 .0201 .0163 .0131 .0105 .0083 .0066 .0051 .0040 .0031 .0024 70-74 ...... 0239 .0195 .0158 .0126 .0100 .0079 .0061 .0047 .0036 .0028 .0021 .0016 .0012 75-79 ...... 0133 .0106 .0083 .0065 .0050 .0038 .0029 .0022 .0017 .0012 .0009 .0007 .0005 \0 80+ ...... 0068 .0052 .0040 .0030 .0023 .0017 .0013 .0009 .0007 .0005 .0003 .0002 .0002 -...I Age Proportion under given age 1 ...... 0205 .0240 .0278 .0319 .0362 .0408 .0455 .0505 .0555 .0607 .0659 .0713 .0767 5 ...... 0867 .1006 .1154 .1311 .1475 .1646 .1820 .1998 .2178 .2359 .2540 .2720 .2899 10 ...... 1654 .1895 .2148 .2412 .2683 .2959 .3237 .3514 .3789 .4059 .4324 .4582 .4832 15 ...... 2444 .2765 .3098 .3438 .3781 .4123 .4461 .4791 .5112 .5422 .5718 .6001 .6270 20 ...... 3237 .3618 .4006 .4394 .4778 .5154 .5518 .5868 .6200 .6514 .6809 .7084 .7339 25 ...... 4022 .4442 .4861 .5272 .5672 .6056 .6421 .6763 .7083 .7379 .7650 .7899 .8124 30 ...... 4792 .5230 .5658 .6071 .6465 .6836 .7182 .7500 .7791 .8055 .8293 .8505 .8694 35 ...... 5541 .5976 .6395 .6791 .7162 .7505 .7818 .8101 .8355 .8580 .8779 .8953 .9105 40 ...... 6261 .6677 .7069 .7434 .7769 .8073 .8345 .8587 .8799 .8983 .9143 .9280 .9397 45 ...... 6949 .7329 .7682 .8004 .8294 .8552 .8779 .8976 .9146 .9291 .9414 .9518 .9604 50 ...... 7601 .7933 .8235 .8505 .8744 .8953 .9132 .9286 .9415 .9524 .9614 .9689 .9750 55 ...... 8207 .8480 .8723 .8937 .9122 .9281 .9415 .9528 .9621 .9697 .9759 .9809 .9850 60 ...... 8750 .8958 .9140 .9297 .9429 .9541 .9634 .9709 .9771 .9821 .9860 .9892 .9916 65 ...... 9208 .9352 .9474 .9578 .9664 .9735 .9792 .9838 .9875 .9904 .9927 .9944 .9958 Parameter ofstable populations Birth rate ... " .. " ...... 0245 .0281 .0333 .0383 .0437 .0493 .0552 .0613 .0676 .0741 .0807 .0874 .0943 Death rate ...... 0345 .0337 .0333 .0333 .0337 .0343 .0352 .0363 .0376 .0391 .0407 .0424 .0443 GRR (27) ...... 1.60 1.82 2.08 2.37 2.70 3.08 3.50 3.98 4.51 5.12 5.80 6.56 7.42 GRR (29) ...... 1.61 1.86 2.15 2.47 2.84 3.26 3.74 4.29 4.91 5.62 6.42 7.33 8.36 GRR (31) ...... 1.63 1.90 2.21 2.57 2.99 3.47 4.01 4.65 5.37 6.21 7.16 8.25 9.50 GRR(33) ...... 1.65 1.94 2.29 2.68 3.15 3.69 4.33 5.06 5.92 6.91 8.06 9.39 10.94 Average age ...... 33.1 31.0 28.9 26.9 25.1 23.4 21.8 20.3 19.0 17.7 16.6 15.5 14.6 Births/population 15-44 ...... 054 .063 .073 .084 .097 .111 .128 .146 .168 .191 .218 .249 .283 "--

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 7 Females (Oeo = 35.00 years)

A1IIlUQl rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0179 .0211 .0246 .0284 .0325 .0368 .0413 .0460 .0508 .0557 .0608 .0659 .0711 1-4 ...... 0605 .0705 .0812 .0926 .1046 .1170 .1297 .1426 .1556 .1686 .1816 .1944 .2071 5-9 ...... 0732 .0834 .0940 .1048 .1157 .1265 .1371 .1474 .1573 .1667 .1755 .1837 .1914 10-14 ...... 0741 .0823 .0905 .0984 .1060 .1130 .1195 .1253 .1304 .1347 .1384 .1413 .1435 15-19 ...... 0750 .0813 .0871 .0924 .0970 .1009 .1041 .1064 .1080 .1089 .1091 .1086 .1076 20-24 ...... 0751 .0794 .0830 .0858 .0879 .0892 .0897 .0894 .0885 .0870 .0850 .0826 .0798 25-29 ...... 0745 .0768 .0783 .0790 .0789 .0781 .0766 .0745 .0719 .0690 .0657 .0622 .0586 30-34 ...... 0733 .0737 .0733 .0721 .0703 .0678 .0649 .0616 .0580 .0542 .0504 .0465 .0428 35-39 ...... , ...... 0716 .0703 .0681 .0654 .0621 .0585 .0546 .0505 .0464 .0423 .0383 .0345 .0310 40-44 ...... 0695 .0665 .0629 .0589 .0546 .0501 .0456 .0411 .0368 .0328 .0290 .0255 .0223 45-49 ...... 0670 .0625 .0576 .0526 .0476 .0426 .0378 .0333 .0291 .0252 .0217 .0186 .0159 50-54 ...... 0634 .0577 .0519 .0462 .0407 .0356 .0308 .0264 .0225 .0191 .0160 .0134 .0111 55-59 ...... 0581 .0516 .0453 .0393 .0338 .0288 .0243 .0203 .0169 .0140 .0114 .0093 .0076 60-64 ...... 0505 .0437 .0374 .0317 .0266 .0221 .0182 .0148 .0120 .0097 .0077 .0062 .0049 65-69 ...... 0405 .0342 .0286 .0236 .0193 .0156 .0125 .0100 .0079 .0062 .0048 .0038 .0029 70-74 ...... 0290 .0238 .0194 .0156 .0125 .0099 .0077 .0060 .0046 .0035 .0027 .0020 .0015 75-79 ...... 0172 .0138 .0110 .0086 .0067 .0052 .0040 .0030 .0023 .0017 .0012 .0009 .0007 \Q 80+ ...... 0097 .0075 .0058 .0044 .0034 .0025 .0019 .0014 .0010 .0007 .0005 .0004 .0003 00 Age Proportion under given age 1 ...... 0179 .0211 .0246 .0284 .0325 .0368 .0413 .0460 .0508 .0557 .0608 .0659 .0711 5 ...... 0783 .0915 .1058 .1210 .1370 .1537 .1709 .1885 .2064 .2244 .2424 .2604 .2782 10 ...... 1515 .1749 .1998 .2258 .2527 .2802 .3081 .3359 .3637 .3910 .4179 .4441 .4696 15 ...... 2256 .2573 .2903 .3242 .3587 .3932 .4275 .4612 .4940 .5258 .5563 .5854 .6131 20 ...... 3006 .3385 .3774 .4166 .4557 .4942 .5316 .5677 .6021 .6347 .6653 .6940 .7207 25 ...... 3757 .4179 .4604 .5025 .5436 .5834 .6213 .6571 .6906 .7217 .7504 .7766 .8005 30 ...... 4501 .4947 .5386 .5814 .6225 .6614 .6978 .7316 .7625 .7907 .8161 .8389 .8592 35 ...... 5235 .5684 .6120 .6536 .6928 .7293 .7627 .7931 .8205 .8449 .8665 .8854 .9019 40 ...... 5951 .6386 .6801 .7190 .7549 .7877 .8173 .8436 .8669 .8872 .9048 .9199 .9329 45 ...... 6646 .7051 .7430 .7779 .8095 .8378 .8629 .8848 .9037 .9200 .9338 .9454 .9552 50 ...... 7316 .7676 .8006 .8305 .8571 .8804 .9007 .9180 .9328 .9452 .9555 .9640 .9711 55 ...... 7950 .8253 .8525 .8767 .8978 .9160 .9314 .9445 .9553 .9642 .9715 .9774 .9822 60 ...... 8531 .8769 .8978 .9160 .9316 .9448 .9557 .9648 .9722 .9782 .9830 .9868 .9898 65 ...... 9036 .9206 .9352 .9477 .9582 .9668 .9739 .9796 .9842 .9879 .9907 .9920 .9946 Parameter ofstable populations Birth rate ...... 0206 .0244 .0286 .0331 .0375 .0430 .0484 .0541 .0599 .0659 .0720 .0783 .0847 Death rate ...... 0306 .0294 .0286 .0281 .0279 .0280 .0284 .0291 .0299 .0309 .0320 .0333 .0347 GRR (27) ...... 1.34 1.59 1.82 2.08 2.37 2.70 3.07 3.49 3.96 4.49 5.09 5.76 6.52 GRR (29) ...... 1.40 1.62 1.87 2.15 2.47 2.84 3.26 3.74 4.29 4.91 5.61 6.41 7.31 GRR (31) ...... 1.41 1.64 1.91 2.23 2.59 3.00 3.48 4.03 4.67 5.39 6.23 7.18 8.27 GRR (33) ...... 1.42 1.67 1.97 2.31 2.71 3.18 3.73 4.37 5.11 5.97 6.97 8.13 9.47 Average age ...... 34.7 32.5 30.3 28.3 26.3 24.5 22.8 21.2 19.8 18.4 17.2 16.1 15.1 Births/population 15-44 ...... 047 .055 .063 .073 .084 .097 .111 .128 .146 .167 .191 .217 .247

.._-'--_. -_."-,._. TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTAUTY (continued) LEVEL 9 Females (Oeo = 40.00 years)

--Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0158 .0188 .0221 .0257 .0295 .0336 .0379 .0424 .0471 .0518 .0567 .0617 .0667 1-4 ...... 0557 .0653 .0758 .0870 .0988 .1111 .1238 .1367 .1498 .1629 .1760 .1890 .2018 5-9 ...... 0684 .0786 .0891 .1000 .1111 .1221 .1330 .1436 .1538 .1636 .1728 .1814 .1894 10-14 ...... 0698 .0781 .0864 .0946 .1024 .1098 .1167 .1229 .1284 .1332 .1372 .1405 .1431 15-19 ...... 0711 .0776 .0838 .0894 .0945 .0988 .1024 .1051 .1071 .1084 .1089 .1087 .1080 20-24 ...... 0718 .0765 .0805 .0838 .0863 .0880 .0890 .0891 .0886 .0874 .0856 .0834 .0808 25-29 ...... 0720 .0747 .0767 .0779 .0783 .0779 .0767 .0750 .0727 .0699 .0668 .0635 .0600 30-34 ...... , ...... 0717 .0726 .0727 .0720 .0705 .0684 .0658 .0627 .0593 .0556 .0518 .0480 .0443 35-39 ...... , ...... 0709 .0701 .0684 .0661 .0632 .0598 .0560 .0521 .0480 .0439 .0400 .0361 .0324 40-44 ...... 0698 .0672 .0640 .0603 .0562 .0519 .0474 .0430 .0387 .0345 .0306 .0270 .0236 45-49 ...... '" .,. .0681 .0640 .0595 .0546 .0497 .0447 .0399 .0352 .0309 .0269 .0233 .0200 .0171 50-54 ...... '" . " ...... 0655 .0600 .0544 .0487 .0432 .0379 .0330 .0284 .0243 .0207 .0174 .0146 .0122 55-59 ...... 0612 .0547 .0484 .0423 .0365 .0313 .0265 .0223 .0186 .0154 .0127 .0104 .0084 60-64 ..... '" .... , ...... 0546 .0476 .0410 .0350 .0295 .0246 .0204 .0167 .0136 .0110 .0088 .0070 .0056 65-69 ...... 0453 .0385 .0324 .0269 .0221 .0180 .0145 .0116 .0092 .0073 .0057 .0044 .0034 70-74 ...... 0338 .0280 .0230 .0186 .0150 .0119 .0093 .0073 .0056 .0043 .0033 .0025 .0019 75-79 ...... " ...... 0213 .0173 .0138 .0109 .0085 .0066 .0051 .0039 .0029 .0022 .0016 .0012 .0009 \0 80+ ...... 0131 .0103 .0080 .0061 .0046 .0035 .0026 .0019 .0014 .0010 .0007 .0005 .0004 \0 Age Proportion under given age 1 ...... 0158 .0188 .0221 .0257 .0295 .0336 .0379 .0424 .0471 .0518 .0567 .0617 .0667 5 ...... 0715 .0842 .0979 .1127 .1284 .1484 .1617 .1791 .1968 .2147 .2327 .2506 .2685 10 ...... 1400 .1627 .1871 .2127 .2394 .2668 .2947 .3227 .3507 .3783 .4055 .4321 .4579 15 ...... 2097 .2408 .2735 .3073 .3419 .3767 .4114 .4456 .4791 .5115 .5427 .5725 .6010 20 ...... 2809 .3185 .3573 .3968 .4363 .4755 .5137 .5507 .5862 .6198 .6516 .6813 .7090 25 ...... 3527 .3949 .4378 .4806 .5227 .5635 .6027 .6399 .6747 .7072 .7372 .7647 .7898 30 ...... 4247 .4697 .5145 .5585 .6009 .6414 .6794 .7148 .7474 .7771 .8040 .8282 .8498 35 ...... 4963 .5423 .5872 .6305 .6715 .7098 .7452 .7775 .8067 .8328 .8559 .8762 .8940 40 ...... 5673 .6124 .6556 .6966 .7346 .7696 .8012 .8296 .8547 .8767 .8958 .9123 .9265 45 ...... 6370 .6796 .7197 .7568 .7908 .8214 .8487 .8726 .8933 .9112 .9264 .9393 .9501 50 ...... 7052 .7436 .7791 .8115 .8405 .8662 .8885 .9078 .9243 .9381 .9497 .9593 .9672 55 ...... 7707 .8036 .8335 .8602 .8837 .9041 .9215 .9363 .9486 .9588 .9671 .9739 .9794 60 ...... 8319 .8583 .8819 .9025 .9202 .9354 .9481 .9586 .9672 .9742 .9798 .9843 .9878 65 ...... 8865 .9059 .9229 .9374 .9497 .9600 .9684 .9753 .9808 .9852 .9886 .9913 .9934 Parameter ofstable populations Birth rate .. '" ...... 0175 .0212 .0250 .0291 .0336 .0383 .0433 .0486 .0540 .0597 .0654 .0713 .0773 Death rate ...... 0278 .0262 .0250 .0241 .0236 .0233 .0233 .0236 .0240 .0247 .0254 .0263 .0273 ORR (27) ...... 1.24 1.42 1.63 1.86 2.12 2.41 2.75 3.12 3.55 4.02 4.56 5.17 5.85 ORR (29) ...... 1.25 1.44 1.66 1.91 2.20 2.53 2.91 3.34 3.83 4.38 5.01 5.73 6.54 ORR,(3l) ...... 1.25 1.46 1.70 1.97 2.30 2.67 3.09 3.58 4.15 4.79 5.54 6.39 7.36 ORR (33) ...... 1.25 1.47 1.73 2.04 2.40 2.81 3.30 3.86 4.52 5.28 6.17 7.20 8.39 Average age ...... " 36.2 33.9 31.6 29.5 27.4 25.5 23.7 22.0 20.5 19.1 17.8 16.7 15.6 Births/population 15-44 ...... ~ ...... 042 .048 .056 .065 .075 .086 .099 .114 .130 .149 .170 .194 .221 TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LBVEL OF MORTALITY (continued) LEVEL 11 Females (Oeo = 45.00 years)

AIUUIOl rate ofincrease

-.010 -.005 .000 .005 .010 .0/5 .020 .025 .030 .035 .(}4() .045 .050

Proportion in age interval Age interval Under 1 ...... 0142 .0170 .0201 .0235 .0272 .oJ 11 .0352 .0396 .0440 .0487 .0534 .0582 .0631 1-4 ...... 0516 .0610 .0712 .0822 .0939 .1061 .1187 .1316 .1447 .1579 .1711 .1842 .1971 5-9 ...... 0643 .0743 .0848 .0957 .1069 .1181 .1292 .1401 .1506 .1606 .1702 .1792 .1875 10-14 ...... 0660 .0743 .0828 .0911 .0992 .1069 .1141 .1206 .1265 .1316 .1360 .1396 .1425 15-19 ...... 0676 .0743 .0807 .0867 .0920 .0967 .1006 .1038 .1061 .1077 .1086 .1087 .1082 20-24 ...... 0688 .0738 .0781 .0818 .0847 .0868 .0881 .0887 .0884 .0875 .0860 .0840 .0816 25-29 ...... 0696 .0727 .0751 .0767 .0175 .0175 .0767 .0753 .0732 .0707 .0677 .0645 .0611 30-34 ...... 0700 .0714 .0719 .0716 .0706 .0688 .0664 .0635 .0603 .0568 .0531 .0493 .0455 35-39 .. , ...... 0700 .0696 .0684 .0665 .0639 .0607 .0572 .0534 .0494 .0453 .0413 .0375 .0337 40-44 ...... 0697 .0676 .0648 .0614 .0575 .0533 .0490 .0446 .0402 .0360 .0320 .0283 .0249 45-49 ...... 0689 .0651 .0609 .0563 .0514 .0465 .0417 .0370 .0326 .0284 .0247 .0213 .0182 50-54 ...... 0671 .0619 .0564 .0508 .0453 .0400 .0349 .0302 .0260 .0221 .0187 .0157 .0131 55-59 ...... 0637 .0574 .0510 .0448 .0390 .0335 .0286 .0241 .0202 .0168 .0138 .0114 .0093 60-64 ...... 0581 .0510 .0442 .0379 .0321 .0270 .0224 .0185 .0151 .0122 .0098 .0079 .0062 65-69 ...... 0496 .0425 .0359 .0300 .0248 .0203 .0165 .0132 .0105 .0083 .0065 .0051 .0040 70-74 ...... 0384 .0321 .0265 .0216 .0174 .0139 .0110 .0086 .0067 .0052 .0039 .0030 .0023 .0167 .0133 .0104 .0081 .0063 ..- 75-79 ...... 0255 .0207 .0048 .0036 .0027 .0020 .0015 .0011 0 80+ ...... 0170 .0134 .0105 .0081 .0062 .0047 .0035 .0026 .0019 .0014 .0010 .0007 .0005 0 Age Proportion under given age 1 ...... 0142 .0170 .0201 .0235 .0272 .0311 .0352 .0396 .0440 .0487 .0534 .0582 .0631 5 ...... 0658 .0780 .0913 .1057 .1210 .1371 .1539 .1711 .1887 .2065 .2245 .2424 .2602 10 ...... 1301 .1523 .1761 .2014 .2279 .2552 .2831 .3112 .3393 .3672 .3947 .4215 .4477 15 ...... 1961 .2266 .2589 .2926 .3271 .3621 .3971 .4318 .4658 .4988 .5306 .5611 .5902 20 ...... 2637 .3009 .3396 .3792 .4191 .4588 .4978 .5356 .5719 .6065 .6392 .6698 .6984 25 ...... 3325 .3747 .4177 .4610 .5038 .5457 .5859 .6242 .6604 .6940 .7252 .7539 .7800 30 ...... 4021 .4474 .4929 .5377 .5814 .6231 .6626 .6995 .7336 .7647 .7930 .8184 .8411 35 ...... 4721 .5188 .5648 .6094 .6519 .6919 .7290 .7630 .7938 .8215 .8460 .8677 .8867 40 ...... 5421 .5884 .6332 .6758 .7158 .7527 .7862 .8164 .8432 .8668 .8874 .9052 .9204 45 ...... 6117 .6559 .6979 .7372 .7733 .8060 .8352 .8610 .8835 .9028 .9194 .9335 .9453 50 ...... 6806 .7211 .7588 .7934 .8247 .8525 .8769 .8980 .9160 .9313 .9441 .9547 .9635 55 ...... 7476 .7829 .8152 .8443 .8700 .8925 .9118 .9282 .9420 .9534 .9628 .9705 .9766 60 ...... 8114 .8403 .8662 .8891 .9090 .9260 .9404 .9523 .9622 .9702 .9766 .9818 .9859 65 ...... 8695 .8913 .9104 .9270 .9411 .9530 .9628 .9708 .9772 .9824 .9865 .9897 .9921 Parameter ofstable populations Birth rate ...... 0156 .0188 .0222 .0260 .0302 .0346 .0393 .0443 .0494 .0547 .0602 .0658 .0715 Death rate ...... 0256 .0238 .0222 .0210 .0202 .0196 .0193 .0193 .0194 .0197 .0202 .0208 .0215 ORR (27) ...... 1.13 1.29 1.48 1.69 1.92 2.19 2.50 2.84 3.23 3.66 4.16 4.71 5.33 ORR (29) ...... 1.13 1.30 1.50 1.73 2.00 2.30 2.64 3.03 3.47 3.98 4.55 5.20 5.94 ORR (31) ...... 1.12 1.31 1.53 1.78 2.07 2.41 2.79 3.24 3.75 4.34 5.01 5.78 6.68 ORR (33) ...... 1.12 1.32 1.56 1.83 2.15 2.53 2.97 3.47 4.07 4.76 5.56 6.48 7.56 Average age ...... 37.6 35.2 32.9 30.6 28.5 26.4 24.6 22.8 21.2 19.7 18.4 17.2 16.1 Births/population 15-44 ...... 038 .044 .051 .059 .068 .078 .090 .103 .118 .135 .155 .177 .201 TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 13 Females (Oeo = 50.00 years)

A1UlIUII rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .()4() .045 .050

Age interval Proportion in age interval Under 1 ...... 0129 .0155 .0185 .0217 .0252 .0290 .0330 .0372 .0415 .0460 .0506 .0553 .0601 1-4 ...... 0481 .0572 .0672 .0780 .0895 .1016 .1l42 .1271 .1402 .1534 .1667 .1799 .1930 5-9 ...... 0607 .0705 .0810 .0919 .1031 .1144 .1257 .1368 .1476 .1579 .1678 .1770 .1857 10-14 ...... 0625 .0709 .0794 .0879 .0962 .1041 .11l6 .1184 .1246 .1301 .1348 .1387 .1419 15-19 ...... 0645 .0713 .0779 .0841 .0897 .0947 .0990 .1025 .1052 .1070 .1082 .1086 .1083 20-24 ...... 0660 .0712 .0759 .0799 .0832 .0856 .0873 .0881 .0882 .0875 .0863 .0845 .0822 25-29 ...... 0673 .0708 .0735 .0755 .0767 .0770 .0765 .0754 .0736 .0712 .0685 .0654 .0620 30-34 ...... '" ...... 0683 .ovoo .0710 .0711 .0704 .0689 .0668 .0642 .061l .0577 .0541 .0504 .0466 35-39 ...... , ...... 0690 .0690 .0682 .0666 .0643 .0615 .0581 .0544 .0505 .0466 .0426 .0387 .0349 40-44 ...... 0693 .0676 .0652 .0621 .0585 .0545 .0503 .0459 .0416 .0374 .0333 .0295 .0260 45-49 ...... 0692 .0659 .0619 .0576 .0529 .0480 .0432 .0385 .0340 .0298 .0259 .0224 .0192 50-54 ...... 0682 .0633 .0581 .0526 .0471 .0418 .0367 .0319 .0274 .0234 .0199 .0168 .0140 55-59 ...... , ... " .... , ...... 0658 .0595 .0533 .0471 .041l .0355 .0304 .0258 .0217 .0180 .0149 .0123 .0100 60-64 ...... 061l .0539 .0470 .0406 .0346 .0291 .0243 .0201 .0165 .0134 .0108 .0087 .0069 65-69 ...... 0535 .0461 .0392 .0330 .0274 .0225 .0183 .0148 .01l8 .0094 .0074 .0058 .0045 70-74 ...... 0428 .0359 .0298 .0244 .0198 .0159 .0126 .0099 .0077 .0060 .0046 .0035 .0026 75-79 ...... 0295 .0242 .0196 .0156 .0124 .0097 .0075 .0057 .0044 .0033 .0025 .0018 .0014 .0214 .0013 .0009 .0007 -0 80+ ...... 0170 .0133 .0103 .0079 .0060 .0045 .0034 .0025 .0018 - Age Proportion under given age 1 ...... 0129 .0155 .0185 .0217 .0252 .0290 .0330 .0372 .0415 .0460 .0506 .0553 .0601 5 ...... 0610 .0727 .0857 .0997 .1l47 .1306 .1471 .1642 .1817 .1994 .2173 .2352 .2531 10 ...... 1217 .1432 .1666 .1916 .2178 .2450 .2728 .3010 .3293 .3574 .3851 .4122 .4387 15 ...... 1842 .2142 .2461 .2795 .3140 .3491 .3844 .4195 .4539 .4874 .5198 .5509 .5806 20 ...... 2487 .2855 .3240 .3636 .4037 .4439 .4834 .5219 .5591 .5945 .6280 .6595 .6889 25 ...... 3147 .3567 .3998 .4435 .4869 .5295 .5707 .6101· .6472 .6820 .7143 .7440 .7711 30 ...... 3820 .4275 .4734 .5190 .5636 .6065 .6472 .6854 .7208 .7533 .7827 .8093 .8331 35 ...... 4503 .4975 .5444 .5901 .6340 .6754 .7141 .7496 .7819 .81l0 .8369 .8597 .8798 40 ...... 5192 .5665 .6126 .6567 .6983 .7369 .7722 .8041 .8325 .8575 .8794 .8984 .9147 45 ...... 5885 .6341 .6778 .7188 .7568 .7914 .8225 .8500 .8741 .8949 .9127 .9279 .9407 50 ...... 6578 .7000 .7397 ..7764 .8097 .8394 .8657 .8885 .9081 .9247 .9386 .9503 .9599 55 ...... 7260 .7634 .7978 .8290 .8568 .8812 .9023 .9203 .9355 .9481 .9585 .9670 .9739 60 ...... 7917 .8229 .8510 .8761 .8979 .9168 .9327 .9461 .9572 .9662 .9735 .9793 .9839 65 ...... 8528 .8768 .8981 .9166 .9325 .9459 .9571 .9662 .9736 .9796 .9843 .9880 .9908 Parameter ofstable populations Birth rate ...... 0139 .0168 .0200 .0236 .0275 .0316 .0361 .0408 .0457 .0507 .0559 .0613 .0667 Death rate ...... 0239 .0218 .0200 .0186 .0175 .0166 .0161 .0158 .0157 .0157 .0159 .0163 .0167 GRR (27) ...... 1.04 1.19 1.36 1.55 1.77 2.02 2.30 2.62 2.98 3.38 3.83 4.35 4.92 GRR (29) ...... 1.03 1.19 1.38 1.59 1.83 2.1l 2.42 2.78 3.19 3.66 4.19 4.79 5.47 GRR (31) ...... 1.03 1.20 1.40 1.63 1.89 2.20 2.56 2.96 3.43 3.97 4.59 5.30 6.12 GRR (33) ...... 1.02 1.20 1.42 1.67 1.96 2.30 2.70 3.17 3.71 4.34 5.07 5.92 6.91 Average age ...... 38.9 36.4 34.0 31.6 29.4 27.3 25.3 23.5 21.8 20.3 18.9 17.6 16.5 Births/population 15-44 ...... 034 .040 .046 .054 .062 .072 .082 .095 .109 .124 .142 .163 .185

,/ TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 15 Females (Oeo = 55.00 years)

Annual rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0118 .0143 .0170 .0201 .0235 .0271 .0310 .0350 .0392 .0436 .0481 .0527 .0573 1-4 ...... 0452 .0540 .0638 .0744 .0857 .0977 .1102 .1230 .1362 .1494 .1627 .1760 .1892 5-9 ...... 0576 .0673 .0777 .0886 .0998 .1112 .1227 .1339 .1449 .1555 .1656 .1751 .1840 10-14 ...... 0597 .0680 .0766 .0852 .0936 .1018 .1094 .1165 .1230 .1287 .1337 .1379 .1413 15-19 ...... 0618 .0687 .0755 .0819 .0877 .0930 .0976 .1013 .1043 .1065 .1078 .1085 .1084 20-24 ...... 0637 .0691 .0740 .0782 .0818 .0846 .0865 .0876 .0880 .0876 .0865 .0849 .0828 25-29 ...... 0653 .0691 .0722 .0744 .0759 .0765 .0764 .0755 .0739 .0717 .0691 .0661 .0629 30-34 ...... 0668 .0689 .0702 .0706 .0702 .0691 .0672 .0648 .0618 .0586 .0550 .0514 .0476 35-39 ...... 0680 .0684 .0680 .0667 .0647 .0621 .0589 .0554 .0516 .0476 .0436 .0397 .0359 40-44 ...... 0689 .0676 .0655 .0627 .0593 .0555 .0514 .0471 .0428 .0385 .0345 .0306 .0270 45-49 ...... 0694 .0664 .0628 .0586 .0541 .0493 .0445 .0398 .0353 .0310 .0270 .0234 .0201 50-54 ...... 0691 .0645 .0594 .0541 .0487 .0433 .0381 .0333 .0287 .0246 .0209 .0177 .0148 55-59 ...... 0674 .0613 .0551 .0489 .0430 .0373 .0320 .0272 .0229 .0192 .0159 .0131 .0107 60-64 ...... 0635 .0564 .0494 .0428 .0366 .0310 .0260 .0215 .0177 .0144 .0117 .0094 .0075 65-69 ...... , ...... 0567 .0491 .0420 .0354 .0296 .0244 .0200 .0161 .0129 .0103 .0081 .0064 .0049 70-74 ...... 0464 .0392 .0327 .0269 .0219 .0176 .0141 .0111 .0087 .0067 .0052 .0040 .0030 75-79 ...... , ...... 0330 .0272 .0221 .0178 .0141 .0111 .0086 .0066 .0050 .0038 .0029 .0021 .0016 .- .0126 .0097 .0074 .0056 .0041 .0031 .0022 .0016 .0012 .0008 0 80+ ...... 0258 .0206 .0162 N Proportion under given age Age I ...... 0118 .0143 .0170 .0201 .0235 .0271 .0310 .0350 .0392 .0436 .0481 .0527 .0573 5 ...... 0570 .0683 .0808 .0945 .1092 .1248 .1411 .1580 .1754 .1930 .2108 .2287 .2465 10 ...... 1146 .1356 .1585 .1831 .2090 .2360 .2638 .2920 .3203 .3485 .3764 .4038 .4305 15 ...... 1742 .2036 .2351 .2683 .3026 .3378 .3732 .4085 .4433 .4772 .5101 .5417 .5718 20 ...... 2361 .2724 .3106 .3501 .3904 .4308 .4708 .5099 .5476 .5837 .6179 .6502 .6803 25 ...... 2998 .3414 .3845 .4283 .4722 .5154 .5573 .5975 .6356 .6713 .7045 .7350 .7630 30 ...... 3651 .4105 .4567 .5028 .5481 .5919 .6337 .6729 .7095 .7430 .7736 .8012 .8259 35 ...... 4319 .4794 .5268 .5734 .6183 .6610 .7009 .7377 .7713 .8016 .8286 .8525 .8736 40 ...... 4999 .5478 .5948 .6401 .6830 .7230 .7598 .7931 .8228 .8492 .8722 .8922 .9095 45 ...... 5688 .6154 .6603 .7028 .7423 .7785 .8111 .8402 .8656 .8877 .9067 .9228 .9365 50 ...... 6383 .6819 .7231 .7614 .7964 .8279 .8557 .8800 .9009 .9187 .9337 .9462 .9566 55 ...... 7073 .7464 .7825 .8155 .8451 .8712 .8938 .9132 .9296 .9433 .9546 .9639 .9714 60 ...... 7747 .8077 .8377 .8645 .8881 .9085 .9259 .9405 .9526 .9625 .9705 .9770 .9821 65 ...... 8382 .8640 .8871 .9073 .9247 .9395 .9518 .9620 .9703 .9769 .9822 .9864 .9896 Parameter ofstable populations Birth rate ...... 0125 .0152 .0182 .0215 .0252 .0291 .0334 .0378 .0425 .0473 .0523 .0574 .0627 Death rate ...... ·· ...... 0225 .0202 .0182 .0165 .0152 .0141 .0134 .0128 .0125 .0123 .0123 .0124 .0127 GRR (27) ...... 0.96 1.10 1.26 1.44 1.64 1.87 2.14 2.43 2.76 3.14 3.56 4.04 4.57 GRR (29) ...... 0.95 1.10 1.27 1.47 1.69 1.95 2.24 2.58 2.96 3.39 3.88 4.44 5.07 GRR (31) ...... 0.95 1.10 1.29 1.50 1.75 2.05 2.36 2.74 3.17 3.67 4.24 4.90 5.66 GRR (33) ...... 0.94 1.11 1.30 1.53 1.80 2.12 2.49 2.92 3.42 4.00 4.67 5.46 6.37 Average age ...... 40.0 37.4 35.0 32.6 30.5 28.1 26.1 24.2 22.4 20.8 19.4 18.0 16.8 Births/population 15-44 ...... 032 .037 .043 .050 .057 .066 .076 .088 .101 .115 .132 .151 .172 "

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 17 Females (Oeo = 60.00 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0108 .0131 .0158 .0187 .0220 .0255 .0292 .0331 .0372 .0415 .0459 .0503 .0549 1-4 ...... 0425 .0511 .0606 .0710 .0823 .0941 .1066 .1194 .1325 .1459 .1592 .1726 .1859 5-9 ...... 0546 .0642 .0745 .0853 .0966 .1081 .1196 .1311 .1422 .1530 .1634 .1731 .1823 10-14 ...... " .. .0569 .0652 .0737 .0824 .0910 .0993 .1072 .1145 .1212 .1272 .1325 .1369 .1406 15-19 ...... 0591 .0661 .0730 .0795 .0856 .0911 .0960 .1000 .1033 .1057 .1073 .1082 .1083 20-24 ...... 0613 .0668 .0719 .0764 .0803 .0833 .0856 .0870 .0876 .0874 .0866 .0851 .0832 25-29 ...... 0633 .0673 .0706 .0732 .0750 .0759 .0761 .0754 .0740 .0721 .0696 .0668 .0636 30-34 ...... 0652 .0676 .0692 .0700 .0699 .0690 .0674 .0652 .0624 .0593 .0558 .0522 .0485 35-39 ...... 0669 .0676 .0675 .0666 .0649 .0625 .0596 .0562 .0525 .0486 .0446 .0407 .0369 40-44 ...... 0683 .0674 .0657 .0632 .0600 .0564 .0524 .0482 .0439 .0396 .0355 .0316 .0279 45-49 ...... 0694 .0668 .0634 .0595 .0552 .0505 .0458 .0411 .0365 .0322 .0281 .0244 .0210 50-54 ...... 0697 .0654 .0606 .0554 .0501 .0448 .0396 .0346 .0300 .0258 .0220 .0186 .0156 55-59 ...... " ...... 0687 .0629 .0568 .0507 .0447 .0389 .0336 .0286 .0242 .0203 .0169 .0139 .0114 60-64 ...... 0657 .0586 .0517 .0450 .0387 .0329 .0276 .0230 .0190 .0155 .0126 .0101 .0081 65-69 ...... 0598 .0520 .0447 .0380 .0318 .0264 .0216 .0176 .0141 .0113 .0089 .0070 .0055 70-74 ...... " ...... 0501 .0425 .0356 .0295 .0241 .0195 .0156 .0123 .0097 .0075 .0058 .0044 .0034 75-79 ...... 0367 .0304 .0248 .0200 .0160 .0126 .0098 .0076 .0058 .0044 .0033 .0025 .0018 .... .0011 0 80+ ...... 0311 .0250 .0198 .0154 .0119 .0091 .0069 .0051 .0038 .0028 .0020 .0015 w Age Proportion under given age 1 ...... 0108 .0131 .0158 .0187 .0220 .0255 .0292 .0331 .0372 .0415 .0459 .0503 .0549 5 ...... 0533 .0642 .0764 .0898 .1042 .1196 .1357 .1525 .1698 .1873 .2051 .2229 .2407 10 ...... 1079 .1284 .1509 .1751 .2008 .2277 .2554 .2836 .3120 .3404 .3685 .3961 .4231 15 ...... 1648 .1936 .2246 .2575 .2918 .3270 .3626 .3981 .4333 .4676 .5009 .5330 .5636 20 ...... 2239 .2597 .2976 .3370 .3774 .4181 .4585 .4981 .5365 .5733 .6082 .6412 .6720 25 ...... 2852 .3265 .3695 .4135 .4577 .5014 .5441 .5851 .6241 .6607 .6948 .7263 .7551 30 ...... 3485 .3938 .4401 .4867 .5327 .5774 .6201 .6605 .6981 .7328 .7644 .7931 .8188 35 ...... 4136 .4614 .5093 .5567 .6026 .6464 .6876 .7257 .7606 .7921 .8203 .8453 .8673 40 ...... 4805 .5290 .5769 .6233 .6675 .7089 .7471 .7818 .8130 .8407 .8649 .8860 .9042 45 ...... 5489 .5964 .6425 .6864 .7275 .7653 .7995 .8300 .8569 .8803 .9004 .9176 .9321 50 ...... 6183 .6632 .7060 .7460 .7827 .8158 .8453 .8711 .8934 .9125 .9285 .9420 .9531 55 ...... 6880 .7286 .7666 .8014 .8238 .8606 .8849 .9057 .9234 .9382 .9505 .9606 .9688 60 ...... 7567 .7915 .8234 .8521 .8775 .8995 .9184 .9344 .9476 .9585 .9674 .9745 .9802 65 ...... 8224 .8501 .8751 .8971 .9161 .9324 .9461 .9574 .9666 .9740 .9800 .9846 .9883 Parameter ofstable populations Birth rate ...... ,. .0113 .0138 .0167 .0198 .0233 .0271 .0311 .0354 .0399 .0445 .0494 .0543 .0594 Death rate ...... 0213 .0188 .0167 .0148 .0133 .0121 .0111 .0104 .0099 .0095 .0094 .0093 .0094 ORR (27) ...... 0.90 1.03 1.18 1.35 1.54 1.76 2.00 2.28 2.59 2.95 3.34 3.79 4.30 ORR (29) ...... 0.89 1.03 1.19 1.37 1.58 1.82 2.10 2.41 2.77 3.17 3.63 4.16 4.75 ORR (31) ...... 0.88 1.03 1.20 1.40 1.63 1.89 2.20 2.55 2.96 3.43 3.96 4.58 5.29 ORR (33) ...... 0.87 1.03 1.21 1.43 1.68 1.97 2.31 2.71 3.18 3.72 4.35 5.08 5.93 Average age ...... 41.1 38.5 36.0 33.5 31.1 28.9 26.8 24.8 23.0 21.3 19.8 18.4 17.2 Births/population 15-44 ...... '" ... .030 .034 .040 .046 .053 .062 .071 .082 .094 .108 .124 .141 .151 TABLE I!. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 19 Females (Oeo = 65.00 years)

Annual rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0100 .0122 .0148 .0176 .0207 .0241 .0278 .0316 .0356 .0398 .0441 .0485 .0530 1-4 ...... 0401 .0484 .0578 .0680 .0791 .0909 .1033 .1161 .1293 .1426 .1561 .1695 .1829 5-9 ...... 0518 .0613 .0714 .0823 .0935 .1051 .1167 .1283 .1397 .1507 .1612 .1712 .1806 10-14 ...... 0542 .0624 .0710 .0797 .0884 .0969 .1050 .1126 .1195 .1257 .1312 .1359 .1398 15-19 ...... , ...... 0566 .0636 .0705 .0772 .0835 .0893 .0944 .0987 .1022 .1048 .1067 .1078 .1081 20-24 ...... 0589 .0646 .0698 .0746 .0787 .0820 .0846 .0862 .0871 .0872 .0865 .0852 .0834 25-29 ...... 0612 .0654 .0690 .0719 .0740 .0752 .0756 .0752 .0741 .0723 .0700 .0672 .0642 30-34 ...... 0634 .0661 .0680 .0691 .0694 .0688 .0674 .0654 .0628 .0598 .0565 .0529 .0493 35-39 ...... 0655 .0666 .0669 .0663 .0649 .0627 .0600 .0568 .0532 .0494 .0455 .0416 .0377 40-44 ...... 0675 .0669 .0655 .0633 .0604 .0570 .0532 .0491 .0448 .0406 .0365 .0325 .0288 45-49 ...... 0691 .0668 .0638 .0601 .0560 .0515 .0468 .0422 .0376 .0332 .0291 .0253 .0218 50-54 ...... 0699 .0660 .0615 .0565 .0513 .0460 .0408 .0358 .0312 .0268 .0229 .0194 .0164 55-59 ...... 0697 .0641 .0583 .0522 .0463 .0405 .0350 .0300 .0254 .0214 .0178 .0147 .0121 60-64 ...... 0676 .0607 .0538 .0470 .0406 .0327 .0292 .0244 .0202 .0165 .0134 .0108 .0087 65-69 ...... " .0627 .0549 .0474 .0404 .0340 .0283 .0233 .0190 .0153 .0122 .0097 .0076 .0060 70-74 ...... 0537 .0459 .0387 .0322 .0264 .0215 .0172 .0137 .0108 .0084 .0065 .0050 .0038 75-79 ...... 0406 .0338 .0278 .0225 .0181 .0143 .0112 .0087 .0066 .0051 .0038 .0029 .0021 .... .0064 .0047 .0035 .0025 .0018 .0013 ~ 80+ ...... 0376 .0303 .0241 .0189 .0146 .0112 .0085 Proportion under given age Age 1 ...... 0100 .0122 .0148 .0176 .0207 .0241 .0278 .0316 .0356 .0398 .0441 .0485 .0530 5 ...... 0501 .0606 .0725 .0856 .0998 .1150 .1310 .1477 .1649 .1824 .2002 .2180 .2358 10 ...... 1019 .1219 .1440 .1679 .1934 .2201 .2478 .2760 .3046 .3331 .3614 .3892 .4165 15 ...... 1561 .1843 .2150 .2476 .2818 .3170 .3528 .3886 .4241 .4589 .4926 .5251 .5563 20 ...... 2126 .2479 .2855 .3249 .3653 .4063 .4471 .4873 .5262 .5637 .5993 .6329 .6644 25 ...... 2715 .3125 .3554 .3995 .4440 .4883 .5317 .5735 .6133 .6509 .6858 .7182 .7478 30 ...... 3327 .3779 .4244 .4714 .5180 .5635 .6073 .6487 .6874 .7231 .7558 .7854 .8120 35 ...... 3961 .4440 .4924 .5405 .5874 .6323 .6747 .7141 .7502 .7830 .8123 .8384 .8613 40 ...... 4616 .5106 .5593 .6068 .6522 .6950 .7347 .7709 .8034 .8323 .8578 .8799 .8990 45 ...... 5291 .5775 .6248 .6701 .7127 .7521 .7879 .8199 .8482 .8729 .8942 .9124 .9278 50 ...... 5981 .6443 .6886 .7302 .7687 .8035 .8347 .8621 .8858 .9061 .9233 .9377 .9497 55 ...... 6681 .7103 .7501 .7867 .8200 .8496 .8755 .8979 .9170 .9330 .9462 .9572 .9660 60 ...... 7378 .7745 .8083 .8390 .8662 .8901 .9105 .9279 .9424 .9543 .9640 .9719 .9781 65 ...... 8054 .8352 .8621 .8860 .9068 .9247 .9398 .9523 .9626 .9709 .9775 .9827 .9868 Parameter ofstable populations Birth rate ...... ···· ...... 0104 .0127 .0154 .0184 .0217 .0253 .0292 .0333 .0377 .0422 .0469 .0517 .0566 Death rate ...... 0204 .0177 .0154 .0134 .0117 .0103 .0092 .0083 .0077 .0072 .0069 .0067 .0066 ORR (27) ...... 0.85 0.97 1.12 1.28 1.46 1.66 1.89 2.16 2.45 2.79 3.17 3.59 4.07 ORR (29) ...... 0.84 0.97 1.12 1.30 1.49 1.72 1.98 2.28 2.61 3.00 3.43 3.93 4.49 ORR (31) ...... 0.83 0.97 1.13 1.32 1.53 1.78 2.07 2.41 2.79 3.23 3.74 4.32 4.99 ORR (33) ...... 0.82 0.96 1.14 1.34 1.57 1.85 2.17 2.55 2.99 3.50 4.09 4.78 5.58 Average age ...... 42.2 39.6 38.0 34.4 32.0 29.7 27.5 25.4 23.5 21.8 20.3 18.8 17.5 Births/population 15-44 ...... 028 .032 .038 .044 .050 .058 .067 .077 .089 .102 .117 .113 .152 "

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTAUTY (continued) LEVEL 21 Females (Oeo = 70.00 years)

Annual rate oflncrease .050 -.010 -.005 .000 .005 .010--- .015 .020 .025 .030 .035 .040 .045

Age interval Proportion in age interval ..under 1 ...... 0093 .0115 .0139 .0167 .0197 .0230 .0266 .0304 .0343 .0385 .0427 .0471 .0515 1-4 ... ~~ ...... 0378 .0460 .0551 .0652 .0762 .0879 .1002 .1131 .1262 .1397 .1532 .1667 .1801 5-9 ...... 0492 .0585 .0686 .0793 .0906 .1022 .1140 .1257 .1373 .1485 .1592 .1694 .1790 10-14 ...... , ... " ...... 0516 .0598 .0684 .0771 .0859 .0946 .1028 .1106 .1178 .1242 .1299 .1349 .1390 15-19 ...... , ...... 0540 .0611 .0681 .0750 .0815 .0874 .0927 .0973 .1010 .1039 .1060 .1073 .1078 20-24 ...... 0565 .0623 .0678 .0727 .0771 .0807 .0834 .0854 .0865 .0868 .0863 .0852 .0835 25-29 ...... 0590 .0634 .0673 .0704 .0728 .0743 .0750 .0748 .0739 .0723 .0702 .0676 .0646 30-34 ...... 0615 .0645 .0667 .0681 .0687 .0684 .0673 .0655 .0631 .0602 .0570 .0535 .0499 35-39 ...... 0640 .0654 .0660 .0658 .0647 .0628 .0603 .0572 .0537 .0500 .0462 .0423 .0385 40-44 ...... 0663 .0662 .0651 .0633 .0607 .0574 .0538 .0498 .0456 .0414 .0373 .0333 .0295 45-49 ...... 0684 .0665 .0639 .0605 .0566 .0523 .0477 .0431 .0385 .0341 .0299 .0261 .0226 50-54 ...... 0699 .0663 .0621 .0573 .0523 .0471 .0420 .0369 .0322 .0278 .0238 .0202 .0171 55-59 ...... , ...... 0704 .0651 .0595 .0536 .0477 .0419 .0364 .0312 .0266 .0224 .0187 .0155 .0127 60-64 ...... 0692 .0625 .0556 .0489 .0424 .0364 .0308 .0258 .0214 .0176 .0143 .0116 .0093 65-69 ...... 0653 .0575 .0500 .0428 .0362 .0303 .0250 .0204 .0165 .0132 .0105 .0083 .0065 70-74 ...... 0574 .0493 .0417 .0349 .0288 .0235 .0189 .0151 .0119 .0093 .0072 .0055 .0042 75-79 ...... , ...... , ..... " ... .0446 .0374 .0309 .0252 .0203 .0161 .0127 .0098 .0076 .0058 .0044 .0033 .0024 0 80+ ...... 0455 .0368 .0294 .0231 .0180 .0138 .0105 .0079 .0058 .0043 .0031 .0023 .0016 -VI Age Proportion under given age 1 ...... 0093 .0115 .0139 .0167 .0197 .0230 .0266 .0304 .0343 .0385 .0427 .0471 .0515 5 ...... 0471 .0574 .0690 .0819 .0959 .1109 .1268 .1434 .1606 .1781 .1959 .2138 .2317 10 ...... 0963 .1159 .1376 .1612 .1865 .2132 .2408 .2692 .2978 .3266 .3551 .3832 .4106 15 ...... 1479 .1757 .2059 .2383 .2724 .3077 .3437 .3798 .4156 .4508 .4850 .5180 .5496 20 ...... 2020 .2368 .2741 .3133 .3539 .3951 .4364 .4770 .5166 .5547 .5910 .6253 .6575 25 ...... 2585 .2990 .3418 .3860 .4310 .4758 .5198 .5624 .6031 .6415 .6174 .7105 .7410 30 ...... 3175 .3625 .4091 .4565 .5038 .5501 .5948 .6373 .6770 .7139 .7476 .7781 .8056 35 ...... 3790 .4270 .4758 .5246 .5724 .6185 .6621 .7028 .7401 .1741 .8046 .8317 .8555 40 ...... 4429 .4924 .5419 .5904 .6371 .6813 .7223 .7599 .7939 .8241 .8507 .8739 .8940 45 ...... 5093 .5586 .6070 .6537 .6978 .7378 .7161 .8091 .8395 .8655 .8880 .9013 .9235 50 ...... 5777 .6251 .6709 .7142 .7544 .7910 .8238 .8528 .8780 .8996 .9180 .9333 .9461 55 ...... 6416 .6914 .1330 .1715 .8061 .8381 .8658 .8898 .9102 .9275 .9418 .9536 .9632 60 ...... 1179 .1566 .7924 .8251 .8543 .8800 .9022 .9210 .9368 .9498 .9605 .9691 .9159 65 ...... 7811 .8190 .8481 .8740 .8967 .9163 .9329 .9468 .9582 .9674 .9148 .9806 .9852 Parameter ofstable populations Birth rate ...... 0095 .0117 .0143 .0172 .0204 .0238 .0276 .0316 .0358 .0402 .0448 .0495 .0543 Death rate ...... 0195 .0167 .0143 .0122 .0104 .0088 .0076 .0066 .0058 .0052 .0048 .0045 .0043 GRR (27) ...... 0.81 0.93 1.06 1.22 1.39 1.58 1.81 2.06 2.34 2.66 3.02 3.43 3.88 GRR (29) ...... 0.80 0.92 1.01 1.23 1.42 1.64 1.88 2.17 2.49 2.85 3.27 3.14 4.28 GRR (31) ...... 0.78 0.92 1.07 1.25 1.45 1.69 1.97 2.28 2.65 3.01 3.55 4.10 4.74 GRR (33) ...... 0.77 0.91 1.07 1.21 1.49 1.75 2.06 2.41 2.83 3.31 3.88 4.53 5.29 Average age ...... 43.4 40.7 38.0 35.4 32.8 30.4 28.2 26.0 24.1 22.3 20.7 19.2 11.9 Births/population 15-44 ...... 026 .031 .036 .041 .048 .055 .064 .073 .084 .097 .111 .127 .145 TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 23 Females (Oeo = 75.00 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0087 .0108 .0132 .0159 .0189 .0221 .0257 .0295 .0334 .0375 .0418 .0461 .0506 1-4 ...... 0355 .0434 .0524 .0624 .0734 .0851 .0975 .1103 .1236 .1371 .1508 .1644 .1780 5-9 ...... 0463 .0555 .0655 .0762 .0875 .0993 .1112 .1231 .1348 .1462 .1572 .1676 .1774 10-14 ...... , ...... 0486 .0568 .0654 .0742 .0832 .0920 .1005 .1085 .1159 .1226 .1285 .1337 .1380 15-19 ...... 0511 .0581 .0653 .0723 .0790 .0852 .0908 .0956 .0996 .1028 .1051 .1066 .1073 20-24 ...... 0536 .0595 .0651 .0704 .0750 .0789 .0820 .0842 .0855 .0861 .0859 .0849 .0834 25-29 ...... 0561 .0608 .0650 .0684 .0711 .0730 .0739 .0741 .0734 .0721 .0701 .0676 .0648 30-34 ...... 0588 .0621 .0647 .0665 .0674 .0674 .0667 .0651 .0630 .0603 .0572 .0538 .0503 35-39 ...... " ...... 0615 .0634 .0644 .0645 .0638 .0623 .0600 .0572 .0539 .0503 .0466 .0427 .0389 40-44 ...... 0642 .0645 .0639 .0625 .0602 .0573 .0539 .0501 .0461 .0419 .0379 .0339 .0301 45-49 ...... 0667 .0654 .0632 .0602 .0566 .0526 .0482 .0437 .0392 .0348 .0306 .0267 .0232 50-54 ...... 0688 .0658 .0620 .0576 .0529 .0479 .0428 .0378 .0331 .0287 .0246 .0210 .0177 55-59 ...... 0702 .0655 .0602 .0546 .0488 .0431 .0376 .0324 .0277 .0234 .0196 .0162 .0134 60-64 ...... 0704 .0640 .0574 .0507 .0443 .0381 .0324 .0273 .0227 .0187 .0153 .0124 .0099 65-69 ...... 0681 .0604 .0528 .0456 .0388 .0326 .0270 .0222 .0180 .0145 .0115 .0091 .0071 70-74 ...... ,. .0620 .0536 .0457 .0385 .0319 .0261 .0212 .0169 .0134 .0105 .0082 .0063 .0048 .... 75-79 ...... 0505 .0426 .0355 .0291 .0236 .0188 .0149 .0116 .0089 .0068 .0052 .0039 .0029 0 80+ ...... 0588 .0479 .0384 .0304 .0237 .0182 .0139 .0104 .0078 .0057 .0042 .0030 .0022 0\ Age Proportion under given age 1 ...... 0087 .0108 .0132 .0159 .0189 .0221 .0257 .0295 .0334 .0375 .0418 .0461 .0506 5 ...... 0442 .0542 .0656 .0783 .0922 .1072 .1231 .1398 .1570 .1747 .1926 .2106 .2285 10 ...... 0905 .1096 .1311 .1545 .1798 .2065 .2343 .2629 .2918 .3209 .3497 .3782 .4060 15 ...... 1391 .1664 .1964 .2288 .2629 .2984 .3348 .3714 .4077 .4435 .4783 .5118 .5440 20 ...... 1902 .2246 .2617 .3011 .3419 .3837 .4255 .4670 .5073 .5463 .5834 .6184 .6513 25 ...... 2437 .2841 .3269 .3714 .4169 .4625 .5075 .5511 .5929 .6323 .6692 .7034 .7347 30 ...... 2999 .3449 .3918 .4399 .4880 .5355 .5814 .6252 .6663 .7044 .7393 .7710 .7995 35 ...... 3587 .4070 .4566 .5064 .5554 .6030 .6481 .6903 .7293 .7647 .7965 .8248 .8497 40 ...... 4202 .4704 .5210 .5709 .6192 .6652 .7081 .7475 .7832 .8150 .8431 .8675 .8887 45 ...... 4844 .5349 .5849 .6334 .6795 .722b .7620 .7976 .8293 .8570 .8809 .9014 .9188 50 ...... 5511 .6003 .6481 .6936 .7361 .7751 .8103 .8413 .8685 .8918 .9115 .9282 .9419 55 ...... 6199 .6661 .7101 .7512 .7890 .8230 .8531 .8792 .9016 .9204 .9362 .9491 .9597 60 ...... 6902 .7315 .7702 .8058 .8378 .8661 .8907 .9116 .9292 .9438 .9557 .9653 .9730 65 ...... 7605 .7955 .8276 .8565 .8821 .9042 .9231 .9389 .9519 .9625 .9710 .9777 .9830 Parameter ofstable populations Birth rate ...... 0086 .0109 .0133 .0161 .0192 .0226 .0263 .0302 .0344 .0387 .0432 .0478 .0526 Death rate ...... 0186 .0159 .0133 .0111 .0092 .0076 .0063 .0052 .0044 .0037 .0032 .0028 .0026 ORR (27) ...... 0.78 0.90 1.03 1.17 1.34 1.53 1.75 1.99 2.26 2.57 2.92 3.32 3.76 ORR (29) ...... 0.71 0.89 1.03 1.19 1.37 1.58 1.82 2.09 2.40 2.76 3.16 3.62 4.14 ORR (31) ...... 0.75 0.88 1.03 1.20 1.40 1.63 1.90 2.20 2.55 2.96 3.43 3.96 4.57 ORR (33) ...... 0.74 0.88 1.03 1.22 1.43 1.68 1.98 2.32 2.72 3.19 3.73 4.36 5.10 Average age ...... 44.9 42.11 39.3 36.46 33.9 31.4 29.0 26.8 24.7 22.8 21.1 19.6 18.2 Births/population 15-44 ...... 025 .030 .034 .040 .046 .053 .062 .071 .082 .094 .107 .123 .140 or

TABl.E II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 1 Males (oeo = 18.03 years)

AII1IIUl1 rate 01inuease -.010 -.005 ooo .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under I ...... 0309 .0352 .0399 .0448 .0499 .0553 .0608 .0665 .0722 .0781 .0841 .0901 .0962 1-4 ...... 0847 .0954 .1067 .1I83 .1303 .1424 .1547 .1670 .1793 .1915 .2035 .2154 .2271 5-9 ...... 0956 .1054 .1I52 .1249 .1345 .1438 .1527 .1611 .1692 .1766 .1836 .1900 .1958 10-14 ...... 0947 .1018 .1085 .1I48 .1205 .1256 .1301 .1339 .1371 .1396 .1415 .1428 .1436 15-19 ...... 0939 .0984 .1024 .1056 .1081 .IC99 .1110 .1I15 .1113 .II06 .1093 .1076 .1055 20-24 ...... 0910 .0930 .0944 .0949 .0948 .0940 .0926 .0907 .0883 .0856 .0825 .0792 .0757 25-29 ...... " ...... 0863 .0861 .0851 .0835 .0814 .0787 .0756 .0722 .0686 .0648 .0610 .0571 .0532 30-34 ...... 0806 .0785 .0757 .0724 .0688 .0649 .0608 .0567 .0525 .0484 .0444 .0405 .0368 35-39 ...... 0739 .0701 .0659 .0615 .0570 .0525 .0479 .0436 .0394 .0354 .0316 .0282 .0250 40-44 ...... 0659 .0610 .0559 .0509 .0460 .0413 .0368 .0326 .0287 .0252 .0220 .0191 .0165 45-49 ...... 0571 .0515 .0461 .0409 .0361 .0316 .0275 .0237 .0204 .0174 .0148 .0126 .0106 50-54 ...... " ...... 0476 .0419 .0366 .0317 .0272 .0232 .0197 .0166 .0139 .0116 .0096 .0080 .0065 55-59 ...... , ...... 0376 .0323 .0275 .0232 .0195 .0162 .0134 .0110 .0090 .0073 .0059 .0048 .0038 60-64 ...... 0275 .0230 .0191 .0157 .0129 .0104 .0084 .0068 .0054 .0043 .0034 .0027 .0021 65-69 ...... " ...... 0177 .0145 .0117 .0094 .0075 .0059 .0047 .0037 .0028 .0022 .0017 .0013 .0010 70-74 ...... , ...... , ...... 0097 .0077 .0061 .0048 .0037 .0029 .0022 .0017 .0013 .0010 .0007 .0005 .0004 .... 75-79 ...... , ...... ,. .0040 .0031 .0024 .0018 .0014 .0011 .0008 .0006 .0004 .0003 .0002 .0002 .0001 ooסס. ooסס. 0001. 0001. 0001. 0002. 0002. 0003. 0004. 0006. 0008. 0011. 0014...... , ...... 80+ 0 -l Age Proportion under given age I ...... 0309 .0352 .0399 .0448 .0499 .0553 .0608 .0665 .0722 .0781 .0841 .0901 .0962 5 ...... 1155 .1307 .1466 .1631 .1802 .1977 .2155 .2335 .2515 .2696 .2876 .3055 .3232 10 ...... 2111 .2361 .2618 .2881 .3147 .3415 .3682 .3946 .4207 .4462 .4712 .4955 .5190 15 ...... 3058 .3378 .3703 .4028 .4352 .4671 .4982 .5285 .5578 .5859 .6127 .6383 .6626 20 ...... 3997 .4363 .4727 .5084 .5433 .5770 .6093 .6400 .6691 .6965 .7221 .7459 .7681 25 ...... 4907 .5293 .5670 .6034 .6381 .6710 .7019 .7307 .7574 .7820 .8046 .8251 .8438 30 ...... 5770 .6154 .6522 .6869 .7195 .7497 .7775 .8030 .8260 .8469 .8655 .8822 .8970 35 ...... ' ...... 6576 .6939 .7278 .7593 .7883 .8146 .8384 .8596 .8785 .8952 .9099 .9227 .9339 40 ...... 7315 .7639 .7938 .8209 .8453 .8671 .8863 .9032 .9179 .9306 .9415 .9509 .9589 45 ...... , ...... 7974 .8249 .8497 .8718 .8913 .9084 .9231 .9358 .9466 .9558 .9635 .9700 .9754 50 ...... 8545 .8764 .8958 .9128 .9274 .9399 .9506 .9595 .9670 .9732 .9784 .9826 .9860 55 ...... , ...... 9021 .9183 .9324 .9444 .9546 .9632 .9703 .9761 .9809 .9848 .9880 .9905 .9925 60 ...... 9397 .9506 .9599 .9676 .9741 .9794 .9837 .9872 .9899 .9922 .9939 .9953 .9964 65 ...... 9672 .9736 .9790 .9834 .9869 .9898 .9921 .9939 .9953 .9964 .9973 .9979 .9985 Parameter ofstable populations Birth rate .. '" ...... 0427 .0489 .0555 .0624 .0698 .0774 .0854 .0936 .1020 .II05 .1I93 .1281 .1371 Death rate ...... , ...... 0527 .0539 .0555 .0514 .0598 .0624 .0654 .0686 .0720 .0755 .0793 .0831 .0871 ORR (27) ...... , ...... 2.49 2.84 3.23 3.68 4.19 4.76 5.41 6.14 6.96 7.88 8.92 10.08 11.39 ORR (29) ...... 2.55 2.94 3.38 3.89 4.46 5.12 5.86 6.71 7.67 8.76 9.99 11.39 12.97 ORR (31) ...... 2.62 3.05 3.54 4.11 4.77 5.52 6.38 7.37 8.51 9.81 11.30 13.00 14.95 ORR (33) ...... , ...... 2.70 3.17 3.72 4.36 5.11 5.98 6.99 8.17 9.53 1I.11 12.94 15.06 17.51 Average age ...... 27.8 26.1 24.5 23.0 21.5 20.2 18.9 17.7 16.7 15.7 14.8 13.9 13.2 Births/population 15-44 ...... 087 .100 .116 .133 .153 .175 .201 .230 .262 .299 .340 .386 .438

~ TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 3 Males (oeo = 22.85 years)

A1UIUal rate ofIncrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0254 .0293 .0335 .0379 .0426 .0474 .0525 .0577 .0631 .0685 .0741 .0797 .0854 1-4 ...... 0759 .0864 .0974 .1090 .1209 .1330 .1454 .1579 .1704 .1828 .1952 .2073 .2193 5-9 ...... 0883 .0983 .1084 .1185 .1285 .1383 .1478 .1569 .1656 .1737 .1813 .1883 .1948 10-14 ...... 0883 .0959 .1031 .1100 .1163 .1221 .1272 .1317 .1356 .1387 .1412 .1431 .1444 15-19 ...... 0884 .0936 .0982 .1022 .1054 .1079 .1097 .1108 .1112 .1110 .1102 .1089 .1071 20-24 ...... 0869 .0897 .0918 .0931 .0937 .0935 .0927 .0913 .0894 .0870 .0843 .0812 .0779 25-29 ...... 0838 .0844 .0843 .0834 .0818 .0797 .0770 .0740 .0706 .0671 .0633 .0595 .0557 30-34 ...... '" ...... 0799 .0785 .0764 .0737 .0706 .0670 .0632 .0592 .0551 .0511 .0470 .0431 .0394 35-39 ...... 0750 .0718 .0682 .0642 .0599 .0555 .0510 .0466 .0423 .0382 .0344 .0307 .0273 40-44 ...... · ... ······ . .0688 .0643 .0595 .0546 .0497 .0449 .0403 .0359 .0318 .0280 .0245 .0214 .0186 45-49 ...... 0615 .0561 .0506 .0453 .0403 .0355 .0310 .0270 .0233 .0200 .0171 .0145 .0123 50-54 ...... 0532 .0473 .0417 .0364 .0315 .0271 .0231 .0196 .0165 .0138 .0115 .0096 .0079 55-59 ...... 0440 .0381 .0327 .0279 .0235 .0197 .0164 .0136 .0112 .0091 .0074 .0060 .0048 60-64 ...... · .. · .. ·. .0339 .0286 .0240 .0199 .0164 .0134 .0109 .0088 .0070 .0056 .0044 .0035 .0028 65-69 ...... 0234 .0193 .0158 .0128 .0103 .0082 .0065 .0051 .0040 .0031 .0024 .0018 .0014 70-74 ...... 0139 .0112 .0089 .0071 .0055 .0043 .0033 .0025 .0019 .0015 .0011 .0008 .0006 75-79 ...... 0066 .0051 .0040 .0031 .0024 .0018 .0013 .0010 .0007 .0006 .0004 .0003 .0002 .... .0027 .0021 .0016 .0012 .0009 .0006 .0005 .0003 .0002 .0002 .0001 .0001 .0001 0 80+ ...... 00 Proportion under given age Age 1 ...... 0254 .0293 .0335 .0379 .0426 .0474 .0525 .0577 .0631 .0685 .0741 .0797 .0854 5 ...... 1013 .1157 .1309 .1468 .1634 .1805 .1979 .2156 .2334 .2513 .2692 .2870 .3047 10 ...... 1896 .2139 .2393 .2653 .2919 .3188 .3457 .3725 .3990 .4250 .4505 .4754 .4995 15 ...... 2779 .3098 .3424 .3753 .4082 .4409 .4729 .5042 .5346 .5638 .5918 .6185 .6438 20 ...... 3063 .4034 .4406 .4774 .5136 .5488 .5826 .6150 .6457 .6747 .7019 .7273 .7509 25 ...... 4532 .4931 .5324 .5705 .6073 .6423 .6754 .7063 .7351 .7617 .7862 .8085 .8289 30 ...... 5370 .5775 .6166 .6539 .6891 .7220 .7524 .7803 .8058 .8288 .8495 .8681 .8846 35 ...... 6170 .6561 .6930 .7276 .7597 .7890 .8156 .8396 .8609 .8799 .8966 .9112 .9240 40 ...... 6920 .7279 .7612 .7918 .8196 .8445 .8666 .8862 .9033 .9181 .9309 .9419 .9513 45 ...... 7607 .7921 .8207 .8464 .8693 .8894 .9069 .9221 .9351 .9461 .9555 .9633 .9699 50 ...... 8232 .8482 .8713 .8917 .9095 .9249 .9380 .9491 .9584 .9661 .9726 .9779 .9822 55 ...... 8755 .8955 .9130 .9281 .9410 .9519 .9611 .9686 .9749 .9800 .9841 .9874 .9901 60 ...... 9195 .9336 .9457 .9560 .9646 .9717 .9775 .9822 .9860 .9891 .9915 .9934 .9949 65 ...... 9534 .9623 .9697 .9759 .9810 .9851 .9884 .9910 .9931 .9947 .9960 .9969 .9977 Parameter ofstable populations Birth rate ...... 0331 .0382 .0438 .0497 .0559 .0625 .0694 .0764 .0837 .0912 .0988 .1066 .1145 Death rate ...... ··.· .. ··.···· .0431 .0432 .0438 .0447 .0459 .0475 .0494 .0514 .0537 .0562 .0588 .0616 .0645 ORR (27) ...... 1.99 2.27 2.59 2.95 3.36 3.82 4.34 4.92 5.58 6.33 7.17 8.11 9.16 ORR (29) ...... 2.02 2.33 2.68 3.09 3.55 4.07 4.69 5.35 6.12 6.99 7.98 9.11 10.38 ORR (31) ...... 2.06 2.40 2.79 3.24 3.76 4.36 5.04 5.83 6.74 7.77 8.96 10.32 11.87 ORR (33) ...... 2.10 2.47 2.91 3.41 4.00 4.68 5.48 6.41 7.48 8.73 10.18 11.85 13.79 Average age ...... 29.8 28.0 26.2 24.5 23.0 21.5 20.1 18.8 17.7 16.6 15.6 14.7 13.8 Births/population 15-44 ...... 069 .079 .091 .105 .121 .139 .160 .183 .209 .239 .272 .309 .351 ·"

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 5 Males (OeD = 27.67 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0217 .0252 .0290 .0331 .0374 .0419 .0467 .0516 .0566 .0618 .0670 .0723 .0777 1-4 ...... 0689 .0791 .0899 .1013 .1131 .1253 .1377 .1503 .1629 .1755 .1881 .2005 .2128 5-9 ...... 0820 .0920 .1023 .1127 .1231 .1333 .1432 .1528 .1620 .1706 .1788 .1863 .1933 10-14 ., ...... 0827 .0905 .0982 .1055 .1123 .1186 .1243 .1294 .1338 .1374 .1404 .1428 .1445 15-19 ...... 0835 .0891 .0943 .0988 .1026 .1057 .1080 .1096 .1106 .1108 .1104 .1095 .1080 20-24 ...... 0829 .0863 .0891 .0910 .0922 .0926 .0923 .0914 .0899 .0879 .0854 .0826 .0795 25-29 . '" ...... 0812 .0824 .0829 .0826 .0817 .0800 .0778 .0751 .0720 .0687 .0651 .0614 .0576 30-34 ...... 0787 .0779 .0764 .0743 .0716 .0684 .0649 .0611 .0572 .0531 .0491 .0452 .0414 35-39 . " ...... '" ...... 0752 .0726 .0695 .0659 .0619 .0577 .0534 .0490 .0447 .0406 .0366 .0328 .0293 40-44 ...... 0706 .0665 .0620 .0574 .0526 .0478 .0431 .0386 .0344 .0304 .0267 .0234 .0204 45-49 ...... 0648 .0595 .0542 .0489 .0437 .0387 .0341 .0298 .0258 .0223 .0191 .0163 .0139 50-54 ...... " ...... 0577 .0518 .0460 .0404 .0352 .0305 .0261 .0223 .0189 .0159 .0133 .0110 .0092 55-59 ...... '" ...... 0494 .0432 .0374 .0321 .0273 .0230 .0192 .0160 .0132 .0108 .0088 .0072 .0058 60-64 ...... 0397 .0338 .0286 .0239 .0198 .0163 .0133 .0108 .0087 .0069 .0055 .0044 .0034 65-69 ...... " ...... 0289 .0240 .0198 .0162 .0131 .0105 .0083 .0066 .0052 .0040 .0031 .0024 .0019 70-74 ...... , ...... , ...... 0184 .0149 .0120 .0095 .0075 .0059 .0046 .0035 .0027 .0021 .0016 .0012 .0009 ..... 75-79 ...... 0095 .0075 .0059 .0046 .0035 .0027 .0020 .0015 .0011 .0008 .0006 .0005 .0003 0 80+ ...... 0044 .0034 .0026 .0020 .0015 .0011 .0008 .0006 .0004 .0003 .0002 .0002 .0001 \0 Age Proportion under given age 1 ...... 0217 .0252 .0290 .0331 .0374 .0419 .0467 .0516 .0566 .0618 .0670 .0723 .0777 5 ...... 0905 .1043 .1189 .1344 .1505 .1672 .1844 .2018 .2195 .2373 .2551 .2729 .2905 10 ...... 1725 .1963 .2212 .2471 .2736 .3005 .3276 .3546 .3815 .4079 .4339 .4592 .4838 15 ...... 2552 .2868 .3194 .3525 .3859 .4191 .4519 .4840 .5152 .5454 .5743 .6020 .6283 20 ...... 3387 .3760 .4137 .4513 .4885 .5248 .5599 .5937 .6258 .6562 .6847 .7115 .7364 25 ...... 4216 .4623 .5027 .5423 .5806 .6174 .6523 .6851 .7157 .7440 .7701 .7941 .8158 30 ...... 5028 .5448 .5856 .6249 .6623 .6974 .7301 .7602 .7877 .8127 .8352 .8555 .8735 35 ...... 5814 .6227 .6621 .6993 .7339 .7658 .7949 .8213 .8449 .8658 .8844 .9007 .9149 40 ...... 6566 .6953 .7316 .7652 .7958 .8235 .8483 .8703 .8896 .9064 .9210 .9335 .9442 45 ...... 7272 .7618 .7936 .8225 .8484 .8714 .8915 .9089 .9240 .9368 .9477 .9569 .9645 50 ...... 7920 .8214 .8478 .8714 .8921 .9101 .9256 .9387 .9498 .9591 .9668 .9732 .9784 55 ...... 8497 .9731 .8938 .9118 .9273 .9406 .9517 .9610 .9687 .9750 .9801 .9842 .9876 60 ...... 8991 .9163 .9312 .9439 .9546 .9636 .9709 .9770 .9819 .9858 .9889 .9914 .9934 65 ...... 9387 .9501 .9597 .9678 .9744 .9799 .9842 .9878 .9906 .9927 .9945 .9958 .9968 Parameter ofstable populations Birth rate ...... 0269 .0313 .0361 .0413 .0469 .0527 .0588 .0651 .0717 .0784 .0853 .0923 .0994 Death rate ...... 0369 .0363 .0361 .0363 .0369 .0377 .0388 .0401 .0417 .0434 .0453 .0413 .0494 GRR (27) ...... 1.67 1.90 2.17 2.48 2.82 3.21 3.65 4.15 4.71 5.34 6.05 6.85 7.75 GRR (29) ...... 1.69 1.95 2.24 2.58 2.97 3.41 3.91 4.48 5.13 5.87 6.71 7.65 8.73 GRR(3l) ...... 1.71 1.99 2.32 2.69 3.13 3.63 4.20 4.86 5.62 6.49 7.48 8.62 9.93 GRR (33) ...... 1.73 2.04 2.40 2.82 3.30 3.87 4.53 5.30 6.20 7.24 8.44 9.83 11.45 Average age ...... 31.6 29.6 27.7 25.9 24.2 22.7 21.2 19.8 18.5 17.4 16.3 15.3 14.4 Births/population 15-44 ...... 057 .066 .076 .088 .101 .116 .134 .153 .175 .200 .228 .260 .296

J TABLB II. "WEST" MODBL STABLE POPULATIONS ARRANGBD BY LEVEL OF MORTALITY (continued) LEVEL 7 Males (oeo = 32.48 years)

Annual raze ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0189 .0221 .0257 .0295 .0335 .0378 .0423 .0470 .0518 .0567 .0617 .0668 .0720 1-4 ...... 0631 .0730 .0837 .0949 .1066 .1187 .1312 .1438 .1565 .1693 .1820 .1947 .2071 5-9 ...... 0765 .0866 .0970 .1075 .1181 .1286 .1389 .1489 .1585 .1676 .1762 .1842 .1917 10-14 ...... 0777 .0858 .0937 .1013 .1086 .1153 .1215 .1270 .1318 .1360 .1394 .1422 .1442 15-19 ...... 0790 .0850 .0906 .0955 .0998 .1034 .1063 .1083 .1097 .1103 .1103 .1097 .1086 20-24 ...... 0792 .0831 .0863 .0888 .0905 .0915 .0917 .0912 .0900 .0883 .0861 .0835 .0806 25-29 ...... 0784 .0803 .0814 .0816 .0811 .0800 .0781 .0758 .0730 .0698 .0664 .0629 .0592 30-34 ...... 0771 .0769 .0760 .0744 .0721 .0693 .0661 .0625 .0587 .0548 .0508 .0469 .0431 35-39 ...... , ...... 0748 .0729 .0702 .0670 .0634 .0594 .0552 .0510 .0467 .0425 .0385 .0346 .0310 40-44 ...... 0716 .0680 .0639 .0595 .0548 .0501 .0455 .0409 .0365 .0324 .0286 .0251 .0219 45-49 ...... 0671 .0622 .0570 .0518 .0466 .0415 .0367 .0322 .0281 .0243 .0209 .0179 .0153 50-54 ...... 0613 .0554 .0495 .0439 .0385 .0335 .0289 .0247 .0210 .0177 .0149 .0124 .0103 55-59 ...... , ...... 0540 .0475 .0415 .0358 .0306 .0260 .0219 .0182 .0151 .0125 .0102 .0083 .0067 60-64 ...... 0448 .0385 .0328 .0276 .0230 .0191 .0156 .0127 .0103 .0083 .0066 .0052 .0041 65-69 .... , ...... 0341 .0286 .0237 .0195 .0159 .0128 .0102 .0081 .0064 .0050 .0039 .0030 .0023 70-74 ...... 0230 .0188 .0152 .0122 .0097 .0076 .0059 .0046 .0035 .0027 .0021 .0015 .0012 75-79 ...... " .. , ...... 0127 .0102 .0080 .0063 .0049 .0037 .0028 .0021 .0016 .0012 .0009 .0007 .0005 80+ ...... 0065 .0050 .0039 .0029 .0022 .0016 .0012 .0009 .0006 .0005 .0003 .0002 .0002 -0 Age Proportion under given age 1 ...... 0189 .0221 .0257 .0295 .0335 .0378 .0423 .0470 .0518 .0567 .0617 .0668 .0720 5 ...... 0820 .0952 .1093 .1244 .1401 .1566 .1735 .1908 .2083 .2260 .2438 .2615 .2791 10 ...... 1585 .1818 .2063 .2319 .2582 .2852 .3124 .3397 .3668 .3936 .4200 .4457 .4708 15 ...... 2363 .2675 .3000 .3332 .3668 .4005 .4339 .4667 .4986 .5296 .5594 .5879 .6150 20 ...... 3153 .3526 .3905 .4287 .4666 .5039 .5401 .5750 .6083 .6399 .6697 .6976 .7236 25 ...... 3945 .4357 .4769 .5175 .5572 .5954 .6318 .6662 .6984 .7283 .7558 .7812 .8042 30 ...... 4729 .5160 .5582 .5992 .6383 .6753 .7099 .7420 .7714 .7981 .8223 .8440 .8634 35 ...... 5500 .5929 .6343 .6736 .7105 .7447 .7760 .8045 .8301 .8529 .8731 .8909 .9065 40 ...... 6248 .6658 .7045 .7406 .7739 .8041 .8313 .8554 .8768 .8954 .9116 .9255 .9375 45 ...... 6964 .7338 .7684 .8001 .8287 .8542 .8767 .8963 .9133 .9278 .9402 .9506 .9594 50 ...... 7635 .7959 .8254 .8519 .8753 .8957 .9134 .9286 .9414 .9522 .9611 .9686 .9747 55 ...... 8249 .8513 .8750 .8957 .9137 .9292 .9423 .9533 .9624 .9699 .9760 .9810 .9850 60 ...... 8788 .8989 .9164 .9315 .9444 .9552 .9641 .9715 .9775 .9823 .9862 .9893 .9917 65 ...... 9236 .9374 .9492 .9591 .9674 .9742 .9798 .9842 .9878 .9906 .9928 .9945 .9958 Parameter ofstable populations Birth rate ...... 0225 .0265 .0308 .0354 .0404 .0457 .0513 .0570 .0630 .0692 .0755 .0820 .0885 Death rate ...... 0325 .0315 .0308 .0304 .0304 .0307 '.0313 .0320 .0330 .0342 .0355 .0370 .0385 GRR (27) ...... 1.45 1.65 1.89 2.16 2.46 2.80 3.18 3.62 4.11 4.66 5.28 5.98 6.76 GRR (29) ...... 1.46 1.68 1.94 2.23 2.57 2.95 3.39 3.89 4.45 5.10 5.83 6.65 7.59 GRR (31) •...... 1.47 1.71 1.99 2.32 2.69 3.12 3.62 4.19 4.85 5.61 6.47 7.46 8.59 GRR (33) ...... 1.48 1.74 2.05 2.41 2.83 3.32 3.89 4.55 5.32 6.21 7.25 8.45 9.85 Average age ...... 33.2 3I.I 29.1 27.2 25.4 23.7 22.1 20.7 19.3 18.1 16.9 15.9 14.9 Births/population 15-44 ...... 049 .057 .066 .076 .088 .101 .116 .133 .152 .174 .198 .226 .257 .}'

TABLE II "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 9 Males (Oeo= 37.30 years)

Annual rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0168 .0198 .0231 .0267 .0305 .0346 .0389 .0433 .0480 .0527 .0576 .0625 .0675 1-4 ...... 0583 .0680 .0783 .0894 .1010 .1131 .1255 .1382 .1510 .1639 .1768 .1895 .2022 5-9 ...... 0718 .0818 .0922 .1028 .1l36 .1244 .1350 .1453 .1552 .1647 .1737 .1821 .1899 10-14 ...... 0733 .0815 .0896 .0975 .1051 .1122 .1l87 .1247 .1299 .1344 .1383 .1414 .1438 15-19 ...... 0750 .0812 .0871 .0925 .0972 .1012 .1045 .1070 .1087 .1097 .1l01 .1098 .1089 20-24 ...... 0757 .0801 .0837 .0867 .0888 .0902 .0908 .0907 .0899 .0885 .0866 .0842 .0815 25-29 ...... 0758 .0781 .0797 .0804 .0804 .0797 .0782 .0762 .0737 .0707 .0675 .0640 .0604 30-34 ...... , ...... " .... .0753 .0757 .0753 .0742 .0723 .0699 .0669 .0636 .0599 .0561 .0522 .0483 .0445 35-39 ...... '" ... .0741 .0727 .0706 .0678 .0644 .0607 .0567 .0525 .0483 .0441 .0400 .0361 .0324 40-44 ...... 0720 .0689 .0652 .0611 .0566 .0520 .0474 .0428 .0384 .0342 .0303 .0267 .0233 45-49 ...... 0688 .0642 .0592 .0541 .0490 .0439 .0390 .0344 .0301 .0261 .0225 .0193 .1165 50-54 ...... 0642 .0584 .0526 .0468 .0413 .0361 .0313 .0269 .0230 .0194 .0164 .0137 .0114 55-59 ...... " .... .0578 .0513 .0451 .0392 .0337 .0287 .0243 .0203 .0169 .0140 .01l5 .0094 .0076 60-64 ...... 0495 .0428 .0367 .0311 .0261 .0217 .0179 .0146 .0119 .0096 .0077 .0061 .0048 65-69 ...... 0390 .0330 .0275 .0228 .0186 .0151 .0121 .0097 .0077 .0060 .0047 .0037 .0028 70-74 ...... 0275 .0226 .0184 .0149 .0119 .0094 .0074 .0057 .0044 .0034 .0026 .0020 .0015 .0162 .0130 .0103 .0081 .0063 .0049 .0037 .0028 .0021 .0016 ..- 75-79 ...... " .... .0012 .0009 .0006 ..- 80+ ...... 0089 .0070 .0054 .0041 .0031 .0023 .0017 .0013 .0009 .0007 .0005 .0003 .0002 ..- Age Proportion under given age 1 ...... 0168 .0198 .0231 .0267 .0305 .0346 .0389 .0433 .0480 .0527 .0576 .0625 .0675 5 ...... 0751 .0877 .1014 .1l61 .1315 .1477 .1644 .1815 .1990 .2166 .2343 .2520 .2597 10 ...... 1468 .1695 .1936 .2189 .2452 .2721 .2994 .3268 .3542 .3813 .4080 .4341 .4596 15 ...... 2202 .2510 .2832 .3164 .3502 .3843 .4181 .4515 .4841 .5158 .5463 .5755 .6034 20 ...... 2951 .3322 .3703 .4089 .4474 .4855 .5226 .5585 .5928 .6255 .6563 .6853 .7123 25 ...... 3709 .4123 .4541 .4956 .5363 .5767 .6134 .6492 .6827 .7140 .7429 .7695 .7938 30 ...... 4466 .4904 .5338 .5760 .6167 .6553 .6916 .7253 .7564 .7847 .8104 .8335 .8542 35 ...... 5219 .5661 .6091 .6502 .6890 .7252 .7585 .7889 .8163 .8409 .8626 .8818 .8986 40 ...... 5961 .6389 .6796 .7179 .7534 .7859 .8152 .8414 .8646 .8850 .9027 .9180 .9311 45 ...... 6681 .7078 .7448 .7790 .8101 .8379 .8626 .8843 .9031 .9192 .9330 .9446 .9544 50 ...... 7369 .7719 .8041 .8331 .8590 .8818 .9016 .9186 .9331 .9453 .9555 .9640 .9709 55 ...... 8011 .8303 .8566 .8800 .9003 .9179 .9329 .9455 .9561 .9648 .9719 .9777 .9824 60 ...... 8589 .8817 .9017 .9191 .9340 .9466 .9572 .9659 .9730 .9788 .9834 .9871 .9900 65 ...... 9084 .9245 .9384 .9502 .9601 .9683 .9751 .9805 .9849 .9883 .9910 .9932 .9948 Parameter ofstable populations Birth rate ...... 0194 .0229 .0268 .0311 .0356 .0405 .0456 .0510 .0566 .0623 .0682 .0742 .0804 Death rate ...... 0294 .0279 .0268 .0261 .0256 .0255 .0256 .0260 .0266 .0273 .0282 .0292 .0304 GRR (27) •...... 1.29 1.47 1.68 1.92 2.19 2.49 2.84 3.22 3.66 4.16 4.71 5.34 6.04 GRR (29) ...... 1.29 1.49 1.72 1.98 2.28 2.62 3.01 3.45 3.96 4.53 5.18 5.92 6.76 GRR (31) ...... 1.29 1.51 1.76 2.05 2.38 2.76 3.20 3.71 4.29 4.96 5.73 6.61 7.62 GRR (33) ...... 1.30 1.53 1.80 2.12 2.49 2.92 3.42 4.00 4.68 5.47 6.39 7.45 8.69 Average age . . • ...... 34.7 32.5 30.4 28.4 26.5 24.7 23.0 21.5 20.0 18.7 17.5 16.4 15.4 Births/population 15-44 ..... '" .... .043 .050 .058 .061 .Oll .089 .103 .1l8 .135 .154 .176 .201 .229

~

..•_-----~.~~------~. TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 11 Males (Oeo = 42.12 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0151 .0179 .0210 .0244 .0281 .0320 .0361 .0404 .0449 .0495 .0542 .0590 .0639 1-4 ...... 0542 .0636 .0738 .0846 .0962 .1082 .1206 .1333 .1462 .1591 .1721 .1850 .1978 5-9 ...... 0676 .0775 .0879 .0986 .1095 .1205 .1313 .1419 .1522 .1620 .1713 .1800 .1882 10-14 ...... , ...... " .... .0694 .0776 .0859 .0940 .1018 .1092 .1161 .1224 .1280 .1329 .1371 .1405 .1432 15-19 ...... 0713 .0778 .0839 .0896 .0947 .0990 .1027 .1055 .1077 .1090 .1096 .1096 .1090 20-24 ...... '" ., ...... 0725 .0772 .0812 .0845 .0871 .0889 .0899 .0901 .0897 .0885 .0869 .0847 .0821 25-29 .... , ...... 0732 .0760 .0780 .0792 .0796 .0792 .0781 .0764 .0741 .0714 .0683 .0649 .0614 30-34 ...... 0735 .0744 .0745 .0737 .0723 .0702 .0675 .0644 .0609 .0572 .0534 .0495 .0457 35-39 ...... 0732 .0723 .0706 .0681 .0651 .0617 .0579 .0538 .0497 .0455 .0414 .0375 .0337 40-44 ...... 0721 .0694 .0661 .0623 .0581 .0536 .0490 .0445 .0400 .0358 .0318 .0280 .0246 45-49 ...... 0700 .0657 .0610 .0561 .0510 .0459 .0410 .0362 .0318 .0277 .0240 .0207 .0177 50-54 ...... 0664 .0608 .0551 .0494 .0438 .0385 .0335 .0289 .0247 .0210 .0178 .0149 .0124 55-59 " ...... 0611 .0546 .0482 .0421 .0365 .0312 .0265 .0223 .0186 .0154 .0127 .0104 .0085 60-64 ...... 0536 .0467 .0402 .0343 .0289 .0242 .0200 .0164 .0134 .0108 .0087 .0069 .0055 65-69 ...... , ..... '" ... .0436 .0370 .0311 .0259 .0213 .0174 .0140 .0112 .0089 .0070 .0055 .0043 .0033 70-74 ...... 0319 .0264 .0216 .0175 .0141 .0112 .0088 .0069 .0053 .0041 .0031 .0024 .0018 .0197 .0159 .0011 .0008 ..... 75-79 " ...... " .. , ...... 0127 .0101 .0079 .0061 .0047 .0036 .0027 .0020 .0015 ..... 80+ ...... 0118 .0092 .0072 .0055 .0042 .0031 .0023 .0017 .0013 .0009 .0007 .0005 .0003 N Age Proportion under given age 1 ...... 0151 .0179 .0210 .0244 .0281 .0320 .0361 .0404 .0449 .0495 .0542 .0590 .0639 5 ...... 0693 .0815 .0948 .1091 .1242 .1402 .1567 .1737 .1910 .2086 .2263 .2440 .2616 10 ...... 1368 .1590 .1827 .2077 .2338 .2607 .2880 .3156 .3432 .3706 .3976 .4240 .4498 15 ...... 2062 .2366 .2686 .3017 .3356 .3699 .4042 .4380 .4712 .5035 .5346 .5645 .5931 20 ...... 2775 .3144 .3525 .3913 .4303 .4689 .5068 .5436 .5789 .6125 .6443 .6742 .7021 25 ...... 3500 .3916 .4337 .4758 .5174 .5578 .5967 .6337 .6685 .7010 .7312 .7589 .7842 30 ...... 4232 .4675 .5117 .5550 .5970 .6370 .6748 .7101 .7426 .7724 .7994 .8238 .8456 35 ...... 4967 .5419 .5861 .6288 .6692 .7072 .7423 .7744 .8035 .8296 .8528 .8733 .8913 40 ...... 5699 .6142 .6567 .6969 .7344 .7689 .8002 .8282 .8532 .8751 .8943 .9109 .9250 45 ...... 6420 .6836 .7228 .7592 .7924 .8225 .8492 .8727 .8932 .9109 .9260 .9388 .9496 50 ...... 7120 .7493 .7838 .8152 .8434 .8684 .8902 .9090 .9251 .9386 .9500 .9595 .9673 55 ...... 7784 .8102 .8389 .8646 .8872 .9068 .9236 .9379 .9498 .9597 .9678 .9744 .9797 60 ...... 8395 .8648 .8872 .9068 .9237 .9381 .9502 .9602 .9684 .9751 .9805 .9848 .9882 65 ...... 8931 .9114 .9274 .9410 .9526 .9622 .9701 .9766 .9818 .9859 .9892 .9917 .9937 Parameter ofstable populations Birth rate ...... 0169 .0202 .0237 .0277 .0319 .0364 .0412 .0463 .0515 .0569 .0625 .0682 .0740 Death rate ...... 0269 .0252 .0237 .0227 .0219 .0214 .0212 .0213 .0215 .0219 .0225 .0232 .0240 GRR (27) ...... 1.16 1.33 1.52 1.74 1.98 2.26 2.57 2.93 3.32 3.77 4.23 4.85 5.49 GRR (29) ...... 1.16 1.34 1.55 1.79 2.06 2.37 2.72 3.12 3.58 4.10 4.69 5.36 6.12 GRR (31) ...... 1.16 1.36 1.58 1.84 2.14 2.48 2.88 3.34 3.87 4.47 5.17 5.96 6.87 GRR (33) ...... 1.16 1.37 1.61 1.89 2.23 2.61 3.06 3.59 4.20 4.91 5.74 6.70 7.81 Average age ...... 36.1 33.8 31.6 29.5 27.5 25.6 23.9 22.2 20.7 19.3 18.1 16.8 15.8 Births/population 15-44 ...... 039 .045 .052 .060 .070 .080 .093 .106 .122 .140 .160 .182 .207 ... "

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 13 Males (Oeo = 47.11 years)

Annual rate 0/increase -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0136 .0163 .0192 .0225 .0260 .0297 .0337 .0379 .0422 .0467 .0512 .0559 .0606 1-4 ...... 0506 .0597 .0696 .0804 .0917 .1037 .1160 .1287 .1417 .1547 .1678 .1808 .1937 5-9 ...... 0638 .0736 .0840 .0948 .1058 .1169 .1279 .1388 .1493 .1594 .1690 .1781 .1865 10-14 ., ...... 0659 .0741 .0825 .0908 .0988 .1065 .1137 .1203 .1262 .1314 .1359 .1397 .1427 15-19 ...... 0680 .0746 .0810 .0869 .0923 .0970 .1010 .1042 .1066 .1083 .1092 .1095 .1091 20-24 ...... 0696 .0745 .0788 .0825 .0855 .0876 .0889 .0895 .0893 .0885 .0871 .0851 .0827 25-29 ...... 0707 .0739 .0763 .0779 .0786 .0786 .0779 .0764 .0744 .0719 .0690 .0657 .0623 30-34 ...... 0716 .0730 .0735 .0732 .0721 .0703 .0679 .0650 .0617 .0581 .0544 .0506 .0467 35-39 ...... 0721 .0716 .0703 .0683 .0656 .0624 .0588 .0549 .0508 .0467 .0426 .0387 .0349 40-44 ...... 0719 .0696 .0667 .0632 .0592 .0549 .0504 .0459 .0415 .0372 .0331 .0293 .0257 45-49 ...... 0707 .0668 .0624 .0576 .0527 .0477 .0427 .0379 .0334 .0292 .0254 .0219 .0188 50-54 ...... 0682 .0629 .0573 .0516 .0460 .0406 .0355 .0307 .0264 .0225 .0190 .0160 .0134 55-59 ...... 0639 .0574 .0510 .0448 .0390 .0335 .0286 .0242 .0202 .0168 .0139 .0114 .0093 60-64 ...... 0572 .0502 .0435 .0373 .0316 .0265 .0220 .0182 .0148 .0120 .0097 .0078 .0062 65-69 ...... 0479 .0409 .0346 .0289 .0239 .0196 .0159 .0127 .0102 .0080 .0063 .0049 .0038 70-74 ...... 0361 .0301 .0248 .0202 .0163 .0130 .0103 .0081 .0063 .0048 .0037 .0028 .0021 75-79 .... , ...... 0233 .0189 .0152 .0121 .0095 .0074 .0057 .0044 .0033 .0025 .0019 .0014 .0010 .... 80+ ...... 0150 .0118 .0092 .0071 .0054 .0041 .0031 .0023 .0017 .0012 .0009 .0006 .0005 \,U Proportion under given age Age I ...... 0136 .0163 .0192 .0225 .0260 .0297 .0337 .0379 .0422 .0467 .0512 .0559 .0606 5 ...... 0642 .0760 .0889 .1028 .1177 .1334 .1498 .1666 .1839 .2014 .2190 .2367 .2543 10 ...... 1280 .1496 .1729 .1976 .2235 .2503 .2777 .3054 .3331 .3607 .3880 .4147 .4408 15 ...... 1938 .2237 .2554 .2884 .3223 .3568 .3914 .4256 .4593 .4922 .5239 .5544 .5835 20 ...... 2618 .2983 .3363 .3753 .4146 .4538 .4923 .5298 .5660 .6004 .6331 .6638 .6926 25 ...... 3313 .3728 .4152 .4578 .5000 .5414 .5813 .6193 .6553 .6889 .7202 .7489 .7753 30 ...... 4021 .4467 .4914 .5356 .5787 .6200 .6591 .6958 .7297 .7608 .7891 .8147 .8376 35 ...... 4737 .5196 .5649 .6088 .6507 .6903 .7270 .7607 .7914 .8189 .8435 .8652 .8843 40 ...... 5458 .5913 .6352 .6771 .7164 .7527 .7858 .8156 .8422 .8656 .8861 .9039 .9192 45 ...... 6177 .6609 .7019 .7403 .7756 .8076 .8362 .8616 .8837 .9028 .9192 .9332 .9449 50 ...... 6884 .7277 .7643 .7979 .8282 .8552 .8789 .8995 .9171 .9320 .9446 .9550 .9637 55 ...... 7566 .7906 .8216 .8495 .8742 .8958 .9144 .9302 .9435 .9545 .9636 .9711 .9771 60 ...... 8205 .8480 .8726 .8943 .9132 .9294 .9430 .9544 .9637 .9714 .9775 .9825 .9864 65 ...... 8777 .8982 .9161 .9316 .9448 .9559 .9650 .9725 .9786 .9834 .9872 .9902 .9926 Parameter ofstable populations Birth rate ...... 0150 .0179 .0212 .0249 .0288 .0331 .0376 .0423 .0472 .0524 .0577 .0631 .0686 Death rate ...... 0250 .0229 .0212 .0199 .0188 .0181 .0176 .0173 .0172 .0174 .0177 .0181 .0186 ORR (27) ...... 1.06 1.22 1.39 1.59 1.81 2.07 2.35 2.68 3.04 3.46 3.92 4.45 5.03 ORR (29) ...... 1.06 1.22 1.41 1.63 1.88 2.16 2.48 2.85 3.27 3.74 4.29 4.90 5.60 ORR (31) ...... 1.05 1.23 1.43 1.67 1.94 2.26 2.62 3.04 3.52 4.07 4.71 5.43 6.27 ORR (33) ...... 1.05 1.24 1.46 1.71 2.02 2.37 2.78 3.25 3.81 4.46 5.21 6.08 7.09 Average age ...... 37.3 35.0 32.7 30.6 28.5 26.5 24.6 22.9 21.3 19.9 18.6 17.4 16.2 Births/population 15-44 ...... 035 .041 .048 .055 .064 .073 .084 .097 .111 .128 .146 .166 .190 , TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 15 Males (Oeo = 51.83 years)

Annual rate 0/ increase -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0125 .0150 .0179 .0210 .0243 .0280 .0318 .0358 .0400 .0444 .0489 .0534 .0581 1-4 ...... 0476 .0565 .0663 .0768 .0881 .0999 .1123 .1249 .1379 .1510 .1641 .1772 .1902 5-9 ...... 0607 .0704 .0807 .0915 .1025 .1138 .1249 .1360 .1467 .1571 .1669 .1762 .1850 10-14 ...... 0629 .0712 .0796 .0880 .0962 .1041 .1115 .1183 .1245 .1300 .1348 .1388 .1421 15-19 ., ...... " ...... 0652 .0719 .0784 .0846 .0902 .0951 .0994 .1029 .1056 .1076 .1087 .1092 .1090 20-24 ...... 0670 .0722 .0768 .0807 .0839 .0864 .0880 .0889 .0890 .0884 .0871 .0853 .0831 25-29 ...... 0686 .0721 .0748 .0767 .0778 .0781 .0776 .0764 .0746 .0722 .0695 .0664 .0630 30-34 ...... 0701 .0717 .0726 .0726 .0718 .0703 .0682 .0655 .0623 .0589 .0552 .0515 .0477 35-39 ...... 0711 .0710 .0701 .0684 .0660 .0630 .0596 .0558 .0518 .0477 .0437 .0397 .0359 40-44 ...... 0716 .0698 .0672 .0639 .0601 .0560 .0516 .0472 .0427 .0384 .0343 .0304 .0267 45-49 ...... , ...... 0713 .0677 .0636 .0590 .0541 .0492 .0442 .0394 .0348 .0305 .0265 .0229 .0197 50-54 ...... 0696 .0645 .0590 .0534 .0478 .0424 .0372 .0323 .0278 .0238 .0202 .0170 .1)143 55-59 ...... ,. .0661 .0597 .0534 .0471 .0411 .0355 .0304 .0258 .0216 .0180 .0149 .0123 .0100 60-64 ...... 0602 .0530 .0462 .0398 .0339 .0285 .0238 .0197 .0161 .0131 .0106 .0085 .0068 65-69 ...... " ..... '" ... .0513 .0441 .0375 .0315 .0261 .0215 .0175 .0141 .0113 .0089 .0070 .0055 .0043 70-74 ...... '" .... .0397 .0333 .0276 .0226 .0183 .0147 .0116 .0091 .0071 .0055 .0042 .0032 .0024 .... 75-79 ...... '" ...... 0263 .0215 .0174 .0139 .0110 .0086 .0066 .0051 .0039 .0029 .0022 .0016 .0012 .J>, 80+ ...... 0181 .0144 .0113 .0087 .0067 .0051 .0038 .0028 .0021 .0015 .0011 .0008 .0006

Age Proportion under given age 1 ...... 0125 .0150 .0179 .0210 .0243 .0280 .0318 .0358 .0400 .0444 .0489 .0534 .0581 5 ...... 0601 .0715 .0841 .0978 .1124 .1279 .1441 .1608 .1779 .1954 .2130 .2306 .2483 10 ...... 1208 .1419 .1648 .1892 .2150 .2416 .2690 .2968 .3246 .3524 .3799 .4069 .4332 15 ...... 1837 .2131 .2444 .2772 .3112 .3457 .3805 .4151 .4492 .4825 .5147 .5457 .5753 20 ...... 2488 .2850 .3228 .3618 .4013 .4409 .4799 .5180 .5548 .5900 .6234 .6549 .6843 25 ...... 3159 .3572 .3996 .4425 .4853 .5273 .5679 .6069 .6438 .6784 .7105 .7402 .7674 30 ...... 3845 .4292 .4743 .5192 .5630 .6053 .6455 .6833 .7184 .7506 .7800 .8066 .8305 35 ...... 4546 .5009 .5469 .5918 .6348 .6756 .7137 .7488 .7807 .8095 .8353 .8581 .8781 40 ...... 5257 .5720 .6170 .6602 .7008 .7386 .7732 .8045 .8325 .8573 .8789 .8977 .9140 45 ...... 5973 .6417 .6842 .7241 .7610 .7946 .8249 .8517 .8753 .8957 .9132 .9281 .9407 50 ...... 6686 .7094 .7477 .7830 .8151 .8438 .8691 .8911 .9101 .9262 .9397 .9511 .9604 55 ...... 7382 .7739 .8068 .8365 .8630 .8862 .9063 .9234 .9379 .9500 .9599 .9681 .9747 60 ...... 8044 .8337 .8601 .8836 .9041 .9217 .9367 .9492 .9595 .9680 .9749 .9804 .9847 65 ...... 8645 .8867 .9063 .9233 .9379 .9502 .9605 .9688 .9757 .9811 .9854 .9888 .9915 Parameter ofstable populations Birth rate ...... · .0135 .0162 .0193 .0227 .0264 .0304 .0347 .0392 .0439 .0488 .0539 .0591 .0643 Death rate ...... 0235 .0212 .0193 .0177 .0164 .0154 .0147 .0142 .0139 .0138 .0139 .0141 .0143 ORR (27) ...... 0.99 1.13 1.29 1.47 1.68 1.92 2.19 2.49 2.83 3.21 3.65 4.13 4.68 ORR (29) ...... 0.98 1.13 1.31 1.51 1.74 2.00 2.30 2.64 3.03 3.47 3.97 4.55 5.19 ORR (31) ...... 0.97 1.13 1.32 1.54 1.79 2.08 2.42 2.81 3.25 3.76 4.35 5.03 5.80 ORR (33) ...... 0.96 1.14 1.34 1.58 1.85 2.18 2.55 3.00 3.51 4.10 4.80 5.60 6.54 Average age ...... 38.4 36.0 33.7 31.4 29.3 27.2 25.3 23.5 21.9 20.4 19.0 17.7 16.6 Births/population 15-44 ...... 033 .038 .044 .051 .059 .068 .078 .090 .103 .118 .135 .154 .176 .\,

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 17 Males (Oeo = 56.47 years)

Annual rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0115 .0139 .0166 .0196 .0229 .0264 .0301 .0340 .0381 .0424 .0467 .0512 .0558 1-4 ...... 0450 .0537 .0632 .0737 .0848 .0966 .1089 .1216 .1346 .1477 .1609 .1741 .1872 5-9 ...... 0578 .0674 .0776 .0884 .0995 .1108 .1221 .1333 .1443 .1548 .1649 .1745 .1834 10-14 ...... 0601 .0684 .0768 .0853 .0937 .1017 .1094 .1164 .1229 .1286 .1336 .1379 .1414 15-19 ...... 0625 .0693 .0759 .0823 .0881 .0933 .0978 .1016 .1046 .1068 .1082 .1089 .1089 20-24 ...... ··· . .0646 .0699 .0747 .0789 .0824 .0851 .0871 .0882 .0885 .0881 .0871 .0855 .0834 25-29 ...... 0665 .0702 .0732 .0754 .0768 .0774 .0772 .0762 .0747 .0725 .0699 .0669 .0636 30-34 ...... 0684 .0704 .0715 .0719 .0714 .0702 .0683 .0658 .0628 .0595 .0559 .0522 .0484 35-39 ...... 0700 .0702 .0697 .0683 .0662 .0634 .0602 .0565 .0526 .0486 .0446 .0406 .0367 40-44 ...... · ...... 0711 .0696 .0674 .0644 .0608 .0569 .0526 .0482 .0438 .0395 .0353 .0313 .0277 45-49 ...... ·. .0715 .0683 .0644 .0601 .0554 .0505 .0455 .0407 .0361 .0317 .0276 .0239 .0206 50-54 ...... 0707 .0658 .0606 .0551 .0495 .0440 .0387 .0338 .0292 .0250 .0213 .0180 .0151 55-59 ...... , ...... 0680 .0618 .0554 .0492 .0431 .0374 .0321 .0273 .0230 .0192 .0159 .0131 .0108 60-64 ...... 0629 .0557 .0488 .0422 .0361 .0305 .0255 .0212 .0174 .0142 .0115 .0092 .0074 65-69 ...... 0547 .0472 .0403 .0340 .0284 .0234 .0191 .0155 .0124 .0098 .0078 .0061 .0047 70-74 ...... 0433 .0365 .0304 .0250 .0203 .0163 .0130 .0103 .0080 .0062 .0048 .0037 .0028 75-79 ...... , ...... 0296 .0243 .0198 .0158 .0126 .0099 .0077 .0059 .0045 .0034 .0026 .0019 .0014 80+ ...... 0219 .0174 .0137 .0107 .0082 .0062 .0047 .0035 .0026 .0019 .0014 .0010 .0007 -VI Proportion under given age Age 1 ...... 0115 .0139 .0166 .0196 .0229 .0264 .0301 .0340 .0381 .0424 .0467 .0512 .0558 5 ...... 0565 .0676 .0799 .0933 .1077 .1229 .1390 .1556 .1727 .1901 .2076 .2253 .2430 10 ...... ,...... 1143 .1350 .1575 .1816 .2072 .2337 .2611 .2889 .3169 .3449 .3726 .3998 .4264 15 ...... 1744 .2033 .2343 .2669 .3008 .3355 .3705 .4054 .4398 .4735 .5062 .5377 .5678 20 ...... 2369 .2726 .3103 .3492 .3889 .4288 .4683 .5070 .5444 .5803 .6144 .6465 .6767 25 ...... 3014 .3425 .3849 .4281 .4713 .5139 .5554 .5952 .6329 .6684 .7015 .7320 .7601 30 ...... 3680 .4127 .4581 .5035 .5481 .5913 ..6325 .6714 .7076 .7409 .7714 .7989 .8237 35 ...... 4363 .4830 .5297 .5754 .6195 .6615 .7008 .7372 .7704 .8004 .8273 .8511 .8721 40 ...... 5063 .5533 .5993 .6437 .6857 .7249 .7610 .7937 .8230 .8491 .8719 .8917 .9089 45 ...... 5774 .6229 .6667 .7081 .7465 .7818 .8136 .8419 .8669 .8885 .9072 .9231 .9365 50 ...... 6490 .6912 .7311 .7681 .8019 .8323 .8592 .8827 .9029 .9202 .9348 .9470 .9571 55 ...... 7196 .7570 .7917 .8232 .8514 .8763 .8979 .9164 .9321 .9452 .9561 .9650 .9722 60 ...... 7877 .8188 .8471 .8723 .8945 .9136 .9300 .9437 .9551 .9644 .9720 .9781 .9830 65 ...... 8506 .8745 .8958 .9145 .9305 .9441 .9555 .9649 .9725 .9786 .9835 .9873 .9903 Parameter ofstable populations Birth rate ...... 0122 .0148 .0177 .0209 .0245 .0283 .0324 .0367 .0412 .0459 .0508 .0558 .0609 Death rate ...... 0222 .0198 .0177 .0159 .0145 .0133 .0124 .0117 .0112 .0109 .0108 .0108 .0109 ORR (27) ...... 0.92 1.06 1.20 1.38 1.58 1.80 2.05 2.34 2.66 3.02 3.43 3.88 4.40 ORR (29) ...... 0.91 1.06 1.22 1.41 1.62 1.87 2.15 2.47 2.84 3.25 3.73 4.26 4.87 ORR (31) ...... 0.90 1.06 1.23 1.44 1.67 1.95 2.26 2.62 3.04 3.52 4.07 4.70 5.42 ORR (33) ...... 0.90 .106 1.25 1.47 1.72 2.03 2.38 2.79 3.27 3.82 4.47 5.22 6.10 Average age ...... 39.5 37.0 34.6 32.3 30.1 28.0 26.0 24.1 22.4 20.9 19.4 18.1 16.9 Births/population 15-44 ...... 030 .035 .041 .047 .055 .063 .073 .084 .097 .111 .127 .145 .165

f TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 19 Males (Oeo = 61.23 years)

Annual rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0107 .0130 .0155 .0184 .0215 .0249 .0286 .0324 .0364 .0406 .0449 .0493 .0537 1-4 ...... 0425 .0510 .0604 .0707 .0817 .0934 .1057 .1184 .1314 .1446 .1579 .1712 .1844 5-9 ...... 0505 .0644 .0746 .0854 .0965 .1079 .1193 .1307 .1418 .1526 .1629 .1727 .1819 10-14 ...... 0574 .0656 .0741 .0827 .0911 .0994 .1072 .1145 .1212 .1272 .1324 .1369 .1407 15-19 ...... 0598 .0667 .0735 .0800 .0860 .0915 .0962 .1002 .1035 .1059 .1076 .1084 .1086 20-24 ...... " ...... 0621 .0676 .0726 .0770 .0808 .0838 .0860 .0874 .0880 .0878 .0870 .0855 .0836 25-29 ...... 0643 .0683 .0715 .0740 .0757 .0766 .0767 .0760 .0746 .0726 .0702 .0673 .0641 30-34 ...... 0666 .0689 .0704 .0710 .0709 .0699 .0683 .0660 .0632 .0600 .0565 .0529 .0491 35-39 .. " ...... , ...... 0686 .0693 .0690 .0680 .0661 .0636 .0606 .0571 .0533 .0494 .0454 .0414 .0375 40-44 ...... 0704 .0693 .0673 .0646 .0614 .0576 .0535 .0492 .0448 .0404 .0362 .0323 .0285 45-49 ...... 0715 .0686 .0650 .0609 .0564 .0516 .0467 .0419 .0372 .0328 .0287 .0249 .0215 50-54 ...... , ...... , .0714 .0669 .0618 .0565 .0510 .0455 .0402 .0352 .0305 .0262 .0223 .0189 .0159 55-59 .. '" ...... " .... " .... .0697 .0636 .0574 .0511 .0450 .0392 .0337 .0288 .0243 .0204 .0169 .0140 .0115 60-64 ...... 0654 .0583 .0512 .0445 .0382 .0325 .0273 .0227 .0187 .0153 .0124 .0100 .0080 65c69 ...... 0580 .0504 .0432 .0366 .0307 .0254 .0208 .0169 .0136 .0108 .0086 .0067 .0052 70-74 ...... 0470 .0398 .0333 .0275 .0225 .0182 .0145 .0115 .0090 .0070 .0054 .0041 .0031 75-79 . '" ...... " ...... 0331 .0274 .0223 .0180 .0143 .0113 .0088 .0068 .0052 .0039 .0030 .0022 .0016 .... 80+ ...... 0265 .0212 .0168 .0131 .0101 .0077 .0058 .0043 .0032 .0024 .0017 .0012 .0009 0'1 Age Proportion under given age 1 ...... 0107 .0130 .0155 .0184 .0215 .0249 .0286 .0324 .0364 .0406 .0449 .0493 .0537 5 ...... 0532 .0640 .0759 .0891 .1033 .1184 .1343 .1508 .1679 .1852 .2028 .2205 .2381 10 ...... 1082 .1284 .1506 .1744 .1998 .2263 .2536 .2815 .3097 .3378 .3657 .3932 .4200 15 ...... 1655 .1940 .2246 .2571 .2909 .3257 .3609 .3961 .4309 .4650 .4982 .5301 .5607 20 ...... 2254 .2607 .2981 .3371 .3769 .4171 .4571 .4963 .5344 .5709 .6057 .6386 .6693 25 ...... 2875 .3283 .3707 .4141 .4577 .5009 .5431 .5837 .6224 .6588 .6927 .7241 .7529 30 ...... 3518 .3965 .4422 .4881 .5334 .5775 .6198 .6597 .6970 .7314 .7629 .7914 .8171 35 ...... 4184 .4654 .5126 .5592 .6043 .6475 .6880 .7257 .7601 .7914 .8194 .8443 .8662 40 ...... 4870 .5347 .5816 .6271 .6705 .7111 .7486 .7828 .8135 .8408 .8648 .8857 .9038 45 ...... 5574 .6039 .6489 .6918 .7318 .7687 .8021 .8319 .8583 .8812 .9010 .9179 .9323 50 ...... 6289 .6725 .7140 .7527 .7882 .8203 .8488 .8738 .8955 .9140 .9297 .9428 .9537 55 ...... 7003 .7394 .7758 .8091 .8392 .8658 .8890 .9090 .9260 .9420 .9520 .9617 .9696 60 ...... 7699 .8030 .8331 .8602 .8842 .9050 .9228 .9378 .9503 .9606 .9690 .9757 .9811 65 ...... 8353 .8612 .8844 .9047 .9224 .9374 .9500 .9605 .9690 .9759 .9813 .9857 .9891 Parameter ofstable populations Birth rate ...... 0112 .0136 .0163 .0184 .0228 .0264 .0304 .0345 .0389 .0434 .0481 .0530 .0579 Death rate ...... 0212 .0186 .0163 .0144 .0128 .0114 .0104 .0095 .0089 .0084 .0081 .0080 .0079 GRR (27) ...... 0.87 1.00 1.14 1.31 1.49 1.70 1.94 2.21 2.51 2.85 3.24 3.67 4.16 GRR (29) ...... 0.86 1.00 1.15 1.33 1.53 1.76 2.03 2.33 2.68 3.07 3.52 4.02 4.60 GRR (31) ...... 0.85 0.99 1.16 1.35 1.57 1.83 2.13 2.47 2.86 3.31 3.83 4.43 5.11 GRR (33) ...... 0.84 0.99 1.17 1.37 1.62 1.90 2.23 2.62 3.07 3.59 4.20 4.90 5.73 Average age ...... 40.6 38.1 35.6 33.2 30.9 28.7 26.7 24.7 23.0 21.3 19.9 18.5 17.3 Births/population 15-44 ...... , ...... 028 .033 .038 .045 .052 .060 .069 .079 .091 .104 .119 .137 .156

, ,,~-

------~------'-') ", _,'_r ..

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 21 Males (Oeo = 66.02 years)

Annual rate 0/increase -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval Under 1 ...... 0099 .0121 .0146 .0174 .0205 .0238 .0274 .0311 .0351 .0392 .0434 .0477 .0522 1-4 ...... 0403 .0485 .0578 .0679 .0788 .0905 .1028 .1155 .1285 .1418 .1551 .1685 .1818 5-9 ...... 0523 .0617 .0718 .0825 .0936 .1051 .1167 .1282 .1395 .1505 .1610 .1710 .1804 10-14 ...... 0548 .0630 .0715 .0801 .0887 .0971 .1051 .1126 .1195 .1258 .1312 .1359 .1399 15-19 ...... 0573 .0642 .0711 .0777 .0839 .0896 .0946 .0989 .1023 .1050 .1069 .1080 .1083 20-24 ...... 0597 .0653 .0705 .0752 .0792 .0824 .0849 .0865 .0874 .0874 .0868 .0855 .0837 25-29 " ...... '" ... " .... .0622 .0663 .0698 .0726 .0746 .0757 .0761 .0756 .0745 .0727 .0704 .0676 .0645 30-34 ...... 0647 .0673 .0691 .0701 .0702 .0695 .0681 .0660 .0634 .0604 .0570 .0534 .0497 35-39 ...... 0671 .0681 .0682 .0675 .0659 .0637 .0609 .0576 .0539 .0500 .0461 .0421 .0383 40-44 ...... 0694 .0686 .0671 .0647 .0617 .0581 .0541 .0499 .0456 .0413 .0371 .0331 .0293 45-49 ...... 0711 .0686 .0653 .0615 .0572 .0525 .0477 .0429 .0383 .0338 .0296 .0257 .0222 50-54 ...... 0718 .0676 .0628 .0576 .0522 .0468 .0415 .0364 .0316 .0273 .0233 .0197 .0166 55-59 ...... , ... .0709 .0651 .0590 .0528 .0467 .0408 .0353 .0302 .0256 .0215 .0179 .0148 .0122 60-64 ...... 0676 .0605 .0535 .0467 .0403 .0343 .0290 .0242 .0200 .0164 .0133 .0107 .0086 65-69 ...... '" ...... 0611 .0534 .0460 .0392 .0330 .0274 .0225 .0183 .0148 .0118 .0094 .0074 .0058 70-74 ...... 0508 .0432 .0364 .0302 .0248 .0201 .0161 .0128 .0101 .0078 .0061 .0046 .0035 75-79 " ...... 0369 .0307 .0251 .0204 .0163 .0129 .0101 .0078 .0060 .0045 .0034 .0026 .0019 .... 80+ ...... 0321 .0258 .0205 .0161 .0124 .0095 .0072 .0054 .0040 .0029 .0021 .0016 .0011 ~ Age Proportion under given age 1 ...... 0099 .0121 .0146 .0174 .0205 .0238 .0274 .0311 .0351 .0392 .0434 .0477 .0522 5 ...... 0502 .0607 .0724 .0853 .0993 .1143 .1301 .1466 .1636 .1809 .1985 .2162 .2340 10 ...... 1025 .1223 .1442 .1678 .1930 .2194 .2468 .2748 .3031 .3314 .3595 .3872 .4143 15 ...... 1573 .1853 .2156 .2479 .2817 .3165 .3519 .3874 .4226 .4571 .4907 .5231 .5542 20 ...... 2146 .2495 .2867 .3256 .3656 .4061 .4465 .4863 .5249 .5621 .5976 .6311 .6625 25 ...... 2743 .3148 .3572 .4007 .4448 .4885 .5314 .5728 .6123 .6496 .6844 .7166 .7462 30 ...... 3364 .3811 .4270 .4733 .5193 .5643 .6075 .6484 .6868 .7223 .7547 .7842 .8108 35 ...... 4011 .4484 .4961 .5434 .5896 .6338 .6756 .7145 .7502 .7826 .8117 .8376 .8605 40 ...... 4682 .5165 .5643 .6109 .6555 .6975 .7365 .7720 .8041 .8327 .8578 .8798 .8988 45 ...... 5376 .5851 .6313 .6756 .7172 .7556 .7906 .8220 .8497 .8740 .8949 .9128 .9280 50 ...... 6087 .6537 .6967 .7371 .7743 .8081 .8383 .8649 .8880 .9077 .9245 .9386 .9503 55 ...... 6805 .7213 .7595 .7947 .8266 .8550 .8798 .9013 .9196 .9350 .9478 .9583 .9669 60 ...... 7514 .7863 .8185 .8475 .8733 .8958 .9151 .9315 .9452 .9565 .9657 .9731 .9791 65 ...... 8190 .8469 .8720 .8942 .9135 .9301 .9441 .9557 .9652 .9729 .9790 .9839 .9877 Parameter ofstable populations Birth rate ...... 0102 .0125 .0152 .0181 .0213 .0248 .0286 .0326 .0369 .0413 .0459 .0506 .0554 Death rate ...... 0202 .0175 .0152 .0131 .0113 .0098 .0086 .0076 .0069 .0063 .0059 .0056 .0054 ORR (27) ...... 0.83 0.95 1.09 1.24 1.42 1.62 1.84 2.10 2.39 2.71 3.08 3.50 3.96 ORR (29) ...... 0.81 0.94 1.09 1.26 1.45 1.67 1.92 2.21 2.54 2.91 3.34 3.82 4.37 ORR (31) ...... 0.80 0.94 1.10 1.28 1.49 1.73 2.01 2.34 2.71 3.14 3.63 4.20 4.84 ORR (33) ...... 0.79 0.93 1.10 1.30 1.53 1.79 2.11 2.47 2.90 3.39 3.97 4.64 5.42 Average age ...... 41.7 39.1 36.6 34.1 31.7 29.5 27.3 25.3 23.5 21.8 20.3 18.9 17.6 Births/population 15-44 .... '" ... " .027 .031 .036 .042 .049 .057 .065 .075 .086 .099 .113 .130 .148

{ TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued) LEVEL 23 Males (Oeo= 71.19 years)

AnllUll1 rate ofincrease -.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Proportion in age interval Age interval Under 1 ...... 0092 .0114 .0138 .0165 .0195 .0228 .0263 .0301 .0340 .0381 .0423 .0466 .0510 1-4 ...... 0377 .0458 .0549 .0649 .0758 .0874 .0997 .1125 .1256 .1390 .1524 .1660 .1794 5-9 ...... 0492 .0584 .0684 .0791 .0903 .1019 .1136 .1253 .1368 .1480 .1588 .1690 .1786 10-14 ...... 0516 .0598 .0683 .0770 .0858 .0943 .1026 .1104 .1175 .1240 .1297 .1347 .1388 15-19 ...... 0541 .0611 .0681 .0749 .0813 .0873 .0926 .0971 .1009 .1038 .1059 .1072 .1078 20-24 ...... 0566 .0624 .0678 .0727 .0770 .0806 .0834 .0853 .0864 .0868 .0863 .0852 .0836 25-29 ...... 0592 .0636 .0674 .0706 .0729 .0744 .0751 .0749 .0740 .0725 .0703 .0677 .0648 30-34 ...... '...... 0619 .0649 .0671 .0685 .0690 .0687 .0676 .0658 .0634 .0605 .0573 .0538 .0502 35-39 ...... 0647 .0661 .0666 .0663 .0652 .0633 .0607 .0576 .0542 .0504 .0466 .0427 .0388 40-44 ...... 0673 .0671 .0660 .0641 .0614 .0581 .0544 .0504 .0462 .0419 .0378 .0337 .0299 45-49 ...... 0697 .0677 .0650 .0615 .0575 .0531 .0485 .0438 .0391 .0347 .0304 .0265 .0230 50-54 ...... 0713 .0676 .0632 .0584 .0532 .0480 .0427 .0376 .0328 .0283 .0243 .0206 .0174 55-59 ...... 0716 .0662 .0604 .0544 .0484 .0425 .0369 .0317 .0270 .0227 .0190 .0157 .0129 60-64 ...... 0698 .0629 .0560 .0492 .0427 .0366 .0310 .0259 .0215 .0177 .0144 .0116 .0093 65-69 ...... 0649 .0571 .0496 .0425 .0359 .0300 .0248 .0203 .0164 .0131 .0104 .0082 .0064 70-74 ...... 0560 .0480 .0406 .0340 .0280 .0228 .0184 .0147 .0116 .0090 .0070 .0054 .0041 75-79 ...... 0427 .0357 .0295 .0240 .0193 .0154 .0121 .0094 .0072 .0055 .0042 .0031 .0023 80+ ...... 0423 .0342 .0273 .0215 .0167 .0128 .0097 .0073 .0054 .0040 .0029 .0021 .0015 -00 Proportion under given age Age 1 ...... 0092 .0114 .0138 .0165 .0195 .0228 .0263 .0301 .0340 .0381 .0423 .0466 .0510 5 ...... 0470 .0571 .0686 .0814 .0953 .1102 .1260 .1425 .1596 .1770 .1947 .2126 .2304 10 ...... 0961 .1155 .1371 .1605 .1856 .2121 .2396 .2678 .2964 .3251 .3535 .3816 .4090 15 ...... 1478 .1753 .2053 .2375 .2714 .3065 .3422 .3782 .4139 .4491 .4832 .5162 .5479 20 ...... 2019 .2364 .2734 .3124 .3527 .3937 .4348 .4753 .5148 .5529 .5892 .6235 .6557 25 ...... 2585 .2988 .3412 .3851 .4298 .4744 .5182 .5607 .6013 .6396 .6755 .7087 .7393 30 ...... 3177 .3624 .4087 .4557 .5027 .5488 .5933 .6356 .6753 .7121 .7458 .7764 .8040 35 ...... 3796 .4273 .4757 .5242 .5716 .6174 .6608 .7013 .7386 .7726 .8031 .8302 .8542 40 ...... 4443 .4933 .5424 .5905 .6368 .6807 .7215 .7590 .7928 .8230 .8496 .8729 .8930 45 ...... 5116 .5604 .6084 .6545 .6982 .7388 .7760 .8094 .8390 .8650 .8874 .9066 .9230 50 ...... 5813 .6282 .6733 .7160 .7557 .7919 .8244 .8531 .8781 .8996 .9178 .9332 .9459 55 ...... 6526 .6958 .7366 .7744 .8090 .8399 .8671 .8907 .9109 .9279 .9421 .9538 .9633 60 ...... 7242 .7620 .7970 .8288 .8573 .8824 .9040 .9224 .9379 .9506 .9611 .9695 .9762 65 ...... 7940 .8249 .8530 .8780 .9000 .9189 .9350 .9484 .9594 .9683 .9755 .9811 .9856 Parameter ofstable populations Birth rate ...... 0094 .0116 .0141 .0169 .0200 .0234 .0271 .0310 .0352 .0395 .0440 .0486 .0534 Death rate ...... 0194 .0166 .0141 .0119 .0100 .0084 .0071 .0060 .0052 .0045 .0040 .0036 .0034 GRR (27) ...... 0.79 0.91 1.04 1.19 1.36 1.55 1.77 2.02 2.30 2.61 2.96 3.36 3.81 GRR (29) ...... 0.78 0.90 1.04 1.21 1.39 1.60 1.85 2.12 2.44 2.80 3.20 3.67 4.20 GRR (31) ...... 0.71 0.90 1.05 1.22 1.42 1.66 1.93 2.24 2.59 3.00 3.48 4.02 4.64 GRR (33) ...... 0.75 0.89 1.05 1.24 1.46 1.71 2.01 2.36 2.77 3.24 3.79 4.43 5.18 Average age ...... 43.1 40.5 37.9 35.3 32.8 30.4 28.2 26.1 24.1 22.4 20.7 19.3 17.9 Births/population 15-44 ...... 026 .030 .035 .040 .047 .054 .062 .072 .083 .095 .109 .125 .142

!fl..) II "

-- , -----, -- AnnexID

TABLES FOR ADJUSTING STABLE ESTIMATES FOR THE EFFECTS OF DECLINING MORTALITY

TABLE III.l. PROPORTIONS TO BE ADDED TO STABLE ESTIMATES OF THE BIRTH RATE AND OF THE GROSS REPRODUCTION RATE TO CORRECT FOR THE EFFECTS OF DECLINING MORTALITY, AT VARIOUS DURATIONS (t YEARS) OF THAT DECLINE AND ASSUMING THAT k = .01

Part (a). For use to correctstable estimates derived from C(x), andfrom ten-year intercensal growth rate

xt 5 10 15 20 25 30 35 40

Birth rate

5 .0 ••••••.••..••••••• -.016 -.030 -.027 -.025 -.028 -.033 -.039 -.043 10 ...... -.008 -.015 -.019 -.023 -.025 -.027 -.029 -.032 15 '0 •••••••••••••••••• .000 .012 .011 .001 -.004 -.006 -.004 -.004

20 •••••••••••••••• 0 ••• .005 .032 .043 .034 .021 .018 .022 .026 25 '0' ••••••••••••••••• .006 .040 .064 .066 .053 .043 .047 .051 30 ...... 006 .043 .073 .085 .081 .072 .069 .073 35 ...... 007 .044 .016 .094 .099 .096 .094 .092 40 ...... 006 .043 .076 .096 .106 .112 .116 .114 Gross reproduction rate

5 ••••••••••• 0 •••••••• -.017 -.032 -.035 -.014 -.005 -.006 -.005 -.010 1 0 ...... -.009 -.017 -.027 -.011 -.002 .005 .006 .001 1 5 .0 .•.••••••..••••••• .000 .010 .013 .012 .019 .026 .032 .031

20 ••••••••••••••••••• 0 .004 .029 .045 .045 .045 .051 .059 .062 25 .0 •••••••••••••••••• .005 .037 .065 .078 .078 .077 .085 .088 30 ...... 005 .040 .074 .096 .106 .107 .108 .111 35 ...... 006 .041 .017 .105 .124 .131 .133 .130

40 ••••••• 0.0 •••••••• 0. .005 .040 .016 .108 .131 .147 .156 .153

Part (b). For use to correctstable estimates derivedfrom C(x), andfromfive-year intercensal growth rate

xt 5 10 15 20 25 30 35 40

Birth rate 5 ...... -.003 -.007 -.010 -.012 -.013 -.014 -.016 -.018 ..I 10 ...... 010 .017 .005 -.002 -.004 -.000 -.001 .000 15 ••••••••••••••• 0 •• 0. .021 .045 .039 .024 .019 .021 .028 .031

20 ••••••••••••• 0 •••••• .025 .062 .070 .055 .045 .045 .054 .059 25 ...... 025 .068 .089 .085 .075 .068 .075 .082 30 ...... 025 .069 .096 .102 .102 .095 .096 .101 35 ...... 023 .066 .098 .109 .118 .117 .118 .119 40 ...... 022 .063 .094 .108 .123 .129 .137 .138 Gross reproduction rate 5 ...... -.005 -.009 -.004 .005 .017 .021 .019 .015 10 ...... , .007 .015 .012 .016 .027 .034 .038 .034 15 ... ,...... 018 .042 .045 .042 .050 .057 .065 .066 20 ...... 022 .058 .076 .073 .076 .081 .092 .096 25 ...... 023 .065 .094 .103 .108 .106 .114 .120 30 ...... 022 .065 .102 .120 .135 .133 .136 .140 35 ...... 021 .063 .103 .127 .150 .155 .158 .158 40 ...... 019 .059 .100 .126 .156 .167 .178 .117

119 Part (d). For use to correct stable estimates derived from C(x), and from average mortality in the five years preceding the census

xt 5 10 15 20 25 30 35 40

Birth rate

5 ...... -.001 -.000 -.000 -.001 -.001 -.000 -.001 -.002 10 ...... 006 .013 .008 .005 .005 .008 .009 .009 15 ...... , .. .0Il .026 .024 .018 .016 .019 .022 .024 x 20 ...... 013 .035 .040 .034 .028 .031 .035 .039 25 ...... 014 .038 .049 .049 .044 .043 .046 .051

30 •••••••••••••• 0 ••••• .014 .040 .054 .059 .059 .058 .059 .063 35 .0 ••••••••••••••• "0 .014 .040 .057 .065 .070 .072 .073 .074 40 ...... 014 .039 .057 .067 .075 .082 .087 .088

I .'Y- Gross reproduction rate

5 ...... -.003 -.003 .004 .016 .026 .032 .032 .030 10 ...... 004 .012 .014 .022 .033 .042 .044 .042 15 ...... 009 .026 .031 .037 .046 .054 .059 .059 20 ...... 012 .035 .048 .054 .060 .057 .073 .076 25 ...... 012 .039 .058 .071 .077 .081 .087 .091 30 ...... 013 .041 .063 .082 .094 .099 .101 .104 35 ...... 013 .041 .066 .088 .105 .Il6 .117 .Il8 40 ...... 013 .040 .066 .090 .Il2 .127 .134 .134

120 TABLB 111.2. VALUES OF TIm INDEX "DBATHS AT AGE 65 AND OVER/DEATHS AT AGB 5 AND OVER" IN "WEST" MODBL STABLB POPULATIONS GIVEN THE RATIO OF ANNUAL BIRTHS TO THE POPULATION AGED 15·44, AND TIm EXPECTATION OF LIFB AT BIRTH

Expectation Ratio ofbirths to population aged 15-44 oflife at birth .040 .050 .060 .070 .080 .090 .100 .110 .120 .130 .140

Females

20 .378 .326 .287 .255 .225 .241 .181 .164 .148 .136 .124 25 .408 .396 .309 .259 .241 .216 .193 .174 .157 .143 .103 30 .439 .378 .333 .294 .259 .232 .207 .186 .167 .152 .139 35 .469 .406 .359 .318 .281 .251 .224 .201 .180 .163 .120 40 .502 .437 .388 .344 .304 .272 .243 .219 .196 .178 .131 45 .536 .470 0418 .373 .332 .298 .266 .240 .216 .196 .144 50 .572 .506 .454 .370 .364 .328 .295 .266 .240 .205 .162 55 .607 .542 .490 .406 .399 .362 .327 .297 .269 .246 .184 60 .650 .589 .538 .492 .447 .410 .373 .342 .312 .287 .264 65 .699 .644 .598 .554 .510 .473 .436 .404 .373 .346 .321 70 .757 .711 .671 .633 .594 .560 .526 .495 .464 .437 .411 75 .830 .797 .768 .739 .709 .682 .654 .627 .601 .577 .554

Males

18.0 ...... 310 .268 .233 .206 .183 .164 .149 .134 .122 .110 .102 22.9 ...... 341 .294 .255 .226 .201 .179 .162 .145 .132 .120 .109 27.7 ...... 375 .323 .282 .247 .219 .179 .177 .159 .145 .130 .115 32.5 ...... 408 .354 .309 .273 .242 .214 .193 .173 .157 .142 .130 37.3 ...... 434 .381 .335 .296 .263 .235 .212 .191 .173 .157 .142 42.1 ...... 471 .414 .366 .325 .291 .261 .234 .211 .191 .173 .157 47.1 ...... 509 .451 .402 .359 .326 .288 .259 .234 .213 .193 .176 51.8 ...... 542 0485 .435 .392 .350 .318 .289 .263 .238 .214 .199 56.5 ...... 581 .526 .477 .433 .391 .357 .326 .298 .272 .246 .228 61.2 ...... 628 .576 .529 .485 .443 .408 .375 .346 .360 .291 .272 66.0 ...... 682 .634 .590 .547 .510 .476 .443 .413 .384 .360 .336 71.2 ...... 759 .718 .682 .622 .615 .582 .599 .524 .498 .472 .447

I •

121 TABLE I1I.3. VALUES OF THE PRODUCT kt IN "WEST" MODEL LIFE TABLES FOR GIVEN INCREASES IN EXPECTATION OF LIFE AT BIRTH DURING A PERIOD OF t YEARS FROM INDICATED LEVELS OF oeO AT THE BEGINNING OF THE PERIOD

Increase Expectation oflife at birth at the beginning ofperiod oft years InOeo 20 24 28 32 36 40 44 48 52 56 60

Females

2 .0575 .0456 .0374 .0314 .0268 .0233 .0204 .0185 .0165 .0144 .0127 4 .1085 .0868 .0715 .0603 .0518 .0451 .0397 .0383 .0318 .0278 .0247 6 .1541 .1242 .1029 .0872 .0751 .0655 .0582 .0548 .0461 .0405 .0360 8 .1953 .1583 .1319 .1121 .0969 .0848 .0780 .0701 .0596 .0525 .0468 10 .2327 .1897 .1587 .1354 .1173 .1033 .0909 .0845 .0723 .0639 .0562 12 .2668 .2187 .1837 .1572 .1366 .1231 .1098 .0979 .0843 .0746 .0636 14 .2982 .2455 .2070 .1777 .1551 .1396 .1242 .1106 .0956 .0840 .0703 16 .3272 .2704 .2287 .1969 .1749 .1548 .1376 .1226 .1064 .0914 18 .3540 .2937 .2492 .2154 .1913 .1692 .1503 .1339 .1158 .0981 20 .3789 .3155 .2684 .2352 .2066 .1827 .1623 .1447 .1232 22 .4022 .3360 .2870 .2517 .2210 .1954 .1736 .1541 .1299 24 .4240 .3552 .3067 .2670 .2345 .2074 .1844 .1615 26 .4445 .3738 .3232 .2814 .2472 .2187 .1938 .1682 28 .4637 .3935 .3385 .2948 .2592 .2295 .2012 30 .4823 .4100 .3529 .3075 .2705 .2389 .2079 32 .5020 .4253 .3663 .3195 .2813 .2462 34 .5185 .4397 .3790 .3308 .2907 .2530 36 .5338 .4531 .3910 .3416 .2980 38 .5482 .4658 .4024 .3510 .3048 40 .5616 .4778 .4131 .3584 42 .5743 .4892 .4225 .3651 44 .5863 .4999 .4299 46 .5977 .5093 .4366 48 .6084 .5167 50 .6178 .5234 52 .6252 54 .6319

Males

2 .0602 .0478 .0392 .0329 .0281 .0244 .0215 .0197 .0171 .0150 .0132 4 .1136 .0909 .0749 .0632 .0542 .0472 .0437 .0381 .0331 .0291 .0257 6 .1613 .1300 .1078 .0913 .0786 .0687 .0634 .0552 .0481 .0423 .0375 8 .2044 .1658 .1381 .1174 .1014 .0909 .0818 .0712 .0622 .0548 .0469 10 .2436 .1987 .1662 .1418 .1230 .1106 .0989 .0862 .0754 .0666 .0550 12 .2794 .2290 .1924 .1646 .1451 .1290 .1149 .1003 .0879 .0759 .0619 14 .3122 .2571 .2167 .1862 .1648 .1461 .1299 .1136 .0997 .0840 .0676 ~ 16 .3426 .2832 .2396 .2083 .1832 .1621 .1440 .1260 .1091 .0909 18 .3707 .3076 .2611 .2280 .2003 .1771 .1572 .1378 .1172 .0967 20 .3968 .3304 .2832 .2464 .2163 .1912 .1697 .1472 .1241 22 .4212 .3520 .3029 .2635 .2313 .2044 .1815 .1553 .1298 24 .4440 .3741 .3213 .2795 .2454 .2169 .1908 .1622 26 .4655 .3938 .3384 .2945 .2586 .2287 .1989 .1679 28 .4876 .4122 .3544 .3086 .2711 .2380 .2058 30 .5074 .4293 .3694 .3218 .2829 .2461 .2116 32 .5257 .4453 .3835 .3343 .2923 .2530 34 .5429 .4603 .3968 .3461 .3003 .2588 36 .5589 .4744 .4093 .3554 .3073 38 .5739 .4876 .4210 .3635 .3130 40 .5880 .5001 .4304 .3704 42 .6012 .5119 .4385 .3762 44 .6137 .5213 .4454 46 .6255 .5294 .4511 48 .6348 .5363 50 .6429 .5420 52 .6498 54 .6555

122 123 Annex IV

TABLES FOR ESTIMATING CUMULATED FERTILITY FROM AGE SPECIFIC FERTILITY RATES

TABLE IV.1. MULTIPLYING FACTORS Wi FOR ESTIMATING THE AVERAGE VALUE OVER FIVE-YEAR AGE GROUPS OF CUMULATED FERTILITY (Fi) ACCORDING TO THE FORMULA

I-I F I = 5 L jj+wJ; }=o

(WHEN/o = o. /l = AGE SPECIFIC FERTILITY RATE FOR AGES 14.5 TO 19.5, fa = FOR AGES 19.5 TO 24.5 ETC.)

Age Interval Exact limits of (i) age interval Multiplying factors we for values off1/fs and iii asIndicated in lower part oftable

1 ...... 15-20 1.120 1.310 1.615 1.950 2.305 2.640 2.925 3.170 2 ...... 20-25 2.555 2.690 2.780 2.840 2.890 2.925 2.960 2.985 3 ...... 25-30 2.925 2.960 2.985 3.010 3.035 3.055 3.075 3.095 4 ...... 30-35 3.055 3.075 3.095 3.120 3.140 3.165 3.190 3.215 5 ...... 35-40 3.165 3.190 3.215 3.245 3.285 3.325 3.375 3.435 6 ...... 40-45 3.325 3.375 3.435 3.510 3.610 3.740 3.915 4.150 7 ...... 45-50 3.640 3.895 4.150 4.395 4.630 4.840 4.985 5.000

/l/f2 .036 .113 .213 .330 .460 .605 .764 .939 iii 31.7 30.7 29.7 28.7 27.7 26.7 25.7 24.7

TABLE IV.2. MULTIPLYING FACTORS W FOR ESTIMATING THE AVERAGE VALUE OVER FIVE-YEAR AGE GROUPS OF CUMULATED FERTILITY (F,) ACCORDING TO THE FORMULA

i-I FI=5 L Jj+wJi }=o

(WHEN/o = O. 11 = AGE SPECIFIC FERTILITY RATE FOR AGES 15 TO 20, 12 = FOR AGES 20 TO 25 ETC.)

'-) Age Interval Exact limits of (i) age Interval Multiplying factors we for values offl/f2 and iii as Indicated In lower part oftable

1 ...... 15-20 .335 .680 1.030 1.390 1.760 2.130 2.460 2.754 2 ...... 20-25 2.025 2.170 2.265 2.330 2.380 2.420 2.455 2.485 3 ...... 25-30 2.420 2.455 2.485 2.510 2.535 2.560 2.580 2.605 4 ...... 30-35 2.560 2.580 2.605 2.625 2.650 2.675 2.700 2.730 5 ...... 35-40 2.675 2.700 2.730 2.760 2.800 2.845 2.895 2.960 6 ...... 40-45 2.845 2.895 2.960 3.040 3.145 3.285 3.470 3.720 7 ...... 45-50 3.195 3.455 3.720 3.980 4.240 4.495 4.750 5.000

/l/h .036 .113 .213 .330 .460 .605 .764 .939 iii 32.2 31.2 30.2 29.2 28.2 27.2 26.2 25.2

124 Annex V

TABLES FOR ESTIMATING MORTALITY FROM CHILD SURVIVORSIllP RATES

TABLE V.2. MULTIPLYING FACTORS FOR ESTIMATING THE PROPORTION OF CHILDREN BORN ALIVE WHO DIE BY AGE a - q(a) - FROM THE PROPORTION DEAD AMONG ClDLDREN EVER BORN REPORTED BY WOMEN CLASSIFIED IN TEN-YEAR AGB INTERVALS

Mortality measure Exacts limits ofage Multiplying factors to obtain q (a) sholD1l in col. 1 from proportion ofchildren estimated Interval ofwomen reported as deadby women ofages specified In col. 2; for values ofPliPI' ffi, (1) (2) and iii' as specified in lower part oftable

{ q(2) ...... 15-25 0.982 1.000 1.021 1.045 1.072 1.l05 1.144 1.193 q(5) ...... 25-35 0.990 1.004 1.018 1.033 1.048 1.064 1.081 1.099 q(l5) ...... 35-45 0.977 0.993 1.009 1.024 1.040 1.056 1.071 1.086 q(25) ...... 45-55 0.990 1.008 1.025 1.043 1.062 1.080 1.099 1.118 q(35) ...... 55-65 0.990 1.007 1.025 1.043 1.061 1.080 1.099 1.119 ...\ Pl/P2 0.387 0.330 0.268 0.205 0.143 0.090 0.045 0.014 m 24.7 25.7 26.7 27.7 28.7 29.7 30.7 31.7 m' 24.2 25.2 26.2 27.2 28.2 29.2 30.2 31.2 ~

125 /'

ADnex VI

A NOTE ON INTERPOLATION

The tables presented in annexes I to V contain entries at small estimated expectation oflife is thus (1.395) (20) -(.395) (25) = 18.03. enough intervals of the argument (e.g., small enough differences in Extrapolation should not ordinarily be attempted more than one­ mortality between model life tables, and small enough differences half interval beyond the tabulated values. in rate of increase in model stable populations incorporating the A more complicated form of interpolation is needed in using same life table) for intervening values to be approximated satis­ the model stable populations given in annex II. Model stable factorily by linear interpolation. This procedure assumes that each populations with mortality not exactly coincident with one of the relevant variable follows a straight line in the interval between model life tables reproduced in annex I can be found by interpolating entries. between two pages, if the rate of increase happens to. be -.005, In the use of most of the tables in the annexes, the interpolation 0.00, 0.005, or one ofthe values at the head ofthe columns on each required is a simple determination ofan intervening value, and there page. Similarly, model stable populations that happen to have is scarcely justification for comment on such a standard procedure. mortality at exactly one of the levels given in annex I, but a rate of However, for completeness the requisite steps will be described. increase not divisible by .005 can be located by interpolation between Consider as an example the use of interpolation in finding a columns on one ofthe pages in annex II. But in general, it is necessary "West" female model life table consistent with the enumerated to locate the model stable population best fitting an observed femalepopulationover ten at the end ofadecade, and the enumerated population by first interpolating between columns on each of two population at all ages at the beginning. Assume that the projection levels of mortality - for example, to locate stable populations ofthe initial population by the survival factors in annex I (table 1.3) with a given rate of increase, but a C(35) that is higher (at one at mortality level 7 (Oeo = 35 years for females) produced a popula­ mortality level) and lower (at the adjacent mortality level) than in tion ten and over 0.8201 times the initial population; that projection the given population. Then an additional interpolation determines by the mortality factors at level 9 (Oeo = 40 years for females) the fractional mortality level in the model stable population agreeing produced a population over ten 0.8416 times the initial population, with the given population in C(35) as well as r. and the recorded female population over ten in the later census was Thus suppose that one wants to find the female model stable 0.8312 times the initial population. How is the level of mortality population with r = 0.0136, and C(35) = .7150. Inspection of the (and the expectation of life at birth) found? tables in annex II indicates that a value in the neighbourhood of The difference between the proportion surviving according to .7150 is found a little more than halfway between the columns model level 9 and model level 7 is 0.8416 - 0.8201, or 0.0215. headed r = .010 and r = .015 at mortality level 7. Interpolation The difference between the proportion surviving according to model then shows that when r = .0136, C(35) is .72 (.7293) + .28 (.6928) level 9 and the recorded proportion surviving is 0.8416 - 0.8312, = .7191 at level 7. This value is slightly higher than .7150; C(35) is or 0.0104. In descending linearly from level 9 towards level 7, one lower in a stable population with the same r and higher oeo' so that must traverse a fraction .0104/.0215 = .484 to arrive at a level that the next step is to find that at level 9, when r = .0136, C(35) is would yield the reported proportion surviving. Thus the estimated .72 (.7098) + .28 (.6715) = .6991. This procedure has found two mortality level is 9 -.484(2), or 8.032. The expectation of life in model stable populations, each with the requisite r, one with a this model life table is 40 -.484(5), or 37.58 years. An equivalent C(35) above .7150, and the other with C(35) below .7150. The f, procedure that is simpler on a desk calculator is to multiply the difference between the two values of C(35) is .0200, and the lower value at one end of the interval for which interpolation is needed subtracted from .7150 is .0159. The interpolation factor (f) is thus by the fraction f (here equal to .484), and the value at the other end .0159/.0200, or .795. Hence the estimated mortality level is (.795) (7) of the interval by 1 -.f(here .516), and add the products. This is + (.205) (9) = 7.41, and 0eo is (.795) (35.0) + (.205) (40.0) = 36.0. simpler because the addition ofthe products is achieved automatically Once these interpolation factors have been determined, they may be through cumulative multiplication. A rule for making what can be used to find any other desired parameters of the model stable a confusing choice is: multiply by the larger number (ofland 1 - f) population with C(35) = .7150 and r = .0136. Consider the estima­ the nearer value. Thus the estimated level ofmortality in the present tion of the birth rate. First the birth rate is found for the stable example is 9(.516) + 7(.484) = 8.032. Nine is multiplied by the population at levels 7 and 9 with r = .0136 (the first is .72 (.04304) larger fraction (by .516) because the recorded proportion surviving + .28 (.03791) = .04160, and the second is.72 (.03832) + .28 (.03357) (0.8312) is closer to the proportion surviving according to level 9 = .03699); then the birth rate at level 7.41 is found «.795) (.04160) (0.8416) than level 7 (0.8201). Once these multipliers have been + (.205) (.03699) = .04065). The estimation ofthe gross reproduction determined (i.e., f and 1 - 0, they may be applied to any of the rate for an estimated mean age of the fertility schedule of, say, life table functions given in annex I for model tables 7 and 9 - to 28.1 years requires additional interpolations. The procedure to be provide estimates of l» or qx at every age, for example. followed is to interpolate (with factors of .55 and .45) between the If the observed quantity is beyond the last tabulated value, so GRR's for m= 27 and 29 in the model stable populations at levels7 that extrapolation instead ofinterpolation is required, the mechanics and 9 with r = .010 and .015 to establish GRR for m = 28.1 in of calculation are not modified. Suppose the fraction surviving these four tabulated stable populations, and then to employ steps according to level 1 were 0.6831, according to level 3 0.7125, and analogous to those used in estimating the birth rate. observed 0.6715. The difference between the given value and that These examples have all been selected from model life table and for level 3 would be 0.0410, between levelland level 3 would be stable populations, but the same principles and procedures apply 0.0294. Hence f is .0410/.0294 = 1.395, and 1 - f is -0.395. The to other tables equally well.

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