Department of International Economic and Social Affairs
Population Studies
ST/ESA/SER.A/107
No. 107
Step-by-Step Guide to the
Estimation of Child Mortality
If~'l
~
United Nations
NewYork,1990
NOTE
The designations employed and the presentation of the material in this publication
do not imply the expression of any opinion whatsoever on the part of the Secretariat
of the United Nations concerning the legal status of any country, territory, city or area
or of its authorities, or concerning the delimitation of its frontiers or boundaries.
ST/ESA/SER.A/ 107
UNITED NAnONS PUBLICAnON
Sales No. E.89.xIII.9
01900
ISBN 92-1-151183-6
Copyright © United Nations 1989
All rights reserved
Manufactured in the United States of America
PREFACE
aimed at promoting widespread understanding and use of
the estimation methods developed in the various population fields. Though the Guide is simpler and less
comprehensive than other manuals covering similar
topics, it does not sacrifice substance to simplification and
thus provides a solid basis for understanding all the
intricacies of the methods available. It is thus useful both
for the demographer wishing to master those methods
and for the non-demographer whose aim is to become
familiar with their main traits.
To accompany the Guide, a program for microcomputers, named QFIVE, was prepared by the Population Division expressly to apply the Brass method as described
here. * Special thanks are due to Kenneth Hill of Johns
Hopkins University, who wrote a preliminary version of
the Guide. It was later expanded by the Population Division, in order to make its contents more accessible to
those who are not familiar with recent demographic techniques.
Acknowledgement is also due to UNICEF for providing part of the financial support that made the Guide possible.
In 1983 estimates of the levels and trends of infant
mortality for all the countries of the world were first
published by the Population Division of the Department
of International Economic and Social Affairs of the
United Nations Secretariat, with the support and
encouragement of the United Nations Children's Fund
(UNICEF) and the assistance of the World Health Organization (WHO) and the United Nations regional commissions (United Nations, 1983a). Since then, those estimates and projections have been updated twice by the
United Nations (United Nations, 1986 and 1988).
Recently, an additional indicator of child mortalitynamely, mortality of children under the age of 5-was
added (United Nations, 1988). Because of the lack of
reliable vital-registration statistics in most developing
countries, the available estimates are often obtained by
the use of indirect estimation methods.
Aware of the possible problems in the interpretation
and use of such methods, UNICEF, which has played an
important role in disseminating information about levels
and trends of mortality in childhood, requested the Population Division of the United Nations Secretariat to
prepare the present guide to the estimation of child mortality in order to familiarize a wide audience with the
estimation methods most commonly used and their
strengths and limitations.
From a demographic perspective, the Guide is closely
related to the series of Population Division manuals
*Inquiries concerning the QFIVE program should be directed to the
Director, Population Division, United Nations, New York, New York
10017.
III
CONTENTS
Page
PREFACE......................................................................................................
111
INTRODUCTION
.
Chapter
I.
INDICATORS OF MORTALITY IN CHILDHOOD.....................................................................
Life tables..........................................................................................................
Measurement of mortality in childhood..................................................................
Cohort versus period measures of mortality in childhood.........................................
Model life tables.................................................................................................
Observed patterns of mortality in childhood and the model life tables........................
II.
DATA REQUIRED FOR THE BRASS METHOD
'"
Nature of the data required...................................................................................
Children ever born and children dead.................................................................
Total female population of reproductive age........................................................
Compilation of the data required...........................................................................
The case of Bangladesh........................................................................................
III.
RATIONALE OF THE BRASS METHOD
Allowing for the age pattern of child-bearing..........................................................
Derivation of the method......................................................................................
Estimating time trends of mortality........................................................................
Limitations of the Brass method associated with its simplifying assumptions...............
IV.
TRUSSELL VERSION OF THE BRASS METHOD....................................................................
Computational procedure......................................................................................
A detailed example..............................................................................................
Compilation of the data required........................................................................
Computational procedure..................................................................................
Estimates of mortality in childhood by sex..........................................................
V.
PALLONI-HELIGMAN VERSION OF THE BRASS METHOD
Data required......................................................................................................
Computational procedure......................................................................................
A detailed example..............................................................................................
Compilation of the data required........................................................................
Computational procedure..................................................................................
Estimations of mortality in childhood by sex.......................................................
VI.
INTERPRETATION AND USE OF THE ESTIMATES YIELDED BY THE BRASS METHOD
Which version of the Brass method should one use?
Use of information on the pattern of mortality in childhood to test the adequacy of
mortality models
What are the consequences of using the "wrong" model?
Analysis of data from successive censuses or surveys..............................................
Overall observations on the use of the Brass method...............................................
VII.
BRASS-MACRAE METHOD..........................................................................................
Nature of the basic data
Derivation of the method and its rationale..............................................................
Limitations of the Brass-Macrae method
Application of the Brass-Macrae method
Data required..................................................................................................
Computational procedure
A detailed example..........................................................................
Comments on the results of the method..................................................................
v
3
3
5
6
7
10
13
13
13
14
14
14
22
22
23
23
23
25
25
27
27
29
32
34
34
34
38
38
38
45
46
46
46
48
50
53
56
56
57
57
57
57
58
58
59
Page
ANNEXES
I.
Coale-Demeny model life table values for probabilities of dying
exact age x, q(x)
II. United Nations model life table values for probabilities of dying
exact age x, q(x)
III. Relationship between infant mortality, q( I), and child mortality,
Demeny mortality models
IV. Relationship between infant mortality, q(l), and child mortality,
Nations mortality models
between birth and
.
61
between birth and
.
68
4qt, in the Coale-
.
79
4ql, in the United
.
80
NOTES ..•......•••...............•.•......................•.....................•..............................................
81
82
83
GLOSSARy •....................................................................................................................
REFERENCES
.
TABLES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Example of a life table
.
Estimates of the probabilities of dying by ages 1 and 5, q(l) and q(5), by major
region, 1950-1955, 1965-1970 and 1980-1985
..
Correspondence between observed proportions of children dead by age group of
mother and estimated probabilities of dying
.
Coefficients for the estimation of child mortality multipliers, k(i), Trussell version of
the Brass method, using the Coale-Demeny mortality models
..
Coefficients for the estimation of the time reference, t(i), for values of q(x), Trussell
version of the Brass method, using the Coale-Demeny mortality models
.
Calculation of the sex ratio at birth from data on children ever born, classified by sex,
from the 1974 Bangladesh Retrospective Survey
.
Application of the Trussell version of the Brass method to data on both sexes from the
1974 Bangladesh Retrospective Survey
.
Application of the Trussell version of the Brass method to data on males from the
1974 Bangladesh Retrospective Survey
.
Application of the Trussell version of the Brass method to data on females from the
1974 Bangladesh Retrospective Survey
.
Coefficients for the estimation of child-mortality multiplers, k(i), Palloni-Heligman
version of the Brass method, using the United Nations mortality models
.
Coefficients for the estimation of the time reference, t(i), for values of q(x), PalloniHeligman version of the Brass method, using the United Nations mortality models
.
Application of the Palloni-Heligman version of the Brass method to data on males
from the 1974 Bangladesh Retrospective Survey
..
Application of the Palloni-Heligman version of the Brass method to data on females
from the 1974 Bangladesh Retrospective Survey
..
Application of the Palloni-Heligman version of the Brass method to data on both sexes
from the 1974 Bangladesh Retrospective Survey
..
Comparison of estimated infant and under-five mortality in Bangladesh with the available mortality models
.
Estimates of infant and under-five mortality in Bangladesh, obtained using the Coa1eDemeny mortality models
.
Estimates of infant and under-five mortality in Bangladesh, obtained using the United
Nations mortality models
.
Estimation of under-five mortality in Tunisia from data from successive censuses,
using model West and the Trussell version of the Brass method
.
Estimation of under-five mortality in Ecuador from data from several sources, using
model West and the Trussell version of the Brass method
.
Estimation of the probability of dying, q(x), for Bamako, Mali, using the BrassMacrae method
.
Estimation of the probability of dying by age 2, q(2), for Solomon Islands, using the
Brass-Macrae method
.
3
6
23
26
27
29
30
32
32
36
37
41
44
45
47
48
49
51
54
58
59
FIGURES
1.
2.
Typical shapes of the l(x)' and Iqx functions of a life table
Relation between cohort and period measures shown by a Lexis diagram
vi
..
.
4
7
I
f
Page
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Relationship between infant mortality, q(l), and child mortality, 4ql, in the CoaleDemeny mortality models
.
Relationship between infant mortality, q( 1), and child mortality, 4ql, in the United
Nations mortality models
·
Comparison of country-specific estimates of infant and child mortality with the Coale..
Demeny mortality models
Comparison of country-specific estimates of infant and child mortality with the United
..
Nations mortality models
Under-five mortality, q( 5), for both sexes in Bangladesh, estimated using model South
and the Trussell version of the Brass method
..
Under-five mortality, q(5), for males and females in Bangladesh, estimated using
.
model South and the Trussell version of the Brass method
Under-five mortality, q( 5), for males in Bangladesh, estimated using the South Asian
model and the Palloni-Heligman version of the Brass method
..
Under-five mortality, q(5), for males and females in Bangladesh, estimated using the
.
South Asian model and the Palloni-Heligman version of the Brass method
Comparison of the estimates of under-five mortality in Bangladesh, obtained using
models South and South Asian
.
Infant and under-five mortality for both sexes in Bangladesh, estimated using the four
..
Coale-Demeny mortality models
Infant and under-five mortality for both sexes in Bangladesh, estimated using the five
United Nations mortality models
..
Range of variation of the possible estimates of infant and under-five mortality for both
sexes in Bangladesh
.
Under-five mortality for both sexes in Tunisia, estimated using model West and the
..
Trussell version of the Brass method
Under-five mortality for both sexes in Ecuador, estimated using model West and the
Trussell version of the Brass method
.
8
9
10
11
31
33
43
44
47
50
51
52
53
54
DISPLAYS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Possible combinations for the compilation of the data on children ever born and children dead by age of mother required by the Brass method........................................
Possible combinations for the compilation of the data on the total number of women
of reproductive age required by the Brass method....................................................
Tabulation of data on children ever born and children surviving as appearing in the
report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality...........
Tabulation of the female population by age group and marital status as appearing in
the report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality.......
First step in the compilation of data on children ever born and children dead for
Bangladesh. .................................. .................................................................... ...
Second step in the compilation of data on children ever born and children dead for
Bangladesh. .................................. .......................................................................
Compilation of data on the total number of women by age group for Bangladesh ........
Alternative compilation of data on the total number of women by age group for Bangladesh (data rejected in the application of the Brass method)
Worksheet for the compilation of data on births in a year by age group of mother for
the Palloni-Heligman version of the Brass method....................................................
Tabulation of data on births in a year as appearing in the report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality..............................................
Compilation of data on births in a year by age group of mother for Bangladesh..........
vii
15
16
17
18
19
20
21
21
35
41
42
INTRODUCTION
this Guide. Another useful source of information on
indirect methods is chapter III of Manual X: Indirect
Techniques for Demographic Estimation (United Nations,
1983b).
The aim of this Guide is to provide the reader with all
the information necessary to apply two methods for the
estimation of child mortality. It does not presuppose familiarity with demography or with basic demographic
measures. The reader is introduced to the basic concepts
encountered in the measurement of mortality, the typical
indicators of mortality in childhood, the rationale underlying the methods described and the data required for
their application. The procedures followed for the actual
application of each method are described in step-by-step
fashion, and some guidance is provided regarding the
interpretation and use of the estimates obtained.
The Guide should be especially useful for persons who
are engaged in programme activities aimed at reducing
levels of infant and child mortality in developing countries and who require measures of such mortality to identify target population groups for whom mortality is high
and to assess programme impact in terms of mortality
reduction.
The conventional measurement of mortality requires
information on the number of deaths and on the population subject to the risk of dying. Typically, the first type
of information is derived from registration systems that
record deaths as they occur; the second is obtained
mostly from censuses. In the majority of developing
countries, either registration systems do not exist or
omission and other errors are so common that measures
based on the data produced fail to reflect properly levels
or trends of mortality.
Over the past twenty years, considerable advances
have been made to compensate for the lack of reliable
vital registration data. A number of methods based on
information obtained exclusively from censuses or surveys have been developed, and census and survey data
have become more commonly available. In this Guide
two methods that use retrospective information on the
children that women have borne will be described. The
first, known as the Brass method (Brass, 1964), has
proved to produce reliable estimates of child mortality in
a variety of circumstances. The second, known as the
Brass-Macrae method (Brass and Macrae, 1984), relies
on information that can be obtained less expensively, and
it promises to be useful in evaluating the impact of local
projects. Because both of these methods rely on information that is only indirectly related to mortality, they are
generally described as indirect estimation methods.
Other methods also exist for estimating child mortality,
but they either require considerably more information
than those described here or have proved less reliable.
The reader interested in obtaining more information
about such methods may consult the list of references in
This Guide contains all the instructions necessary to
apply two versions of the Brass method, named the
Trussell and Palloni-Heligman versions after the persons
who derived them, and another method, developed by
Brass and Macrae. Since the Guide is intended for use by
persons who need not have formal demographic training,
it also includes an introduction to the basic demographic
concepts involved in the estimation of mortality in childhood and detailed descriptions of the nature of the data
required for each method.
The Guide is divided into seven chapters. The first
discusses mortality measurement in general; the next five
are devoted to different aspects of the Brass method; and
the last focuses on the procedure proposed by Brass and
Macrae. The Brass-Macrae method is treated in a single
chapter because of both its relative simplicity and its
recency. At the time of writing, the Brass-Macrae procedure is still in the process of being tested, and its
efficacy cannot be guaranteed in all cases. The Brass
method, on the other hand, has been used for more than
two decades, has already given rise to numerous variations or refinements of the original procedure and,
despite its known limitations, has performed well under a
variety of' circumstances. It is therefore recommended
that every reader become acquainted with at least one
version of the Brass method.
Although the material in this Guide has been presented
in the simplest way possible, the Guide's content is not
necessarily simple, and the reader should not expect to
master it in a single reading. As with any learning process, an understanding of the intricacies of estimating
mortality in childhood can be acquired only incrementally, by working and reworking through examples and
by consulting several times the chapters discussing the
rationale behind the different methods and their limitations.
To master the basics of the Brass method, it is recommended that the reader work through chapters I to IV in
order. The aim should be to master chapters II and IV,
on the data requirements of the Brass method and the
application of one of its variants, while becoming familiar with mortality measurement in general as discussed in
chapter I and with the strengths and limitations of the
Brass method as presented in chapter III. Only after
becoming thoroughly familiar with those chapters should
the reader proceed to chapter V, in which a second ver1
Chapter III. Rationale of the Brass method
This chapter describes heuristically the theoretical
underpinnings of the Brass method and explains why the
method works even under conditions of changing mortality. The limitations of the method and the possible biases
that may result from violations of its basic assumptions
are also discussed.
Chapter IV. Trussell version of the Brass method
This chapter describes, in step-by-step fashion, the
application of one version of the Brass method, that proposed by Trussell. This version uses the Coale-Demeny
model life tables, those most widely used to date. A
detailed example and a brief discussion of the results provide the reader with practical information on the use of
the method.
Chapter V. Palloni-Heligman version of the Brass
method
This chapter describes a second version of the Brass
method, that proposed by Palloni and Heligman. This
version uses the United Nations model life tables for
developing countries. Aside from describing the additional data that this procedure requires, the chapter provides a detailed example and briefly discusses the results
obtained.
Chapter VI. Interpretation and use of the estimates
yielded by the Brass method
This chapter compares the estimates obtained by using
the Trussell and Palloni-Heligman versions of the Brass
method. The problem of selecting an appropriate model
life table is discussed, and the possible biases introduced
by selecting the "wrong" model are assessed. In addition,
the chapter considers the problem of comparing and
assessing estimates obtained from different sources.
Examples are given of how the estimates yielded by the
Brass method can be used to determine mortality trends
in childhood when data from difference sources are available.
Chapter VII. Brass-Macrae method
This chapter describes the basic data needed to apply
the Brass-Macrae method, explains why it works and
discusses its possible limitations. The procedure to apply
the method is then described and illustrated with a
detailed example. The results obtained are discussed in
the light of the limited information available on the general performance of the method.
sion of the Brass method is described. Chapter VI, dealing with the interpretation and use of the estimates
obtained, may be read early on, but the reader will find it
more useful after chapters IV and V have been mastered.
Chapter VII, describing the Brass-Macrae method, may
be studied almost independently from the rest, though it
should not be read without some familiarity with the concepts presented in chapter I.
The Guide is accompanied by a program for microcomputers, named QFIVE, that applies both the Trussell
and the Palloni-Heligman versions of the Brass method.
Although the program can be used without a complete
understanding of the Brass method, it is recommended
that it be used only after mastering, at the very least,
chapters II and IV. The computerized application of the
estimation method is meant to free the analyst from the
drudgery of longhand calculations, but it cannot replace
the analyst's insight into the use and interpretation of the
estimates obtained. Such insight can only be gained by
understanding how a method works and why. The text of
this Guide is meant to lead the reader to that understanding.
For the benefit of those interested in a more detailed
description of the Guide, an annotated outline of its
chapters follows.
Chapter I. Indicators of mortality in childhood
This chapter presents the demographic concepts used
in the estimation of mortality in childhood. The life table,
the basic demographic instrument for the measurement of
mortality, is described in detail. Attention is then focused
on the main indicators of mortality in childhood. The
importance of considering mortality levels over the age
range 0 to 5, rather than the age range 0 to 1, is
explained. Through the discussion of model life table systems, the reader is introduced to different patterns of
mortality in childhood. Examples of estimates for specific
countries are used to illustrate the variety of existing patterns.
Chapter II. Data required for the Brass method
This chapter discusses in detail the data required to
estimate mortality in childhood by using the Brass
method. It provides worksheets to aid the user in compiling the data needed. By working through a detailed
example, the user becomes familiar with possible variations of the basic data.
2
Chapter I
INDICATORS OF MORTALITY IN CHILDHOOD
The population for which the effects of mortality are
represented by a life table is an example of a cohort. A
cohort is a group of persons experiencing the same event
during a given period. For instance, all persons marrying
in 1974 constitute a marriage cohort. Similarly, all persons born in 1923 are a birth cohort. A life table
represents the survivors of a birth cohort. However, the
birth cohort represented by most life tables is not a real
one, since it would not be possible to complete the life
table until all the members of the birth cohort had died.
For instance, a life table representing the survivors of the
1923 birth cohort could not be completed until some time
around 2013 or 2023. Consequently, demographers resort
to hypothetical birth cohorts, that is, cohorts that are a
theoretical fabrication and that do not really exist. Thus,
a period life table represents the effects of mortality on a
hypothetical cohort that is assumed to be subject during
its entire life to the mortality conditions prevalent during
a given period.
Given the number of survivors of a hypothetical birth
cohort by age-the l(x) values-it is easy to calculate the
number of deaths occurring from one age to the next.
LIFE TABLES
A life table is the demographer's way of representing
the effects of mortality. A complete life table consists of
several functions or sets of numbers, each representing a
different aspect of the impact of mortality. 1 Table 1 provides an example of a life table. The core of the life table
is the set of values shown in column 3 under the heading
I (x). Letting x denote age, those values represent the
number of survivors by age out of an initial number of
births (100,000 in table 1). Thus, according to the life
table in table 1, out of 100,000 births, 86,874 persons
survive to age 10 and 71,074 to age 50. These ages
represent "exact ages", that is, the 71,074 survivors are
persons alive at the exact moment at which they reach
age 50: they are not a day older or a day younger than
exact age 50.
The typical shape of the 1(x) function is displayed in
the upper panel of figure 1. The number of survivors
decreases markedly from birth (exact age 0) to ages 2
and 3 and then declines fairly slowly until around age 60,
after which the decline accelerates:
TABLE
0 ................
1 ................
5 ................
10 ................
15 ................
20 ................
25 ................
30 ................
35 ................
40 ................
45 ................
50 ................
55 ................
60 ................
65 ................
70 ................
75 ................
80 ................
EXAMPLE OF A LIFE TABLE
lex)
ndx
qx
"Lx
nmx
ex
(2)
(3)
(4)
(5)
(6)
(7)
(8)
1
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
20
100 000
91 823
88041
86874
85980
84703
83052
81 194
79 119
76798
74175
71074
67015
61765
54539
45 151
33352
20545
8 177
3781
1 167
894
1277
1651
1 858
2075
2321
2624
3100
4059
5250
7226
9389
11 799
12807
20545
.0818
.0412
.0133
.0103
.0149
.0195
.0224
.0256
.0293
.0342
.0418
.0571
.0783
.1170
.1722
.2613
.3840
1.0000
94238
357424
437289
432 136
426708
419388
410 617
400784
389794
377 432
363 122
345 223
321 951
290761
249225
196255
13 474
102911
.0868
.0106
.0027
.0021
.0030
.0039
.0045
.0052
.0060
.0070
.0085
.0118
.0163
.0249
.0377
.0601
.0951
.1996
57.50
61.59
60.18
55.96
51.51
47.25
43.14
39.07
35.03
31.01
27.02
23.09
19.34
15.77
12.53
9.61
7.13
5.01
Age x
(1)
1.
Source: Ansley J. Coale and Paul Demeny, Regional Model Life Tables and Stable Populations
(Princeton. Princeton University Press. 1966. p.17
number of survivors to exact age x
number of deaths between exact ages x and x + n
nqx
probability of dying between exact ages x and x + n
nLx number of person-years lived between exact ages x and x
nmx age-specific mortality rate for age group x to x + n
ex expectation of life at exact age x
l(x)
ndx
3
+n
Figure 1. Typical shapes of the l(x) and ,qx functions of a life table
Survivors of 100,000 female births by age, 1(x)
Thousands of
survivors
100
90
I-
""
80 ~ "-_
70
-------_.
...............
f-
----------
60 -
--
50 -40
I-
30
~
20
~
10
~
"-
""-
""
""
""
""
"
""..
,
,
I
I
_ _--L_
_...L-_----lL.--_.......L_
_.....L.._
_..L..-_----lL.--_---'
I
I
I
10
20
30
40
50
60
70
80
90
I
OL.-_~
Age
Probability of dying at each age, 1q x
Probability
of dying
.20
.18
~
.16
I-
.14
~
.12
I-
.08 ..
.06 ..
.04
.02
l
II
,f-\:,
~
o "-- 10
~I
I
20
"""
I _ - - . . . - - -..........,
30
40
50
Age
4
l
/
l
l
l
,
l
.10 -
l
l
II
I
I
I
60
70
80
90
lived by each member of that cohort will then be that
total divided by the initial cohort size (the radix, denoted
by 1(0). Such an average is known as the expectation of
life at birth, eo, an index that summarizes mortality conditions at all ages. In table 1 the last column shows
values of the ex function, that is, the expectation of life at
exact age x. The first value is eo; the others represent the
average number of additional years of life expected by
each of the survivors to exact age x.
The nLx function also allows the calculation of another
important set of mortality measures: age-specific death or
mortality rates. Death rates measure the velocity at which
deaths occur in a given population through time. Their
numerator is the number of deaths observed at a given
age or for a given age group during a certain period, and
their denominator is the time or duration of exposure to
the risk of dying experienced during that period by the
population being considered. In the case of a life-table
cohort, the time of exposure to the risk of dying is provided by the number of person-years lived between one
exact age and another, that is, by the nLx function.
Hence, the death rate between ages x and x + n, denoted
by nmX' is defined as
Consider ages 20 and 25. In the life table displayed in
table 1, 1(20)-the number of survivors to age 20-is
84,703, and 1(25)-the number of survivors to age 25-is
83,052. Hence, the number of persons dying between
ages 20 and 25 equals the difference between those
numbers, that is, 84,703 - 83,052 = 1,651. Note that
the number 1,651 appears in the line corresponding to
age 20 under the column headed ndx' a notation that
stands for the number of deaths occurring between ages x
and x + n. Thus, sdzo = 1,651. All the other numbers in
column 4 of table 1 are calculated in the same way.
Once the number of deaths occurring in the hypothetical birth cohort in each age interval is known, the probability of dying in each age interval can be calculated. For
instance, if 1,651 deaths occur among the 84,703
survivors to age 20 during the next five years of their
lives, the probability of each dying before reaching age
25 is 1,651/84,703 = .0195. The number .0195 appears
in the line for age 20 of the life table under the column
headed nqx, which is the actuarial notation for the probability of dying between ages x and x + n.
The lower panel of figure 1 illustrates the typical shape
of the probability of dying at each age, /qx. Note that the
probability of dying is generally high among children
under age 5, and especially among children under age 1
(i.e. infants). It falls to a minimum around age 10 and
rises gradually up to age 50 or so. Thereafter it rises
steeply until very high levels are reached in old age.
Although this Guide is concerned mainly with the estimation of probabilities of dying between birth and certain
ages in childhood, nqo, it is worth defining here the rest
of the life-table functions, which are displayed in table 1.
Their derivation requires the introduction of a new concept: the time lived by survivors between exact ages.
Consider again the interval between exact ages 20 and
25 and note that 1(20) is 84,703 and 1(25) is 83,052 (that
is, out of 100,000 persons born alive, 84,703 survive to
exact age 20 and 83,052 survive to exact age 25).
Clearly, each of the 83,052 survivors to age 25 lives five
years between exact ages 20 and 25, for a total of 5 x
83,052 = 415,260 years lived. However, the 1,651 persons who die also contribute some years lived to the
total. Assuming, for simplicity's sake, that all those who
die do so at the midpoint of the interval-that is, at exact
age 22.5-each one therefore contributes 2.5 years of
life, for an additional 2.5 x 1,651 = 4,128 years lived.
Hence, the total number of years lived between exact
ages 20 and 25 by the hypothetical cohort under consideration is the sum of those quantities-419,388. This
value is denoted by s~o and, in general, the nLx function
represents the number of person-years lived by the
hypothetical life-table cohort between exact ages x and
x + n.
The nLx function is the basis for the calculation of a
valuable summary measure of mortality conditions, the
expectation of life at birth. If the nLx values are cumulated from birth to the highest age to which anyone
survives-say 100-the resulting sum will be the total
number of years lived by the hypothetical life-table
cohort during its lifetime. The average number of years
(1.1)
Put another way, nmx is the number of deaths of persons
aged x to x + n per person-year lived by the hypothetical
life-table cohort between those ages. Note that this measure is intrinsically different from nqx, which represents
the number of deaths of persons aged x to x + n per surviving person at age x. In other words, death rates are
measures of deaths per unit of time of exposure, whereas
probabilities of dying are measures of deaths per person
exposed. As columns 5 and 7 of table 1 show, the quantitative difference between the two measures is substantial,
largely because nmx is a rate per person-year while nqx is
generally a probability over a period of five years.
MEASUREMENT OF MORTALITY IN CHILDHOOD
The estimation of mortality in childhood has traditionally focused on mortality below age 1 because, as shown
in figure 1, mortality at early ages is highest among
infants (persons under age 1) and because measures of
mortality for the age range 0 to 1 can be obtained solely
from registration data when those data are reliable.
However, given the lack of reliable registration data in
most developing countries and the widespread use of
indirect methods to estimate mortality in childhood, attention has slowly shifted to the measurement of mortality
over an expanded range in childhood. Thus, UNICEF has
recently been publishing sets of estimates of mortality in
childhood that include not only infant mortality, 1qo, but
also under-five mortality, sqo, for all the countries of the
world (see, for instance, UNICEF, 1986, 1987a, 1987b,
1988a,and 1988b). Such a shift has come about mainly
for two reasons: first, the realization that in many countries mortality levels among children older than 1 can be
substantial, and, second, the fact that the most widely
used indirect method of estimating mortality in child-
5
span of interest corresponds to childhood only or, more
specifically, is the age range 0 to 5, the drawbacks of
dealing with real cohorts are less serious. Consequently,
measures of mortality in childhood referring to real
cohorts, called cohort measures, are relatively common
in the literature. In order to interpret such measures
correctly, the reader should be aware of how cohort
measures differ from period measures, which refer to
hypothetical cohorts that reflect the mortality conditions
prevalent during a given period (a year in most
instances) .
Consider the problem of counting the deaths occurring
before age 1 to the cohort born in 1979. Since the dates
of birth of the members of that cohort are likely to span
the whole range of dates between 1 January 1979 and 31
December 1979, in order to count all deaths before age
1, it is necessary to observe the cohort from 1 January
1979, when its first members are born, to 31 December
1980, when its last members become 1. In other words, a
two-year observation period is necessary to estimate the
incidence of mortality among cohort members aging, on
average, one year.
Now suppose that, rather than being interested in the
deaths occurring in a particular birth cohort, one wants to
know how mortality affects persons under age I in a particular year, say 1980. Note that persons under 1 in 1980
include not only those born between 1 January and 31
December 1980, who will clearly be under age 1 during
the whole year, but also those born in 1979 who will be
under age 1 during at least part of 1980. Thus, to measure mortality among persons under age 1 in 1980, information is needed on deaths occurring in 1980 among
members of two birth cohorts: that born in 1979 and that
born in 1980.
The Lexis diagram (figure 2) provides a graphic illustration of the relation between cohort and period measures. The horizontal axis of the diagram represents time
in calendar years, while the vertical axis represents age.
Then, the diagonal lines in the diagram represent the trajectory of persons as they age. Thus, the line AE
hood, the Brass method, produces more reliable estimates
of under-five mortality than of infant mortality.
Note that the indices used by UNICEF are probabilities of dying between certain ages: infant mortality is the
probability of dying between birth and exact age 1, I qo ;
child mortality is the probability of dying between exact
ages I and 5, 4ql; and under-five mortality is the probability of dying between birth and exact age 5, sqo.
Throughout this Guide probabilities are used as indicators
of mortality in childhood. To simplify notation, probabilities of dying between birth and exact age x, instead of
being denoted by the standard notation, xqo, are denoted
by q(x). Note, however, that in referring to the probability of dying between exact ages 1 and 5, also known as
child mortality, the traditional notation 4ql will be used,
since the age span in this case does not start at birth.
To give the reader an idea of the values that infant and
under-five mortality estimates may take, table 2 shows
average estimates and projections of mortality in childhood for the major regions of the world during the
periods 1950-1955, 1965-1970 and 1980-1985. Note that
the more developed regions exhibit consistently lower
infant and under-five mortality than do developing
regions. Africa, in particular, is characterized by very
high mortality in childhood. Recent estimates prepared by
the United Nations Population Division (United Nations,
1988) show that infant mortality, q( 1), currently varies
from a low of 6 deaths per 1,000 live births to a high of
over 150 deaths per 1,000. Under-five mortality, q(5),
varies between 7 and over 250 deaths per 1,000 live
births. Hence, in countries with the highest mortality,
slightly more than one out of every six children dies
before the age of 1 and one out of every four dies before
reaching age 5.
COHORT VERSUS PERIOD MEASURES OF MORTALITY IN CHILDHOOD
It was stated earlier that, in constructing life tables,
demographers often use hypothetical cohorts because of
the practical constraints inherent in following a real birth
cohort through its entire life. However, when the age
TABLE 2.
ESTIMATES OFTHE PROBABILITIES OFDYING BY AGES 1 AND 5, q( 1) AND q(5).
BY MAJOR REGION. 1950-1955, 1965-1970 AND 1980-1985
Probability of dying by age I
(per 1,000 births)
1950-1955
1965-1970
World ............................ ,.,., ........
More developed regions ................
Less developed regions ..................
156
56
180
103
26
117
Africa .........................................
Latin America ..............................
Northern America .................... " ...
East Asia .....................................
South Asia, ............ ,.....................
Europe ..................... ,..................
Oceania ...................... ,................
Union of Soviet Socialist Republics.
191
125
29
182
180
62
67
73
158
91
22
76
135
30
48
26
Major region
Probability of dying by age 5
(per 1,000 births)
1980-1985
1950-1955
1965-1970
1980-1985
78
16
88
240
73
281
161
32
184
118
19
134
112
62
11
36
103
15
31
25
322
189
34
248
305
77
96
102
261
131
26
106
219
35
67
36
182
88
13
50
157
17
40
31
Source: Mortality of Children under age 5: World Estimates and Projections, 1950-2025, Population
Studies, No. 105 (United Nations publication, Sales No. E.88.XIII.4).
6
represents persons born on 1 January 1979 who reach
age 1 on 1 January 1980, while line BF represents persons born on 31 December 1979 who reach age 1 on 31
December 1980. That is, the parallelogram AEFB
represents the cohort born in 1979 as it ages from to 1.
Since squares of the type BEFC represent yearly periods,
it can be seen that, as the 1979 cohort ages, it spans parts
of the 1979 and 1980 periods. Conversely, in the 1980
period (BEFC), parts of two cohorts find themselves in
to 1: the one born in 1979 and
the age range
represented by the triangle BEF and that born in 1980,
whose triangle is BFC. Thus, the deaths of children
under age 1 in 1980 comprise deaths of some children
born in 1979 and of some born in 1980.
°
°
Figure 2. Relation between cohort and
period measures shown by a Lexis diagram
Age
;-------,-------,------:1
2
1
OIL..1979
o
c
B
A
----J
.....I'...
~:....._
1980
1981
Year
This example illustrates how period measures represent
the combined experience of different birth cohorts,
whereas cohort measures represent the combined experience of cohort measures during different periods. The
Lexis diagram illustrates this by showing how the cohort
diagonals cut across periods while the vertical bars
representing periods cut across different cohorts.
In this Guide, the aim is to obtain period measures of
mortality in all instances. However, to understand how
those measures are derived, it is often necessary to consider the interplay between cohort and period indices.
MODEL LIFE TABLES
In many countries where death registration is incomplete or non-existent, adequate life tables cannot be constructed from the available data. Indeed, little may be
known about the actual age pattern of mortality of their
populations. Model life tables, which represent expected
age patterns of mortality, have been developed for use in
such cases.
A number of model-life-table systems exist, but in this
Guide only two will be used, the Coale-Demeny regional
model life tables (Coale and Demeny, 1983) and the
United Nations model life tables for developing countries
(United Nations, 1982).
The Coale-Demeny life tables consist of four sets or
models, each representing a distinct mortality pattern.
Each model is arranged in terms of 25 mortality levels,
associated with different expectations of life at birth for
females in such a way that eo of 20 years corresponds to
levelland eo of 80 years corresponds to level 25. The
four underlying mortality patterns of the Coale-Demeny
models are called "North", "South", "East" and "West".
They were identified through statistical and graphical
analysis of a large number of life tables of acceptable
quality, mainly for European countries.
The United Nations models encompass five distinct
mortality patterns, known as "Latin American",
"Chilean", "South Asian", "Far Eastern" and "General".
Life tables representative of each pattern are arranged by
expectations of life at birth ranging from 35 to 75 years.
The different patterns were identified through statistical
and graphical analysis of a number of evaluated and
adjusted life tables for developing countries.
Model life tables play a crucial role in the estimation
of mortality in childhood. They underlie the derivation of
the estimation methods themselves and serve in evaluating the results obtained. Thus, to make proper use of
model life tables in estimating child mortality, it is necessary to be familiar with the characteristic patterns that
they embody, especially at younger ages. Figures 3 and 4
show one way of comparing those patterns. In both
graphs the values of infant mortality, q( 1) (the probability of dying by exact age 1), have been plotted against
the values of child mortality, 4ql (the probability of
dying between exact ages 1 and 5), for each model.
Thus, each curve in figures 3 and 4 represents the typical
relationship between infant and child mortality in a given
life table model.
With respect to the Coale-Demeny models for both
males and females, figure 3 shows that for any given
value of q(l) above .15, model East produces the lowest
mortality between ages 1 and 5, 4ql, followed by West
and then North. Model South's pattern at young ages
overlaps that of East for very low values of q( 1), crosses
that of West for intermediate values and goes beyond that
of North at high values of infant mortality. In other
words, model East is appropriate for populations where
the risks of dying between ages 1 and 5 are low with
respect to those of dying in infancy, whereas model
North is appropriate when the former are high with
respect to the latter. Model West, falling in between
those two, is a good compromise as an "average" model.
Figure 4 shows the equivalent comparisons for the
United Nations models. Notice that, in contrast with the
Coale-Demeny models, the curves do not intersect and
that the order of the models varies by sex. However, the
proximity of the curves corresponding to all the patterns
7
Figure 3. Relationship between infant mortality, q(l), and
child mortality, 4q., in the Coale-Demeny mortality models
4q1
Males
...------------------------------,
.35
f-
.30
~
.25
I-
.20
~
.15
f-
.10
I-
.05
4
I
I
I
I
.20
.30
.40
.50
q(1)
Females
q1
.-----------------------------,
.35
.30
.25
.20
.15
.10
.05
O~-..::~---L
.10
L..._
.20
__L
.30
8
........
.40
__L
.50
___l
q(1)
Figure 4. Relationship between infant mortality, q(l), and
child mortality, sq«, in the United Nations mortality models
Males
South Asian
.20
"
""",
.18
,',1" Latin
.16
,~
,,;'/
American
,~
.14
,'/
,'~
,,~
,'~
,j~
.12
...
...
...
,~
,'/
.10
..
,~
~~
,~...
.08
Far Eastern
L~···
.....
..
.......
~
;
.06
.04
.02
OL.......oa:::..Jc:;;....--1_---1._---L._.....L.._....L-_.J.-_.J.-_.L...-_L.---.l_--..._.....
.02
.04
.06
.08
.10
.12
.14
.16
.18
.20
.22
.24
q(1)
Females
.20
Latin American,
.18
.16
I
/
I
I
.14
.12
.10
.08
.06
.04
.02
0L.----L:.:...-.....L._..J-_..L...----1_--'-_...L-_..J..-----I_---I-_--I-_..J..-----I
.02
.04
.06
.08
.10
.12
.14
9
.16
.18
.20
.22
.24
q(1)
in general may consult chapter I of Manual X: Indirect
Techniques for Demographic Estimation (United Nations,
1983b), chapters 1 and 2 of Regional Model Life Tables
and Stable Populations (Coale and Demeny, 1983) and
chapters I to IV of Model Life Tables for Developing
Countries (United Nations, 1982).
except the Chilean one implies that the United Nations
models are less differentiated at younger ages than the
Coale-Demeny models. The marked differences existing
between the Chilean pattern and all the others should be
noted, especially since the Chilean pattern is "more East
than East", in that it represents the experience of a population whose mortality risks between ages 1 and 5 are
very low compared with those below age 1.
The importance of these models and their distinctive
traits will become evident during the description and use
of the methods presented below. Those interested in
obtaining a more detailed description of model life tables
Figure 5.
OBSERVED PATTERNS OF MORTALITY IN CHILDHOOD
AND THE MODEL LIFE TABLES
To give the reader a sense of how well the mortality
models available reflect the actual experience of different
populations, figures 5 and 6 compare the relationship
Comparison of country-specific estimates of infant and child mortality
with the Coale-Demeny mortality models
q1
4
.20
r--------------------------------,
.... South
....
...
..
..
••••
.15
...
~
I
North
//
.... /~
...~/
/
..
.~/
, ~..
..
//...
/
.
..
....
X
/
,.,.
,.
,.,.,.
west
Bangladesh/ .... Nepal,'
..
.
.10 -
/
Indonesia //
/
/
//
..,"
.'.'
/'.
..j / Haiti,.,.'"
. , ',.,
/East
,.
//
.,."
//
~,/
-:
Kenya ,;lPeru
,.,-'"
///
Guatemala*. / /
.. ,.,.'" Lesotho...:
•
,/,/ Turkey *
.05
~
Colombia ~/
Thailand
JI
/
..."~'
'
""",,--./
,.
~
Panama -:;4 .~'
;
,_'
/ ...".s:~
.AI~'!:.:.~
I
.05
...,"
,.'..
..
,/
//
,/
,/
",./
• Barbados*
,/,/.
//,/
Turkey
//
.."Barbados*
¥"'"
,.
fIIII"""
.~'
.,~:."
.,.....
»: .... ,//
y.
/"
Venezuela Korea./
o
..
,/
~
..,/'/
Costa Rica
I
.10
I
I
.15
.20
q(1)
left); for Guatemala the estimates used refer to 1975-1980 (see Mortality of. Children under Age 5: World Estimates and Projections, 19502025, Population Studies, No. 105 (United Nations publication, Sales
No. E.88'xm.4). For the mortality models (North, South, East. West),
see Ansley J. Coale and Paul Demeny, Regional Model Life Tables and
Stable Populations (Princeton, Princeton University Press, 1966). ..
*Estimates obtained from sources other than the World Fertility
Survey.
Sources: For most countries, the estimates shown are those referring
to the period 0-9 years before the World Fertility Survey and published
in Shea O. Rutstein. Infant and Child Mortality: Levels, Trends and
Demographic Differentials, Comparative Studies, No. 24 (London,
World Fertility Survey, 1983); the upper right estimate for Turkey was
obtained from a multiround survey (1966-1967 Turkish Demographic
Survey); the estimates for Barbados were calculated from vital registration data referring to 1945-1947 (upper right) and 1959-1961 (lower
10
q( l) and 4ql used here were obtained from the births
and deaths of children under age 5 occurring during the
10 years preceding each survey.
Bearing in mind that the accuracy and reliability of the
country estimates displayed in figures 5 and 6 may vary,
it is useful to consider the degree to which they can be
approximated by existing mortality models. Figure 5
shows that the Coale-Demeny models provide excellent
approximations for a few countries: Colombia is North,
Thailand and Venezuela are West, Bangladesh is South,
and Costa Rica and the upper point for Turkey are East.
For a few other countries, the models provide very good
between the infant and child mortality typical of the
different model life tables with that observed in selected
countries. The data for most of the countries depicted
were derived from the set of World Fertility Surveys carried out during the second half of the 1970s. The estimates for Barbados and Guatemala and one of the estimates for Turkey were obtained from other sources.
For the most part, the estimates of q( l) and 4ql plotted in figures 5 and 6 were derived from the fertility histories of women interviewed. A fertility history is the
series of dates of birth and, if appropriate, dates of death
of all the children a woman has had. The estimates of
Figure 6.
Comparison of country-specific estimates of infant and child mortality
with the United Nations mortality models
q1
4
.20 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,
/
Latin American
,/
South Asian
.15
/
II
//,l III
General
/
II
1
//:11
••••••• Far Eastern
- - - Chilean
/
,
"
/
/
,/ /
,/
I
/
I
I
I
I
Bangladesh " II / •
.10
....../
,,........ I
,
..
"
.
"
,,' /
Indonesia
"
Kenya
Guatemala*.
.05
.
.
""
"
"
,/
",
",,-
,,""
"/....
/
/.
Lesotho
/
/
/.
,,'."
Turkey*~"
.,,"
,,"
"
r"
Turkeya/. Barbados*
/'
,'/.
,,'"h h
,,",,"
Thailand
"""
,,"
.,,'"
."."."
!-.....,...:; Barbados*
"
........~.... Costa Rica
OL..-....====-_----.... .L..:..:.:..:.;:;,:;:.:;..:...;.:,;,.
Venezuela
.05
,I" /(.
/.
,,' /....
Colombia /:/./.
Korea
"
'
..L..-
..J-
.15
.10
....L..-
.20
~
q(1)
ity of Children under Age 5: World Estimates and Projections. 19502025. Population Studies, No. 105 (United Nations publication, Sales
No. E.88.XIII.4). For the mortality models (Latin American, Chilean,
South Asian, Far Eastern and General), see Model Life Tables for
Developing Countries, Population Studies, No. 77 (United Nations publication, Sales No. E.81.XllI.7).
*Estimates obtained from sources other than the World Fertility
Survey.
Sources: For most countries, the estimates shown are those referring
to the period 0-9 years before the World Fertility Survey and published
in Shea O. Rutstein, Infant and Child Mortality: Levels. Trends and
Demographic Differentials. Comparative Studies, No. 24 (London,
World Fertility Survey, 1983); the upper right estimate for Turkey was
obtained from a multiround survey (1966-1967 Turkish Demographic
Survey); the estimates for Barbados were calculated from vital registration data referring to 1945-1947 (upper right) and 1959-1961 (lower
left); for Guatemala the estimates used refer to 1975-1980 (see Mortal-
11
though in particular instances one set of models provides
a better fit than the other. Thus, while Costa Rica
appears to be exactly East and Venezuela and Thailand
are clearly West, Panama is Latin American. There
remain, however, countries that are not adequately
approximated by any of the models used here. Indonesia,
Guatemala, Lesotho and the upper point for Barbados are
some examples. To a lesser extent, Peru and Haiti also
belong to that group, although they are closer to the
United Nations models than to the Coale-Demeny
models.
These comparisons show that the models generally
provide good fits for the data observed on actual populations. However, and perhaps not surprisingly, the real
world exhibits greater variety than is captured by available models. Even though some of the differences
between the models and country-specific estimates may
arise from errors in the basic data, it is certain that
different mortality patterns exist. As will be seen, one of
the challenges in using the Brass method is to make
allowance for the appropriate mortality pattern in each
case.
approximations: Korea, Panama and Kenya are close to
North, and the lower point for Barbados is close to East.
There are other countries, however, for which the
approximation provided by the Coale-Demeny models is
less satisfactory, though compromises may be reached if
a single model needs to be selected to represent each of
them. Thus, for instance, Guatemala and Indonesia may
be approximated by model North; Peru and Nepal by
South; Haiti by West; and Lesotho, the lower point for
Turkey and the upper point for Barbados by East.
Figure 6 shows the same comparison between the
United Nations models and country-specific estimates. It
is interesting to note that some of the countries that were
fit very well by the Coale-Demeny models are no longer
close to any United Nations model (for instance, Korea,
Colombia and Kenya). On the other hand, some countries
whose estimates are relatively far from the CoaleDemeny models are close to certain United Nations
models (e.g., Nepal fits the General pattern and Turkey is
close to the Chilean). Among the rest of the countries
considered, the majority can be fit relatively well by
either the Coale-Demeny or the United Nations models,
12
Chapter II
DATA REQUIRED FOR THE BRASS METHOD
At least since the 1940s, demographers working in
developing countries have been aware that the proportion
of children dead among those ever borne by women in a
given age group is an indicator of mortality in childhood.
The actual proportion observed is known to be largely
determined by two factors: the mortality risks to which
children are exposed and the duration of exposure to
those risks. In 1964 William Brass proposed a method
that permitted the estimation of mortality risks by making
allowance for the duration of exposure, and thus it
became possible to derive estimates of various values of
q(x)-the probability of dying between birth and exact
age x-from the observed proportions of children dead.
As suggested above, data on children ever born and
children dead were available in some developing countries long before an estimation method was devised. Even
today, the power of the Brass method stems not only
from its theoretical underpinnings but also from its use of
data that are relatively easy to obtain and whose reliability is generally acceptable. As with any other estimation
method, the quality of the basic data used as input largely
determines the quality of the resulting estimates. It is
essential, therefore, to ensure that the highest standards
are adhered to at all stages of data-gathering.
Although this Guide does not purport to be a manual
for data collection, it is important for the analyst to have
a clear grasp of what is being measured and of how
different questions can be used to best advantage. Not
only is such understanding necessary to avoid errors in
the application of the estimation method, it is also an
asset in evaluating the estimates obtained.
How many are still alive?
How many living children do you have?
How many children have you had who were born
alive and later died?
Set 4. How many children do you have who live with
you?
How many children do you have who live elsewhere?
How many children have you had who were born
alive and later died?
Notice that the answers to each set of questions will
yield, in some cases by addition or subtraction, the
number of children ever born and the number of children
dead that each woman has had. Notice also that only
those children who were born alive should be counted as
"children ever born". Abortions and, particularly, stillbirths should not be included.
Among the sets of questions presented, set 4 is generally considered to produce the best results, because, by
focusing attention on both the children present and those
absent, it leads to a lower level of omission. Set 3 is also
recommended because, by avoiding the direct use of the
concept of "children ever born", it may be easier for the
respondent to grasp. Sets 1 and 2 may yield imperfect
data when respondents fail to understand that children
dead should also be reported as ever born. They may
also lead to errors in societies where the qualifier "born
alive" is construed to mean "still living". However, in
societies where explicit mention of dead children is not
acceptable, set 2 may provide the best means of gathering
the required information.
In some surveys or censuses, the sets of questions
presented here are posed separately with respect to male
and female children. Data on children ever born and dead
by sex are useful not only because they allow the estimation of mortality by sex, but also because they permit
further evaluation of the quality of the data, as will be
explained later on.
Certain surveys may produce data on children ever
born and dead by gathering information on the fertility
history of each woman. Roughly speaking, such information consists of the dates of birth of all children a woman
has had and the dates of death of all deceased children.
Complete fertility histories allow the calculation of the
total number of children ever born and dead for each
woman and thus permit the application of the Brass
method. They also allow the application of other procedures to estimate child mortality and consequently provide several opportunities to check the internal consistency of the data. However, fertility histories involve conSet 3.
NATURE OF THE DATA REQUIRED
In its simplest variant, the Brass method requires three
pieces of information: the number of children ever born,
the number of children ever born who have died (children dead) and the total female population of reproductive age.
Children ever born and children dead
The information on children ever born and dead is normally obtained as follows. In a surveyor census, women
in a given age range (15 to 49, usually) are asked a certain number of questions about their child-bearing experience. The sets of questions that may be used include:
Set 1. How many children, who were born alive, have
you ever had?
How many of those children have died?
Set 2. How many children, who were born alive, have
you ever had?
13
siderable data-collection efforts and are very costly. For
that reason, they are not considered typical data sources
for the information needed to apply the Brass method.
Being aware of the variety of ways in which information on children ever born and dead may be gathered is
important because the analyst must be prepared to convert existing tabulations of the actual items of information collected by censuses or surveys into the format
required for estimation. Although those data are often
tabulated according to such characteristics as labour-force
participation or education of women-which may allow
the analysis of differentials of mortality in childhood by
socio-economic indicators-the Brass method requires
only that data on children ever born and dead be
classified by age of mother. The traditional five-year age
groups-15-19, 20-24, 25-29, 30-34, 35-39, 40-44 and
45-49-are typically used for tabulation purposes and
produce the data needed for the estimation method.
calculating the ratios of male to female children ever
born. Those ratios are estimates of the sex ratio at birth,
that is, the average number of male births per female
birth. That number is a biological constant that varies little from population to population and is generally found
within the range of 1.03 to 1.08 male births per female
birth. As the example in chapter IV will show, values of
the sex ratio at birth that deviate markedly from the
range of expected values indicate possible deficiencies in
the basic data.
THE CASE OF BANGLADESH
The case of Bangladesh will be used as an example in
the application of two versions of the Brass method, so it
is appropriate to consider here the nature of the data
available for that country. In 1974 a Retrospective Survey of Fertility and Mortality conducted in Bangladesh
included questions on children ever born and children
dead. Display 3 reproduces the tabulation of such information appearing in the published report. From that tabulation it can be inferred that questions such as those constituting set 4 (see p. 13) were used to gather the basic
information and that they were posed only to evermarried women (that is, single women were not even
asked the questions). Consequently, although the second
column is labelled "total women", those numbers should
not be used in applying the estimation method.
Note that even if the table failed to indicate that only
ever-married women were involved, the analyst should
raise questions about the total female population shown,
since, in a country like Bangladesh where fertility has
hardly changed, it would not be expected that the number
of women aged 15-19 would be smaller than the number
of women aged 20-24. The very small size of age group
10-14 would also be highly suspicious and should prompt
further clarification of the true meaning of the data
presented.
Display 4 presents the published tabulation of the
female population by age group and marital status. The
numbers of women appearing in the second column,
labelled "total", should be used in applying the Brass
method. Note that, as expected, the number of women
declines steadily with age, at least until age group 55-59
is reached.
The worksheets (displays 1 and 2) can be used to compile the information necessary for the application of the
Brass method. Consider first the data on children ever
born and dead. At first sight, it is unclear whether the
tabulations in display 3 include the numbers of children
ever born needed to apply the Brass method. The
required numbers can, however, be calculated from the
available data on "children at home", "children away"
and "children dead". The first step is to copy the available numbers onto a reproduction of the worksheet in
display 1, as shown in display 5. The next step is to calculate the missing data, children ever born, as the sum of
columns 2, 4 and 5 of the worksheet in display 5. A
completed worksheet, containing all the data required, is
Total female population of reproductive age
Turning now to the third item of information needed
for the estimation procedure-the total female population
of reproductive age (l5-49)-the reader must be warned
that it is a source of multiple errors. Problems arise
because the method assumes that the data used are
representative of all women aged 15 to 49, irrespective
of their child-bearing or marital status. In practice, some
women fail to provide the information sought, thus
becoming cases of "non-response". More importantly,
some women are purposefully excluded from providing
information, as in countries where it is considered inappropriate to ask single women about their child-bearing
experience. Yet, tabulations of the data on children ever
born and dead usually include a column headed "total
number of women", often without explicitly indicating
that only women actually providing information are
included. Mistakes arise when those numbers of women
are used in the application of the method, which requires
that all women, irrespective of whether they provided
information, be considered.
COMPILATION OF THE DATA REQUIRED
Although, as stated earlier, the Brass method requires
a minimum of information-the number of children ever
born, the number of children dead and the total female
population of reproductive age-that information is collected and published in a variety of ways.2 To aid the
analyst in compiling and organizing the data required,
two worksheets have been prepared (displays 1 and 2).
The worksheets show items of information often found in
actual tabulations. The analyst can obtain the information
necessary to apply the Brass method by addition and subtraction (the appropriate combination or combinations of
items are indicated in the heading of each column).
Note that the worksheets make allowance for the availability of information by sex, although data for both
sexes combined are all that is needed. As indicated
above, when data by sex are available, not only can mortality be estimated for each sex separately, but also the
internal consistency of the information can be checked by
14
Display 1. Possible combinations for the compilation of the data on children ever born and children dead
by age of mother required by the Brass method
Children
Age group
a/mother
Children
ever born
Children
Children
dead
surviving
Jiving
at home
Children
living
elsewhere
(/)
(2)
(3)
(4)
(5)
(2)+(3)
(/)-(3)
(2) + (4) + (5)
(1)-(4)-(5)
(4)+(5)
15-19
20-24
25-29
30-34
35-39
40-44
45-49
15-19
20-24
25-29
30-34
35-39
40-44
45-49
15-19
20-24
25-29
30-34
35-39
40-44
45-49
15
Display 2. Possible combinations for the compilation of the data on the total number of women
of reproductive age required by the Brass method
7htal
number
of women
Ever-married
women
Single
women
lIbmen of
stated
parity
lIbmen of
not-stated
parity
(I)
(2)
(3)
(4)
(5)
(2)+(3)
Age group
of women
(4)+(5)
15-19
20-24
25-29
30-34
35-39
40-44
45-49
number of women as derived from information contained
in the original tabulations (see displays 3 and 4). Display
8 illustrates how the worksheet in display 2 may be used
to compile data on the total number of women by adding
the number of ever-married women copied from display
3 to the number of never-married (single) women copied
from display 4. Note that the resulting total numbers of
women differ, albeit slightly, from those copied directly
from display 4 and shown in display 7. Such differences
arise because although all ever-married women were
asked about their child-bearing experience, some failed to
provide the information requested and were therefore
excluded from the numbers presented in display 3. Since
it is suggested that all women, irrespective of reporting
status, be used in applying the Brass method, the
numbers in display 7 will be used in the examples
presented in chapters IV and V.
presented in display 6. Note that the computed numbers
of children ever born shown in column 1 of display 6 are
the same as those appearing in the original table (display
3) under the heading "total births". Although the data on
children ever born and dead could have been copied
directly from the published tabulation, it is sound practice
to check the internal consistency of published data by
carrying out calculations such as those illustrated in
displays 5 and 6, especially when one is unsure of the
meaning of certain labels ("total births" in this instance).
Turning now to the data on the total number of women
by age group, recall that they should be obtained from
display 4. Display 7 illustrates the compilation of those
data using the worksheet shown in display 2. The
worksheets in displays 6 and 7 now contain the basic data
required to apply the Brass method. It is of interest, however, to explore here the consistency of the data on total
16
Display 3. Tabulation of data on children ever born and children surviving as appearing in the report on the
1974 Bangladesh Retrospective Survey of Fertility and Mortality
BANGLADESH
BANGLADESH CENSUS 1974 RETROSPECTIVE SURVEY OF FERTILITY AND MORTALITY
DE FACTO
TABLE 8 : EVER-MARRIED ~EN BY AGE GROUP, WITH TOTAL CHIlDREN EVER BORNE, NUMBER AT HOME,
NUMBER ELSEWHERE AND NUMBER DEAD, BY SEX OF CHILDREN
ALL EVER-MARRIED
AGE
GROUP
OF
WPlEN
TOTAL
0-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60+
N.S.
TOTAL
MLE BIRTHS ONLY
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
SO-54
55-59
60+
N.S.
I
TOTAL
'O'EN
CHILDREN
AWAY
CHILDREN
DEAD
0
327
349
989
587
S66
846
801
061
559
615
0
1 811
215 365
997 384
1 937 955
2 261 196
2 490 168
2 415 023
1 959 544
1 920 863
1 137 721
3 233 015
0
9 483 700
18 570 045
109
036
220
904
859
071
724
149
032
419
869
0
0
11 047
38 921
82 780
124 046
176 698
268 130
291 071
352 615
263 461
998 608
0
1 003
117 165
529 877
1 047 294
1 204 582
1 333 957
1 291 745
1 030 737
1 018 383
591 871
1 667 762
0
TOTAL
BIRTHS
CHIlDREN
AT HOME
204
6 671
1 160 919
4 901 382
9 085 852
9 910 256
10 384 001
9 164 329
6 905 673
5 963 087
3 257 428
8 136 608
0
4 866
921 227
3 820 649
6 927 908
7 126 473
6 974 267
5 472 460
3 664 328
2 601 163
1 206 148
2 102 978
0
17 018 632
68 876 212
40 822 467
259
2 019
2 521
2 573
2 003
1 766
1 473
1 128
1 040
601
1 631
24
83
219
522
919
1 276
1 281
1 441
913
2 800
4 112
597 248
2 S07 018
4 675 978
5 109 487
5 435 726
4 883 599
3 714 957
3 211 030
1 769 751
4 410 239
0
12 983 983
36 319 145
23 877 392
2 607 377
9 834 376
FEMLE BIRTHS ONLY
10-14
2 565
15-19
479 678
1 526 643
20-24
25-29
2 063 505
1 759 823
30-34
35-39
1 601 696
40-44
1 330 442
45-49
992 793
888 514
SO-54
496 594
55-59
60+
1 303 670
N.S.
0
2 565
563 671
2 394 364
4 409 874
4 800 769
4 948 275
4 280 730
3 190 716
2 752 057
1 487 677
3 726 369
0
1 757
452 191
1 882 429
3 382 004
3 345 614
3 049 196
2 148 736
1 271 179
761 131
291 729
359 109
0
0
13 280
44 428
137 209
398 541
742 868
1 008 716
990 730
1 088 446
6S0098
1 802 007
0
98 200
467 507
890 661
1 056 614
1 156 211
1 123 278
928 807
902 480
545 850
1 565 253
0
32 557 067
16 945 075
6 876 323
8 735 669
TOTAL
4
SOl
1 557
2 106
1 792
1 635
1 369
1 047
955
545
1 468
104
436
318
496
082
100
382
791
877
625
217
111
448
199
614
767
S07
842
262
899
164
170
0
TOTAL
I
~EN
12 445 923
3
469
1 938
3 545
3 780
3 925
3 323
2 393
1 840
914
1 743
808
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), p. 37.
17
Display 4. Tabulation of the female population by age group and marital status as appearing in the report on the
1974 Bangladesh Retrospective Survey of Fertility and Mortality
BANGLADESH
BANGLADESH CENSUS 1974 RETROSPECTIVE SURVEY OF FERTILITY AND MORTALITY
DE FACTO
TABLE 3. POPULATION BY SEX, AGE GROUP. MARITAL STATUS AND NUI'BER OF P1ARRIAGES
F E P1 ALE S
WI[)(IlED
DIVORCED
AGE
GROUP
TOT A L
NEVER
PlARRIED
P1ARRIED
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
10-7.
15-79
5 490 429
6 199 640
4 615 449
3 014 106
2 653 155
2 601 009
2 015 663
1 711 680
1 419 515
1 135 129
1 048 558
601 412
697 117
325 222
329 326
122 956
113 871
18 411
1 410
5 423 801
6 114 626
4 353 919
912 130
121 364
28 426
8 365
4049
3 511
3 555
4 183
2 987
6 614
5008
4 156
1 519
3 121
12 949
256 611
1 928 736
2 415 049
2 462 803
1 811 682
1 519 415
1 204 957
827 838
617 085
288 310
227 089
90506
48 907
18 157
10202
4943
408
5961
5 937
4 141
31 412
48 242
19 334
115629
177 037
262 790
296 957
422 448
312 740
460 656
227 922
273 108
103 037
101 696
71 515
411
15356
68653
62586
33 422
15 426
8 936
6 764
6 196
3665
1940
1917
1 192
594
183
411
207
TOTAL
34 366 124
17 061 120
13 868 828
3 001 633
227 448
MARRIED 0-4
5-9
ONCE
ONLY 10-14
15-19
20-24
Z5-Z9
30-34
35-39
40-44
45-49
SO-54
55-59
60-64
65-69
70-14
15-79
80-84
85+
N.S.
8 815
18 132
213 551
1 952 378
2 366 653
2 383 749
1 810 194
1 580 278
1 289 195
1 003 347
923 454
538 306
621 909
289 210
293 183
110 013
102 649
11 890
819
3 121
12 166
253 839
1 859 246
2 Z66 382
2 282 639
1 693 882
1 414 313
1 051 554
736 408
541 170
251 196
Z04 649
81 399
43 514
16 592
9608
4 736
408
5 754
5366
4 741
Z9 878
45335
12 445
104 368
158431
232 574
262 710
373 598
219 945
415 710
201 009
249 015
93 421
92 630
66 947
411
15 637 185
12 739 422
2 700 348
TOTAL
~
85+
N.S.
TOTAL
805
968
408
14 971
62864
54356
28 073
11 166
6 759
4 860
42Z9
2686
766
1 5SO
802
594
EVER-P1ARRIED NOT STATED
OR EVER-P1ARRIED BUT PRESENT
P1ARITAL STATUS NOT STATED
51 540
66 128
44 8ZZ
13 115
5 914
3 024
4561
2 183
1 553
583
1 177
1 435
781
594
2 561
163
718
183
207 695
390
580
592
718
715
207
399
411
201
194 Z94
3 721
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), p. 28.
18
Display 5. First step in the compilation of data on children ever born
and children dead for Bangladesh
Children
ever born
ChiJdren
dea4
Children
surviving
(1)
(2)
(3)
=
=
(2)+(3)
Age group
of momer
~
'"
-S
=
(2) + (4) + (5)
Children
living
at IUJme
(4)
Children
living
elsewhere
(5)
=
(1)-(3)
=
(1)-(4)-(5)
(4)+(5)
15-19
215 365
921 227
24327
20-24
997384
3 820649
83 349
25-29
I 937 955
6927908
219989
30-34
2 261 196
7 126473
522 587
35-39
2490 168
6974267
919566
40-44
2415 023
5472 460
1 276 846
45-49
I 959544
3664 328
1 281 801
15-19
117 165
469036
11 047
20-24
529877
I 938220
38921
25-29
I 047294
3545904
82 780
30-34
I 204 582
3 780 859
124046
35-39
I 333 957
3925071
176 698
40-44
I 291 745
3 323 724
268 130
45-49
I 030737
2393 149
291 071
15-19
98200
452 191
13 280
20-24
467507
1 882 429
44 428
25-29
890661
3382004
137 209
30-34
1 056614
3345614
398541
35-39
I 156 211
3049 196
742 868
40-44
I 123 278
2 148736
I 008 716
45-49
928807
1 271 179
990730
~
~
~
~
~
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Iertility and Mortality (Dacca, 1977), table 8, p. 37
(reproduced in display 3 above).
19
Display 6. Second step in the compilation of data on children ever born
and children dead for Bangladesh
Children
ever born
Children
dead
Children
surviving
at home
Children
living
elsewhere
(1)
(2)
(3)
(4)
(5)
=
=
(2)+(3)
~
~
~
~
=
=
of mother
(2) + (4) + (5)
(1)-(4)-(5)
=
(4)+(5)
15-19
I 160919
215 365
921 227
24327
20-24
4901 382
997 384
3 820649
83349
25-29
9085852
I 937 955
6927 908
219989
30-34
9 910 256
2 261 196
7 126473
522 587
35-39
10 384 001
2 490 168
6974267
919 566
40-44
9 164 329
2 415 023
5472 460
I 276846
45-49
6905673
I 959544
3664 328
I 281 801
15-19
597 248
117 165
469036
11 047
20-24
2 507 018
529877
I 938220
38921
25-29
4675978
I 047 294
3545904
82780
30-34
5 109487
I 204 582
3780859
124046
35-39
5435726
I 333 957
3925071
176698
40-44
4883599
I 291 745
3323724
268 130
45-49
3714957
I 030737
2393 149
291 071
15-19
563 671
98200
452 191
13 280
20-24
2394364
467 507
I 882 429
44 428
25-29
4409874
890661
3 382004
137 209
30-34
4800 769
I 056614
3 345 614
398541
35-39
4948275
I 156 211
3049 196
742 868
40-44
4280780
I 123 278
2 148 736
I 008 716
45-49
3 190 716
928807
I 271 179
990730
~
~
(1)-(3)
Age group
Children
living
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), table 8, p. 37
(reproduced in display 3 above).
20
Display 7. Compilation of data on the total number of women by age group
for Bangladesh
Total
number
of women
Ever-married
women
Single
women
Ufmten of
stated
parity
not-stated
parity
(1)
(2)
(3)
(4)
(5)
Uhmen of
=
(2)+(3)
=
Age group
of women
(4)+(5)
15-19
3014706
20-24
2 653 155
25-29
2607 009
30-34
2 015 663
35-39
1 771 680
40-44
1 479575
45-49
1 135 129
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), table 3, p. 28
(reproduced in display 4 above).
Display 8. Alternative compilation of data on the total number of women by age group for Bangladesh
(data rejected in the application of the Brass method)
Total
number
a/women
Ever-married
women
Single
women
lIbmen of
stated
parity
lIbmen of
not-stated
parity
(1)
(2)
(3)
(4)
(5)
=
(2)+(3)
Age group
=
of women
(4)+(5)
15-19
2992 166
2019436
972 730
20-24
2642682
2 521 318
121 364
25-29
2601 922
2573496
28426
30-34
2 011 447
2003082
8365
35-39
1 770 149
1 766 100
4049
40-44
1 476 893
1 473 382
3511
45-49
1 132 346
1 128 791
3555
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), tables 3 and 8,
pp. 28 and 37 (reproduced in display 4 above). Total number of women calculated as (2) +(3).
21
Chapter III
RATIONALE OF THE BRASS METHOD
The Brass method derives estimates of q(x)-the probability of dying between birth and exact age x-from the
proporti?n ~f children dead among those ever borne by
women III dIfferent age groups by allowing for the duration of exposure to the risk of dying. The duration of
exposure is related to the age of the woman and to the
timing of child-bearing. On average, the older the
women, the longer ago their children would have been
born and the longer the children would· have been
exposed to the risk of dying.
The basic relationship between the proportion of children dead by age group of mother and the probability of
dying in childhood can be illustrated by a very simple
example. Suppose that all women have all their children
at exactly age 20. Suppose further that mortality risks are
constant over time. The children of women of exact age
22 will have been exposed to the risk of dying for
exactly two years, so that the proportion dead will be
exactly the probability of dying by age 2, q(2). Proportions dead of children ever born for women of exact ages
23 and 25 will similarly estimate q( 3) and q( 5). Note
that it is the age of the women in relation to the time of·
child-bearing that determines the children's duration of
exposure to the risk of dying and thus the relationship
between the proportion dead and some q(x) value.
If mortality is assumed constant, the measures of q(2),
q(3) and q(5) will be applicable to any point in time
during the period preceding enumeration. Now let us
consider what would happen if mortality were falling.
Assume again that all women have all their children at
age 20 and that we are considering women aged 22 in
1978. Then the proportion dead of the children borne by
those women would estimate q(2) for the cohort of children born in 1976. With mortality falling, this cohort
value would be somewhat lower than the q(2) in effect
during 1976 (which would reflect the experience of children born during 1974, 1975 and 1976), but somewhat
higher than the value for 1978 (which would reflect the
experience of children born in 1976, 1977 and 1978).
The q(2) for the 1976 birth cohort would thus estimate
the value of q(2) for some time point between 1976 and
1978 (somewhat closer to 1976 than 1978 because most
of the child deaths for the cohort born in 1976 would
have occurred close to birth). Similarly, the proportion of
children dead for women aged 23 in 1978 would estimate
q(3) for the cohort of children born in 1975, approximating a value of q( 3) for some time point close to that
date. Thus, when mortality has been changing,
information on the proportion of children dead can yield
not only estimates of child mortality but also estimates of
its trends.
The above example is, of course, greatly oversimplified. In practice, women start to bear children at
different ages and have subsequent children after variable
birth intervals. The proportion of children dead for
women of a five-year age group represents a complex
average of different probabilities of dying for varying
periods. That complex average, however, depends largely
on the age pattern of fertility of the women consideredwhich determines the distribution of children by duration
of exposure to risk-and on the level and age pattern of
mortality risks affecting the children.
ALLOWING FOR THE AGE PATTERN OF CHILD-BEARING
The age pattern of child-bearing plays an important
role in determining the relationship between the proportion of children dead among those borne by women of a
particular age group and the children's probability of
dying. To return to the simplified example above, if all
women had their children at age 18 (instead of age 20)
the proportion of children dead of women aged 22 would
estimate the probability of dying by age 4, not age 2.
Since in a life table the probability of dying by age 4,
q( 4), must be greater than the probability of dying by
age 2, q(2), a given proportion of children dead for
women aged 22 would indicate lower mortality risks in
an early-fertility population than in a late-fertility population. Put another way, the children of women of a given
age in an early-fertility population would, on average,
have been born longer ago and would therefore have
been exposed longer to the risks of dying than children
of women of the same age in a late-child-bearing population. It is for this reason that fertility patterns must be
taken into account in converting proportions of children
dead into probabilities of dying.
The Brass method makes allowance for the pattern of
fertility in a population by considering the lifetime fertility of women in different age groups. The measure of
lifetime fertility used is called average parity, which is
defined as the average number of children ever born per
woman of a given age group. (It is calculated by dividing
the total number of children ever borne by women of a
given age group by the total number of women in that
age group.)
If fertility has been constant, the average parity of
women now aged, say, 15 to 19 will be the same as the
average parity five years ago of women who are now
aged 20 to 24. Thus, the current distribution of average
parities by age group can be used as an indicator of the
shape of the lifetime fertility distribution that needs to be
considered in converting proportions of children dead
22
was later relaxed through the work of Feeney (1980),
Coale and Trussell (1978) and others. Those authors
showed that if the rate of change of mortality over time
was approximately constant, the reference date of each
q(x) could be estimated by making allowance for the age
pattern of fertility by means of the P( 1)/P(2) and P(2)/
P(3) ratios.
Since the measurement of child mortality trends is a
major objective of this Guide, the use of the Brass
method to detect or quantify changes in such trends
resulting from programme activities requires further discussion. The procedure for estimating the reference dates
of q(x) values assumes that mortality has been changing
steadily over time. A programme that speeds mortality
decline will invalidate this assumption. Such a change in
trend will have a greater effect on the proportions of children dead of younger women-a higher proportion of
whose children will have been exposed to the recently
lower mortality risks-than on the proportions of children
dead of older women. It will also have some effect on
such proportions for women of any age having significant
numbers of recent births. Consequently, the Brass method
will smooth out any sharp change in trend, transforming
it into a gradually accelerating decline and thus making it
harder to associate a decline with programme activities.
It should also be noted that the Brass method is
unlikely to indicate any change in trend until at least two
years after the change occurs. This lag arises from the
observation, discussed in greater detail below, that mortality estimates based on reports of women aged 15-19which reflect the most recent mortality-are generally
unreliable.
into probabilities of dying in childhood. Specifically,
using P(l), P(2) and P( 3) to denote the average parities
of women in age groups 15-19, 20-24 and 25-29, respectively, allowance for the pattern of fertility is made by
using the parity ratios P(l)/P(2) and P(2)/P(3). Note
that, because parity ratios are used, the actual level of
fertility does not matter, only its age pattern. For example, the greater P(l)/P(2), the earlier the pattern of
child-bearing.
DERIVATION OF THE METHOD
The actual derivation of a Brass-type estimation procedure involves the use of simulation to generate proportions of children dead, the probabilities of dying that they
are related to, and the parity ratios P(l )/P(2) and
P(2)/P(3) that link them. Regression analysis is used to
derive estimation equations, which make the application
of the procedure straightforward.
It is worth nothing that there are several versions of
the Brass method. They differ mostly in the type of
models used to simulate the quantities of interest. The
two versions described in this Guide are those proposed
by Trussell (1975) and by Palloni and Heligman (1986).
They differ mainly in that the former uses the CoaleDemeny regional model life tables to simulate mortality,
while the latter uses the United Nations model life tables
for developing countries. Both versions are presented
here in order to provide the analyst with a wide choice of
possible mortality models.
Table 3 indicates how all Brass-type estimation procedures link estimated q(x) values with the observed proportions of children dead by age of mother, denoted by
D(i). As indicated in the table, an estimate of the probability of dying by age I, q( I ), can be derived from the
proportion of children dead reported by women aged 1519, D(l); the probability of dying by age 2, q(2), can
be obtained from the proportion of children dead for
women aged 20-24, D(2), and so on.
TABLE
3.
LIMITATIONS OF THE BRASS METHOD ASSOCIATED
WITH ITS SIMPLIFYING ASSUMPTIONS
The Brass method is based on certain simplifying
assumptions that may not be entirely satisfied in practice
and that have implications for the interpretation of the
estimates obtained.
The Brass method assumes that the mortality risks of
children of women who do not report their child-bearing
experience are the same as those of children whose mothers do. Aside from the problem of women who do not
provide information, women may have moved away from
the survey area before being interviewed or may have
died. Thus, the estimates obtained assume, among other
things, that the survivorship of children is independent of
that of their mothers.
The method also assumes that fertility has remained
constant during the 30 or 35 years preceding the survey
or census. If fertility has been changing, the parity ratios
will be affected, and the allowance made for the age pattern of child-bearing will not be correct. In particular, if
fertility has been declining, the parity ratios will be too
low, indicating a later age pattern of fertility than the
actual one and leading to an overestimate of child mortality.
Perhaps the most basic assumption of the method is
that the reported proportions of children dead are correct.
CORRESPONDENCE BETWEEN OBSERVED PROPORTIONS OF CHIL-
DREN DEAD BY AGE GROUP OF MOTHER AND ESTIMATED PROBABILITIES OF
DYING
Age group
Age group
index
a/mother
15-19
20-24
25-29
30-34
35-39
40-44
45-49
.
..
.
.
.
.
.
1
2
3
4
5
6
7
Proportion of
children dead
Estimated probability
afdYing
by age x
D(i)
q(x)
D(l)
D(2)
D(3)
D(4)
D(5)
D(6)
D(7)
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
ESTIMATING TIME TRENDS OF MORTALITY
The method originally developed by Brass assumed
that mortality was constant, so that cohort and period
probabilities of dying were identical. That assumption
23
aged 15-19 would be too high and therefore would
If some of the children that have died are reported as
being alive or if dead children are omitted to a greater
extent than living children, the mortality estimates
obtained will be too low. If, as is generally the case,
omissions and other errors are more prevalent among
older than younger women, certain patterns in the data
may indicate that errors have occurred. For example, the
mortality estimates based on reports of women aged 40
and over may fail to increase with age of mother at the
same pace as those based on the reports of younger
women. Furthermore, the average parities of women
aged 35 and over may not increase with age (or may
actually decrease at older ages), indicating possible omissions of dead children.
Lastly, the Brass method assumes that mortality risks
among children depend solely on their age and not on
other factors, such as age of mother or birth order. If,
for example, the mortality risks of children borne by
young mothers (under age 20) were higher than average,
then the proportion of children dead reported by women
overestimate overall mortality levels. In addition, the estimates derived from data on women aged 20-24 might
also be upwardly biased, since a significant proportion of
the children reported by women aged 20-24 would have
been born when the women were 15-19. By age group
25-29, however, a high proportion of all children would
have been borne by mothers aged 20 and over, so that
the effect of the abnormally high risks experienced by
children of young mothers on the estimate of q( 5) would
be small.
In practice, child mortality estimates based on reports
of women aged 15-19 and, to a lesser extent, on those of
women aged 20-24 are generally unreliable, often being
higher than estimates based on reports of older women.
The detailed example presented in the next chapter illustrates this effect and shows why the Brass method is not
capable of providing reliable estimates of very recent
mortality conditions (those prevalent during the two or
even three years preceding interview).
24
Chapter IV
TRUSSELL VERSION OF THE BRASS METHOD
Step 3. Calculation of the multipliers, k(i)
The basic estimation equation for the Trussell method
is
This version of the Brass method was developed during the late 1970s by T. J. Trussell (United Nations,
1983b, chap. III). It is based on the Coale-Demeny model
life tables, and it has the advantage over earlier versions
of producing estimates both of probabilities of dying
from birth to different ages in childhood and of the time
point to which each probability refers.
The following section presents step by step the computational procedure to be followed in applying the Trussell
version. The case of Bangladesh is then used to give a
detailed example of its application.
q(x)
k(i)
CEB(i)
FP(i)
(4.1)
these women, and FP(i) is the total number of women in
the age group irrespective of their marital or reporting
status. Although parity values are needed only for age
groups 15-19, 20-24 and 25-29-P(l), P(2) and P(3),
respectively-it is worth calculating the whole set up to
age group 45-49 in order to check the quality of the basic
data. Note that the denominator, FP(i), should include
even those women who did not respond to the questions
on children ever born (those of not-stated parity). Their
inclusion is based on the assumption that they are childless, an assumption supported by, evidence from a large
number of surveys showing that the vast majority of
younger women reported as being of not-stated parity
are, in fact, childless.
q(x)
P(2)
P(3)
(4.4)
= k(i)D(i)
Step 5. Calculation of the reference dates for q(x), to)
As explained earlier, under conditions of steady mortality change over time, a reference time, t(i), can be
estimated for each q(x) estimated in step 4. This reference time is expressed in terms of number of years
before the surveyor census and is estimated through the
use of coefficients applied to parity ratios. As before, the
coefficients were estimated by regression analysis of
simulated model cases. The estimating equation is
t(i)
Step 2.
Calculation of the proportions dead among children ever born
The proportion of children dead is given simply by the
ratio of the total number of dead children to the total
number of children ever born (including those who have
died) for each age group. Thus,
CD(i)
CEB(i)
+ c(i)
Calculation of the probabilities of dying by age
x, q(x)
Once D(i) and k(i) have been calculated for each age
group i, estimates of q(x) are obtained simply as their
product, as already indicated in equation 4.3:
where P(i) is the average parity of women of age group
=
P(l)
P(2)
Step 4.
i, CEB(i) is the total number of children ever borne by
D(i)
= a(i) + b(i)
Thus, the mortality measure q(x), the probability of
dying by exact age x, is related to the proportion dead
D(i) by a multiplying factor k(i) that is determined by
the parity ratios P(l)IP(2) and P(2)IP(3) and three
coefficients a(i), b(i) and c(i). These coefficients were
estimated by regression analysis of simulated model
cases. Table 4 shows the coefficients for the seven age
groups of women, from ages 15-19 through ages 45-49
(i = 1, . . . ,7), and for the four regional families of the
Coale-Demeny model life tables.
COMPUTATIONAL PROCEDURE
=
(4.3)
where
Step 1. Calculation of average parity per woman
Average parity is the average number of children ever
borne by women in a given five-year age group. It is calculated as
P(i)
= k(i) D(i)
= e(i) + f(i) ~g~ + g(i) ~i~~
(4.5)
Table 5 shows the coefficients e(i), f(i) and g(i) for use
in equation 4.5 for each age group of women and for
each of the four Coale-Demeny model-life-table families.
Once values of t(i) are obtained, they can be converted into actual dates by subtracting them from the
reference date of the surveyor census expressed in
decimal terms, as illustrated in the detailed example
below.
(4.2)
where D(i) is the proportion of children dead for women
of age group i, CD(i) is the number of children dead
reported by those women, and CEB(i) is the total
number of children ever borne by those women.
Step 6. Conversion to a common index
Steps 4 and 5 provide estimates of q(x) for ages x of
1,2,3,5, 10, 15 and 20 and of t(i), the number of years
25
TABLE 4.
COEFFICIENTS FOR THE ESTIMATION OF CHILD-MORTALITY MULTIPLIERS, k(i), TRUSSELL
VERSION OF THE BRASS METHOD, USING THE COALE-DEMENY MORTALITY MODELS
Age group
a/mother
Age
group
Index
Coefficients
Age x of
children
a (i)
(/)
(2)
(3)
(4)
b(i)
(5)
e(i)
Model
North ......................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
1.1119
1.2390
1.1884
1.2046
1.2586
1.2240
1.1772
-2.9287
-0.6865
0.0421
0.3037
0.4236
0.4222
0.3486
0.8507
-0.2745
-0.5156
-0.5656
-0.5898
-0.5456
-0.4624
South ......................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
1.0819
1.2846
1.2223
1.1905
1.1911
1.1564
1.1307
-3.0005
-0.6181
0.0851
0.2631
0.3152
0.3017
0.2596
0.8689
-0.3024
-0.4704
-0.4487
-0.4291
-0.3958
-0.3538
East ........................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
1.1461
1.2231
1.1593
1.1404
1.1540
1.1336
1.1201
-2.2536
-0.4301
0.0581
0.1991
0.2511
0.2556
0.2362
0.6259
-0.2245
-0.3479
-0.3487
-0.3506
-0.3428
-0.3268
West .......................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
1.1415
1.2563
1.1851
1.1720
1.1865
1.1746
1.1639
-2.7070
-0.5381
0.0633
0.2341
0.3080
0.3314
0.3190
0.7663
-0.2637
-0.4177
-0.4272
-0.4452
-0.4537
-0.4435
(6)
Estimation equations:
= aU) + b(i) P(1) + c(il P(2)
P(2)
P(3)
q(x) = k(i) D(i)
k(i)
Source: Manual X: Indirect Techniques for Demographic Estimation, Population Studies, No. 81
(United Nations publication, Sales No. E.83'xm.2), p. 77.
ried out by linear interpolation between tabulated values,
as explained below,
Suppose that an estimated value of q(x), denoted by
qe(x), is to be converted to the corresponding qC(5)
where, of course, x =1= 5. For a given model-life-table
family and sex, it is first necessary to identify the mortality levels with q(x) values that enclose the estimated
value, qe(x). Thus, the task is to identify in the appropriate table of annex I levels j and j + 1 such that
before the survey to which each estimate applies. In
order to analyse trends and facilitate comparison both
within and between data sets, each estimated q(x) is converted to a single measure. Although any index from the
model-life-table family can be used for that purpose, it is
suggested that a measure of mortality in childhood that is
not particularly sensitive to the pattern of mortality be
selected. The common index recommended is the probability of dying by age 5, q( 5), also called under-five
mortality. The use of infant mortality as the common
index is not recommended because, as will be seen, the
estimates of q( 1) obtained from the conversion are very
sensitive to the mortality pattern underlying the different
models.
The q(x) values corresponding to the mode1-life-tab1e
family being considered can be used to carry out the
required conversions. The tables in annex I contain the
necessary values of q(x) ordered by mortality level and
expectation of life for each of the Coale-Demeny families
of model life tables (see chap, I) and for males, females
and both sexes separately. The actual conversion is car-
(4,6)
where qj(x) and qj + 1(x) are the model values of q(x)
for levels j and j + 1, respectively, and qe(x) is the
estimated value. Then, the desired common index qC(5)
is given by
where h is the interpolation factor calculated in the following way:
26
TABLE 5.
COEFFICIENTS FORTHE ESTIMATION OF THE TIME REFERENCE,
t(i)a, FORVALUES OF q(X), TRUSSELL
VERSION OF THE BRASSMETHOD, USING THE COALE·DEMENY MORTALITY MODELS
Age
Age group
of mother
group
Index
Coefficients
Estimated
f(;)
(5)
g(i)
(6)
Model
(I)
(2)
(3)
e(i)
(4)
North ... ,.. ,... ,.. " .. " .. ,
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1.0921
1.3207
1.5996
2.0779
2.7705
4,1520
6,9650
5.4732
5.3751
2,6268
-1.7908
-7.3403
-12.2448
-13.9160
-1.9672
0.2133
4.3701
9.4126
14,9352
19,2349
19.9542
South.. ... ,.. ,... ,.. ,.. ,...
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1.0900
1.3079
1.5173
1.9399
2.6157
4.0794
7,1796
5.4443
5.5568
2.6755
-2.2739
- 8.4819
-13.8308
-15.3880
-1.9721
0.2021
4.7471
10.3876
16,5153
21.1866
21.7892
East, .. "." .. ,... ,.. ,.. ,...
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1.0959
1.2921
1.5021
1.9347
2.6197
4,1317
7.3657
5.5864
5,5897
2.4692
-2.6419
- 8.9693
-14.3550
-15.8083
-1.9949
0.3631
5.0927
10,8533
17,0981
21.8247
22.3005
West." .. ,.. " .. ,... ,..... ,
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1,0970
1.3062
1.5305
1.9991
2.7632
4.3468
7,5242
5,5628
5,5677
2.5528
-2.4261
- 8.4065
-13.2436
-14.2013
-1.9956
0.2962
4.8962
10.4282
16,1787
20,1990'
20.0162
q(x)
Estimation equation:
(')+f(,)P(l)
(,)P(2)
tt( t' ) =el
l--+g
l-P(2)
P(3)
Source: Manual X: Indirect Techniques for Demographic Estimation, Population Studies, No, 81
(United Nations publication, Sales No. E.83.XIII.2), p. 78.
a Number of years prior to the survey.
h
=
qe(x) - qj(x)
qj+I(X) - qj(x)
assess the consistency and general trend of the estimates,
as the example below illustrates.
(4.8)
A
If the data on children ever born and children dead are
for both sexes combined, the model qj(x) values should
be taken from the tables for both sexes combined in
annex I. If, however, the data are for male and female
children separately, the estimated values of q(x) will be
sex-specific, and the conversion to a common index
should use the model qj(x) values from the tables for the
relevant sex, also presented in annex I.
Step 7. Interpretation and analysis of results
Once seven estimates (one for each age group i of
women) of the selected common index-q C( 5 ) as suggested above-have been obtained, it is recommended
that they be plotted against time. As noted in step 5, the
t( i) values can be converted into reference dates by subtracting them from the surveyor census reference date
(or the approximate midpoint of the field-work), and the
qC(5) estimates can then be plotted against the resulting
dates. Graphical presentation of the results is essential to
DETAILED EXAMPLE
Compilation of the data required
The data gathered by the 1974 Bangladesh Retrospective Survey of Fertility and Mortality will be used to
illustrate the application of the Trussell version of the
Brass method. In chapter II, the data necessary to apply
the method were compiled in displays 6 and 7 (they are
shown here again for convenience). Since the basic data
are available by sex, they will be used to check the internal consistency of the information on children ever born.
Table 6 illustrates how the sex ratio at birth is calculated from the data on children ever born for different
age groups of mother by dividing the number of male
children by the number of female children ever born.
The sex ratio at birth is biologically determined and
varies relatively little, ranging usually between 1.03 and
1.08 male births per female birth. Table 6 shows that up
27
Display 6. Second step in the compilation of data on children ever born
and children dead for Bangladesh
Children
ever born
(/)
=
(2)+(3)
Agegroup
of mother
~
-s
=
(2) + (4)+ (5)
Children
dead
Children
surviving
Children
living
at home
Children
living
elsewhere
(2)
(3)
(4)
(5)
=
=
(/)-(3)
=
(1)-(4)-(5)
(4)+(5)
15-19
I 160919
215 365
921 227
24327
20-24
4901 382
997384
3 820649
83349
25-29
9085 852
I 937955
6927908
219989
30-34
9910256
2 261 196
7 126473
522587
35-39
10384001
2 490 168
6974267
919566
40-44
9 164 329
2415023
5472 460
I 276 846
45-49
6905673
I 959544
3664 328
I 281 801
15-19
597 248
117 165
469036
11 047
20-24
2 507 018
529877
I 938220
38921
25-29
4675978
1047294
3 545904
82 780
30-34
5 109487
I 204 582
3 780 859
124046
35-39
5435726
I 333 957
3925071
176 698
40-44
4 883 599
I 291 745
3 323 724
268 130
45-49
3714957
I 030737
2393 149
291 071
15-19
563 671
98200
452 191
13 280
20-24
2394364
467507
I 882429
44428
25-29
4409874
890661
3 382004
137 209
30-34
4800769
1056614
3 345 614
398 541
35-39
4948275
I 156 211
3049 196
742868
40-44
4280730
I 123278
2 148 736
I 008 716
45-49
3 190 716
928807
I 271 179
990730
~
~
~
~
~
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Iertility and Mortality (Dacca, 1977), table 8, p. 37
(reproduced in display 3 above).
28
Display 7. Compilation of data on the total number of women
by age group for Bangladesh
lOral
number
of women
Ever-married
women
Single
IWmen
flbmen of
stated
parity
(I)
(2)
(3)
(4)
flbmen of
not-stated
parity
(5)
=
(2)+(3)
Age group
=
of women
(4)+(5)
15-19
3014706
20-24
2 653 155
25-29
2607009
30-34
2 015 663
35-39
1771680
40-44
1 479575
45-49
1 135 129
Source: Bangladesh. Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Iertility and Mortality (Dacca, 1977), table 3, p. 28
(reproduced in display 4 above).
In this example mortality will be estimated for both
sexes combined, so parities should be calculated using the
data on children ever born of both sexes appearing in the
top panel of display 6. Each parity P(i) will therefore be
obtained by dividing those numbers by the total number
of women shown in display 7, age group by age group.
Thus, for women aged 25-29 (i=3),
to age group 30-34 the reported numbers of male and
female children ever born yield sex ratios at birth within
the expected range. Over age 35, however, the sex ratios
at birth are too high, implying that too many males were
reported relative to females. Such a pattern suggests that
the data on children ever born corresponding to older
women (aged 35 and over) are probably affected by the
omission of female children, though misreporting of the
children's sex also could give rise to the same pattern.
P(3)
= 9,085,852 = 3.4852
2,607,009
TABLE
6.
CALCULATION OF THE SEX RATIO AT BIRTH FROM DATA ON CHIL-
DREN EVER BORN. CLASSIFIED BY SEX, FROM THE
1974
BANGLADESH
Column 3 of table 7 shows the complete set of average
parities by age group. Notice that they do not increase
steadily with age. In particular, the average parity for
women aged 45-49 is lower than that for women aged
40-44, which is not consistent with the existence of constant fertility in the past. The increase in the average parity from age group 35-39 to age group 40-44 also seems
too small. Such a pattern of change with age in the average parities, coupled with the high sex ratios at birth
noticed among the children of older women, strongly
suggests that there are omission errors in their reports of
lifetime fertility. Because dead children are more likely
to be omitted than live ones, evidence of omission
requires that the resulting mortality estimates be interpreted with caution.
RETROSPECTIVE SURVEY
Age group
of mother
MaLe children
FemaLe children
ever born
ever born
Sex ratio
at birth
(1)
(2)
(3)
(4) = (2)/(3)
15-19......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
597248
2 507 018
4675978
5 109 487
5435726
4883599
3714957
563 671
2394364
4409 874
4800 769
4948275
4280730
3 190716
1.060
1.047
1.060
1.064
1.099
1.141
1.164
Computational procedure
Step 1. Calculation of average parity per woman
To apply the method, average parities are used to calculate the parity ratios P(I)/P(2) and P(2)/P(3). Thus,
parities need to be calculated only for women aged 1519, 20-24 and 25-29. However, it is good practice to calculate parities for all seven age groups of women,
because they can reveal problems with the basic data.
Calculation of the proportions dead among children ever born
The proportions dead, denoted by D(i), are calculated
as the ratios of the number of children dead to the
number of children ever borne by women of each age
group i. Such ratios are obtained in this case by dividing
the entries in column 2 of display 6 by those of column 1
Step 2.
29
TABLE 7.
ApPLICATION OF THE TRUSSELL VERSION OF THE BRASS METHOD TO DATA ON BOTH SEXES
FROM THE 1974 BANGLADESH RETROSPECTIVE SURVEY
Age group
of mother
Age
group
index
(i)
Average
parity
P(i)
Proportion
dead
D(i)
Multiplier
k(i)
Age
x
Probability
ofdying by
age x, q(x)
reference
t(i)
Reference
date
(I)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(/0)
15-19......................
20-24......................
25-29......................
30-34......................
35-39......................
40-44......................
45-49......................
1
2
3
4
5
6
7
0.3851
1.8474
3.4852
4.9166
5.8611
6.1940
6.0836
.1855
.2035
.2133
.2282
.2398
.2635
.2838
0.9169
0.9954
0.9907
1.0075
1.0294
1,0095
0.9973
1
2
3
5
10
15
20
.170
.203
.211
.230
.247
.266
.283
1.2
2.6
4.6
7.0
9.6
12.4
15.5
1973.1
1971.7
1969.7
1967.3
1964.7
1961.9
1958.8
.294
.248
.230
.230
.229
.237
.239
<t(5)
Step 5. Calculation of the reference dates for q(x), t(i)
The time reference t(i) for each estimated q(x) in
number of years before the survey is calculated according
to equation 4.5 from the two parity ratios P(1)/P(2) and
P(2)/P(3) and the three coefficients e(i), g(i) and h(i)
that correspond to the model-life-table pattern selected.
In the case of Bangladesh, the coefficients shown in the
second panel of table 5, corresponding to the South
model, will be used. Considering once more age group 3
(women aged 25-29), one gets
for both sexes combined. Thus, for women aged 25-29
and i = 3,
D(3) = 1,937,955 = .2133
9,085,852
Results for all age groups, i = 1, ... ,7, are shown in
column 4 of table 7.
Step 3. Calculation of the multipliers, k(i)
The multipliers k(i) are calculated for each age group
i using a set of coefficients from table 4 and the ratios of
average parities P(l)/P(2) and P(2)/P(3). Those ratios
can be computed from column 3 of table 7:
P(l)
P(2)
=
.3851
1.8474
= .2085
P(2)
P(3)
=
1.8474
3.4852
= .5301
t(3)
= 1.5173 + (2.6755)(.2085) + (4.7471)(.5301)
= 4.59
That is, under conditions of steady mortality decline,
the estimate of q( 3) obtained from the proportion of children dead among those ever borne by women aged 25-29
would refer to a period nearly four and a half years
before the survey. Since, in this particular case, the
survey's field-work was carried out mostly during April
1974, the survey's reference date can be taken to be
1974.3-15 April corresponds to day number 105 in the
year, which, divided by the total number of days in a
year, is 105/365 = 0.29, a figure that is rounded to 0.3
in decimal terms. Hence, the reference date for the
estimated q( 3) is
The regression coefficients a(i), b(i) and c(i) in table
4 are specified for each family of the Coale-Demeny
model life tables. The South family has been selected for
use in this example, so that the coefficients in the second
panel of table 4 will be used. For each age group i, k(i)
is computed using equation 4.4. Thus, for age group 3, in
which women are aged 25-29,
= 1.2223 + (.0851)(.2085) + (-.4704)(.5301)
= .9907
1974.3 - 4.6
= 1969.7
or towards the end of the calendar year 1969. The other
values of t(i) and the reference dates calculated from
them are shown in columns 8 and 9 of table 7.
Step 6. Conversion to a common index
In order to study trends in child mortality, the q(x)
values obtained in step 4 need to be converted to a common index. Under-five mortality, q(5), will be used here
as the common index. The conversion is carried out by
interpolating between the q(x) values of the model life
tables presented in annex I. The table used for
interpolation-table A.I.lO-is that for model South and
both sexes combined (the values it displays were derived
assuming a sex ratio at birth of 1.05 male births per
female birth).
As an example, consider the conversion of the
estimated qe(3) to a qC(5) according to model South.
The complete set of k(i) values is shown in column 5 of
table 7.
Step 4. Calculation of the probabilities of dying by age
x, q(x)
The q(x) is computed for each age group i by multiplying the proportion of children dead, D(i), by the multiplier k(i). The correspondence between x values and i
values is shown in table 3. Hence, for age group 3, in
which women are aged 25-29, x is equal to 3, and q(3)
is given as
q(3)
Common
index
Multipliers based on South model.
Sex ratio at birth = 1.05
P(l)/P(2) = .2085
P(2)/P(3) = .5301
k(3)
Time
= k(3) D(3) = (.9907)( .2133) = .2110
indicating a probability of dying by age 3 of 21.1 per
cent. The probabilities of dying, q(x), for all seven age
groups of mother are shown in column 7 of table 7.
30
qC( 5) values are fairly similar for most of the period
The estimated value of qe(3) is .211. According' to table
A.I.lO, this value falls between the q(3) of level 12,
q12(3) = .22564, whose qI2(5) equivalent is .24592, and
that of level 13, ql\3) = .20640, whose q13(5)
equivalent is .22430. Substituting the qj(3) and qe(3)
values in equation 4.8 to find h, the interpolation factor,
one obtains
h
=
.21100 - .22564
.20604 - .22564
= .7469
The qC(5) equivalent for the estimated qe(3)
then derived using equation 4.7 as follows:
qC(5)
= (1.0
- .7469)(.24592)
1959-1970 and that they increase markedly after 1970.
Therefore, taken at face value, those estimates imply that
in Bangladesh mortality in childhood increased during the
early 1970s after varying within a narrow range during
the preceding decade. Such an interpretation is not
correct. As figure 7 indicates, the estimates referring to
recent periods (1972 onwards) are those derived from the
reports of younger women (age groups 20-24 and 15-19)
and hence reflect the higher than average risks of dying
to which the children of those women are subject. Furthermore, given the evidence available on the existence
of omissions of children ever born in the reports of older
women (aged 35 and over), it is likely that the estimates
referring to periods furthest in the past (1959-1965 or
thereabouts) may be biased downward. As a result of
both biases, the curve displayed in figure 7 probably fails
to reflect the actual trend of under-five mortality, q( 5),
in Bangladesh. Unfortunately, the true trend cannot be
derived with certainty from the data available. However,
it can be established with a high degree of confidence
that during the 1965-1970 period under-five mortality in
Bangladesh was approximately .230, that is, during the
late 1960s slightly more than one out of every five children born would die before reaching the fifth birthday.
= .211
is
+ (.7469)(.22430)
= .230
That is, in the South model life tables, the qC(5)
corresponding to a qe(3) of .211 is .230. The complete
set of qC(5) values equivalent to the estimated q(x)
values is shown in column 10 of table 7.
Step 7. Interpretation and analysis of results
Once the values of the common index qC(5) have been
derived for all age groups of women, they can be plotted
against the reference date for each estimate, as shown in
figure 7. That figure shows clearly that the estimated
Figure 7. Under-live mortality, q(5), for both sexes in Bangladesh, estimated
using model South and the Trussell version of the Brass method
•
•
1964
1966
Reference date
Source:
Table 7.
31
1968
1970
1972
1974
Another conclusion that can be drawn from the estimates available so far is that mortality in childhood in
Bangladesh probably did not change much during the
1960s. Of course, given the downward bias that probably
affects earlier estimates, such an assertion cannot be
made with absolute certainty. As the next section will
show, the availability of further evidence may modify
this preliminary conclusion.
That is, the average parities are calculated by dividing
the number of male children ever born by the total
female population, as in equation 4.9, and the proportion
of male children dead, Dm(i), for each age group is calculated by dividing the number of male children dead by
the number of male children ever born. Steps 3 to 7 are
then carried out as indicated in the computational procedure. Female mortality in childhood is estimated in the
same way, substituting female children ever born and
dead instead of the corresponding male children.
Tables 8 and 9 show the results of applying the
Trussell version of the Brass method to the Bangladesh
data classified by sex. Figure 8 plots the estimated
under-five mortality, q(5), for males and females. Note
that for most age groups of mother the male estimates of
q(5) are higher than those for females. Since higher male
than female mortality is the rule in most countries, the
1961-1974 estimates for Bangladesh seem acceptable.
However, the reversal of the relationship between male
and female mortality for the estimates derived from age
group 45-49 is suspect, being in all probability caused by
errors in the basic data rather than by the actual reversal
of mortality differentials by sex. For that reason, the estimates derived from the reports of women aged 45-49
should be disregarded, even when they refer to both
sexes combined.
Estimates of mortality in childhood by sex
As shown in display 6, the data on children ever born
and surviving for Bangladesh are available by sex of
child. It is therefore possible to estimate mortality by sex.
To estimate male mortality, for instance, the procedure to
be followed is essentially the same as that described
above, except that parities and the proportions dead are
calculated only on the basis of male children. In algebraic
terms, letting the subindex m denote male, in step 1
equation 4.1 becomes
CEBm(i)
J>m(i) ==
(4.9)
l'J>(i)
and in step 2 equation 4.2 becomes
co.u,
D (i) - - - m
- CEBm(i)
TABLE
8.
(4.10)
ApPLICATION OF THE TRUSSELL VERSION OF THE BRASS METHOD TO DATA ON MALES
FROM THE
1974 BANGLADESH
RETROSPECTIVE SURVEY
Age
group
Average
Proportion
Age group
0/ mother
index
(i)
parity
Pm(i)
dead
Dm(i)
Multiplier
k(i)
(I)
(2)
(3)
(4)
(5)
15-19......................
20-24......................
25-29................
30-34......................
35-39......................
40-44......................
45-49......................
1
2
3
4
5
6
7
0.1981
0.9449
1.7936
2.5349
3.0681
3.3007
3.2727
.1962
.2114
.2240
.2358
.2454
.2645
.2775
0.9104
0.9957
0.9923
1.0093
1.0312
1.0112
0.9988
Age
x
(6)
1
2
3
5
10
15
20
Probability
of dying by
Time
reference
Reference
index
age x, qm(x)
(7)
t(i)
date
(8)
(9)
q~(5)
(10)
1973.1
1971.7
1969.7
1967.4
1964.8
1962.0
1958.9
.301
.256
.241
.238
.236
.240
.236
.179
.210
.222
.238
.253
.268
.277
1.2
2.6
4.6
6.9
9.5
12.3
15.4
Common
Pm(l)/Pm(2) = .2097
P m(2)/Pm(3) = .5268
Multipliers based on South model.
TABLE
9.
ApPLICATION OF THE TRUSSELL VERSION OF THE BRASS METHOD TO DATA ON FEMALES
FROM THE 1974 BANGLADESH RETROSPECTIVE SURVEY
Age
group
Average
Proportion
index
parity
dead
P/i)
(1)
(i)
(2)
(3)
Dt)
4)
15-19......................
20-24 ......................
25-29 ......................
30-34......................
35-39......................
40-44 ......................
45-49 ......................
1
2
3
4
5
6
7
0.1897
0.9025
1.6916
2.3817
2.7930
2.8932
2.8109
.1742
.1953
.2020
.2201
.2337
.2624
.2911
Age group
of mother
~(l)/P/(2)
Age
x
Probability
of dying by
age .r, q/x)
Time
reference
Reference
Common
index
k(i)
t(i)
date
q/(5)
(5)
(6)
(7)
(8)
(9)
(10)
0.9238
0.9952
0.9890
1.0056
1.0275
1.0078
0.9957
1
2
3
5
10
15
20
.161
.194
.200
.221
.240
.264
.290
1.2
2.6
4.6
7.0
9.7
12.5
15.6
1973.1
1971.7
1969.7
1967.3
1964.6
1961.8
1958.7
.286
.240
.218
.221
.222
.234
.243
Multiplier
= .2072
t (3) = .5335
Uultiphers based on South model.
(2)/P
32
Figure 8. Under-five mortality, q(5), for males and females in Bangladesh, estimated using
model South and the Trussell version of the Brassmethod
q(5)
.35 ~-------------------------------
.30
e. ....
.25
•
.20
.15
-Males
••••• Females
.10
.05
OL....._.L-_.L-_.L-_.L-_....L..._........_
1958
1960
1962
1964
........._
........._
........_ --"-_--'"_---I._................_ L . . . - _ L . . . - - - I
1966
1968
1970
1972
1974
Reference date
Sources: Tables 8 and 9.
underestimate mortality and hide or minimize the decline
that has taken place. It is likely that the same biases may
be operating on the estimates for males, especially if one
accepts as improbable the reversal of the sex differential
implied by the estimates derived from the reports of
women aged 45-49.
If the conclusion that female mortality declined is
accepted on the basis of the arguments presented above,
the earlier conclusions regarding mortality levels for both
sexes combined must be revised to reflect a decline in
those levels, albeit slight, between the early and the late
1960s.
This example demonstrates the importance of analysing
all available evidence before reaching definitive conclusions about the reliability of the different estimates
yielded by the Brass method.
As in the case of the data referring to both sexes, the
estimates derived from information provided by younger
women (age groups 15-19 and 20-24) seem to be biased
upward and must be rejected. From the remaining estimates, it can be concluded that during the 1962-1970
period under-five mortality among males in Bangladesh
remained mostly constant at a level of 238 or 239 deaths
per 1,000 births, while that among females might have
decreased slightly, from about 234 deaths per 1,000
births around 1962 to about 220 towards the end of the
decade. The existence of such a decline may be further
supported by the fact that our earlier analysis of sex
ratios at birth showed that female children were underreported among older women. If, as is likely, the omitted
female children were dead children, the estimates for
females derived from women aged 40-44 or 45-49 would
33
Chapter V
PALLONI-HELIGMAN VERSION OF THE BRASS METHOD
Another version of the Brass method was developed in
the early 1980s by A. Palloni and L. Heligman (1986). It
is based on the United Nations model life tables for
developing countries and, like the Trussell version, produces estimates of the probabilities of dying from birth
and of the time to which those probabilities refer. It
differs from the Trussell version in that it uses information on births in a year in addition to the data required by
the Brass method. The additional data are used to compute the mean age at maternity, an indicator of the average age difference between mothers and their children.
This chapter starts by describing the additional data
requirements of the Palloni-Heligman version. It then
describes the steps of the computational procedure to be
followed in applying it and ends by providing a detailed
example of its application to the case of Bangladesh.
DATA REQUIRED
The data required to apply the Palloni-Heligman version of the Brass method are essentially the same as those
needed for the application of the Trussell version,
namely:
1. Number of children ever born classified by age
group of mother;
2. Number of children dead classified by age group
of mother;
3. Total number of women (irrespective of marital or
reporting status) classified by age.
However, the Palloni-Heligman version requires an
additional set of information:
4. Number of births occurring in a given year
classified by age group of mother
Information on births in a year is used to estimate an
indicator of the timing of child-bearing, namely, the
mean age at maternity (that is, the mean age of the mothers of the children born in a particular period), denoted
by M. When information on births is not available, a
rough estimate of M may be used in applying the
Palloni-Heligman version (a convenient estimate is
M = 27).
The first three items of data listed above can be compiled using the worksheets presented in displays 6 and 7
of chapter II. To compile the data on births in a given
year, display 9 has been prepared. Note that the information on births need not be available by sex of child and
that it is important to establish whether the information
was derived from a registration system or from a survey
or census. Generally, it is assumed that a registration system records births as they occur and consequently that
the age of mother recorded by the registration system is
her age at the time of the birth. Surveys, on the other
hand, usually obtain the necessary information on births
by asking whether or not a woman gave birth during the
year preceding interview. When such data are later tabulated by a woman's age at the time of interview, a systematic bias is introduced in the timing of births. To
correct that bias, it is assumed that births occur uniformly throughout the year. Hence, on average, women
aged 17, say, at the time of interview would have been
aged 16.5 at the time they gave birth. For that reason the
exact-age groups listed in the lower panel of display 9
referring to census or survey data have been shifted back
half a year.
COMPUTATIONAL PROCEDURE
The application of the Palloni-Heligman version of the
Brass method is very similar to that of the Trussell version. To maintain comparability with the latter in the
numbering of steps, step 3 is divided here into two parts:
3 (a) and 3 (b).
Step 1. Calculation of average parity per woman
Average parity is the average number of children ever
borne by women in a given five-year age group. It is calculated as
PC)
I
=
CEB(i)
FP(i)
(5.l)
where P(i) is the average parity of women of age group
i, CEB(i) is the total number of children ever borne by
these women, and FP(i) is the total number of women in
the age group irrespective of their marital or reporting
status. Although parity values are needed only for age
groups 15-19, 20-24 and 25-29-P(1), P(2) and P(3),
respectively-it is worth calculating the whole set up to
age group 45-49 in order to check the quality of the basic
data. Note that the denominator, FP(i), should include
even those women who did not respond to the questions
on children ever born (those of not-stated parity). Their
inclusion is justified on the assumption that they are
childless.
Step 2.
Calculation of the proportions dead among children ever born
The proportion of children dead is given simply by the
ratio of the total number of dead children to the total
number of children ever born (including those who have
died) for each age group of women. Thus,
.
CD(i)
(5.2)
D(I) = CEB(i)
34
Display 9. Worksheet for the compilation of data on births in a year by age group of mother
for the Palloni-Heligman version of the Brass method
Source aOO
"
.s
g
.~
~
-
reponed
Exact-age group
Midpoin: of
age group
of nwther
of mother at
birth of child a
exact-age
Births in
a year
group
15-19
[15,20)
17.5
20-24
[20,25)
22.5
25-29
[25,30)
27.5
30-34
[30,35)
32.5
35-39
[35,40)
37.5
40-44
[40,45)
42.5
45-49
[45,50)
47.5
15-19
[14.5,19.5)
17
20-24
[19.5,24.5)
22
25-29
[24.5,29.5)
27
30-34
[29.5,34.5)
32
35-39
[34.5,39.5)
37
40-44
[39.5,44.5)
42
45-49
[44.5,49.5)
47
S!
~
f:::
'"es
~
a
«The notation [x,y) indicates that exact ages at the time women gave birth range from x to y, exclusive of the latter, that is, age y is not quite reached.
where D(i) is the proportion of children dead for women
of the age group i, CD(i) is the number of dead children
reported by those women and CEB(i) is the number of
children ever borne by those women.
the sum by the total number of births (excluding those to
women of not-stated age). Thus,
7
E (B(i)
M
Calculation of the mean .age at maternity, M
(Palloni-Heligman version only)
The value of the mean age at maternity, M, is
estimated from the number of births occurring in a given
year classified by age group of mother. As indicated earlier, when those data are compiled for use in the estimation procedure, it is important to establish whether they
were obtained from vital registration (a registration system) or from a census or survey.
M is calculated by multiplying the midpoint of each
age group by the number of births to women in that age
group, summing the resulting products, and then dividing
Step 3.
mpti ;
= _i=_1---::;-7
E B(i)
_
i=1
where the symbol E denotes sum, B(i) denotes the
number of births to women in age group i and mp(i) is
the midpoint in years of age group i.
The values of mp(i) are shown in display 9 and
clearly depend on the type of exact-age group being dealt
with. The term "exact-age group" is used here to denote
the true range of variation of the ages of mother in each
reported age group. Generally, women belonging to a
given age group, say 20-24, are all those whose age at
35
last birthday was 20, 21, 22, 23 or 24, that is, women
whose exact ages may vary anywhere from 20.0 to
24.99 . " without quite reaching 25. The notation
[20,25) is used to denote that range of exact ages. The
midpoint of that range is 22.5 years.
[20,25). As indicated in that display, the midpoints of
those intervals are also moved back by half a year.
Step 3 (b). Calculation of the multipliers, k(J)
The basic estimation equation for the Palloni-Heligman
version is the same as for the Trussell version shown in
equation 4.3:
As noted above, when the data on births in a year are
obtained from a vital registration system, it can be
assumed that the reported age of mother is her age at the
time she gave birth. In contrast, in surveys or censuses
gathering information on the births occurring during the
year immediately preceding interview, mothers were on
average half a year younger at the birth of their reported
children than at the time of interview. Hence, as shown
in display 9, the exact-age groups of mothers are shifted
back by half a year-for example [19.5,24.5) rather than
TABLE
10.
q(x)
= k(i) D(i)
(5.4)
but the equation to calculate k(i) now includes M as
input:
k(i)
= a(i) + b(i) ~g~ + c(i) ~~~~ + d(i)M
(5.5)
Table 10 shows the coefficients a(i), b(i), c(i) and d(i)
for the seven age groups of women, from ages 15-19
COEFFICIENTS FOR THE ESTIMATION OF CHILD-MORTALITY MULTIPLIERS.
k(i),' PALLONI-HELIGMAN
VERSION OF
THE BRASS METHOD, USING THE UNITED NATIONS MORTALITY MODELS
Age
Coefficients
group
Age group
Index
of mother
Model
Age x of
children
a (i)
h(i)
(5)
eli)
d{i)
(6)
(7)
(I)
(2)
(3)
(4)
Latin American .. "" .. ".".""
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
0.6892
1.3625
1.0877
0.7500
0.5605
0.5024
0.5326
-1.6937
-0.3778
0.0197
0.0532
0.0222
0.0028
0.0052
0.6464
-0.2892
-0.2986
-0.1106
0.0170
0.0048
0.0256
0.0106
-0.0041
0.0024
0.0115
0.0171
0.0180
0.0168
Chilean """." .. " ..... "" .. ",,.
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
0.8274
1.3129
1.0632
0.8236
0.6895
0.6098
0.5615
-1.5854
-0.2457
0.0196
0.0293
0.0068
-0.0014
0.0040
0.5949
-0.2329
-0.1996
-0.0684
0.0032
0.0166
0.0073
0.0097
-0.0031
0.0021
0.0081
0.0119
0.0141
0.0159
South Asian .".""" .. "" .. "".
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
0.6749
1.3716
1.0899
0.7694
0.6156
0.6077
0.6952
-1.7580
-0.3652
0.0299
0.0548
0.0231
0.0040
0.0018
0.6805
-0.2966
-0.2887
-0.0934
0.0298
0.0573
0.0306
0.0109
-0.0041
0.0024
0.0108
0.0149
0.0141
0.0109
.,Far Eastern.""".""""""""
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
0.7194
1.2671
1.0668
0.7833
0.5765
0.4115
0.3071
-1.3143
-0.2996
0.0017
0.0307
0.0068
0.0014
0.0111
0.5432
-0.2105
-0.2424
-0.1103
-0.0202
0.0083
0.0129
0.0093
-0.0029
0.0019
0.0098
0.0165
0.0213
0.0251
General .............................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
1
2
3
5
10
15
20
0.7210
1.3115
1.0768
0.7682
0.5769
0.4845
0.4760
-1.4686
-0.3360
0.0109
0.0439
0.0176
0.0034
0.0071
0.5746
-0.2475
-0.2695
-0.1090
0.0038
0.0036
0.0246
0.0095
-0.0034
0.0021
0.0105
0.0165
0.0187
0.0189
Estimation equations:
k(i)
= a(i) + b(i) ~g~ + c(i) ~g~ + d(i)M
q(x)
= k(i)D(i)
Source: Alberto Palloni and Larry Heligman, "Re-estimation of
structural parameters to obtain estimates of mortality in developing
countries", Population Bulletin of the United Nations, No. 18 (United
Nations publication, Sales No. E.85.xm.6), table 2B, p. 17; figures 10
column 4 have been corrected.
36
=
through ages 45-49 (i
1, ... ,7), and for the five
regional patterns of the United Nations models.
However, in the Palloni-Heligman version, the values of
e(i), f(i) and g(i) appearing in equation 5.6 are based
on the United Nations models (see table 11). Once the
t( i) values are calculated, they can be converted into
reference dates by subtracting them from the reference
date of the census or survey, as illustrated below.
Step 4.
Calculation of the probabilities of dying by age
x, q(x)
Once D(i) and k(i) have been calculated for each age
group i, estimates of q(x) are obtained simply as their
product, as already indicated in equation 5.4:
q(x)
Step 6.
= k(i) D(i)
Steps 4 and 5 provide estimates of q(x) for ages x of
1, 2, 3, 5, 10, 15 and 20 and of t(i), the number of years
before the surveyor census to which each estimate
applies. In order to analyse trends and facilitate comparison both within and between data sets, each estimated
q(x) is converted to a single measure. Although any
index from the model-life-table family can be used, it is
suggested that a measure of child mortality that is not
particularly sensitive to the pattern of mortality be
Step 5. Calculation of the reference dates for q(x), t(i)
An estimate of the time reference, t(i), of each
estimated q(x) value is calculated by applying the
coefficients in table 11 to the parity ratios P(l)/P(2) and
P(2)/P(3) in the same way as for the Trussell version:
t(i)
= e(i) + f(i)
TABLE 11.
P(l)
P(2)
+ g(i)
P(2)
P(3)
Conversion to a common index
(5.6)
COEFFICIENTS FOR THE ESTIMATION OF THE TIME REFERENCE. t(i): FOR VALUES OF q(x), PALLONI-HELIGMAN VERSION OF
THE BRASS METHOD, USING THE UNITED NATIONS MORTALITY MODELS
Age
Coefficients
group
index
Estimated
J(')
g(i)
(2)
q(x)
(3)
e(i!
(I)
(4)
(5)
(6)
Latin American " ..... ".""""
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(15)
q(20)
1.1703
1.6955
1.8296
2.1783
2.8836
4.4580
6.9351
0.5129
4.1320
2.9020
-2.5688
-10.3282
-17.1809
-19.3871
-0.3850
-0.1635
3.4707
9.0883
15.4301
20.4296
23.4007
Chilean ,," .... " .. "" .. " .. ,," ...
15-19
20-24
25-29
30-34
35-39
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(15)
q(20)
1.3092
1.6897
1.8368
2.2036
2.9955
4.7734
7.4495
1.9474
4.6176
2.6370
-3.3520
-11.4013
-17.8850
-19.0513
-0.7982
-0.0173
4.0305
9.9233
16.3441
20.8883
23.0529
South Asian ."."." .. " ... """.
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1.1922
1.7173
1.8631
2.1808
2.7654
4.1378
6.4885
0.7940
4.3117
2.8767
-2.7219
-10.8808
-18.6219
-22.2001
-0.5425
-0.1653
3.5848
9.3705
16.2255
22.2390
26.4911
Far Eastern" .. "" .... """" ....
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(10)
q(l5)
q(20)
1.2779
1.7471
1.9107
2.3172
3.2087
5.1141
7.6383
1.5714
4.2638
2.7285
-2.6259
-9.8891
-15.3263
-15.5739
-0.6994
-0.0752
3.5881
9.0238
14.7339
18.2507
19.7669
General .. " .. "" ... " .. "" .. " ....
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1
2
3
4
5
6
7
q(l)
q(2)
q(3)
q(5)
q(lO)
q(15)
q(20)
1.2136
1.7025
1.8360
2.1882
2.9682
4.6526
7.1425
0.9740
4.1569
2.8632
-2.6521
-10.3053
-16.6920
-18.3021
-0.5247
-0.1232
3.5220
9.1961
15.3161
19.8534
22.4168
Age group
of mother
Model
40-4t
Source: Alberto Palloni and Larry Heligman, "Re-estimation of
structural parameters to obtain estimates of mortality in developing
countries", Population Bulletin of the United Nations, No. 18 (United
Nations publication, Sales No. E.85.XIII.6), table 5A, p. 19.
"Number of years prior to the survey.
Estimation equation:
t(i)
= e(i) + f(i) PO) + g(i) P(2)
P(2)
P(3)
37
compiled in displays 6 and 7 (reproduced again here). In
this example, estimates of the risks of dying in childhood
will be estimated for male children. Hence, the data of
interest are those referring to male children ever born
and male children dead in display 6.
To apply the Palloni-Heligman version, it is also
necessary to compile information on births occurring during a given year. Display 10 shows the published tabulation on that topic. Note that the 1974 Bangladesh survey
gathered information on the time of occurrence of the
most recent live birth of each ever-married woman and
that such information was tabulated by year of
occurrence. The analyst is thus apparently faced with a
choice of which time period to use. Note that a woman
who had a child in April 1972 and another one in March
1974 would report only the latter birth. Consequently the
data for the period April 1972 to March 1973 do not
cover all the births during that year. It is therefore necessary to use the most recent period-in this case April
1973 to March 1974, the year immediately preceding the
survey. Because of the type of information gathered (the
date of the most recent birth), only the most recent time
period will cover all possible events (births) in a year.
The worksheet presented in display 9 may be used to
compile the data on births in the year preceding the survey. Display 11 illustrates a completed worksheet. Note
that data on births in a year for both sexes combined are
adequate for the application of the method even when
estimates of mortality by sex are desired.
selected. The common index recommended is under-five
mortality, q( 5) .
The q(x) values corresponding to the model-life-table
family being considered can be used to carry out the
required conversions. The tables in annex II contain the
necessary values of q(x) ordered by mortality level and
expectation of life for each of the United Nations models
and for males, females and both sexes separately. The
actual conversion is carried out by linear interpolation
between tabulated values, as explained below.
Suppose that an estimated value of q(x), denoted by
qe(x), is to be converted to the corresponding qC(5)
where x =1= 5. For a given model-life-table family and
sex, it is first necessary to identify the mortality levels
with q(x) values that enclose the estimated value, qe(x).
Thus, the task is to identify in the appropriate table of
annex II levels j and j + I such that
qj(x)
> qe(x) > qj+l(x)
(5.7)
where qj(x) and qj+l(X) are the model values of q(x)
for levels j and j + 1, respectively, and qe(x) is the
estimated value. Then, the desired common index qC(5)
is given by
qC(5)
= (1.0 -
h) qj(5)
+ hqj+l(S)
(5.8)
where h is the interpolation factor calculated in the following way:
h
=
qe(x) - qj(x)
qj+l(x) -qj(x)
(5.9)
Computational procedure
If the data on children ever born and children dead are
for both sexes combined, the model qj(x) values should
be taken from the tables for both sexes combined in
annex II. If, however, the data are for male and female
children separately, the estimated values of q(x) will be
sex-specific, and the conversion to a common index
should use the model qj(x) values from the tables for the
relevant sex also presented in annex II.
Step 1. Calculation of average parity per woman
Average parities are computed by dividing the number
of children ever born by the total number of women, age
group by age group. In this case, only male children ever
born will be used, since male mortality is being
estimated. As an example, the average parity of women
aged 35-39, Pm(5), where the subindex m indicates that
only data on male children are used, is calculated below:
Step 7. Interpolation and analysis of results
Once seven estimates (one for each age group i of
women) of the selected common index-qC(5)-have
been obtained, it is recommended that they be plotted
against time. As noted in step 5, the t(i) values can be
converted into reference dates by subtracting them from
the surveyor census reference date (or the approximate
midpoint of the field-work), and the qC(5) estimates can
then be plotted against the resulting dates. Graphical
presentation of the results is essential to assess the consistency and general trend of the estimates, as the
example below illustrates.
A
P
m(5)
= 5,435,726 = 3.0681
1,771,680
The full set of parities in respect of male children is
shown in column 3 of table 12. At the foot of the table,
the values of the relevant parity ratios, Pm(l)/Pm(2) and
Pm(2)/Pm(3) also are displayed. Note that, as in the case
of both sexes combined, the average parity decreases
from age group 40-44 to age group 45-49, which suggests the existence of omission errors.
Step 2.
Calculation of the proportions dead among children ever born
As in the Trussell version, the proportions of children
dead are calculated by dividing the number of children
dead by the number of those ever born, for each age
group of women. Again in this example, only male children are considered, and hence the subindex m is used.
Thus, D m(5), the proportion of male children dead
among those ever borne by women aged 35-39 is calculated as follows:
DETAILED EXAMPLE
Compilation of the data required
The 1974 Bangladesh Retrospective Survey of Fertility
and Mortality will be used once more to provide an
example. The data on children ever born and children
dead and the total number of women have already been
38
Display 6. Second step in the compilation of data on children ever born
and children dead for Bangladesh
Children
ever born
Children
dead
Children
surviving
(1)
(2)
(3)
=
(2)+(3)
Age group
cf maher
~
""
'1S
=
(2)+ (4) + (5)
=
Children
living
Children
at home
elsewhere
(5)
(4)
living
=
(1)-(3)
=
(1)-(4)-(5)
(4)+(5)
15-19
1 160919
215 365
921 227
24327
20-24
4 901 382
997384
3 820649
83349
25-29
9085852
1 937 955
6927908
219 989
30-34
9 910 256
2 261 196
7 126473
522587
35-39
10 384 001
2490 168
6974267
919566
40-44
9 164 329
2415023
5472 460
1 276846
45-49
6905673
1 959544
3664 328
1 281 801
15-19
597248
117 165
469036
11 047
20-24
2 507 018
529877
1 938 220
38921
25-29
4675978
1 047294
3545904
82780
30-34
5 109487
1 204 582
3 780 859
124046
35-39
5435726
1 333 957
3925071
176 698
40-44
4883599
1 291 745
3 323 724
268 130
45-49
3 714957
1 030737
2393 149
291 071
15-19
563 671
98200
452 191
13 280
20-24
2394364
467507
1 882 429
44428
25-29
4409874
890661
3382004
137 209
30-34
4800 769
1 056 614
3345614
398 541
35-39
4948275
1 156 211
3049 196
742 868
40-44
4280780
1 123 278
2 148 736
1008716
45-49
3 190 716
928807
1 271 179
990730
~
~
~
1
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Iertility and Mortality (Dacca, 1977), table 8, p. 37
(reproduced in display 3 above).
39
Display 7. Compilation of data on the total number of women by age group
for Bangladesh
Total
number
of women
Ever-married
women
Single
women
flbmen of
staled
parity
(1)
(2)
(3)
(4)
flbmen of
not-stated
parity
(5)
=
(2)+(3)
=
Age group
oj women
(4)+(5)
15-19
3 014 706
20-24
2 653 155
25-29
2607 009
30-34
2015663
35-39
1 771 680
40-44
I 479575
45-49
1 135 129
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Iertility and Mortality (Dacca, 1977), table 3, p. 28
(reproduced in display 4 above).
D
m(5)
Step 3 (b).
= 1,333,957 = .2454
5,435,726
Table 10 shows the values of the regression
coefficients a(i), b(i), c(i) and d(i) needed to calculate
the multipliers k(i) for each regional pattern of the
United Nations model life tables. In this example, the
South Asian pattern will be used. As equation 5.5 shows,
for each age group i, k(i) is computed as the sum of
a(i), the product of b(i) and P(1)/P(2), that of c(i) and
P(2)/P(3), and that of d(i) and M. Thus, for age group
5, in which women are aged 35-39,
The complete set of proportions dead is displayed in
column 4 of table 12. Note that, as expected, the proportion dead increases with age of mother.
Calculation of the mean age at maternity, M
(Palloni-Heligman version only)
Step 3 (a).
The mean age at maternity is calculated using the data
compiled in display 11. As equation 5.3 states, M is the
ratio of two quantities: the sum of the products of births
times the midpoints of the age groups of mothers, and the
sum of births. Both are calculated below:
7
E B(i) mp(i)
i=1
= (320,406)(17)
k(5)
Values of k( i) for all age groups are shown in column 5
of table 12. Note that, as in the Trussell version, all
values of k(i) are close to 1.0.
+ (237,297)(37) + (95,356)(42)
+ (38,124)(47) =60,358,600
E B(i)
i=\
= 320,406
Step 4.
Calculation of the probabilities of dying by age
x, q(x)
The probabilities of dying by exact age x are computed
by multiplying the proportions dead D m (i) by the
corresponding multipliers k(i). To estimate qm(10), for
example,
+ 609,271 + 561,493
+ 367,833 + 237,297 + 95,356
+ 38,124 =
+ (.0231)(.2097)
+ (.0298)(.5268) + (.0149)(27.07)
= .6156
= 1.0395
+ (609,271)(22)
+ (561,493)(27) + (367,833)(32)
7
Calculation of the multipliers, k(i)
2,229,780
qm(10) = (1.0395)(0.2454) = .255
Then
The full set of qm(x) estimates is shown in column 7 of
table 12.
M = 60,358,600 = 27.07
2,229,780
40
Display 10. Tabulation of data on births in a year as appearing in the report on the
1974 Bangladesh Retrospective Survey of Fertility and Mortality
IlAHGLAllESH CENSUS 191.. RETROSPECTIVE SURVEY Of fERTILITY AIIll ItlRTALITY
OE fACTO
BANGlADESH
TABLE 10.
CHILDLESS
\OlEH
APRIL 13MIlCH , ..
APRIL 12MIlCH 13
BEfORE
APRIL 12
1973
H.II.
1912
H.II.
NOT
STATED
..5-'9
50-Sot
55+
H.S.
315 530
2 031 6O.t
2 530 026
2 511 51..
2 ODS 083
1 161 .t.tO
1 ..,5 299
1 130 992
1 00 611
2 2..9 2M
819
211 22..
1 166 887
Cl 225
15993..
19049
51 6..9
..9513
39 ..53
.to 848
103 126
204
6842
320 .t06
609 211
561 ..93
361 833
231 291
95 356
38 12..
8 102
6 599
3 896
219 069
..92 384
503986
326 551
223 ..28
105 316
36184
1.. 115
6 115
16563
261 832
818 863
1 222 501
1 131 139
1 181 018
1 151 209
9.t1 263
891 732
1 845 939
615
G
19 169
33 199
36 961
21 99..
10852
6681
2910
1 233
1 139
151
166..9
31 22..
.., 792
31 909
18633
11 651
.. 119
.. W
2049
IS 1l.tO
26992
.., 860
.... 895
.to 008
.... 503
"9 ....,
61 219
76 17..
283 031
TOTAL
11 121 2..2
2 393 112
2 251 923
1 933 02..
9 5.t2 3.tO
1..1 212
996
689 975
CHILD UNDER 15
15-19
ALIVE
20-2"
25-29
30-3"
35-39
..5-'9
50-Sot
55+
H.S.
2.. 863
159 221
1 90S 11..
2 21.. 326
1 , ..5901
1 53.. 253
1 236 6O.t
901 .tiS
801 333
1 528 511
204
5635
280716
552 ..53
513 568
33.. 935
213951
86 .t6..
33 2..1
1 138
.. 881
3 097
200 ..11
.t6.. 519
..,3 360
306 ....5
201 ..10
96 C"
32210
13 313
5 219
13 191
236 600
808 529
133 113
036 ..13
015 109
02.. 122
816 039
150 316
..10 130
204
.t08
11M3
29 221
35018
21 039
10038
6 .t89
2 910
1 233
935
751
15637
35 8.tO
.t66OO
30816
18050
10042
3 561
3 9.t3
1 635
1 169
821..
15 206
12007
10 193
9095
13053
19 .t04
30 610
105 111
TOTAL
1266.. 5..1
2 033 588
1 802 S38
8 305 092
131 000
166 9..1
225 382
UNDER 15
15-19
20-2"
25-29
30-3"
35-39
1 201
39 690
56 818
..7 925
32898
23 3.t6
8 892
.. 883
96..
1 712
199
18658
27 865
30 626
20106
16018
8 9.t2
",51"
1 .t02
1556
3366
2.. 821
7033..
88 551
95 119
105369
132691
131 848
W 16..
C5 ..22
..11
2 126
3972
I got9
955
8'"
192
..5-'9
50-5..
55+
H.S.
6 588
99 582
188 309
200 113
111 766
119 181
186 209
181 285
192 331
598 805
..11
204
1 012
1384
1 192
1 033
S83
1 615
618
204
..1..
1 216
13 215
21 936
29930
27 655
33 051
33 811
39 ..22
..2603
159 ..97
TOTAL
2 010 652
218 335
130 .t86
1 235 096
10212
8055
G
GROUP
TOTAL
UIIO£R 15
15-19
20-2"
25-29
30-3"
35-39
~
.to......
CHILD
IlEAO
~
CHILD
H.S.
UIIO£R 15
IS-19
20-2..
25-29
30-3..
3S-39
.to......
..5-'9
50-Sot
55+
H.S.
TOTAL
Source:
EVER-M/lRIEO lOlEH 8Y fIa GIlOlJ', DATE Of LAST 81RTH NIO SURVIVAL Of CHILD
TOTAL
lOlEH
AGE
1
1
1
1
1
12 855
591..
.. 118
3 1..1
2361
2351
2 913
2 169
3093
18 156
m
396
316
192
381
2 152
56125
..11
183
201
58217
.t68
12 855
5503
.. 118
2958
2 160
2351
2 511
2 393
2901
11 169
Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), p. 39.
TABLE
12.
ApPLICATION OF THE PALLONI-HELIGMAN VERSION OF THE BRASS METHOD TO DATA ON MALES
FROM THE
Age
group
index
Age group
of mother
(1)
15-19......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
Pm(l)/Pm(2)
P m(2)/Pm(3 )
Average
parity
1974 BANGLADESH
RETROSPECTIVE SURVEY
Proportion
dead
(i)
(2)
Pm(i)
(3)
Dm(i)
(4)
1
2
3
4
5
6
7
0.1981
0.9449
1.7936
2.5349
3.0681
3.3007
3.2727
.1962
.2114
.2240
.2358
.2454
.2645
.2775
Multiplier
k(i)
Age
(5)
(6)
0.9598
1.0278
1.0091
1.0240
1.0395
1.0204
1.0069
1
2
3
5
10
15
20
= .2097
= .5268
Probability
ofdying by
age .r, qm(x)
Time
Reference
index
date
q~(5)
(7)
I(i)
(8)
(9)
(/0)
.188
.217
.226
.241
.255
.270
.279
l.l
2.5
4.4
6.5
9.0
11.9
15.8
1973.2
1971.8
1969.9
1967.8
1965.3
1962.4
1958.5
.311
.268
.249
.241
.237
.244
.245
M = 27.07 years
Multipliers based on South Asian model.
41
Common
reference
Display 11. Compilation of data on births in a year by age group of mother
for Bangladesh
Source and
reported
age group
Exact-age group
of mother at
Midpoint of
exact-age
of mother
birth of childa
group
15-19
[15,20)
17.5
20-24
[20,25)
22.5
25-29
[25,30)
27.5
.~
~
30-34
[30,35)
32.5
s:
35-39
[35,40)
37.5
40-44
[40,45)
42.5
45-49
[45,50)
47.5
15-19
[14.5,19.5)
17
320406
20-24
[19.5,24.5)
22
609271
25-29
[24.5,29.5)
27
561 493
30-34
[29.5,34.5)
32
367 833
35-39
[34.5,39.5)
37
237297
40-44
[39.5,44.5)
42
95356
45-49
[44.5,49.5)
47
38 124
.g
Births in
a year
l::
"
S
~
:;
'"
....
0
'"
:;
~
Source: Bangladesh, Census Commission, Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality (Dacca, 1977), table 10, p. 29
(reproduced in display 10 above).
a The notation [x,y) indicates that exact ages at the time women gave birth range from x to y, exclusive of the latter, that is, age y is not quite reached.
Step 6. Conversion to a common index
The estimates of qm(x) for different values of x are
now converted into equivalent estimates of q:;' (5) using
the South Asian family of United Nations model life
tables for males. Consider, for example, the estimated
probability of dying by age 10, q~(lO) = .225. Using
table A.II.3, one can identify the two life-table levels
having q( 10) values such that:
Step 5. Calculation of the reference dates for q(x), t (i)
Again using the South Asian model, the coefficients
needed for the estimation of t(i) are obtained from the
third panel of table 11. Following equation 5.6, t(5) is
calculated below:
t(5)
= 2.7654 + (-10.8808)( .2097)
+ (16.2255)( .5268) = 9.01
q~(10)
That is, the estimated value of qm(10) refers to a period
approximately 9 years before the survey. A more
illuminating reference date is obtained by subtracting
each t(i) value from the survey's reference date
expressed in decimal form-1974.3 in this case (see step
5 of the detailed example for the Trussell version).
Hence, the reference date for qm( 10) is:
1974.3 - 9.0
> q~(10) > q~+1(10)
Those values are q~\ 10), which equals .25794 and
whose q~\5) equivalent is .23986, and ~~\1O), which
equals .24715 and has a corresponding q~ (5) of .22989.
Using equation 5.9, the interpolation factor h is obtained
as follows:
= 1965.3
h
All estimated values of t(i) and the reference dates
derived from them are presented, respectively, in
columns 8 and 9 of table 12.
=
.25500 - .25794
.24715 - .25794
= .2725
Then, making use of equation 5.8, the desired qC(5) is
calculated in the following way:
42
= (1.0 = .237
q~ (5)
The full set of
of table 12.
(.2725)( .23986)
q;;, (5)
+
On the other hand, estimates for the earliest period
may be biased downward, since they are derived from
information provided by older women. But the estimates
for males considered in isolation do not provide clear
evidence of the existence of such biases. If no further
information were available, it would be relatively safe to
conclude that male mortality changed little during the
1960s and to adopt as a reasonable estimate of its level
the mean of the under-five mortality estimates associated
with age groups 25-29, 30-34 and even 35-39, namely a
qm(5) equal to .242 for the period 1965-1970. It should
be noted that such a value is very close to that estimated
in chapter IV using the Trussell version-a qm(5) of
approximately .238 or .239 (see pp. 32-33). The following section will show that the sex differentials in mortality estimated using the United Nations South Asian model
lead to the same conclusions reached when analysing the
sex-specific estimates produced by the Coale-Demeny
South model.
(.2725)( .22989).
equivalents is shown in column 10
Step 7. Interpretation and analysis of results
The estimated probabilities of surviving by age 5,
q~ (5), are plotted against their respective reference dates
in figure 9. Note that the general trend of the resulting
curve is very similar to that obtained using the Trussell
version with model South (see table 8 and figure 8).
Once more, under-five mortality, q( 5), varies within a
fairly narrow range between 1958 and 1970, only to rise
sharply for the most recent period (1970-1973). Again,
this apparent increase is probably spurious, since it is
likely to be caused by the higher-than-average mortality
characterizing the children of younger women.
Figure 9. Under-five mortality, q( 5), for males in Bangladesh, estimated using
the South Asian model and the Palloni-Heligman version of the Brass method
q(5)
.35
.30
1:
5- 49 \
140~441
.25
.20
.15
.10
.05
0
1960
1970
1965
Reference date
Source: Table 12.
43
1975
TABLE
\3.
ApPLICATION OF THE PALLONI-HELIGMAN VERSION OF THE BRASS METHOD TO DATA ON FEMALES
FROM THE
1974
BANGLADESH RETROSPECTIVE SURVEY
Age
group
Average
Proportion
index
(i)
parity
P,li)
dead
D,1i)
Multiplier
of mother
k(i)
Age
x
(I)
(2)
(3)
(4)
(5)
(6)
1
2
3
4
5
6
7
0.\897
0.9025
1.69\6
2.3817
2.7930
2.8932
2.8109
.\742
.\953
.2020
.220\
.2337
.2624
.291\
0.9688
1.0267
\.0070
1.0233
1.0396
1.0208
1.0070
2
3
5
10
15
20
Age group
15-\9 ......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
Pf(l)/~(2)
Pf (2)/ f(3)
Probability
by
age x, q/t)
Time
reference
Reference
Common
index
I (i)
date
q/(5)
(7)
(8)
(9)
(/0)
.169
.20\
.203
.225
.243
.268
.293
2.5
4.4
6.6
9.2
12.\
16.0
\973.2
\971.8
1969.9
1967.7
1965.1
1962.2
1958.3
.296
.253
.226
.225
.225
.240
.25\
of dying
\
\.I
M = 27.07 years
Multipliers based on South Asian model
= .2072
= .5335
Figure 10. Under-five mortality, q(5), for males and females in Bangladesh, estimated using
the South Asian model and the Palloni-Heligman version of the Brass method
q(5)
.35 r-----------------------------115~191
.30h---~
140;44
.
.25
I 35; 39 1
I
•••••••
I 30; 34 I
······~
~
I 25;29
·.···tt·
[ 20-2~
I
I
...
•11
..• ••
.20
.15
-Males
.10
••••• Females
.05
_
O~_"__
1958
_I___....&...._......._
1960
1962
........_
__L._
1964
___"'_
___"'_
_ " '_ _L _ _ L __
1966
Reference date
Sources: Tables \2 and 13.
44
1968
_I___....&...._....&...._
1970
1972
_ ' _ _.....
1974
caused by errors in the basic data and should be disregarded.
In both sets, the estimates for females show a somewhat clearer declining trend during the 1958-1970 period
than those for males. According to the Palloni-Heligman
estimates, under-five mortality among females may have
declined from approximately .240 in 1962 to around
.225-.226 in 1970. According to the Trussell version, the
decline during the same period would have been from
.234 to .218. The magnitude of both declines is nearly
the same, though the starting and ending points differ
slightly. Such consistency is determined both by the basic
data and by the similarity of the South and South Asian
patterns used to derive the estimates considered here. In
the next chapter, the effects of choosing different mortality models in estimating mortality in childhood will be
considered in some detail.
Estimates of mortality in childhood by sex
Tables 13 and 14 present the estimates of q(5)
obtained by applying the Palloni-Heligman version of the
Brass method to the Bangladesh data referring to females
and to both sexes combined (using in all cases the South
Asian model). In addition, figure 10 presents a graphical
comparison of estimated under-five mortality by sex. Figure 10 should be compared with figure 8, which shows
the estimates by sex yielded by the Trussell version. Both
sets of estimates have the same overall characteristics:
with the exception of estimates derived from the reports
of women aged 45-49, the estimated under-five mortality
of males is, as is generally the case in most countries of
the world, higher than that of females. The reversal in
the trend observed for age group 45-49 is likely to be
TABLE
14.
ApPLICATION OF THE PALLONl·HELIGMAN VERSION OF THE BRASS METHOD TO DATA ON BOTH SEXES
FROM THE
1974
BANGLADESH RETROSPECTIVE SURVEY
Age
Age group
of mother
(/)
15-19 ......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
P(1)/P(2)
P(2)/P(3)
M
group
index
Average
parity
Ii)
(2)
1
2
3
4
5
6
7
Proportion
Common
index
Multiplier
kli)
Age
P(i)
dead
Dli)
Probability
ofdying by
age x, qlx)
Time
reference
Iii)
Reference
date
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
0.3851
1.8474
3.4852
4.9166
5.8611
6.1940
6.0836
.1855
.2035
.2133
.2282
.2398
.2635
.2838
0.9642
1.0272
1.0081
1.0273
1.0396
1.0206
1.0069
1
2
3
5
10
15
20
.179
.209
.215
.234
.249
.269
.286
1.1
2.5
4.4
6.6
9.1
12.0
15.9
1973.2
1971.8
1969.9
1967.7
1965.2
1962.3
1958.4
.304
.261
.238
.234
.231
.242
.248
= .2085
= .5301
Multipliers based on South Asian model
Sex ratio at birth = 1.05
= 27.07 years
45
q'(5)
Chapter VI
INTERPRETATION AND USE OF THE ESTIMATES YIELDED
BY THE BRASS METHOD
Through selected examples, this chapter discusses
some of the most common problems encountered in using
the estimates yielded by the Brass method and offers
guidance regarding possible solutions to them. The reader
must be forewarned, however, about the tentative nature
of the solutions proposed, since they respond to problems
arising from the lack of reliable information on the
incidence of mortality over time. In other words, the
strategies described attempt to reduce remaining gaps in
knowledge but cannot eliminate them altogether.
models in estimating mortality in childhood for the same
population.
Given this state of affairs, it is nevertheless important
to provide the reader, first, with tools to test the adequacy of the different models when information on the
pattern of mortality in childhood is available and,
secondly, with some sense of the magnitude of the errors
that may arise if the best model is not used during the
estimation process. These points will be considered in the
next two sections.
WHICH VERSION OF THE BRASS METHOD SHOULD ONE USE?
Use of information on the pattern of mortality in
childhood to test the adequacy of mortality models
The reader who has become familiar with the Trussell
and the Palloni-Heligman versions of the Brass method
presented in chapters IV and V above is probably asking
which version to use. To answer that question, it is
necessary to determine the best model to use for a given
population.
Consider again figures 5 and 6 in chapter I. Those
figures illustrate how the mortality patterns in childhood
of different populations vary and how the patterns of certain populations are close to a given mortality model and
not to others. Thus, if one knew, for example, that the
mortality pattern in childhood of a given country was
close to the Chilean pattern, the answer to the question
posed above would be immediate: use the PalloniHeligman version of the estimation method with the
Chilean model. Unfortunately, for most populations
whose mortality in childhood needs to be estimated
indirectly, it is not easy to establish the model that most
closely approximates the prevailing pattern.
Demographers faced with this quandary have
developed certain rationalizations to justify the use of
different mortality models and, hence, of different
methods. For instance, it has been argued that in populations practising prolonged breast-feeding, mortality in
infancy-below age I-is likely to be relatively low with
respect to mortality at older childhood ages-from 1 to
5-and, consequently, that it would be appropriate to use
model North (that is, the Trussell version). It has also
been suggested that, in the absence of reliable information on the pattern of mortality in childhood, the use of
"average" models such as West (Trussell version) or the
General pattern (Palloni-Heligman version) would be adequate.
In practice, determining the best model to use for a
given population is still more an art than a science, and it
is common to find that different analysts use different
Available information on the pattern of mortality in
childhood may take different forms, but only two possible
variants will be considered here: (1) reasonably reliable
estimates of infant and under-five mortality and (2)
infant- and child-mortality estimates.
First, note that the two combinations mentioned above
are equivalent, since under-five mortality, q( 5), may be
derived from infant mortality, q( 1), and child mortality,
4ql, as follows:
q(5)
= q(1) + (1.0 -
q(1»4qt
(6.1)
For example, in the case of Bangladesh, estimates of
infant and child mortality may be obtained from the
1975-1976 survey carried out as part of the World Fertility Survey programme. The estimates referring to the 0-9
years preceding the survey and derived from the information gathered in the fertility histories of interviewed
women are q(1) = .13275 and 4ql = .09130. Under-five
mortality is then obtained as
q(5)
= .13275 + (1.0 -
.13275)(.09130)
= .21193
The adequacy of available mortality models for
representing mortality in Bangladesh can now be assessed
algebraically, as well as graphically (see figures 5 and 6
in chapter I and annexes III and IV). The algebraic
approach makes use of the tables in annexes I and II. It
consists of using equations 5.7, 5.8 and 5.9 to find the
q(5) values associated with the estimated qe(1) = .13275
for each of the mortality models available. For instance,
according to model North for both sexes combined (see
table A.1.9), the two values of ~(1) surrounding qe( 1)
are qll(1) = .14076 and ql (1) = .12744, whose
corresponding values of under-five mortality are
46
qll(5) = .23793 and qI2(5)
equation 5.9,
h
=
= .21533.
.13275 - .14076
.12744 - .14076
TABLE 15. COMPARISON OF ESTIMATEDINFANT AND UNDER-FIVE
MORTALITY IN BANGLADESHa WITH THE AVAILABLE MORTALITY MODELS
Hence, using
= .60135
and, using equation 5.8,
qN(5)
with q"(I) = ,13275
Ratio of model q(5)
to q"(5)b
Model q(5) associated
Mode/
= (1.0 - .60135)(.23793) + (.60135)(.21533)
= .22434
Values of under-five mortality corresponding to the
estimated qe( 1) for all the available mortality models are
presented in table 15, together with the ratios of those
values to the estimated qe(5) = .21193. Those ratios
indicate how well the models fit the estimated values. A
ratio of 1.0 indicates a perfect fit. In this case, both the
South and the South Asian models provide an excellent
fit, a conclusion already reached through the graphical
comparison presented in chapter I.
Figure 11 presents a comparison of the estimates of
mortality in childhood obtained in chapters IV and V by
Coale-Demeny models
North
South
East.
West
.
.
.
..
.22434
.21315
.17619
.20009
1.059
1.006
0.831
0.944
United Nations models
Latin American
Chilean
South Asian
Far Eastern
General
.
.
.
..
.
.22109
.16863
.21304
.20727
.20922
1.043
0.796
1.005
0.978
0.987
"Estirnates of qe(i) and qe(5) were obtained from the 1975-1976
World Fertility Survey.
b A ratio of 1.0 indicates a perfect fit between the country estimates
and the mortality model.
Figure 11. Comparison of the estimates of under-five mortality in Bangladesh, obtained using
models South and South Asian
q(5)
.35 , . . . - - - - - - - - - - - - - - - - " - - - - - - - - - - - - - 1 1 5 ; 1 9 1
.30
.25
I~5-49 I
140~441
1
130~341
35; 391
e···················~··........... ~
I 25;29
~........
120-2~ I ....•
1
..••..
••
.....
------
.20
.15
-
.10
Coale·Demeny South
••••• United Nations South Asian
.05
o
L...-
.........
1960
....a...-
.........
1965
1970
Reference date
Sources: Tables 7 and 14.
47
~
1975
x 9) estimates of under-five mortality. If data for males,
females and both sexes combined are available, the
number of estimates of under-five mortality is multiplied
by 3, yielding 189 values. If, as in tables 16 and 17,
more than one common index is used for exploratory or
evaluation purposes, the number of estimates increases
again, in this case doubling, to 378. It is useful for the
reader to become familiar with the many possible estimates, since, with the widespread use of computers,
arrays of the type presented in tables 16 and 17 are common: In fact, in the case at hand, those tables display
only a subset of the estimates routinely produced by the
QFIVE program.
Generally, graphical displays offer the best tool to
analyse the various estimates available. Figures 12 and
13 display graphically the sets of estimates for both sexes
combined produced by the Trussell and Palloni-Heligman
versions, respectively. It is apparent from those figures
that the estimates of under-five mortality, q( 5), are generally concentrated within a narrower range than those of
infant mortality, q( 1). The possible variability of the
different estimates is illustrated in figure 14, in which the
highest and lowest estimates associated with each age
group of mother are plotted. The grey areas in the
display represent, albeit roughly, the range of variation
of possible estimates.
using models South and South Asian in applying the
Brass method. Note that, as expected, the estimates are
very similar: they display a similar trend and are usually
within 0.01 points of one another. This example shows
that, if the appropriate model is used, it makes little
difference whether estimates are derived using the
Trussell or the Palloni-Heligman version of the Brass
method.
What are the consequences of using the "wrong" model?
To assess the consequences of using the "wrong"
model, let us suppose that the nine mortality models used
in this Guide represent all possible mortality patterns in
childhood. Although, strictly speaking, that assumption is
false, since there are patterns of mortality in childhood
that do not fit existing models (see chapter I), it nevertheless allows us to explore the sensitivity of different estimates to the choice of model.
To explore the variability of estimates with respect to
the use of different models, we consider again the case of
Bangladesh. Tables 16 and 17 present the complete set of
infant and under-five mortality estimates for that country
derived by applying the Trussell and Palloni-Heligman
versions of the Brass method to the 1974 data classified
by sex. Application of the estimation procedures discussed in this Guide potentially yields a total of 63 (or 7
TABLE
16.
ESTIMATES OF INFANT AND UNDER-FIVE MORTALITY IN BANGLADESH. OBTAINED
USING THE COALE-DEMENY MORTALITY MODELS
Male
Both sexes
Female
Age group
Reference
of
mother
Reference
date
q(5)
q(l)
date
q(5)
q(l)
Reference
date
q(5)
q(l)
North ......................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.1
1971.7
1969.8
1967.6
1965.1
1962.5
1959.7
.297
.254
.228
.221
.214
.218
.215
.177
.151
.135
.131
.127
.129
.127
1973.1
1971.7
1969.8
1967.6
1965.2
1962.6
1959.7
.304
.261
.238
.229
.221
.221
.212
.186
.158
.145
.139
.135
.135
.130
1973.1
1971.8
1969.8
1967.6
1965.1
1962.4
1959.6
.290
.248
.217
.213
.207
.214
.218
.167
.142
.125
.122
.119
.123
.125
South ......................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.1
1971.7
1969.7
1967.3
1964.7
1961.9
1958.8
.294
.248
.230
.230
.229
.237
.239
.170
.149
.141
.141
.140
.144
.145
1973.1
1971.7
1969.7
1967.4
1964.8
1962.0
1958.9
.301
.256
.241
.238
.236
.240
.236
.179
.157
.150
.149
.147
.150
.148
1973.1
1971.7
1969.7
1967.3
1964.6
1961.8
1958.7
.280
.240
.218
.221
.222
.234
.243
.161
.141
.131
.133
.133
.138
.142
East ........................
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.1
1971.6
1969.6
1967.2
1964.5
1961.6
1958.4
.254
.235
.225
.228
.228
.238
.241
.187
.174
.167
.169
.169
.176
.178
1973.1
1971.6
1969.6
1967.2
1964.6
1961.7
1958.5
.261
.242
.235
.235
.235
.241
.238
.197
.183
.178
.179
.178
.183
.180
1973.1
1971.7
1969.6
1967.1
1964.4
1961.5
1958.3
.248
.229
.214
.219
.221
.234
.244
.176
.164
.154
.158
.159
.167
.174
West ...................•...
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.1
1971.7
1969.6
1967.3
1964.7
1962.0
1959.1
.273
.244
.228
.227
.224
.230
.229
.182
.163
.152
.152
.150
.154
.153
1973.1
1971.7
1969.7
1967.3
1964.8
1962.1
1959.2
.279
.250
.238
.235
.231
.234
.227
.192
.172
.164
.161
.159
.161
.156
1973.1
1971.7
1969.6
1967.2
1964.6
1961.9
1959.0
.266
.237
.217
.218
.216
.226
.230
.172
.153
.140
.141
.140
.146
.149
Model
48
TABLE
17.
ESTIMATES OF INFANT AND UNDER·FlVE MORTALITY IN BANGLADESH. OBTAINED
USINGTHE UNITED NATIONS MORTALITY MODELS
Both sexes
Male
Female
ARe group
'1
Reference
Reference
date
q(5)
q(/)
Reference
dare
q(5)
q(l)
.307
.268
.248
.239
.232
.230
.230
.189
.167
.157
.152
.148
.147
.146
1973.2
1971.8
1970.0
1967.8
1965.3
1962.5
1958.9
.264
.229
.223
.218
.223
.235
.142
.127
.124
.122
.124
.129
Model
mother
date
q(5)
q(l)
Latin American ........
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.2
1971.8
1970.0
1967.8
1965.4
1962.6
1959.0
.316
.266
.239
.231
.226
.227
.232
.179
.155
.142
.138
.135
.136
.138
1973.2
1971.8
1970.0
1967.9
1965.5
1962.7
1959.1
.
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.0
1971.7
1969.8
1967.5
1965.0
1962.2
1958.6
.268
.245
.232
.231
.229
.237
.238
.199
.185
.176
.175
.174
.179
.180
1973.0
1971.7
1969.8
1967.6
1965.1
1962.3
1958.7
.272
.251
.242
.239
.235
.240
.235
.210
.196
.189
.187
.184
.188
.184
1973.0
1971.7
1969.8
1967.5
1964.9
1962.1
1958.5
.262
.239
.220
.223
.223
.235
.241
.188
.174
.162
.164
.164
.171
.175
South Asian
..
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.2
1971.8
1969.9
1967.7
1965.2
1962.3
1958.4
.304
.261
.238
.234
.231
.242
.248
.179
.157
.146
.143
.142
.148
.151
1973.2
1971.8
1969.9
1967.8
1965.3
1962.4
1958.5
.311
.268
.249
.241
.237
.244
.245
.188
.166
.155
.151
.149
.153
.153
1973.2
1971.8
1969.9
1967.7
1965.1
1962.2
1958.3
.296
.253
.226
.225
.225
.240
.251
.169
.149
.136
.136
.135
.143
.148
Far Eastern
.
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.1
1971.7
1969.9
1967.7
1965.3
1962.7
1959.4
.304
.260
.236
.227
.218
.218
.210
.183
.160
.148
.144
.139
.139
.134
1973.1
1971.7
1969.9
1967.8
1965.4
1962.8
1959.5
.325
.271
.247
.235
.223
.219
.209
.193
.166
.160
.153
.146
.144
.138
1973.1
1971.7
1969.9
1967.7
1965.3
1962.6
1959.3
.284
.249
.225
.219
.213
.218
.212
.172
.155
.136
.134
.131
.133
.130
..
15-19
20-24
25-29
30-34
35-39
40-44
45-49
1973.2
1971.8
1970.0
1967.8
1965.4
1962.6
1959.1
.302
.261
.237
.229
.223
.224
.226
.181
.160
.147
.143
.140
.141
.142
1973.2
1971.8
1970.0
1967.8
1965.4
1962.7
1959.2
.297
.263
.246
.237
.231
.228
.228
.191
.172
.162
.157
.153
.152
.152
1973.2
1971.8
1970.0
1967.8
1965.3
1962.5
1959.0
.286
.259
.227
.221
.216
.220
.227
.171
.147
.132
.130
.127
.129
.132
Chilean
General
"Not available.
These examples demonstrate the relative "robustness"
of q( 5) as an indicator of mortality in childhood when it
is estimated by the Brass method. An estimate is said to
be robust when its value is not severely affected by deviations from the assumptions on which it is based. For the
Brass method, q(5) is more robust to the choice of mortality model than q( 1). This observation allows us to
provide a partial answer to the question posed at the
beginning of this section. The immediate consequence of
choosing the wrong model is that the estimates obtained
of under-five mortality will be biased. However, so long
as only q(5) is used as the common index, the magnitude
of those biases is likely to be small, particularly for age
groups above 20-24. If a different common index is opted
for, robustness may decline. In particular, estimates of
infant mortality, a very popular indicator, are especially
sensitive to the choice of model and may be severely
biased when the model used deviates markedly from the
mortality pattern actually prevalent in the population
under study.
In conclusion, if reliable evidence allowing an educated
Figure 14 illustrates an important point: no matter
which mortality model is chosen in applying the Brass
estimation method, the errors that may affect the resulting
estimates of q( 5) are likely to be smaller in both absolute
and relative terms than those affecting q( 1). To give an
example, suppose that the Chilean model is chosen to
estimate mortality in childhood in Bangladesh, instead of
the South or South Asian model. Then, considering only
the most reliable estimates-those associated with age
groups 25-29 to 35-39-one would obtain an estimate of
q( 5) for both sexes combined close to .231 and a q( 1) of
approximately .175, instead of the .230 and .141 values
yielded by model South. If the South estimates are indeed
correct, then the Chilean model will still estimate q( 5)
fairly accurately but will overestimate q( 1) by some .034
points, or 24 per cent. At the other extreme, if model
North were selected, q( 5) would be about .221 and q( 1)
about .131, so that although both values would underestimate the South values by some .009 or .010 points, the
error in q( 5) would be 4 per cent, as opposed to 7 per
cent in q(l).
49
Figure 12. Infant and under-five mortality for both sexes in Bangladesh,
estimated using the four Coale-Demeny mortality models
q(x)
.35 . - - - - - - - - - - - - - - - - - - - - - - - - - . . . ,
..,..
.30
,25
;~
-.:;..,. ......~
_.._...-----
.,..;'
.....
~.::....~==:-=---------
..........................................................
"
-_.----
.«,~'
Iq(1)l
.20
----------.15
~I
~/
..
---------------.. .
--_--/
, ~;-;,
._-------------------_
... _-----------------~~~~~
......................................................................
.10
.,I.';"
........ North
----. South
- - - East
.05
--West
Ol....-_ _.L-_ _....L.. _ _.......L
1958
1960
1962
.L-_ _....L.. _ _.......L_ _- - l l - - - _ - - . . J
1964
1966
1968
1970
1972
1974
Reference date
Source: Table 16.
estimates derived from the reports of older women (aged
45-49) in 1975 (yielding an estimated q(5) of .240 for
early 1963) and those obtained from the reports of
women aged 25-29 in 1966 (producing a q(5) estimate of
.245 for mid-1962). Such consistency is responsible for
the relative smoothness of the trend suggested by the two
curves in figure 15. Tunisia is thus a fairly exceptional
case, in which the high consistency of the estimates
derived from independent sources allows the analyst to
adopt them at face value as indicators of the evolution of
mortality in childhood. That is, disregarding the q(5)
estimates associated with younger age groups of women
(15-19 and 20-24), it can be concluded that under-five
mortality in Tunisia declined from 283 deaths per 1,000
births in late 1951 to 245 deaths per 1,000 births in the
early 1960s, reaching a level of about 192 deaths per
1,000 births in early 1972.
The second example is provided by the case of Ecuador, where data on children ever born and surviving are
available from three sources: the 1974 and 1982 censuses
and the 1979 National Fertility Survey carried out as part
of the World Fertility Survey programme. Table 19
presents the data available and the estimates they yield
choice of mortality model is lacking, errors in the estimates obtained are likely to result. Such errors, however,
may be minimized by using q(5) as a common index, as
has been suggested in previous chapters of this Guide.
ANALYSIS OF DATA FROM SUCCESSIVE CENSUSES OR SURVEYS
In this section, the cases of two countries having more
than one source of data on children ever born and surviving are discussed, in order to provide the reader with
some guidance on how to evaluate and use the estimates
obtained.
The first example is the case of Tunisia, whose 1966
and 1975 censuses gathered the information necessary to
apply the Brass method. Table 18 presents the basic data
and the estimates of under-five mortality obtained by
using the Trussell version with model West. The
estimated q(5) values are plotted in figure 15. The most
striking feature of those estimates is the declining trend
they display. Since the estimates derived from the reports
of younger women (mainly age group 15-19) are clearly
biased upward, they should be disregarded.
Another noteworthy feature of the estimates for Tunisia is the high degree of consistency noticeable in the
50
Figure 13. Infant and under-five mortality for both sexes in Bangladesh,
estimated using the five United Nations mortality models
q(x)
.35 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,
.30
.25
..__
........
._._-----~
~~~~~~~~~~~~~~~~~~~
------------------
.20
~
.15
......... Latin American
.;
.10
•••••• South Asian
---- General
.05
------ Far Eastern
- - Chilean
OL..__ _
1958
~
__
1960
~
_ ___L
1962
1964
L.._ _ __J
. . _ ._ ____L__ _- - '
1966
1968
1970
1972
1974
Reference date
Source: Table 17.
TABLE
18.
ESTIMATION OF UNDER-FIVE MORTALITY IN TUNISIA FROM DATA FROM SUCCESSIVE CENSUSES.
USING MODEL WEST AND THE TRUSSELL VERSION OF THE BRASS METHOD
Age of
mother
Number of
Children
Children
ever born
surviving
Reference
dale
Estimated
women
15-19......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39......................
40-44 ......................
45-49 ......................
188751
151018
154431
147782
130005
99455
83371
24212
165 738
377 961
532099
560830
444 159
357 875
1965.5
1964.3
1962.4
1960.1
1957.5
1954.7
1951.7
.275
.250
.245
.245
.252
.271
.283
15-19......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
307400
244 010
168 800
136650
151 910
135 830
113 030
9800
145010
355750
471 590
674 140
686760
573820
1974.5
1973.7
1972.3
1970.6
1968.6
1966.2
1963.1
.282
.193
.192
.197
.208
.227
.240
q(5)
/966 census
28716
205527
485203
701 786
763285
639682
544345
/975 census
11620
168780
425400
578450
852 160
908660
796250
Note: The census dates were 3 May 1966 and 8 May 1975.
Sources: Tunisia, Ministere du plan, Institut national de la statistique, Recensement general de la population et des logements du 3 mai /966, vol. I, tables 17, 27 and 28; unpublished tables from the 8 May
1975 census.
51
Figure 14. Range of variation of the possible estimates of infant and
under-five mortality for both sexes in Bangladesh
q(x)
,.---------------------......
.30
.25
.20
.15
1958
1960
1962
1964
1966
1968
1970
1972
1974
Reference date
Source: Tables 16 and 17.
directly comparable with those obtained from the reports
of older women in 1979 (age groups 40-44 and 45-49).
The latter, however, may be biased downward because of
the omission of dead children that tends to affect the
reports of older women. Therefore, the noticeable
difference between the 1974 census estimates obtained
from women in age groups 30-34 and 35-39 and those
derived from women in age groups 40-44 and 45-49 at
the time of the 1979 survey does not invalidate the
former.
It is noteworthy that in Ecuador the degree of upward
bias affecting the estimates derived from the reports of
younger women is minor. In fact, for the 1982 census it
is hardly noticeable, though this outcome may be the
when model West is used. The different sets of estimates
are displayed graphically in figure 16.
Note that, for Ecuador, the estimates obtained from
different sources cover overlapping periods and are therefore more directly comparable with one another. As
figure 16 shows, there is relatively good agreement
between the estimates derived from the 1979 National
Fertility Survey (from women in age groups 25-29 to
35-39) and those obtained from the 1982 census (from
women in age groups 30-34 to 40-44). The 1974 census
estimates covering the same period are noticeably higher,
but some are derived from younger women (age groups
15-19 and 20-24) and may be biased upward. For earlier
periods (1965-1968), the 1974 census estimates are
52
Figure 15. Under-five mortality for both sexes in Tunisia, estimated using
model West and theTrussell version of the Brass method
q(5)
.28
r-------------------------------------...,
• ••
••
••
•••
•
••
•
••
••
.26
••
••
-----~,--..
11966 census
I ••••••
•••••••
.24
••
.
•....
"
,...-----...,>,
I"
11975 census
.22
'
.L.-
1956
--'-
.......
1960
1964
I
"
"
""
.20
.18 ' - - _ - L 1952
,,
,,
,,
,,
,
'--
1968
,
I
.............-. J'
--L-
1972
---'
1976
Reference date
Source: Table 18
result of preliminary adjustments to the data (see table
19).
Although the estimates for Ecuador display considerably less consistency than those obtained in the case of
Tunisia, it is still possible to infer from them the likely
trend that mortality in childhood has followed through
time. Estimates that could be used to determine that trend
include the 1974 census estimates derived from age
groups 30-34 to 40-44, the 1979 National Fertility Survey estimates derived from age groups 25-29 to 35-39
and the 1982 census estimates derived from age groups
25-29 and 30-34. The trend determined by those estimates implies that under-five mortality in Ecuador
declined from about 173 deaths per 1,000 births in late
1962 to some 103 deaths per 1,000 births in mid-1978.
First, in populations for which reliable information
about the prevalent pattern of mortality in childhood is
lacking, there is always uncertainty about which mortality
model to use in applying the Brass method. Although the
use of q(5)-under-five mortality-as a common index
reduces possible biases in the estimates obtained, they
cannot be guaranteed to be accurate.
Second, the availability of a variety of mortality
models gives the Brass method flexibility dealing with
cases where independent evidence on the mortality pattern of the population involved can be obtained.
Third, given the upward biases that usually affect the
estimates derived from the reports of younger women,
estimates referring to recent periods usually have to be
rejected as inaccurate. Thus, the most reliable estimates
produced by the Brass method usually refer to a period
between three and ten years preceding the time of interview, which limits their usefulness for the timely evaluation of the effects of health or development programmes.
Fourth, the power of the Brass method is increased
when it can be applied to several data sets referring to
the same population. The availability of independent estimates covering overlapping periods allows the analyst to
OVERALL OBSERVATIONS ON THE USE OF THE BRASS METHOD
To sum up, the five chapters describing the nature of
the Brass method, the latest procedures available for its
application and the problems faced in making use of
those procedures and in using or interpreting the estimates obtained highlight the following features of the
methodology as it now exists.
53
TABLE 19.
ESTIMATION OF UNDER-FIVE MORTALITY IN ECUADOR FROM DATA FROM SEVERAL SOURCES.
USING MODEL WEST AND THE TRUSSELL VERSION OF THE BRASS METHOD
Age of
mother
Number of
Children
ever born
women
Children
Reference
date
Estimated
surviving
52040
309499
518 758
628909
729996
686 136
547994
1973.5
1972.2
1970.4
1968.1
1965.6
1962.9
1959.9
.177
.160
.157
.160
.165
.173
.176
262
1422
2364
3084
3350
3078
2624
1978.8
1977.5
1975.6
1973.3
1970.8
1968.0
1965.1
.142
.127
.125
.137
.135
.153
.156
82995
412 349
679249
799253
826430
815408
716214
1981.8
1980.5
1978.6
1976.4
1974.0
1971.4
1968.5
.077
.081
.103
.112
.128
.136
.135
q(5)
1974 census
15-19 ......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
353 781
295702
225738
180190
164 258
139074
109861
15-19 ......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ......................
40-44 ......................
45-49 ......................
1680
1377
1074
883
717
586
480
15-19 ......................
20-24 ......................
25-29 ......................
30-34 ......................
35-39 ..................... ,
40-44 ......................
45-49 ......................
440255
394682
316908
252622
204 310
168940
137524
58368
354693
605308
746534
884760
853 736
700675
1979 National Fertility Survey
288
1585
2672
3568
3913
3728
3255
1982 census'
87940
442309
750573
898 869
955762
965 187
861 021
Note: The census dates were 8 June 1974 and 28 November 1982. The reference date for the National
Fertility Survey was taken to be 1979.75.
apreliminary data. The number of children surviving declared by women aged 15-19 has been adjusted.
Source: Ecuador, Ministerio de Salud Publica, Encuesta Nacional de Salud Materno 1nfanti! y Variables Demogrdficas-Ecuador 1982, 1nforme Final, Torno II (Quito, 1984), pp. 65 and 93-94.
Figure 16. Under-five mortality for both sexes in Ecuador, estimated using
model West and the Trussell version of the Brass method
q(5)
.20
1 4;- 49 1
•
140~441
•
.15
.10
1
.
35 39
;
1
I2O=24l
r3O=34l
.....·......l
.••••. ·
~
,------.J,
~
GH9l
~~~~
ck
I
~0-341
~ .---::.:.....,.:9'.. -.::.:.~. ••..
135-3~ I
~
.-----------------,
45-49
40-44
1
25,29 [
1 20 -;4
-i~<=y~:~······
35-39
. .---
1
••••
..
I!B!J
~---~,
~
20-24
~~,
~
~ ""---e~
--1974 Census
.05
······1979 National Fertility Survey
-----1982 Census
OLL-
1960
L-
1965
----l
L-
1975
1970
Reference date
Source: Table 19.
54
~=_----=
1980
1985
check their consistency and select those less likely to be
affected by extraneous biases.
These observations underscore both the strengths and
the limitations of the Brass method. They do not reflect,
however, the successful record that the method has had
in allowing the estimation of mortality in childhood in
populations with poor or defective data sources. During
the more than twenty years of its existence, the Brass
method has been instrumental in permitting, for the first
time in history, a world-wide assessment of levels and
trends of the mortality of children (see United Nations,
1988, and Bucht, 1988).
55
Chapter VII
BRASS-MACRAE METHOD
its performance under a variety of circumstances. Hence,
the conclusions presented in this chapter are necessarily
tentative and may err on the side of caution.
This chapter describes a method proposed only
recently by W. Brass and S. Macrae (1984) to estimate
mortality in childhood. The Brass-Macrae method is in
some ways complementary to the original Brass method.
It has the advantage of producing estimates for recent
periods (about two years before the time of interview)
and is based on data that may be collected at relatively
low cost. Its main drawback is that the data it uses may
not be representative. Indeed, since the Brass-Macrae
procedure envisages that only women about to give birth
will be asked about the survivorship of their previous
child, the children whose mortality is being measured
may not be representative of all children. In cases where
only women giving birth in hospitals or in government
clinics are interviewed, the data will be even less representative of the total population. Strategies to make the
data used more representative and efforts to understand
the possible biases to which they are subject or to devise
means of eliminating such biases are currently being pursued.' Yet, it is likely that those efforts may result in a
considerably more complex estimation procedure than the
one presented in this Guide.
Simplicity is an important advantage of the BrassMacrae method, and thus may be worth maintaining even
at the expense of full representativeness or perfect accuracy of the data. So far, the method has been largely
applied to data gathered in specific areas-a given town
or city or even a given hospital. 4 Consequently, it has
been clear from the start that the population under study
was not representative of the whole population of a country. In certain circumstances, nationally representative
data may not be required, as, for instance, when a programme to improve child nutrition operates only in a
given region. In such cases, the Brass-Macrae method
may be used by programme managers to monitor changes
in child mortality only in the region of interest. It can
also be argued that, for purposes of monitoring change,
biases affecting the Brass-Macrae estimates may be
discounted as long as they remain constant. The
researcher must, however, be aware of the possible
sources of bias, in order to be able to ascertain the likelihood of biases remaining constant through time.
Owing to its recency, the Brass-Macrae method is still
in the process of being tested. Several organizations,
including UNICEF, the Latin American Demographic
Centre (CELADE) and the London School of Hygiene
and Tropical Medicine, are currently engaged in experimental applications of the method and are trying to
determine how best to gather the basic information
needed for its use. 5 Although the method seems promising, it is still difficult to judge its efficacy or to evaluate
NATURE OF THE BASIC DATA
The Brass-Macrae method derives measures of child
mortality from information obtained at or near the birth
of a child about the survival of the mother's previous
child (and sometimes about the child born prior to that
one also). Unlike the Brass method, the information used
in applying the Brass-Macrae method is derived from
administrative sources (such as health-centre records)
rather than from population censuses or surveys. The
additional costs of data collection are thus small, since all
that is generally required is the addition of two or three
questions to an existing administrative form. The questions could be:
1. Have you been pregnant before? Yes/No
2. If yes: Was the child of your previous pregnancy
born alive? Yes/No
3. If yes: Is that child still alive? Yes/No
An advantage of these questions is that they require
very simple answers that do not even involve numbers.
In that respect, they are likely to elicit more reliable
information than the questions used to gather the data
required for the Brass method.
As already mentioned, the Brass-Macrae method uses
data that do not necessarily reflect the mortality experience of all children in a given population. In particular,
deaths among the last-born children of all women in the
population remain unrecorded. Also, because women are
interviewed only at the time they give birth, the experience of the highly fecund is more likely to be recorded.
Biases arising from such selectivity will always be in
operation, even if all women giving birth in a country are
interviewed. However, there is reason to believe that in
populations where fertility is still moderate to high, the
resulting biases would be minimal.
In general, it is not expected that the data necessary
for the application of the Brass-Macrae method would be
obtained from a nationally representative sample. A hospital or clinic is the typical setting envisaged for the collection of the basic information, although it has been suggested that midwives aiding in home deliveries might be
trained to gather the necessary data. Such an approach to
data collection would be mandatory in countries or
regions where a majority of women give birth at home.
Although nationally representative data may not be necessary for monitoring and evaluating local interventions
aimed at reducing mortality, it is still important to define
56
the target population and take steps to ensure that the
data gathered are in effect representative of that population.
LIMITATIONS OF THE BRASS-MACRAE METHOD
Because of the data it uses and the simplifying assumptions made in its application, the Brass-Macrae method
has certain limitations, which are discussed in some detail
below.
First, the data used exclude information on all lastborn children. Such exclusion is not serious in highfertility populations, but in those populations where completed family size is small (two or three children per
woman), a large proportion of first-born children will be
included in the data used for estimation purposes, and
their mortality experience may not be representative of
average mortality levels in the total population of children. However, in contrast with the Brass method, the
Brass-Macrae approach is based on data on recent births
of all orders to women of all ages and is therefore less
likely to display the biases associated with the increased
mortality of children whose mothers are very young.
Secondly, the Brass-Macrae method as described below
makes no explicit allowance for changing fertility and
mortality conditions. Fertility declines due to significant
increases in average birth intervals will lead to overestimates of mortality when the proportion dead of previous
children is simply equated to q(2). Mortality declines, on
the other hand, will startto be reflected in the q(2) estimates only two or three years after they occur and will
be fully reflected in those estimates only five years after
their inception. Thus, as with the Brass method, sharp
declines in mortality will appear as a smoothed downward curve, but, in contrast with the Brass method, the
downward trend will not pre-date the inception of mortality change.
Thirdly, a single application of the Brass-Macrae
method does not allow the estimation of trends in child
mortality. Data relating to several years of observation
are required to obtain some indication of trends.
Fourthly, since the Brass-Macrae method does not rely
on data obtained through surveys based on probabilistic
samples, the estimates it yields often do not refer to the
total population of a country or even of a region. Biases
due to the selectivity of the population under consideration (that using health clinics, for example) are likely to
affect the estimates obtained and should be taken into
account by the analyst in making comparison with estimates yielded by other methods.
DERIVATION OF THE METHOD AND ITS RATIONALE
It is easy to understand intuitively how the method
works. Let us assume that all live-birth intervals are 2.5
years long and that every woman is asked at the time of
delivery whether she has previously given birth and, if
so, whether the previous-born child is still alive. Then,
each previous child will have been exposed to the risk of
dying for exactly 2.5 years, and the proportion dead will
equal the cohort probability of dying by age 2.5, q(2.5),
provided that all last-born children have the same mortality experience as children whose birth is followed by that
of a sibling.
In practice, of course, birth intervals are not all exactly
2.5 years long, though almost all are within the range of
one to five years. Thus, the proportion dead of previous
children represents a weighted average of the probabilities of dying between birth and ages 1-5 for cohorts born
between one and five years before interview, the weights
being the proportions of women having each length of
birth interval.
Using models of birth intervals and mortality in childhood, Brass and Macrae found that the proportion dead
of previous children is, in most high-fertility populations,
a close approximation to the probability of dying by
exact age 2, q(2), and that the proportion dead of nextto-previous children is a close approximation to the probability of dying by exact age 5, q( 5); Since the proportions dead are strongly influenced by the age pattern of
mortality in childhood-that is, most child deaths occur at
very early ages-the q(x) estimates obtained from them
are reasonably robust to variations in birth-interval distributions. Adjustments can be made for the effects of such
variations if the necessary information on average birthinterval lengths is available.
Brass and Macrae found that the proportion dead of
previous children approximates q(0.8z), where z is the
length in years of the average birth interval. For average
birth intervals of 30 months (2.5 years), the proportion
dead corresponds to q( 2). Because the distribution of
birth intervals by length varies little with respect to fertility level, the method is not overly sensitive to changes in
fertility, especially because fertility declines are mainly
due to a reduction of completed family size rather than to
the substantial lengthening of birth intervals.
The estimates of q( 2) and q( 5) yielded by the BrassMacrae method refer to different birth cohorts and may
be equated to period estimates provided that some
allowance is made for the timing of deaths. Aguirre and
Hill (1987) have estimated, on the basis of declared dates
of birth and death for previous and next-to-previous children, that q(2) refers to a point approximately two years
preceding interview and q( 5) refers to a point between
three and four years preceding interview. Although those
estimates of timing refer to a particular case, they provide a good indication of the rough reference dates of the
q(x) estimates obtained.
ApPLICATION OF THE BRASS-MACRAE METHOD
Data required
The Brass-Macrae method estimates probabilities of
dying in childhood from proportions dead of previous
and, if data are available, next-to-previous children, the
information being collected from women at or just after
the birth of the most recent child.
Hence, the only data required are the following:
1. The number of women giving birth who reported
having had a previous child, irrespective of their age,
marital status etc.;
2. The number of women giving birth who, having
had a previous child, reported that that child had died.
57
If questions on the survival of the next-to-previous
child are also posed, then the following will also be
needed:
3. The number of women giving birth who reported
having had a child before the previous one;
4. The number of women giving birth who, having
had a child before the previous one, reported that that
child had died.
Note that there is no need for information on the
number of children ever born or the total number of
women. The data needed can be easily collected at the
time of delivery, and it is not strictly necessary to compile a year's worth of data to apply the method. Since
any seasonal variation in child mortality will be largely
smoothed out by variability in birth intervals, as long as
a sufficient number of events are recorded, there is no
minimum length of time for which the data-collection
exercise must be maintained. As noted earlier, however,
data collection must be maintained over a period of years
to estimate trends in child mortality.
mortality pattern prevalent in early childhood in the
population being studied (see chapter I).
Having selected an appropriate model-life-table family
from annex I or II, one needs to identify two mortality
levels, j and j + I, whose q(2) values enclose the
estimated qe(2) so that
qJ(2)
=
NPCD(l)
NPC(I)
qC(5)
h
=
NPCD(2)
NPC(2)
h)qi(5)
+ h qi+ I(5)
(7.4)
=
qe(2) - qi(2)
(7.5)
qi+ I(2) - qi(2)
Once qC(5) is calculated, it can be compared with the
qe(5) obtained in step 1. Since the latter refers to a
slightly earlier period than qC (5), under conditions of
declining mortality qC (5) should be lower than qe(5).
The example presented below illustrates such a comparison.
A detailed example
The data for this detailed example were collected in
five health facilities in Bamako, Mali, starting in January
1985 (Hill and others, 1985). A specially designed form
was used to gather information about all deliveries and
the previous two live births that each mother might have
had. Table 20 shows the number of previous births and
next-to-previous births recorded, and the number of children who had died by the time of observation.
(7.1)
TABLE
reported a previous child. Cases of non-response, where
the mother does not report the survival status of the
child, should be excluded from both the numerator and
the denominator.
The probability of dying by age 5, q( 5), is similarly
calculated by dividing the number of women reporting a
next-to-previous child dead by the number of women
reporting a next-to-previous child. Thus,
e(5)
= (1.0 -
where qi (5) and qi + 1(5) are the model values of q( 5) at
levels j and j + I, respectively, in the selected family of
model life tables, and h is an interpolation factor calculated as follows:
where qe(2) is the estimated probability of dying by age
2, NPCD( I) is the number of women with a previous
child dead, and NPC( I) is the number of women who
q
(7.3)
The desired value of qC (5) is then obtained as
Computational procedure
The computational procedure is very simple, consisting
of, at most, two steps.
Step 1. Calculation of the probabilities of dying, q(2)
and q(5)
The probability of dying by age 2, q( 2), is calculated
by dividing the number of women reporting a previous
child who has died by the total number of women reporting a previous child. Thus,
qe(2)
> qe(2) > qJ+l(2)
20.
ESTIMATION OF THE PROBABILITY OF DYING q(x),
FOR BAMAKO. MALI. USING THE BRASS-MACRAE METHOD
Previous
Next to previous
.
..
Number of
births
Number
deod
Probability
of dying
(1)
(2)
(3)
4775
3737
679
620
.1422
.1659
Source: A. G. Hill and others, "L'enquete pilote sur la rnortalite aux
jeunes ages dans cinq rnaternites de 1a ville de Bamako, Mali", in Estimation de La mortalite du jeune enfant (0-5) pour guider les actions de
sante dans les pays en deveioppement, Seminaire INSERM, vol. 145,
(Paris, 1985), pp. 107-130.
(7.2)
where the symbols have the same meanings as before,
but refer to next-to-previous rather than to previous
births.
Step 2. Conversion to a common index
In cases where estimates of both q(2) and q(5) can be
obtained, conversion to a common index is necessary to
compare them. For consistency's sake, it is recommended
that q( 5) be used as the common index, just as in the
application of the Brass method. It is therefore necessary
to find a value qC(5) equivalent to the qe(2) estimate
obtained above. As with the Brass method, a model-lifetable family must be used to perform the necessary
conversion. Ideally, the family used should reflect the
Calculation of the probabilities of dying, q(2)
and q(5)
For each class of previous births, the probabilities of
dying are calculated according to equations 7.1 and 7.2
by dividing the entries of column 2 of table 20 by those
of column I, as shown below:
Step 1.
q
q
58
e(2)
=
"(S)
=
679
4,775
= .1422
620
= .1659
3,737
The resulting qe(2) and qe(5) estimates are displayed in
column 3 of table 20.
Step 2. Conversion to a common index
Using q(5) as the common index, qe(2) needs to be
converted into an equivalent measure of q(5). For both
sexes and a sex ratio at birth of 1.05 male births per
female birth, the Coale-Demeny North model-life-table
values in table A.I.9 show that the estimated qe(2) value
is enclosed by q13(2) = .14701, whose q13(5) = .19235,
and by q14(2) = .13158, for which qI4(5) = .17113.
U sing equation 7.5 to estimate the interpolation factor h,
h
=
.14220 - .14701
.13158 - .14701
Mention has been made of the potential use of the
Brass-Macrae method to assess trends in child mortality
in specific subpopulations. Perhaps one of the best examples so far of that use is provided by the data gathered
through the birth-notification scheme operating in Solomon Islands during the period 1968-1975 and analysed
by Brass and Macrae in their 1984 article. Table 21
presents the basic data and the q(2) estimates derived
from them. Although for earlier years the q(2) estimates
vary in an unexpected fashion, the full set of estimates
shows sustained mortality decline. Indeed, if the likely
reference dates of the estimates are taken into account,
mortality in childhood seems to have been cut almost in
half between 1968 and 1975.
= .3247
This value of h is then used in equation 7.4 to find the
corresponding qC (5) as follows:
qC(5)
= (1.0 = .185
.3247)( .19235)
TABLE
+ (.3247)( .17113)
21.
ESTIMATION OF THE PROBABILITY OF DYING BY AGE
Number of
previous
Note that this qC(5) value is considerably higher than the
qe(5) value obtained in step 1, namely, .166. Such estimates would imply that under-five mortality among the
children of interviewed women increased from .166 in
1981-1982 (three to four years before interview) to .185
around 1983 (about two years before interview). The
unlikelihood of such an increase suggests that the estimates obtained may be subject to selection biases that
affect their accuracy. Aguirre and Hill (1987) argue that
qe(5) is more likely to be affected by those biases, since
it is based on children whose mothers are older on average and who seem to represent only the better educated
and socially advantaged group of the population (older
Bamako women who have already had several children
tend not to give birth in clinics). For that reason, qe(5)
is lower than expected, and it would have to be rejected
as a valid estimate of under-five mortality for 1981-1982.
On the other hand, there is no reason to reject qC(5),
which, at 185 deaths per 1,000 live births, would be an
estimate of the 1983 probability of dying by age 5 among
the children born to women who attended health clinics
in Bamako.
COMMENTS ON THE RESULTS OF THE METHOD
As stated earlier, experience in the use of the BrassMacrae method is limited, so no firm assertions can as
yet be made about the validity of the estimates it provides. In the example presented above, the inconsistencies discovered should raise doubts about the accuracy of
the estimates obtained. However Aguirre and Hill (1987)
note that the estimated q(2) value of .142 is in line with
estimates obtained, using the original Brass method, from
a large survey conducted by the Sahel Institute among a
representative sample of Bamako households. Note that
estimates obtained by applying the original Brass method
are needed to validate the Brass-Macrae results. The
assessment of new techniques in terms of the old is. to be
expected and will probably continue for a few years. For
that reason, the reader should .become familiar with both
methods.
2, q(2),
FOR SOLOMON ISLANDS, USING THE BRASS-MACRAE METHOD
Year of
notification
1968
1969
1970
1971
1972
1973
1974
1975
..
.
.
.
.
.
..
.
Probability
of dying
births
Number
dead
(I)
(2)
(3)
1769
2400
2796
3030
3861
3369
2390
4472
209
255
331
281
333
228
159
.1181
.1063
.1184
.0927
.0862
.0677
.0665
.0590
264
q(2)
Source: W. Brass and S. Macrae, "Childhood mortality estimated
from reports on previous births given by mothers at the time of a
maternity: 1. Preceding-births technique", Asian and Pacific Census
Forum, vol. 11, NO.2 (November 1984), p. 7.
By permitting the estimation of child mortality over a
period of years, the data for Solomon Islands provide an
example of a more realistic use of the Brass-Macrae technique than does the case of Mali. Even if the estimated
q( 2) values are not perfect, their declining trend is indicative of the changes taking place and might serve as an
adequate evaluation tool. It is important, however, to
confirm that the representativeness of the basic data has
remained constant through time, especially in view of the
variability evident in the numbers of women providing
information (see column 1 of table 21). Brass and Macrae
noted that "some selection bias is likely since the births
notified may be to women who are better educated or
otherwise socially advantaged" ,6 but they discounted the
effects of that bias. The existence of such selectivity
could be assessed by gathering additional information on
the women giving birth, including their age, parity, educational level, work status and usual place of residence.
The distribution of reporting women according to those
variables would shed light on the likelihood and possible
extent of biases stemming from changes in selectivity.
Clearly, because of its simplicity and flexibility, the
Brass-Macrae method deserves attention. The tests it is
being subjected to will, it is hoped, show that the selectivity and other biases that may affect the estimates it
yields may reasonably be discounted by analysts
59
interested in assessing programme impact. It is unlikely,
however, that the Brass-Macrae method will replace the
original Brass method as the main technique providing
national estimates of mortality in childhood. It is in providing estimates for selected subpopulations that the
Brass-Macrae method has the greatest potential.
60
ANNEX I
Coale-Demeny model life table values for probabilities of dying
between birth and exact age x, q(x)
TABLES
Page
A.I.1.
A.I.2.
A.I.3.
A.I.4.
A.I.5.
A.I.6.
A.I.7.
A.I.8.
A.I.9.
A.I.10.
A.I.ll.
A.I.12.
North model values for male probabilities of dying, q(x)
South model values for male probabilities of dying, q(x)
East model values for male probabilities of dying, q(x)
West model values for male probabilities of dying, q(x)
North model values for female probabilities of dying, q(x)
South model values for female probabilities of dying, q(x)
East model values for female probabilities of dying, q(x)
West Asian model values for female probabilities of dying, q(x)
North model values for probabilities of dying, q(x), both sexes combined
South model values for probabilities of dying, q(x), both sexes combined
East model values for probabilities of dying, q(x), both sexes combined
West model values for probabilities of dying, q(x), both sexes combined
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
A.I.I.
.
.
.
.
.
.
.
.
.
.
.
.
NORTH MODEL VALUES FOR MALE PROBABILITIES OF DYING.
61
62
62
63
63
64
64
65
65
66
66
67
qi x)
eo
q(l)
q(2)
q(3)
q(5)
q(10)
q(15)
q(20)
17.6
19.9
22.3
24.7
27.2
29.6
32.0
34.5
36.9
39.3
41.8
44.3
46.7
49.1
51.5
53.9
56.3
58.8
61.3
63.9
66.4
68.9
71.6
74.4
77.3
.37117
.33923
.31056
.28459
.26089
.23913
.21904
.20041
.18306
.16686
.15167
.13744
.12411
.11228
.10074
.08955
.07874
.06833
.05836
.04885
.03981
.03130
.02353
.01606
.01056
.45216
.41659
.38401
.35397
.32613
.30021
.27596
.25322
.23181
.21161
.19251
.17444
.15672
.14080
.12544
.10995
.09527
.08145
.06844
.05622
.04476
.03405
.02516
.01690
.01096
.50142
.46363
.42867
.39617
.36581
.33735
.31058
.28533
.26146
.23883
.21734
.19694
.17656
.15814
.14046
.12293
.10632
.09063
.07584
.06189
.04874
.03635
.02659
.01766
.01134
.56587
.52518
.48711
.45138
.41773
.38595
.35588
.32735
.30024
.27443
.24983
.22639
.20251
.18084
.16010
.14031
.12145
.10348
.08638
.07011
.05462
.03979
.02876
.01883
.01193
.62135
.58040
.54145
.50437
.46899
.43520
.40287
.37191
.34222
.31373
.28635
.26008
.23311
.20839
.18463
.16186
.14006
.11924
.09935
.08037
.06224
.04490
.03219
.02084
.01303
.64586
.60516
.56616
.52877
.49289
.45842
.42527
.39338
.36266
.33306
.30452
.27703
.24884
.22285
.19778
.17366
.15051
.12831
.10706
.08673
.06727
.04863
.03487
.02255
.01408
.66959
.62944
.59070
.55331
.51722
.48238
.44871
.41617
.38470
.35426
.32481
.29635
.26736
.24042
.21429
.18903
.16466
.14120
.11865
.09700
.07621
.05622
.04062
.02674
.01702
61
TABLEA.I.2.
Level
1 ,.. ,.. ,..".",."".",.,
2 "."."."""."""""
3 ".".".""".""."",
4 .".""""""."."""
5 .".".""""""."""
6 "."""."."""""".
7 ".".".".".""""",
8 ".".".".".""""."
9 ".".".".""".""",
10 ".".".""".".""",
11 "".""".".""""."
12 ".".".".",.".",.,.,
13 "".".""""""."."
14,."." .. ,.. "."" .. "."
15,.".".".",."., .. "."
16 .. ".".".",.".,." .. "
17 .. '." .. ,..""".,."""
18."."."""""".".".
19 """"""""."""".
20 "."""."."""""".
21 ".".",.".",.".".".
22 "."."""."""""".
23 "."."""."""."""
24 """.".".""".".".
25 ".".""".""""",,,
SOUTH MODEL VALUES FOR MALEPROBABILITIES OF DYING,
eo
q(i)
q(2)
q(3)
q(5)
q(IO)
q(15)
q(20)
19.9
22.3
24.7
27,0
29.3
31,6
33,9
36,2
38,5
40,6
42,9
45,1
47.4
49.6
51,9
54.1
56.3
58,6
61.2
63,7
66,1
68,5
71.0
73,6
76,0
.33555
.31122
.28925
,26924
,25089
,23396
.21828
,20368
,19004
,17752
,16642
,15559
.14502
.13475
.12478
.11513
.10581
,09676
,08618
,07610
,06605
,05605
,04617
,03648
,02822
.46045
.42918
.40024
,37330
.34813
.32449
.30224
.28122
,26132
,24318
,22532
,20812
,19155
,17558
.16022
.14545
.13125
,11733
,10112
,08735
,07420
,06167
,04980
,03863
,02943
.51806
.48359
.45143
.42131
,39298
.36625
.34097
.31699
.29420
,27346
.25249
.23235
.21300
.19442
,17657
,15944
,14299
,12693
,10835
,09300
,07845
,06472
,05184
,03988
,03016
,56599
,52885
.49402
.46124
.43029
.40099
,37319
.34674
,32155
,29866
,27510
,25251
.23086
.21099
,19017
,17107
,15275
,13502
,11472
,09817
,08248
,06770
,05391
,04118
,03094
,60074
,56317
,52756
.49374
.46154
.43083
.40148
.37339
.34647
.32152
,29600
,27145
,24786
.22518
,20338
,18242
,16228
.14275
.12079
,10291
,08604
,07024
,05561
,04223
,03155
,61600
,57843
,54267
.50857
.47599
,44480
,41491
,38622
,35865
,33299
,30675
,28147
.25711
.23364
,21104
,18928
,16833
,14798
,12520
,10655
.08896
.07251
.05727
,04336
,03229
.63801
,60052
,56461
,53016
,49707
.46525
.43460
.40507
.37659
,35007
.32276
.29637
,27086
.24261
.22242
.19946
.17730
,15574
,13164
,11186
,09320
,07575
,05962
,04494
,03332
TABLE A,I.3.
Level
1 """.".""".""",,,
2 """"""."""""",
3 "."."."""."""""
4 """".".""""".".
5 ".""""""""""".
6 "".".""""""."",
7 ."."."."""""".".
8 "."."""."""".",,
9 "."."."""""".""
10 ",,,,,,,,,,,,,.,,,,,,,,,,
11.""".".""""""."
12 .".""".",.""""."
13 " .. ,.. "." .. " ..""",.
14 ".".".",."""""",
15 ".".".""".",."",.
16".".".",.".",."",.
17"".",.".",.".",.,.,
18 ".".""""""""",.
19",."."."""""."",
20 """."."""""".",
21 "." ..".",."",.".",
22."""".""""""""
23 """"""""""".",
24."""""""""""."
25 """"".".""""""
q(X)
EAST MODEL VALUES FOR MALEPROBABILITIES OF DYING, q(X)
eo
q(l)
q(2)
q(3)
q(5)
q(lO)
q(15)
q(20)
17,4
19,9
22.4
24.9
27.4
29,9
32,3
34,8
37,2
39,6
42,1
44.4
46,7
49,0
51.3
53,7
56,0
58.4
60,7
63,0
65,3
67,7
70,2
72,7
75.3
,50506
.46449
,42751
.39356
.36222
.33314
.30604
,28069
,25691
,23453
,21359
,19438
,17584
,15796
,14077
.12426
,10842
,09328
,07883
,06506
,05143
,03892
,02760
,01782
,01011
,57037
,52896
.49039
.45430
.42040
,38844
,35822
.32958
,30238
,27648
,25130
,22804
,20561
,18401
,16324
,14329
,12414
,10558
,08773
,07087
,05534
,04135
,59752
.55576
.51654
,47956
,44459
.41143
,37992
.34991
.32128
,29392
.26698
,24203
,21799
,19484
,17258
,15121
,13067
,11082
,09173
,07363
,05729
,04262
,02971
,01885
,01052
,62736
,58522
,54527
.50731
.47117
.43670
,40376
.37225
,34206
.31309
,28421
,25741
,23159
,20674
,18285
,15991
,13786
,11671
,09643
,07700
,05977
,04429
,03072
,01936
,01074
,65321
,61153
.57165
.53345
,49679
.46159
.42775
.39517
.36379
,33353
,30305
.27477
,24744
,22104
,19557
,17104
,14741
,12467
,10282
,08182
,06351
,04692
,03241
,02033
,01120
,66440
,62306
,58336
,54519
,50846
.47308
.43897
.40605
.37427
.34357
.31267
.28390
,25602
,22903
,20295
,17776
,15345
,13003
,10748
,08578
,06675
,04944
,03424
,02153
,01190
,68063
,63989
,60054
,56253
,52579
.49025
.45586
.42256
.39030
.35904
,32767
,29830
,26973
,24198
,21506
,18899
,16375
,13936
,11583
,09312
,07290
,05438
,03800
.02416
.01354
,0~896
,01847
,01037
62
WEST MODEL VALUES FOR MALE PROBABILITIES OF DYING, q(X)
TABLE A.I.4.
Level
1 ,.........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
eo
q(l)
q(2)
q(3)
q(5)
q(JO)
q(i5)
q(20)
18.0
20.4
22.9
25.3
27.7
30.1
32.5
34.9
37.3
39.7
42.1
44.5
47.1
49.6
51.8
54.1
56.5
58.8
61.2
63.6
66.0
68.6
71.2
73.9
76.6
.41907
.38343
.35132
.32215
.29546
.27089
.24817
.22706
.20737
.18895
.17165
.15537
.13942
.12453
.11136
.09857
.08621
.07430
.06287
.05193
.04091
.03075
.02144
.01332
.00711
.49692
.45848
.42310
.39033
.35985
.33135
.30463
.27948
.25575
.23329
.21200
.19178
.17088
.15167
.13477
.11836
.10210
.08666
.07204
.05821
.04492
.03325
.02281
.01395
.00734
.53102
.49135
.45454
.42020
.38805
.35783
.32936
.30244
.27693
.25272
.22968
.20772
.18466
.16356
.14502
.12708
.10944
.09264
.07668
.06153
.04715
.03469
.02364
.01434
.00748
.56995
.52888
.49043
.45429
.42024
.38806
.35758
.32865
.30112
.27489
.24985
.22592
.20039
.17713
.15673
.13707
.11816
.09999
.08256
.06585
.05011
.03666
.02479
.01490
.00769
.59898
.55789
.51907
.48231
.44742
.41425
.38263
.35246
.32361
.29599
.26952
.24412
.21685
.19200
.17012
.14897
.12855
.10888
.08996
.07177
.05464
.03996
.02697
.01615
.00829
.61840
.57742
.53849
.50142
.46607
.43230
.40000
.36905
.33936
.31084
.28343
.25704
.22853
.20266
.17982
.15766
.13623
.11553
.09556
.07634
.05826
.04266
.02881
.01727
.00886
.64324
.60260
.56369
.52640
.49062
.45625
.42320
.39138
.36074
.33118
.30266
.27511
.24544
.21833
.19429
.17088
.14813
.12609
.10476
.08416
.06469
.04766
.03242
.01959
.01015
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
A.I.5.
NORTH MODEL VALUES FOR FEMALE PROBABILITIES OF DYING.
q(X)
eo
q(i)
q(2)
q(3)
q(5)
q(iO)
q(J5)
q(20)
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
60.0
62.5
65.0
67.5
70.0
72.5
75.0
77.5
80.0
.31973
.29202
.26715
.24461
.22405
.20517
.18774
.17158
.15653
.14247
.12930
.11695
.10549
.09502
.08488
.07508
.06565
.05663
.04801
.03982
.03205
.02465
.01823
.01218
.00781
.40293
.37069
.34121
.31408
.28896
.26560
.24378
.22332
.20409
.18595
.16881
.15261
.13681
.12190
.10753
.09395
.08113
.06904
.05764
.04689
.03677
.02713
.01964
.01287
.00813
.45415
.41911
.38680
.35684
.32892
.30280
.27827
.25517
.23336
.21271
.19313
.17456
.15609
.13860
.12187
.10612
.09129
.07733
.06417
.05178
.04007
.02889
.02065
.01338
.00837
.52217
.48342
.44735
.41363
.38199
.35220
.32408
.29747
.27223
.24825
.22543
.20372
.18169
.16094
.14136
.12290
.10550
.08907
.07355
.05886
.04492
.03150
.02217
.01415
.00873
.58336
.54353
.50584
.47010
.43613
.40378
.37292
.34343
.31521
.28818
.26225
.23741
.21178
.18739
.16436
.14264
.12213
.10276
.08444
.06706
.05054
.03455
.02402
.01509
.00918
.61121
.57130
.53325
.49691
.46215
.42885
.39692
.36627
.33680
.30846
.28117
.25493
.22788
.20191
.17730
.15401
.13197
.11110
.09132
.07253
.05463
.03728
.02573
.01607
.00971
.63767
.59794
.55977
.52307
.48775
.45374
.42095
.38933
.35880
.32932
.30083
.27334
.24517
.21784
.19180
.16705
.14354
.12121
.09996
.07973
.06043
.04166
.02860
.01789
.01080
63
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
A.I.6.
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
q(X)
eo
q(1)
q(2)
q(3)
q(5)
q(/O)
q(/5)
q(20)
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
60.0
62.5
65.0
67.5
70.0
72.5
75.0
77.5
80.0
.30700
.28449
.26415
.24563
.22865
.21298
.19847
.18496
.17234
.16065
.15044
.14047
.13075
.12130
.11214
.10327
.09471
.08639
.07715
.06801
.05890
.04987
.04096
.03226
.02486
.44469
.41367
.38496
.35824
.33324
.30978
.28769
.26682
.24705
.22823
.21077
.19397
.17781
.16226
.14732
.13295
.11899
.10546
.09199
.07930
.06717
.05564
.04472
.03450
.02614
.50808
.47315
.44059
.41008
.38140
.35435
.32876
.30450
.28145
.25935
.23855
.21860
.19948
.18112
.16351
.14661
.13025
.11411
.09906
.08475
.07122
.05849
.04661
.03564
.02679
.56057
.52240
.48664
.45301
.42128
.39125
.36278
.33571
.30993
.28511
.26155
.23900
.21741
.19674
.17692
.15793
.13971
.12209
.10526
.08964
.07493
.06117
.04842
.03676
.02745
.60018
.56140
.52462
.48966
.45635
.42456
.39415
.36504
.33712
.30969
.28367
.25873
.23481
.21185
.18982
.16865
.14832
.12869
.11051
.09359
.07778
.06311
.04965
.03746
.02783
.61944
.58060
.54354
.50812
.47421
.44169
.41407
.38046
.35158
.32315
.29599
.26990
.24483
.22073
.19756
.17529
.15385
.13312
.11408
.09637
.07985
.06458
.05063
.03805
.02817
.64448
.60571
.56844
.53256
.49799
.46464
.43246
.40137
.37132
.34185
.31318
.28556
.25896
.23334
.20866
.18489
.16198
.13978
.11924
.10038
.08285
.06672
.05205
.03891
.02867
TABLE
Level
SOUTH MODEL VALUES FOR FEMALE PROBABILITIES OF DYING.
A.I.7.
EAST MODEL VALUES FOR FEMALE PROBABILITIES OF DYING.
q(X)
eo
q(/)
q(2)
q(3)
q(5)
q(/O)
q(/5)
q(20)
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
60.0
62.5
65.0
67.5
70.0
72.5
75.0
77.5
80.0
.42785
.39330
.36180
.33288
.30618
.28141
.25833
.23674
.21648
.19742
.17960
.16300
.14703
.13169
.11698
.10290
.08942
.07657
.06433
.05268
.04093
.03063
.02144
.01360
.00755
.50168
.46469
.43029
.39814
.36796
.33955
.31270
.28728
.26315
.24019
.21787
.19683
.17668
.15741
.13897
.12135
.10431
.08809
.07270
.05811
.04450
.03284
.02266
.01418
.00777
.53306
.49504
.45941
.42588
.39422
.36426
.33582
.30876
.28298
.25837
.23414
.21121
.18929
.16834
.14832
.12920
.11074
.09316
.07646
.06058
.04616
.03389
.02325
.01446
.00788
.56794
.52877
.49177
.45671
.42341
.39172
.36151
.33264
.30503
.27858
.25223
.22720
.20330
.18048
.15871
.13792
.11806
.09908
.08095
.06362
.04824
.03522
.02401
.01483
.00802
.59930
.56000
.52252
.48671
.45243
.41958
.38806
.35776
.32861
.30054
.27232
.24536
.21956
.19487
.17126
.14867
.12705
.10635
.08653
.06754
.05101
.03704
.02508
.01538
.00825
.61558
.57636
.53877
.50269
.46803
.43468
.40256
.37159
.34172
.31287
.28388
.25599
.22923
.20357
.17899
.15543
.13284
.11119
.09043
.07051
.05316
.03852
.02602
.01590
.00849
.63645
.59744
.55981
.52350
.48843
.45452
.42173
.38998
.35923
.32943
.29964
.27058
.24262
.21573
.18990
.16509
.14124
.11834
.09633
.07519
.05651
.04090
.02757
.01678
.00892
64
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
II ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
a With sex ratio at birth of
WEST MODEL VALUES FOR FEMALE PROBABILITIES OF DYING. q(X)
eo
q(1)
q(2)
q(3)
q(5)
q(IO)
q(/5)
q(20)
20.0
22.5
25.0
27.5
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
52.5
55.0
57.5
60.0
62.5
65.0
67.5
70.0
72.5
75.0
77.5
80.0
.36517
.33362
.30519
.27936
.25573
.23398
.21386
.19518
.17774
.16143
.14612
.13171
.11831
.10548
.09339
.08177
.07066
.06004
.04994
.04034
.03093
.02262
.01516
.00894
.00445
.45000
.41443
.38171
.35144
.32329
.29700
.27235
.24916
.22729
.20660
.18700
.16837
.15061
.13280
.11636
.10064
.08581
.07180
.05857
.04608
.03441
.02740
.01623
.00939
.00460
.48801
.45064
.41601
.38375
.35357
.32524
.29885
.27335
.24949
.22685
.20532
.18481
.16508
.14504
.12676
.10934
.09291
.07740
.06276
.04891
.03615
.02575
.01679
.00963
.00467
.53117
.49176
.45494
.42042
.38795
.35730
.32831
.30082
.27470
.24983
.22611
.20346
.18152
.15894
.13873
.11959
.10146
.08429
.06799
.05251
.03840
.02714
.01752
.00994
.00478
.56544
.52555
.48794
.45237
.41865
.38661
.35611
.32702
.29922
.27263
.24715
.22271
.19900
.17441
.15227
.13126
.11132
.09238
.07439
.05725
.04165
.02928
.01877
.01055
.00501
.59028
.55022
.51217
.47596
.44144
.40847
.37693
.34671
.31773
.28989
.26313
.23737
.21229
.18613
.16260
.14020
.11890
.09864
.07935
.06094
.04426
.03102
.01981
.01107
.00522
.62507
.58051
.54214
.50535
.47002
.43606
.40339
.37192
.34158
.31231
.28404
.25673
.23010
.20251
.17716
.15297
.12990
.10789
.08689
.06683
.04842
.03386
.02154
.01197
.00560
TABLE
Level
A.I.8.
A.I.9.
NORTH MODEL VALUES FOR PROBABILITIES OF DYING. q(X), BOTH SEXES COMBINED"
eo
q(1)
q(2)
q(3)
q(5)
q(/O)
q(/5)
q(20)
18.8
21.2
23.6
26.1
28.6
31.0
33.5
36.0
38.4
40.9
43.4
45.9
48.3
50.8
53.2
55.7
58.1
60.6
63.1
65.7
68.2
70.7
73.3
75.9
78.6
.34608
.31620
.28938
.26509
.24292
.22256
.20377
.18635
.17012
.15496
.14076
.12744
.11503
.10386
.09300
.08249
.07235
.06262
.05331
.04445
.03602
.02806
.02094
.01417
.00922
.42815
.39420
.36313
.33451
.30800
.28333
.26026
.23863
.21829
.19909
.18095
.16379
.14701
.13158
.11670
.10215
.08837
.07540
.06317
.05167
.04086
.03067
.02247
.01493
.00958
.47836
.44191
.40825
.37698
.34782
.32050
.29482
.27062
.24775
.22609
.20553
.18602
.16657
.14861
.13139
.11473
.09899
.08414
.07015
.05696
.04451
.03271
.02369
.01557
.00963
.54455
.50481
.46772
.43297
.40030
.36949
.34037
.31277
.28658
.26166
.23793
.21533
.19235
.17113
.15096
.13182
.11367
.09645
.08012
.06462
.04989
.03575
.02555
.01655
.01037
.60282
.56241
.52408
.48765
.45296
.41987
.38826
.35802
.32904
.30127
.27459
.24902
.22271
.19815
.17474
.15248
.13131
.11120
.09208
.07388
.05653
.03985
.02820
.01804
.01115
.62896
.58864
.55011
.51323
.47790
.44400
.41144
.38016
.35005
.32106
.29313
.26625
.23862
.21264
.18779
.16407
.14147
.11991
.09938
.07980
.06110
.04309
.03041
.01939
.01195
.65402
.61407
.57561
.53856
.50284
.46841
.43517
.40308
.37207
.34209
.31311
.28513
.25654
.22941
.20332
.17831
.15436
.13145
.10953
.08858
.06851
.04912
.03476
.02242
.01399
1.05.
65
TABLEA.I.IO.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
a
SOUTH MODEL VALUES FOR PROBABILITIES OF DYING. q(X), BOTHSEXESCOMBINED"
eo
q(l)
q(2)
q(3)
q(5)
q(10)
q(15)
q(20)
19.9
22.4
24.8
27.2
29.6
32.0
34.4
36.8
39.2
41.5
43.9
46.3
48.7
51.0
53.4
55.8
58.1
60.0
63.1
65.6
68.0
70.5
73.0
75.5
78.0
.32162
.29818
.27701
.25772
.24004
.22373
.20862
.19455
.18141
.16929
.15862
.14821
.13806
.12819
.11861
.10934
.10040
.09170
.08178
.07215
.06256
.05304
.04363
.03442
.02658
.45276
.42161
.39279
.36595
.34087
.31731
.29514
.27420
.25436
.23589
.21822
.21022
.18485
.16908
.15393
.13935
.12527
.11154
.09667
.08342
.07077
.05873
.04732
.03662
.02783
.51319
.47850
.44614
.41583
.38733
.36045
.33501
.31090
.28798
.26658
.24569
.22564
.20640
.18793
.17020
.15318
.13678
.12082
.10382
.08898
.07492
.06168
.04929
.03781
.02852
.56335
.52570
.49042
.45723
.42589
.39624
.36811
.34136
.31588
.29205
.26849
.24592
.22430
.20358
.18371
.16466
.14639
.12871
.11011
.09401
.07880
.06451
.05123
.03902
.02924
.60047
.56231
.52613
.49175
.45901
.42777
.39790
.36932
.34191
.31575
.28999
.26525
.24149
.21868
.19677
.17570
.15547
.13589
.11578
.09836
.08201
.06676
.05270
.03990
.02974
.61768
.57949
.54309
.50835
.47512
.44328
.41274
.38341
.35520
.32819
.30150
.27583
.25112
.22734
.20446
.18246
.16127
.14073
.11978
.10158
.08452
.06864
.05403
.04077
.03028
.64117
.60305
.56648
.53133
.49752
.46495
.43356
.40327
.37402
.34606
.31809
.29110
.26506
.23993
.21571
.19235
.16983
.14795
.12559
.10626
.08815
.07135
.05593
.04200
.03105
With sex ratio at birth of 1.05.
TABLE A.I.ll.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
a With
EAST MODELVALUES FOR PROBABILITIES OF DYING. q(x), BOTHSEXESCOMBINED"
eo
q(l)
q(2)
q(3)
q(5)
q(lO)
q(15)
q(20)
18.7
21.2
23.7
26.2
28.7
31.2
33.6
36.1
38.6
41.0
43.5
45.9
48.3
50.7
53.1
55.6
58.0
60.4
62.8
65.2
67.6
70.0
72.5
75.0
77.6
.46740
.42976
.39546
.36396
.33488
.30791
.28277
.25925
.23719
.21643
.19701
.17907
.16179
.14515
.12917
.11384
.09915
.08513
.07176
.05902
.04631
.03488
.02460
.01576
.00886
.53686
.49761
.46107
.42691
.39482
.36459
.33602
.30895
.28324
.25878
.23499
.21282
.19150
.17103
.15140
.13259
.11447
.09705
.08040
.06465
.05005
.03720
.02589
.01638
.00910
.56608
.52614
.48867
.45337
.42002
.38842
.35841
.32984
.30260
.27658
.25096
.22700
.20399
.18191
.16075
.14047
.12095
.10221
.08428
.06726
.05186
.03836
.02656
.01671
.00923
.59837
.55768
.51917
.48263
.44787
.41476
.38315
.35293
.32400
.29626
.26861
.24267
.21779
.19393
.17107
.14918
.12820
.10811
.08888
.07047
.05415
.03987
.02745
.01715
.00941
.62691
.58639
.54768
.51065
.47515
.44110
.40839
.37692
.34663
.31744
.28806
.26042
.23384
.20827
.18371
.16013
.13748
.11573
.09487
.07485
.05741
.04210
.02883
.01792
.00976
.64059
.60028
.56161
.52446
.48874
.45435
.42121
.38924
.35839
.32859
.29863
.27029
.24295
.21661
.19126
.16687
.14340
.12084
.09916
.07833
.06012
.04411
.03023
.01878
.01024
.65908
.61918
.58067
.54349
.50757
.47282
.43921
.40667
.37514
.34460
.31400
.28478
.25651
.22918
.20279
.17733
.15277
.12911
.10632
.08437
.06490
.04780
.03291
.02056
.01129
sex ratio at birth of 1.05.
66
TABLE A.I.l2.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
a With
WEST MODEL VALUES FOR PROBABILITIES OF DYING, q(X), BOTH SEXES COMBINED
a
eo
q(l)
q(2)
q(3)
q(5)
q(IO)
q(15)
q(20)
19.0
21.4
23.9
26.4
28.8
31.3
33.7
36.2
38.6
41.1
43.5
46.0
48.5
51.0
53.4
55.8
58.2
60.6
63.1
65.5
68.0
70.5
73.1
75.7
78.3
.39278
.35913
.32882
.30128
.27608
.25289
.23143
.21151
.19292
.17553
.15920
.14383
.12912
.11524
.10259
.09037
.07862
.06734
.05656
.04628
.03604
.02678
.01838
.01118
.00581
.47403
.43699
.40291
.37136
.34202
.31459
.28888
.26469
.24187
.22027
.19980
.18036
.16099
.14247
.12579
.10972
.09415
.07941
.06547
.05229
.03979
.02908
.01960
.01173
.00594
.51004
.47149
.43575
.40242
.37123
.34193
.31433
.28825
.26354
.24010
.21780
.19654
.17511
.15453
.13611
.11843
.10138
.08521
.06989
.05537
.04178
.03033
.02030
.01204
.00611
.55103
.51077
.47312
.43777
.40449
.37306
.34330
.31507
.28823
.26267
.23827
.21496
.19119
.16826
.14795
.12854
.11001
.09233
.07545
.05934
.04440
.03202
.02124
.01248
.00627
.58262
.54211
.50388
.46771
.43339
.40077
.36969
.34005
.31171
.28459
.25861
.23368
.20814
.18342
.16141
.14033
.12015
.10083
.08236
.06469
.04830
.03475
.02297
.01342
.00669
.60468
.56415
.52565
.48900
.45406
.42068
.38875
.35815
.32881
.30062
.27353
.24744
.22061
.19460
.17142
.14914
.12778
.10729
.08765
.06883
.05143
.03698
.02442
.01425
.00708
.63218
.59182
.55318
.51613
.48057
.44640
.41354
.38189
.35139
.32198
.29358
.26614
.23796
.21061
.18593
.16214
.13924
.11721
.09604
.07571
.05675
.04093
.02711
.01587
.00793
sex ratio at birth of 1.05.
67
ANNEX II
United Nations model life table values for probabilities of dying
between birth and exact age x, q(X)
TABLES
Page
A.II.1.
A.II.2.
A.II.3.
A.IL4.
A.IL5.
A.IL6.
A.IL7.
A.IL8.
A.IL9.
A.II.IO.
A.II.II.
A.ILI2.
A.II.I3.
A.II.14.
A.II.15.
Latin American model values for male probabilities of dying, q(x)
Chilean model values for male probabilities of dying, q(x)
South Asian model values for male probabilities of dying, q(x)
Far Eastern model values for male probabilities of dying, q(x)
General model values for male probabilities of dying, q(x)
Latin American model values for female probabilities of dying, q(x)
Chilean model values for female probabilities of dying, q(x)
South Asian model values for female probabilities of dying, q(x)
Far Eastern model values for female probabilities of dying, q(x)
General model values for female probabilities of dying, q(x)
Latin American model values for probabilities of dying, q(x), both sexes combined
Chilean model values for probabilities of dying, q(x), both sexes combined
South Asian model values for probabilities of dying, q(x), both sexes combined
Far Eastern model values for probabilities of dying, q(x), both sexes combined
General model values for probabilities of dying, q(x), both sexes combined
TABLE A.II.I.
Level
I ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
II ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
.
..
..
..
..
.
.
.
..
.
..
.
.
.
.
68
69
70
70
71
72
72
73
74
74
75
76
76
77
78
LATIN AMERICAN MODEL VALUES FOR MALE PROBABILITIES OF DYING. q(x)
eo
q(1)
q(2)
q(3)
q(5)
q(IO)
q(15)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
.20429
.19839
.19260
.18690
.18129
.17577
.17033
.16497
.15969
.15448
.14934
.14427
.13927
.13433
.12945
.12464
.11988
.11518
.11055
.10597
.10144
.09697
.09256
.08821
.08391
.07967
.07550
.07138
.06733
.06335
.05943
.05559
.05181
.04812
.04452
.26825
.25985
.25161
.24352
.23556
.22774
.22005
.21249
.20506
.19775
.19056
.18348
.17652
.16967
.16293
.15630
.14978
.14337
.13706
.13086
.12476
.11877
.11289
.10711
.10144
.09588
.09043
.08510
.07988
.07478
.06980
.06495
.06022
.05563
.05117
.30174
.29204
.28251
.27316
.26397
.25494
.24607
.23735
.22878
.22036
.21209
.20395
.19596
.18810
.18039
.17280
.16535
.15804
.15086
.14381
.13689
.13010
.12345
.11692
.11054
.10429
.09818
.09221
.08639
.08071
.07518
.06980
.06458
.05952
.05463
.33663
.32562
.31481
.30420
.29376
.28352
.27345
.26355
.25383
.24428
.23490
.22568
.21663
.20774
.19901
.19043
.18202
.17377
.16568
.15774
.14996
.14234
.13487
.12757
.12042
.11345
.10663
.09999
.09352
.08722
.08110
.07516
.06940
.06384
.05848
.36840
.35637
.34454
.33291
.32147
.31022
.29916
.28829
.27760
.26709
.25676
.24661
.23664'
.22684
.21722
.20777
.19850
.18940
.18048
.17173
.16316
.15476
.14653
.13849
.13063
.12295
.11546
.10816
.10106
.09415
.08744
.08094
.07464
.06857
.06271
.38433
.37187
.35961
.34754
.33567
.32399
.31249
.30118
.29006
.27912
.26836
.25778
.24737
.23715
.22711
.21724
.20755
.19804
.18871
.17956
.17059
.16180
.15320
.14478
.13654
.12850
.12066
.11301
.10557
.09833
.09130
.08448
.07789
.07152
.06539
.40543
.39248
.37973
.36716
.35478
.34258
.33057
.31874
.30709
.29561
.28432
.27321
.26227
.25151
.24093
.23053
.22031
.21027
.20041
.19073
.18124
.17194
.16282
.15389
.14516
.13662
.12829
.12017
.11255
.10455
.09708
.08982
.08280
.07603
.06950
68
TABLE
Level
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
I ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
II ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
(continued)
eo
q(/)
q(2)
q(3)
q(5)
q(!O)
q(/5)
q(20)
70
71
72
73
74
75
.04100
.03758
.03426
.03105
.02795
.02499
.04686
.04270
.03870
.03486
.03120
.02771
.04991
.04537
.04101
.03685
.03289
.02914
.05331
.04836
.04362
.03910
.03481
.03077
.05709
.05170
.04655
.04165
.03702
.03265
.05950
.05386
.04847
.04335
.03849
.03393
.06322
.05721
.05147
.04601
.04084
.03598
TABLE
Level
A.II.!.
A.n.2.
CHILEAN MODEL VALUES FOR MALE PROBABILITIES OF DYING. q(x)
eo
q(1)
q(2)
q(3)
q(5)
q(!O)
q(/5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.23869
.23135
.22413
.21702
.21003
.20314
.19636
.18968
.18310
.17661
.17022
.16393
.15772
.15161
.14558
.13964
.13379
.12803
.12236
.11677
.11128
.10587
.10055
.09532
.09019
.08515
.08021
.07537
.07063
.06601
.06149
.05710
.05282
.04868
.04466
.04079
.03706
.03348
.03006
.02681
.02374
.27532
.26628
.25741
.24870
.24015
.23176
.22352
.21542
.20748
.19967
.19200
.18446
.17706
.16979
.16265
.15564
.14875
.14200
.13537
.12887
.12249
.11624
.11012
.10412
.09826
.09252
.08692
.08146
.07614
.07096
.06592
.06104
.05632
.05175
.04735
.04312
.03906
.03519
.03151
.02802
.02474
.29384
.28395
.27426
.26475
.25542
.24628
.23730
.22850
.21986
.21139
.20307
.19491
.18691
.17906
.17135
.16380
.15640
.14914
.14202
.13506
.12824
.12156
.11503
.10865
.10241
.09632
.09039
.08461
.07898
.07352
.06822
.06309
.05814
.05335
.04875
.04434
.04012
.03609
.03227
.02867
.02527
.31352
.30277
.29223
.28190
.27178
.26185
.25212
.24258
.23323
.22406
.21507
.20625
.19761
.18915
.18085
.17273
.16477
.15697
.14934
.14188
.13458
.12745
.12048
.11368
.10704
.10056
.09426
.08814
.08219
.07642
.07083
.06542
.06021
.05519
.05037
.04575
.04134
.03714
.03317
.02942
.02590
.33309
.32155
.31025
.29917
.28832
.27767
.26724
.25701
.24698
.23715
.22752
.21807
.20882
.19976
.19089
.18219
.17369
.16536
.15722
.14925
.14148
.13388
.12646
.11922
.11216
.10529
.09861
.09211
.08581
.07971
.07381
.06811
.06261
.05733
.05226
.04741
.04279
.03840
.03425
.03034
.02667
.34705
.33505
.32328
.31173
.30042
.28931
.27843
.26775
.25729
.24702
.23696
.22710
.21744
.20797
.19869
.18961
.18072
.17202
.16351
.15519
.14706
.13912
.13137
.12381
.11644
.10927
.10229
.09552
.08895
.08258
.07643
.07049
.06477
.05927
.05400
.04896
.04416
.03960
.03529
.03124
.02744
.36922
.35656
.34413
.33193
.31996
.30820
.29666
.28533
.27421
.26331
.25261
.24211
.23182
.22173
.21185
.20216
.19267
.18338
.17429
.16541
.15672
.14823
.13995
.13186
.12398
.11631
.10885
.10161
.09458
.08778
.08120
.07485
.06874
.06287
.05724
.05186
.04674
.04189
.03729
.03298
.02895
69
TABLE A.II.3.
Level
I ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
I ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
q(X)
eo
q(I)
q(2)
q(3)
q(5)
q(IO)
q(l5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.22680
.22045
.21419
.20803
.20194
.19594
.19002
.18416
.17838
.17267
.16702
.16143
.15591
.15044
.14503
.13968
.13439
.12914
.12396
.11882
.11375
.10872
.10375
.09884
.09398
.08918
.08444
.07977
.07517
.07063
.06617
.06179
.05750
.05329
.04919
.04519
.04131
.03755
.03393
.03045
.02713
.30531
.29613
.28709
.27818
.26939
.26072
.25217
.24373
.23540
.22719
.21908
.21108
.20318
.19539
.18771
.18013
.17265
.16527
.15800
.15083
.14378
.13683
.12998
.12325
.11664
.11014
.10376
.09751
.09139
.08540
.07955
.07386
.06831
.06293
.05772
.05269
.04785
.04320
.03876
.03454
.03056
.34498
.33437
.32390
.31357
.30339
.29335
.28344
.27367
.26404
.25454
.24516
.23592
.22681
.21782
.20897
.20025
.19165
.18319
.17486
.16666
.15860
.15068
.14289
.13525
.12775
.12040
.11320
.10616
.09929
.09258
.08605
.07971
.07355
.06760
.06184
.05631
.05100
.04592
.04109
.03652
.03221
.38328
.37132
.35952
.34788
.33640
.32507
.31389
.30286
.29199
.28126
.27069
.26026
.24999
.23986
.22989
.22007
.21041
.20089
.19154
.18234
.17331
.16444
.15574
.14720
.13884
.13066
.12266
.11485
.10724
.09982
.09262
.08564
.07887
.07234
.06605
.06001
.05423
.04873
.04350
.03856
.03393
.41247
.39965
.38699
.37448
.36213
.34993
.33789
.32600
.31427
.30269
.29127
.28000
.26889
.25794
.24715
.23653
.22607
.21577
.20564
.19568
.18590
.17630
.16687
.15764
.14859
.13974
.13109
.12265
.11442
.10642
.09865
.09112
.08383
.07680
.07004
.06355
.05736
.05146
.04587
.04060
.03566
.42365
.41055
.39761
.38482
.37219
.35970
.34737
.33519
.32317
.31130
.29958
.28802
.27662
.26537
.25429
.24337
.23261
.22203
.21161
.20137
.19130
.18141
.17172
.16221
.15289
.14377
.13486
.12617
.11769
.10944
.10144
.09368
.08617
.07893
.07196
.06528
.05889
.05282
.04706
.04164
.03656
.43606
.42271
.40950
.39643
.38352
.37075
.35813
.34566
.33333
.32116
.30914
.29727
.28556
.27400
.26261
.25137
.24030
.22940
.21867
.20811
.19774
.18754
.17753
.16772
.15809
.14868
.13947
.13048
.12172
.11320
.10491
.09688
.08911
.08162
.07441
.06749
.06088
.05459
.04863
.04301
.03775
TABLE
Level
SOUTH ASIAN MODEL VALUES FOR MALE PROBABILITIES OF DYING.
A.II.4.
FAR EASTERN MODEL VALUES FOR MALE PROBABILITIES OF DYING.
q(X)
eo
q(l)
q(2)
q(3)
q(5)
q(IO}
q(l5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
.16455
.15873
.15304
.14746
.14200
.13664
.13139
.12624
.12119
.11623
.11137
.10660
.10192
.09733
.09282
.08840
.08470
.07982
.07566
.07159
.06760
.20614
.19819
.19044
.18287
.17548
.16827
.16122
.15434
.14762
.14105
.13464
.12838
.12226
.11629
.11046
.10477
.09923
.09382
.08855
.08342
.07843
.22919
.22008
.21120
.20254
.19409
.18586
.17782
.16999
.16234
.15489
.14762
.14053
.13362
.12689
.12033
.11394
.10772
.10167
.09579
.09007
.08453
.25567
.24528
.23515
.22528
.21566
.20628
.19714
.18824
.17955
.17110
.16286
.15484
.14702
.13942
.13202
.12482
.11783
.11104
.10445
.09805
.09186
.28513
.27346
.26208
.25098
.24015
.22960
.21931
.20928
.19950
.18997
.18070
.17166
.16287
.15431
.14598
.13789
.13003
.12240
.11500
.10784
.10090
.30681
.29434
.28217
.27028
.25868
.24735
.23630
.22552
.21500
.20475
.19475
.18502
.17553
.16630
.15731
.14857
.14008
.13183
.12383
.11608
.10858
.33721
.32377
.31063
.29777
.28519
.27289
.26086
.24911
.23763
.22642
.21547
.20479
.19438
.18422
.17433
.16469
.15532
.14622
.13737
.12880
.12048
70
~
(continued)
TABLE A.II.4.
Level
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
eo
q(l)
q(2)
q(3)
q(5)
q(/O)
q(/5)
q(20)
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.06370
.05989
.05617
.05254
.04901
.04557
.04224
.03901
.03588
.03287
.02998
.02720
.02455
.02203
.01965
.01740
.01530
.01335
.01155
.00990
.07358
.06887
.06430
.05988
.05559
.05145
.04746
.04363
.03994
.03641
.03304
.02983
.02680
.02393
.02123
.01872
.01638
.01422
.01224
.01045
.07915
.07394
.06890
.06403
.05932
.05479
.05044
.04626
.04226
.03844
.03480
.03135
.02809
.02503
.02216
.01948
.01701
.01473
.01265
.01077
.08586
.08007
.07447
.06907
.06388
.05888
.05409
.04950
.04512
.04095
.03699
.03325
.02973
.02642
.02333
.02046
.01782
.01539
.01319
.01120
.09419
.08771
.08146
.07544
.06965
.06410
.05878
.05369
.04885
.04425
.03989
.03578
.03191
.02829
.02493
.02181
.01894
.01632
.01394
.01180
.10132
.09431
.08755
.08104
.07479
.06878
.06303
.05754
.05231
.04734
.04264
.03820
.03404
.03015
.02652
.02318
.02010
.01729
.01475
.01247
.11244
.10466
.09715
.08992
.08296
.07629
.06989
.06378
.05796
.05242
.04719
.04225
.03762
.03328
.02926
.02553
.02212
.01900
.01618
.01365
TABLEA.II.5.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
GENERAL MODEL VALUES FOR MALE PROBABILITIES OFDYING. q(X)
eo
q(l)
q(2)
q(3)
q(5)
q(JO)
q(/5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.20001
.19387
.18785
.18192
.17609
.f7036
.16471
.15915
.15368
.14828
.14297
.13773
.13256
.12747
.12244
.11749
.11261
.10779
.10305
.09837
.09376
.08922
.08475
.08035
.07602
.07177
.06759
.06349
.05947
.05554
.05170
.04795
.04430
.04075
.03731
.03399
.03079
.02772
.02478
.02200
.01937
.25388
.24542
.23713
.22900
.22101
.21318
.20549
.19793
.19052
.18323
.17608
.16905
.16215
.15538
.14872
.14219
.13577
.12948
.12330
.11725
.11131
.10549
.09978
.09421
.08875
.08342
.07821
.07314
.06820
.06339
.05873
.05420
.04983
.04561
.04155
.03765
.03393
.03039
.02703
.02386
.02089
.28242
.27275
.26326
.25396
.24485
.23590
.22713
.21852
.21008
.20179
.19366
.18569
.17787
.17020
.16269
.15531
.14809
.14101
.13408
.12729
.12064
.11414
.10779
.10159
.09553
.08963
.08388
.07829
.07286
.06759
.06249
.05756
.05280
.04822
.04383
.03963
.03563
.03184
.02825
.02488
.02173
.31326
.30233
.29161
.28110
.27080
.26069
.25078
.24107
.23154
.22220
.21304
.20406
.19526
.18663
.17819
.16991
.16181
.15388
.14613
.13855
.13113
.12389
.11683
.10994
.10323
.09670
.09035
.08418
.07820
.07242
.06683
.06143
.05625
.05127
.04650
.04196
.03764
.03355
.02970
.02610
.02274
.34387
.33184
.32003
.30845
.29708
.28592
.27498
.26424
.25371
.24337
.23324
.22331
.21357
.20402
.19468
.18552
.17656
.16779
.15921
.15083
.14264
.13465
.12685
.11925
.11185
.10466
.09767
.09090
.08434
.07799
.07187
.06597
.06031
.05488
.04969
.04475
.04007
.03565
.03149
.02761
.02400
.36121
.34866
.33633
.32422
.31233
.30065
.28919
.27793
.26688
.25604
.24540
.23496
.22472
.21469
.20485
.19521
.18578
.17654
.16751
.15867
.15004
.14161
.13338
.12537
.11757
.10998
.10261
.09546
.08854
.08185
.07539
.06917
.06320
.05748
.05202
.04682
.04189
.03724
.03287
.02879
.02501
.38482
.37165
.35870
.34597
.33345
.32114
.30903
.29713
.28544
.27395
.26266
.25157
.24069
.23001
.21954
.20927
.19920
.18934
.17968
.17023
.16100
.15197
.14316
.13456
.12619
.11805
.11014
.10246
.09502
.08782
.08088
.07419
.06777
.06161
.05573
.05014
.04484
.03983
.03514
.03075
.02669
71
TABLEA.II.6.
LATIN AMERICAN MODEL VALUES FOR FEMALE PROBABILITIES OF DYING. q(X)
Level
e"
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.16750
.16339
.15935
.15537
.15144
.14757
.14375
.13998
.13625
.13256
.12891
.12530
.12172
.11818
.11466
.11118
.10772
.10429
.10089
.09751
.09415
.09081
.08749
.08420
.08093
.07767
.07444
.07123
.06804
.06488
.06174
.05862
.05552
.05246
.04943
.04643
.04347
.04055
.03767
.03485
.03209
.24014
.23350
.22697
.22055
.21424
.20802
.20189
.19585
.18989
.18402
.17823
.17252
.16688
.16132
.15582
.15040
.14504
.13975
.13452
.12936
.12426
.11923
.11426
.10935
.10451
.09973
.09502
.09037
.08579
.08128
.07684
.07248
.06819
.06398
.05986
.05582
.05187
.04803
.04428
.04064
.03713
.28091
.27276
.26475
.25688
.24913
.24152
.23403
.22665
,21939
.21224
.20519
.19825
.19142
.18468
.17804
.17150
.16506
.15871
.15245
.14628
.14020
.13421
.12832
.12252
.11681
.11119
.10566
.10023
.09490
.08967
.08454
.07951
.07459
.06977
.06508
.06050
.05604
.05172
.04753
.04348
.03958
.32422
.31447
.30488
.29546
.28620
.27710
.26815
.25935
.25069
.24217
.23379
.22554
.21742
.20944
.20158
.19385
.18624
.17876
.17139
.16415
.15703
.15003
.14315
.13640
.12976
.12325
.11687
.11061
.10448
.09848
.09261
.08688
.08129
.07584
.07054
.06539
.06040
.05557
.05091
.04643
.04213
.36255
.35150
.34063
.32995
.31944
.30911
.29895
.28895
.27911
.26943
.25991
.25054
.24132
.23226
.22333
.21456
.20593
.19745
.18911
.18091
.17285
.16494
.15717
.14955
.14207
.13474
.12756
.12054
.11367
.10695
.10040
.09400
.08778
.08173
.07586
.07016
.06466
.05935
.05424
.04934
.04465
.38118
.36955
.35810
.34685
.33577
.32487
.31415
.30360
.29321
.28299
.27293
.26303
.25329
.24371
.23428
.22500
.21588
.20691
.19810
.18943
.18091
.17255
.16435
.15630
.14840
.14067
.13310
.12569
.11844
.11137
.10447
.09774
.09120
.08483
.07867
.07270
.06693
.06137
.05602
.05090
.04601
.40680
.39437
.38212
.37007
.35820
.34652
.33501
.32368
.31253
.30155
.29073
.28009
.26962
.25931
.24917
.23919
.22938
.21973
.21025
.20093
.19178
.18280
.17398
.16533
.15686
.14856
.14044
.13250
.12475
.11718
.10980
.10262
.09564
.08886
.08230
.07595
.06983
.06393
.05828
.05286
.04771
TABLE A.II.7.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
CHILEAN MODEL VALUES FOR FEMALE PROBABILITIES OF DYING. q(X)
eu
q(1)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
.21831
.21287
.20750
.20220
.19697
.19180
.18669
.18163
.17663
.17168
.16677
.16192
.15710
.15233
.14759
.14290
.13824
.13363
.12904
.12449
.11998
.26540
.25805
.25083
.24373
.23673
.22985
.22307
.21639
.20980
.20330
.19690
.19057
.18433
.17818
.17210
.16610
.16017
.15432
.14854
.14283
.13720
.28914
.28081
.27263
.26459
.25669
.24892
.24128
.23375
.22635
.21906
.21188
.20481
.19784
.19097
.18421
.17754
.17097
.16449
.15810
.15181
.14561
.31380
.30445
.29527
.28627
.27742
.26873
.26019
.25180
.24354
.23543
.22744
.21959
.21186
.20426
.19677
.18941
.18216
.17504
.16802
.16112
.15433
.33696
.32669
.31662
.30673
.29702
.28749
.27813
.26894
.25990
.25102
.24229
.23371
.22528
.21699
.20883
.20082
.19294
.18520
.17759
.17011
.16276
.35412
.34321
.33251
.32201
.31170
.30158
.29164
.28187
.27227
.26285
.25358
.24447
.23552
.22673
.21809
.20959
.20125
.19305
.18500
.17709
.16932
.38324
.37126
.35950
.34795
.33660
.32547
.31452
.30377
.29321
.28284
.27264
26263
.25279
.24313
.23363
.22431
.21516
.20618
.19736
.18871
.18022
72
TABLE A.II.7.
Level
eo
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
q(1)
.11550
.11106
.10665
.10228
.09795
.09365
.08938
.08517
.08099
.07686
.07278
.06875
.06478
.06086
.05702
.05324
.04954
.04591
.04238
.03895
TABLE A.II.8.
!.Rvel
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
.39 ..........................
40 ..........................
41 ..........................
q(2)
(continued)
q(3)
.13164
.12615
.12073
.11538
.11011
.10490
.09978
.09473
.08976
.08488
.08008
.07537
.07075
.06623
.06181
.05750
.05330
.04922
.04526
.04144
.13950
.13349
.12756
.12173
.11599
.11034
.10478
.09932
.09396
.08871
.08356
.07851
.07358
.06877
.06408
.05951
.05507
.05077
.04661
.04260
q(5)
.14765
.14109
.13463
.12829
.12206
.11595
.10995
.10406
.09830
.09266
.08714
.08176
.07650
.07138
.06640
.06157
.05688
.05235
.04798
.04378
q(IO)
.15554
.14846
.14150
.13468
.12798
.12142
.11499
.10870
.10254
.09653
.09066
.08494
.07936
.07395
.06869
.06360
.05867
.05392
.04934
.04496
q(J5)
q(20)
.16170
.15422
.14688
.13969
.13264
.12574
.11898
.11237
.10592
.09961
.09347
.08749
.08167
.07602
.07055
.06525
.06013
.05520
.05046
.04592
.17190
.16375
.15576
.14794
.14029
.13280
.12549
./1835
.11138
.10460
.09799
.09157
.08535
.07931
.07347
.06784
.06241
.05719
.05219
.04741
SOUTH ASIAN MODEL VALUES FOR FEMALE PROBABILITIES OF DYING. q(x)
eo
q(l)
q(2)
q(3)
q(5)
q(JO)
q(J5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.20275
.19788
.19307
.18831
.18362
.17897
.17438
.16983
.16532
.16085
.15641
.15202
.14765
.14332
.13901
.13474
.13049
.12626
.12206
.11788
.11372
.10959
.10548
.10138
.09731
.09326
.08923
.08522
.08124
.07729
.07337
.06947
.06562
.06180
.05803
.05430
.05063
.04702
.04348
.04002
.03664
.28410
.27649
.26897
.26155
.25423
.24699
.23985
.23278
.22579
.21888
.21204
.20528
.19859
.19197
.18542
.17893
.17251
.16616
.15987
.15365
.14750
.14141
.13539
.12944
.12356
.11774
.11200
.10634
.10076
.09526
.08985
.08453
.07930
.07418
.06917
.06427
.05949
.05484
.05033
.04597
.04177
.32720
.31804
.30901
.30009
.29129
.28260
.27403
.26555
.25719
.24892
.24075
.23268
.22471
.21684
.20905
.20136
.19377
.18627
.17886
.17155
.16433
.15721
.15018
.14325
.13642
.12969
.12307
.11656
.11016
.10387
.09771
.09167
.08577
.08000
.07437
.06890
.06359
.05844
.05347
.04868
.04409
.36998
.35929
.34874
.33833
.32806
.31792
.30791
.29804
.28829
.27867
.26917
.25980
.25054
.24141
.23240
.22351
.21474
.20609
.19757
.18917
.18089
.17274
.16471
.15681
.14904
.14141
.13391
.12655
.11934
.11228
.10538
.09864
.09206
.08566
.07943
.07340
.06756
.06192
.05650
.05130
.04633
.40323
.39145
.37982
.36834
.35701
.34582
.33477
.32387
.31310
.30247
.29198
.28162
.27140
.26132
.25137
.24156
.23189
.22235
.21296
.20370
.19459
.18563
.17681
.16814
.15961
.15124
.14304
.13500
.12713
.11943
.11192
.10459
.09746
.09053
.08380
.07729
.07101
.06496
.05915
.05359
.04829
.41669
.40450
.39246
.38057
.36883
.35724
.34579
.33449
.32332
.31230
.30142
.29068
.28007
.26961
.25929
.24911
.23908
.22918
.21944
.20984
.20039
.19109
.18194
.17296
.16413
.15545
.14696
.13863
.13049
.12253
.11476
.10718
.09981
.09265
.08572
.07901
.07253
.06630
.06032
.05461
.04918
.43606
.42327
.41064
.39815
.38582
.37363
.36159
.34969
.33794
.32634
.31488
.30357
.29241
.28139
.27052
.25980
.24923
.23882
.22855
.21845
.20851
.19872
.18911
.17966
.17038
.16127
.15235
.14362
.13508
.12674
.11861
.11069
.10298
.09551
.08827
.08128
.07454
.06806
.06186
.05594
.05031
73
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
A.H.9.
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
q(X)
eo
q(1)
q(2)
q(3)
q(5)
q(10)
q(/5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.14593
.14183
.13781
.13387
.13000
.12619
.12244
.11875
.11511
.11153
.10800
.10451
.10107
.09767
.09431
.09100
.08772
.08448
.08128
.07811
.07498
.07188
.06882
.06580
.06281
.05985
.05694
.05406
.05122
.04842
.04566
.04295
.04028
.03767
.03511
.03260
.03016
.02779
.02548
.02325
.02110
.19139
.18529
.17934
.17351
.16782
.16224
.15678
.15142
.14617
.14103
.13598
.13103
.12616
.12139
.11669
.11208
.10756
.10311
.09874
.09445
.09024
.08610
.08203
.07804
.07413
.07029
.06653
.06284
.05923
.05571
.05226
.04889
.04561
.04242
.03933
.03632
.03342
.03062
.02792
.02534
.02288
.21636
.20913
.20207
.19517
.18844
.18186
.17543
.16914
.16298
.15696
.15106
.14529
.13963
.13409
.12866
.12333
.11812
.11300
.10800
.10309
.09828
.09358
.08897
.08446
.08004
.07573
.07151
.06740
.06338
.05946
.05565
.05194
.04834
.04485
.04147
.03821
.03506
.03204
.02915
.02639
.02376
.24413
.23563
.22734
.21926
.21138
.20369
.19618
.18885
.18168
.17468
.16784
.16115
.15461
.14821
.14195
.13583
.12984
.12399
.11827
.11268
.10721
.10187
.09665
.09156
.08659
.08175
.07703
.07243
.06796
.06361
.05939
.05530
.05134
.04752
.04383
.04028
.03687
.03361
.03050
.02753
.02473
.27285
.26310
.25361
.24435
.23532
.22651
.21792
.20953
.20134
.19335
.18554
.17791
.17047
.16319
.15607
.14913
.14234
.13572
.12925
.12294
.11678
.11078
.10492
.09921
.09366
.08825
.08299
.07788
.07292
.06812
.06346
.05896
.05462
.05043
.04641
.04255
.03885
.03533
.03197
.02879
.02579
.29366
.28307
.27275
.26268
.25286
.24329
.23394
.22481
.21591
.20721
.19872
.19043
.18233
.17442
.16670
.15915
.15179
.14461
.13760
.13076
.12409
.11759
.11126
.10510
.09911
.09328
.08762
.08212
.07680
.07164
.06665
.06184
.05720
.05274
.04845
.04435
.04043
.03670
.03316
.02981
.02665
.33631
.32403
.31205
.30035
.28892
.27777
.26688
.25624
.24585
.23570
.22579
.21612
.20667
.19745
.18844
.17965
.17108
.16272
.15458
.14664
.13891
.13139
.12407
.11696
.11005
.10335
.09685
.09056
.08448
.07860
.07294
.06749
.06224
.05722
.05241
.04783
.04346
.03932
.03541
.03172
.02826
TABLE
Level
FAR EASTERN MODEL VALUES FOR FEMALE PROBABILITIES OF DYING,
A.n.lO.
GENERAL MODEL VALUES FOR FEMALE PROBABILITIES OF DYING,
qt x)
eo
q(/)
q(2)
q(3)
q(5)
q(IO)
q(I5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
.16449
.16030
.15617
.15210
.14810
.14415
.14025
.13640
.13260
.12884
.12512
.12144
.11780
.11419
.11061
.10707
.10356
.10007
.09661
.09318
.08977
.22558
.21905
.21265
.20636
.20018
.19410
.18812
.18224
.17645
.17074
.16513
.15959
.15414
.14876
.14346
.13823
.13308
.12799
.12298
.11803
.11314
.25960
.25172
.24399
.23641
.22897
.22166
.21448
.20743
.20050
.19368
.18697
.18038
.17389
.16751
.16123
.15505
.14897
.14299
.13710
.13130
.12560
.29698
.28762
.27845
.26945
.26063
.25197
.24348
.23514
,22695
.21890
.21100
.20324
.19562
.18813
.18077
.17354
.16644
.15946
.15261
.14588
.13927
.33384
.32313
.31262
.30231
,29220
.28228
.27524
.26297
.25358
.24436
.23531
.22643
.21770
.20913
.20072
.19245
.18435
.17639
.16858
.16091
.15340
.35454
.34312
.33192
.32092
.31012
.29953
,28912
.27890
.26886
.25901
.24933
.23983
.23049
.22133
.21233
.20349
.19482
.18632
.17796
.16978
.16175
.38628
.37379
.36152
.34946
.33762
.32598
.31454
.30331
.29227
,28142
.27077
.26030
.25002
.23993
.23001
.22028
.21073
.20137
.19217
.18316
.17434
74
TABLE
Level
e"
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
..........................
..........................
..........................
..........................
..........................
..............•...........
..........................
..........................
..........................
..........................
................•.........
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
TABLE A.II.11.
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
q(J)
.08639
.08304
.07971
.07640
.07312
.06987
.06664
.06344
.06027
.05713
.05402
.05094
.04790
.04490
.04195
.03905
.03620
.03341
.03069
.02804
A.II.lO. (continued)
q(2)
.10833
.10359
.09891
.09430
.08976
.08529
.08089
.07656
.07230
.06813
.06403
.06001
.05607
.05223
.04848
.04483
.04128
.03785
.03453
.03133
q(3)
.11999
.11448
.10906
.10373
.09850
.09337
.08833
.08339
.07855
.07382
.06919
.06467
.06026
.05597
.05180
.04776
.04385
.04008
.03645
.03297
q(5)
.13278
.12642
.12018
.11407
.10808
.10221
.09647
.09086
.08538
.08004
.07483
.06976
.06483
.06005
.05542
.05095
.04664
.04251
.03854
.03476
«to,
.14604
.13882
.13175
.12484
.11807
.11146
.10501
.09871
.09257
.08659
.08078
.07513
.06966
.06437
.05926
.05434
.04962
.04509
.04077
.03666
q(/5)
.15388
.14618
.13864
.13126
.12405
.11701
.11014
.10343
.09691
.09056
.08439
.07841
.07262
.06702
.06163
.05644
.05146
.04670
.04217
.03786
q(20)
.16569
.15723
.14895
.14086
.13296
.12525
.11774
.11042
.10330
.09638
.08968
.08318
.07690
.07085
.06502
.05943
.05408
.04898
.04412
.03953
LATIN AMERICAN MODEL V ALVES FOR PROBABILITIES OFDYING. q(X), BOTH SEXES COMBINED a
e"
q(l)
q(2)
q(3)
q(5)
q(/Oj
q(/5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.18634
.18132
.17638
.17152
.16673
.16202
.15737
.15278
.14826
.14379
.13938
.13502
.13071
.12645
.12224
.11807
.11395
.10987
.10584
.10184
.09788
.09397
.09009
.08625
.08246
.07870
.07498
.07131
.06768
.06409
.06055
.05706
.05362
.05024
.04691
.04365
.04045
.03733
.03428
.03132
.02845
.25454
.24700
.23959
.23232
.22516
.21812
.21119
.20437
.19766
.19105
.18455
.17813
.17182
.16559
.15946
.15342
.14747
.14160
.13582
.13013
.12452
.11900
.11356
.10820
.10294
.09776
.09267
.08767
.08276
.07795
.07324
.06862
.06411
.05970
.05541
.05123
.04718
.04325
.03946
.03581
.03230
.29158
.28263
.27385
.26522
.25673
.24839
.24019
.23213
.22420
.21640
.20872
.20117
.19374
.18643
.17924
.17217
.16521
.15836
.15163
.14501
.13850
.13211
.12582
.11965
.11360
.10765
.10183
.09613
.09054
.08508
.07974
.07454
.06946
.06452
.05973
.05508
.05058
.04624
.04206
.03806
.03423
.33058
.32018
.30997
.29993
.29007
.28039
.27086
.26150
.25230
.24325
.23436
.22561
.21702
.20857
.20026
.19210
.18408
.17620
.16847
.16087
.15341
.14609
.13891
.13187
.12498
.11823
.11163
.10517
.09886
.09271
.08671
.08088
.07520
.06969
.06436
.05920
.05423
.04945
.04486
.04048
.03631
.36555
.35399
.34263
.33146
.32048
.30968
.29906
.28861
.27834
.26823
.25830
.24853
.23892
.22948
.22020
.21108
.20212
.19333
.18469
.17621
.16788
.15972
.15172
.14388
.13621
.12870
.12137
.11420
.10721
.10039
.09376
.08731
.08105
.07499
.06912
.06347
.05802
.05279
.04779
.04303
.03850
.38279
.37074
.35887
.34720
.33572
.32442
.31330
.30236
.29160
.28101
.27059
.26034
.25026
.24035
.23060
.22103
.21161
.20237
.19329
.18438
.17563
.16705
.15864
.15040
.14233
.13444
.12673
.11919
.11185
.10469
.09772
.09095
.08438
.07802
.07187
.06594
.06023
.05476
.04953
.04455
.03982
.40610
.39340
.38090
.36858
.35645
.34450
.33274
.32115
.30974
.29851
.28745
.27656
.26585
.25531
.24495
.23475
.22473
.21489
.20521
.19571
.18638
.17723
.16826
.15947
.15087
.14245
.13422
.12618
.11835
.11071
.10328
.09607
.08906
.08229
.07574
.06943
.06336
.05755
.05199
.04671
.04170
a With sex ratio at birth of 1.05.
75
TABLE A.I1.12.
CHILEAN MODEL VALUES FOR PROBABILITIES OF DYING. q(X), BOTH SEXES COMBINED
a
LRvef
eo
q(l)
q(2)
q(3)
q(5)
q(lO)
q(l5)
q(20)
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.22875
.22233
.21602
.20979
.20366
.19761
.19164
.18575
.17994
.17421
.16854
.16295
.15742
.15196
.14656
.14123
.13597
.13076
.12562
.12054
.11552
.11057
.10568
.10085
.09609
.09139
.08676
.08220
.07772
.07332
.06899
.06475
.06059
.05653
.05257
.04870
.04495
.04131
.03780
.03441
.03116
.27048
.26227
.25420
.24627
.23848
.23083
.22330
.21589
.20861
.20144
.19439
.18744
.18061
.17388
.16726
.16074
.15432
.14801
.14179
.13568
.12967
.12375
.11794
.11222
.10661
.10110
.09569
.09040
.08521
.08013
.07517
.07033
.06561
.06102
.05656
.05224
.04806
.04403
.04015
.03643
.03289
.29155
.28242
.27346
.26467
.25604
.24757
.23924
.23106
.22303
.21513
.20737
.19974
.19224
.18487
.17762
.17050
.16350
.15663
.14987
.14323
.13671
.13031
.12404
.11787
.11183
.10591
.10012
.09445
.08890
.08349
.07822
.07308
.06808
.06322
.05852
.05397
.04958
.04535
.04129
.03742
.03372
.31366
.30359
.29371
.28403
.27453
.26521
.25606
.24708
.23826
.22960
.22110
.21276
.20456
.19652
.18862
.18086
.17325
.16578
.15845
.15126
.14422
.13731
.13053
.12390
.11741
.11105
.10484
.09878
.09286
.08709
.08148
.07602
.07072
.06558
.06062
.05582
.05121
.04677
.04253
.03847
.03462
.33497
.32406
.31336
.30286
.29256
.28246
.27255
.26283
.25328
.24392
.23472
.22570
.21685
.20816
.19964
.19128
.18308
.17504
.16715
.15943
.15186
.14445
.13719
.13009
.12314
.11636
.10973
.10327
.09697
.09085
.08489
.07911
.07350
.06808
.06284
.05779
.05294
.04829
.04384
.03961
.03559
.35050
.33903
.32778
.31675
.30592
.29530
.28487
.27464
.26460
.25474
.24507
.23558
.22626
.21712
.20815
.19936
.19073
.18228
.17399
.16587
.15792
.15014
.14252
.13507
.12778
.12067
.11373
.10696
.10037
.09397
.08774
.08170
.07585
.07020
.06474
.05949
.05444
.04962
.04500
.04062
.03646
.37606
.36373
.35163
.33974
.32808
.31662
.30537
.29433
.28348
.27283
.26238
.25212
.24205
.23217
.22247
.21297
.20364
.19450
.18555
.17677
.16818
.15978
.15156
.14352
.13567
.12801
.12054
.11326
.10617
.09929
.09261
.08614
.07988
.07383
.06801
.06240
.05703
.05190
.04700
.04235
.03795
a
With sex ratio at birth of 1.05.
TABLEA.II.13.
SOUTH ASIAN MODEL VALUES FOR PROBABILITIES OF DYING. q(x), BOTH SEXES COMBINED a
Level
eo
q(l)
q(2)
q(3)
q(5)
q(IO)
q(15)
q(20)
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
.21507
.20944
.20389
.19841
.19300
.18766
.18239
.17717
.17201
.16690
.16185
.15684
.15188
.14697
.14210
.13727
.13248
.12774
.12303
.29496
.28655
.27825
.27007
.26199
.25402
.24616
.23839
.23071
.22314
.21565
.20825
.20094
.19372
.18659
.17954
.17258
.16570
.15891
.33631
.32640
.31663
.30700
.29749
.28811
.27885
.26971
.26070
.25180
.24301
.23434
.22579
.21734
.20901
.20079
.19269
.18469
.17681
.37679
.36545
.35426
.34322
.33233
.32158
.31098
.30051
.29018
.28000
.26995
.26004
.25026
.24062
.23111
.22175
.21252
.20343
.19448
.40797
.39565
.38349
.37149
.35963
.34792
.33637
.32496
.31370
.30258
.29161
.28079
.27012
.25959
.24921
.23898
.22891
.21898
.20921
.42025
.40760
.39510
.38275
.37055
.35850
.34660
.33485
.32324
.31179
.30048
.28932
.27830
.26744
.25673
.24617
.23577
.22552
.21543
.43606
.42298
.41005
.39727
.38464
.37215
.35982
.34763
.33558
.32369
.31194
.30035
.28890
.27761
.26647
.25548
.24466
.23399
.22349
76
TABLE
Level
('()
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
a With
q(1)
.11836
.11374
.10914
.10459
.10008
.09560
.09117
.08678
.08243
.07813
.07388
.06968
.06554
.06146
.05744
.05350
.04964
.04586
.04217
.03859
.03512
.03177
A.II.13.
(continued)
q(2)
.15221
.14559
.13906
.13262
.12627
.12001
.00385
.10778
.10182
.09596
.09021
.08458
.07906
.07368
.06842
.06330
.05834
.05353
.04888
.04441
.04012
.03603
q(3)
q(5)
q(IO)
.16905
.16140
.15386
.14645
.13915
.13198
.12493
.11801
.11123
.10459
.09809
.09174
.08554
.07951
.07365
.06796
.06245
.05714
.05203
.04713
.04245
.03800
.18567
.17701
.16849
.16011
.15189
.14382
.13590
.12815
.12056
.11314
.10590
.09885
.09198
.08531
.07884
.07258
.06654
.06073
.05516
.04984
.04478
.03998
.19959
.19014
.18085
.17172
.16276
.15397
.14535
.13692
.12867
.12062
.11277
.10512
.09769
.09048
.08350
.07675
.07026
.06402
.05804
.05234
.04693
.04182
q(15)
q(20)
.20550
.19573
.18613
.17671
.16745
.15837
.14947
.14076
.13225
.12393
.11583
.10793
.10026
.09282
.08562
.07867
.07197
.06555
.05939
.05353
.04797
.04271
.21316
.20299
.19300
.18318
.17354
.16409
.15482
.14576
.13689
.12824
.11980
.11159
.10362
.09588
.08840
.08117
.07422
.06754
.06116
.05508
.04932
.04388
sex ratio at birth of 1.05.
TABLE
Level
1 ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
A.ll.14.
FAR EASTERN MODEL VALUES FOR PROBABILITIES OF DYING.
q(x), BOTH SEXES COMBINED"
eo
q(I)
q(2)
q(3)
q(5)
q(IO)
q(15)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
.15546
.15049
.14561
.14083
.13614
.13154
.12702
.12259
.11823
.11394
.10973
.10558
.10151
.09750
.09355
.08967
.08585
.08209
.07840
.07477
.07120
.06769
.06424
.06086
.05755
.05430
.05112
.04800
.04496
.04200
.03911
.03630
.03358
.03095
.02841
.02597
.02363
.19895
.19190
.18503
.17831
.17174
.16533
.15905
.15292
.14691
.14104
.13529
.12967
.12417
.11878
.11350
.10834
.10329
.09835
.09352
.08880
.08419
.07969
.07529
.07101
.06683
.06276
.05881
.05497
.05124
.04763
.04414
.04077
.03753
.03442
.03144
.02859
.02589
.22294
.21474
.20674
.19895
.19134
.18391
.17666
.16957
.16266
.15590
.14930
.14285
.13656
.13040
.12439
.11852
.11279
.10720
.10174
.09642
.09124
.08619
.08127
.07649
.07184
.06733
.06295
.05871
.05461
.05065
.04683
.04316
.03964
.03627
.03305
.02999
.02708
.25004
.24057
.23134
.22235
.21357
.20502
.19667
.18853
.18059
.17285
.16529
.15791
.15072
.14371
.13686
.13019
.12369
.11736
.11119
.10519
.09935
.09367
.08816
.08281
.07762
.07259
.06773
.06304
.05851
.05414
.04995
.04593
.04208
.03841
.03491
.03160
.02847
.27914
.26841
.25795
.24775
.23780
.22809
.21863
.20940
.20040
.19162
.18306
.17471
.16657
.15864
.15090
.14337
.13604
.12890
.12195
.11520
.10865
.10228
.09610
.09012
.08433
.07872
.07331
.06810
.06308
.05825
.05362
.04919
.04497
.04095
.03713
.03352
.03012
.30040
.28884
.27757
.26657
.25584
.24537
.23515
.22518
.21544
.20595
.19669
.18766
.17885
.17026
.16189
.15373
.14579
.13806
.13055
.12324
.11615
.10926
.10258
.09611
.08985
.08381
.07797
.07234
.06693
.06174
.05676
.05201
.04747
.04316
.03908
.03522
.03159
.33677
.32390
.31132
.29903
.28701
.27527
.26380
.25259
.24164
.23095
.22051
.21032
.20038
.19067
.18121
.17199
.16301
.15427
.14577
.13750
.12947
.12168
.11413
.10681
.09974
.09291
.08632
.07997
.07388
.06803
.06243
.05709
.05200
.04718
.04262
.03832
.03428
77
TABLE
(continued)
A.II.14.
Level
e"
q(1)
q(2)
q(3)
q(5)
q(10)
q(15)
q(20)
38
39
40
41
72
73
74
75
.02139
.01927
.01726
.01537
.02332
.02090
.01863
.01651
.02434
.02177
.01935
.01711
.02552
.02276
.02019
.01780
.02693
.02395
.02119
.01863
.02820
.02503
.02210
.01939
.03051
.02700
.02376
.02078
..........................
..........................
..........................
..........................
a With
sex ratio at birth of 1.05.
TABLE
Level
I ..........................
2 ..........................
3 ..........................
4 ..........................
5 ..........................
6 ..........................
7 ..........................
8 ..........................
9 ..........................
10 ..........................
11 ..........................
12 ..........................
13 ..........................
14 ..........................
15 ..........................
16 ..........................
17 ..........................
18 ..........................
19 ..........................
20 ..........................
21 ..........................
22 ..........................
23 ..........................
24 ..........................
25 ..........................
26 ..........................
27 ..........................
28 ..........................
29 ..........................
30 ..........................
31 ..........................
32 ..........................
33 ..........................
34 ..........................
35 ..........................
36 ..........................
37 ..........................
38 ..........................
39 ..........................
40 ..........................
41 ..........................
a With
A.II.15.
GENERAL MODEL VALVES FOR PROBABILITIES OF DYING,
q(x), BOTH SEXES COMBINED a
e"
q(l)
q(2)
q(3)
q(5)
q(10)
q(J5)
q(20)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
.18268
.17750
.17239
.16738
.16244
.15757
.15278
.14806
.14340
.13880
.13426
.12978
.12536
.12099
.11667
.11241
.10819
.10403
.09991
.09584
.09182
.08784
.08391
.08004
.07621
.07243
.06870
.06503
.06141
.05785
.05435
.05091
.04754
.04424
.04101
.03787
.03482
.03185
.02899
.02624
.02360
.24007
.23526
.22519
.21795
.21085
.20387
.19701
.19028
.18365
.17714
.17074
.16444
.15824
.15215
.14616
.14026
.13446
.12875
.12314
.11763
.11220
.10687
.10164
.09650
.09146
.08651
.08167
.07692
.07228
.06774
.06331
.05900
.05479
.05071
.04676
.04294
.03925
.03570
.03230
.02906
.02598
.27129
.26249
.25386
.24540
.23710
.22895
.22096
.21311
.20540
.19783
.19040
.18310
.17593
.16889
.16198
.15519
.14852
.14197
.13555
.12924
.12306
.11699
.11105
.10523
.09953
.09396
.08851
.08319
.07800
.07294
.06802
.06323
.05859
.05410
.04975
.04557
.04155
.03770
.03402
.03052
.02722
.30531
.29515
.28519
.27542
.26584
.25644
.24722
.23817
.22930
.22059
.21204
.20366
.19543
.18736
.17945
.17168
.16407
.15661
.14929
.14212
.13510
.12823
.12151
.11494
.10852
.10225
.09614
.09018
.08438
.07874
.07327
.06797
.06284
.05788
.05311
.04852
.04413
.03994
.03595
.03217
.02860
.33898
.32759
.31642
.30545
.29470
.28414
.27379
.26362
.25365
.24386
.23425
.22483
.21558
.20651
.19762
.18890
.18036
.17198
.16378
.15575
.14789
.14020
.13269
.12535
.11819
.11120
.10440
.09778
.09135
.08510
.07905
.07319
.06754
.06209
.05685
.05183
.04703
.04246
.03813
.03403
.03018
.35796
.34596
.33417
.32261
.31125
.30010
.28915
.27840
.26785
.25749
.24732
.23733
.22754
.21793
.20850
.19925
.19019
.18131
.17261
.16409
.15575
.14759
.13962
.13184
.12425
.11684
.10963
.10262
.09581
.08919
.08279
.07660
.07062
.06486
.05934
.05404
.04899
.04418
.03962
.03532
.03128
.38533
.37269
.36007
.34767
.33548
.32350
.31172
.30014
.28877
.27759
.26661
.25583
.24524
.23485
.22465
.21464
.20483
.19521
.18578
.17654
.16750
.15866
.15002
.14158
.13335
.12532
.11751
.10991
.10253
.09537
.08844
.08174
.07529
.06907
.06311
.05740
.05196
.04678
.04189
.03728
.03295
sex ratio at birth of 1.05.
78
ANNExm
Relationship between infant mortality, q(l), and child mortality,
in the Coale-Demeny mortality models
200
4Q.,
..
.
.
.
..
.
.
.
.
...
ISDUIQ · · //
.....
.....
//
//~
//
~
./
.... / /
..- /
150
..../ /
..-/
.~
/
//
/
//
100
/
//
/
//
//
//
/
/
/1'
..
.....
..-
..-
.....
..
..-
.....
..-
.....
.....
.....
50
//
//
/
//
//
//
o
.i
,~
L __
.. .
~
=1:;;.-===_-_-_-_-
....
--
..L
50
--l
L
l-
.J..
100
150
200
250
...J
q(1)
q
4 1
ANNEX IV
Relationship between infant mortality, q(l), and child mortality,
in the United Nations mortality models
4q\,
,----------------------------------,
200
150
00
o
100
50
50
NOTES
I For a description of how a life table is constructed in practice, see
Henry S. Shryock, Jacob S. Siegel and others, The Methods and
Materials of Demography (Washington, D.C., U.S. Department of
Commerce, 1973), vol. 2, pp. 429-461.
2 To understand the type of data collected, it is recommended that the
analyst refer to the questionnaire used to gather the data and even to the
instructions given to interviewers or enumerators. Unfortunately, those
documents are often not published together with the tabulations of the
data. However, whenever they are accessible, it is recommended that
the analyst indicate how the information on children ever born and surviving was collected when presenting the results obtained from applying
the Brass method.
3 See, for instance, Alejandro Aguirre and Allan G. Hill, "Childhood
mortality estimates using the preceding birth technique: some applications and extensions", CPS Research Paper 87-2 (Centre for Population
Studies, London School of Hygiene and Tropical Medicine, September
1987).
4 See, for instance, A. G. Hill and others, "L'enquete pilote sur la
mortalite aux jeunes ages dans cinq maternites de la ville de Bamako,
Mali", in Estimation de la mortalite du jeune enfant (0-5) pour guider
les actions de sante dans les pays en developpement. Seminaire
INSERM, vol. 145, 107-130 (Paris, 1985); and Alejandro Aguirre and
Allan G. Hill, op. cit; W. Brass and S. Macrae, "Childhood mortality
estimated from reports on previous births given by mothers at the time
of a maternity: 1. Preceding-births technique", Asian and Pacific
Census Forum, vol. II, No.2 (November 1984); and Miguel Irigoyen
and Sonia M. Mychaszula, "Estimacion de la mortalidad infantil mediante el metodo del hijo previo: Aplicacion en el Hospital Rural de Junin
de los Andes", paper presented at the IUSSP Seminar on Collection and
Processing of Demographic Data in Latin America, Santiago, 23-27
May 1988.
5 See Irigoyen and Mychaszula, op. cit., and Aguirre and Hill, op.
cit.
6 Brass and Macrae, op. cit., p. 7.
81
GLOSSARY
hypothetical cohort: A construct representing a cohort that does not
number of births in a given year to women in age group i
number of deaths between exact ages x and x
proportion of children dead
expectation of life at exact age x
really exist. Most life tables represent the effects of mortality on
hypothetical birth cohorts (see cohort).
+n
infant: Person under age 1.
infant mortality: The probability of dying between birth and exact age
expectation of life at birth
1. Denoted by q(l) or by 1qo .
age group of mother or age group of women (see table 3
for the correspondence between different values of i and
specific age groups)
number of survivors to exact age x
infant mortality rate: The number of deaths of children aged 0 to 1 per
person-year lived by those children. Denoted by
ity. It consists of several sets of numbers, or functions, each
representing one particular aspect of the incidence of mortality in a
population.
mean age at maternity: The mean age of mothers at the birth of a group
of children, usually those born in a given year. It is denoted by M
and used in the application of the Palloni-Heligman version of the
Brass method.
not-stated parity: Refers to women who do not report the number of
children they have had.
parity: Number of children ever borne by a woman. Abortions and stillbirths are not counted.
parity ratio: The ratio of average parities for women in different age
groups. Those used in the Brass method are P(I)/P(2) and P(2)/
number of person-years lived between exact ages x and
x+n
age-specific mortality rate
mp(i)
M
P(i)
nqx
q(x)
q(l)
\qo
midpoint in years of age group i (used in calculating the
mean age at maternity)
mean age at maternity
average parity of women in age group i
probability of dying between exact ages x and x + n
probability of dying between birth and exact age x
infant mortality
infant mortality
sqo
q(5)
under-five mortality
4ql
child mortality
t(i)
reference date
P(3).
proportion of children dead: The ratio of children dead to children ever
under-five mortality
born, usually calculated for different age groups of women. Denoted
by D(i).
radix: Initial number of births in a life table subject to the mortality
conditions it represents. The radix is denoted by 1(0) and it is usually
100,000.
reference date: Date to which estimates of mortality in childhood refer.
Expressed in number of years before the surveyor census gathering
the basic information. Denoted by t(i).
reproductive age: The age-span during which women are able to conceive. The reproductive-age span is usually set to range between
exact ages 15 and 50, that is, 15-49 in completed years.
robustness: Characteristic of estimates that are not greatly affected by
deviations from the assumptions upon which their derivation is based.
An estimate is said to be robust to assumption A, when deviations
from that particular assumption bring about only small changes in the
value of the estimate.
sex ratio at birth: Average number of male births per female birth. It
varies between 1.03 and 1.08 male births per female birth.
standard five-year age groups: Age groups of the following type: 0-4,
5-9, 10-14, 15-19 etc.
under-five mortality: The probability of dying between birth and exact
age 5. Denoted by q(5) or sqo.
weighted average: The weighted average A of quantities Z(I),
Z(2), ... ,Z(n), using weights W(1), W(2), ...• W(n), is calculated as follows:
age-specific death rate: See age-specific mortality rate.
age-specific mortality rate: Number of deaths of persons aged x to
x + n per person-year lived by the population in that age group. Usually denoted by
nmX"
average parity: Average number of children ever born per woman.
Denoted by P(i).
birth interval: The time elapsed between the births of any two consecutive children of a given women.
child mortality: The probability of dying between exact ages 1 and 4.
Denoted by 4qt. The equation relating 4qt to infant and under-five
mortality is
4qt
= (q(5)
lmo.
life table: The demographer's way of representing the effects of mortal-
- q(l»
"* (1.0 -
q(l»
cohort: A group of persons experiencing the same event during a given
period. A birth cohort is the group of persons born during the same
period (usually a year).
exact age: A person's age at the exact moment of reaching a certain
age, i.e., not one day younger or older than that age.
expectation of life at age x: Average number of additional years that a
person aged x is expected to live under the mortality conditions
represented by a life table. Denoted by eX"
expectation of life at birth: Average number of years that a newly born
n
person is expected to live under the mortality conditions represented
by a life table. Denoted by eo.
r; W(j)Z(j)
A
fertility history: Set of dates of birth and, where appropriate, dates of
= .:.j_=_l::--
r; W(j)
j=!
death of all children borne by a woman.
82
_
REFERENCES
Aguirre, A., and A. G. Hill (1987). Childhood mortality estimates using
the preceding birth technique: some applications and extensions. CPS
Research Paper 87-2. Centre for Population Studies, London School
of Hygiene and Tropical Medicine.
Bangladesh, Census Commission (1977). Report on the 1974 Bangladesh Retrospective Survey of Fertility and Mortality. Dacca.
Brass, W. (1964). Uses of census and survey data for the estimation of
vital rates. Paper prepared for the African Seminar on Vital Statistics,
Addis Ababa, 14-19 December 1964. United Nations document E/
CN.14/CAS .4IVS/7 .
Brass, W., and S. Macrae (1984). Childhood mortality estimated from
reports on previous births given by mothers at the time of a maternity: I. Preceding-births technique. Asian and Pacific Census Forum,
vol. 11, No.2 (November).
Bucht, Birgitta (1988). Adequacy of aggregate child survival data for
estimating mortality levels and trends. Paper presented at the IUSSP
Seminar on Collection and Processing of Demographic Data in Latin
America, Santiago, 23-27 May 1988.
Coale, A. J., and P. Demeny, with B. Vaughan (1983). Regional Model
life Tables and Stable Populations. 2nd ed. New York: Academic
Press.
Coale, A. J., and T. J. Trussell (1978). Estimating the time to which
Brass estimates apply. Annex I to Fine-tuning Brass-type mortality
estimates with data on ages of surviving children, by S. H. Preston
and A. Palloni. Population Bulletin of the United Nations, No. 10.
Sales No. E.78.xIII.6.
Feeney, G. (1980). Estimating infant mortality trends from child survivorship data. Population Studies. vol. 34, No.1.
Hill, A. G., and others (1985). L'enqute pilote sur la mortalite aux
jeunes ages dans cinq maternites de la ville de Bamako, Mali. In
Estimation de la mortalite du jeune enfant (0-5) pour guider les
actions de sante dans les pays en developpement. Seminaire
INSERM, vol. 145. Paris.
Palloni, A., and L. Heligman (1986). Re-estimation of structural parameters to obtain estimates of mortality in developing countries. Population Bulletin of the United Nations, No. 18. Sales No. E.85.XIII.6.
Trussell, T. J. (1975). A re-estimation of the multiplying factors for the
Brass technique for determining childhood survivorship rates. Population Studies, vol. 29, No. I, pp. 97-108.
United Nations (1982). Model life Tables for Developing Countries.
Population Studies, No. 77. Sales No. E.81.XIII.7.
___ (l983a). Infant mortality: world estimates and projections,
1950-2025. Population Bulletin of the United Nations. No. 14. Sales
No. E.82.XIII.6.
___ (l983b). Manual X: Indirect Techniques for Demographic Estimation. Population Studies, No.8!. Sales No. E.83.XIII.2.
___ (1986). World Population Prospects: Estimates and Projections
as Assessed in 1984, Population Studies, No. 98. Sales No.
E.85.XIII.3.
___ (1988). Mortality of Children under Age 5: World Estimates
and Projections, 1950-2025. Population Studies, No. 105. Sales No.
E.88.XIII.4.
United Nations Children's Fund (UNICEF) (1986). Statistics on Children in UNICEF Assisted Countries.
_ _ (1987a). The State of the World's Children, 1987. Oxford and
New York: Oxford University Press.
___ (1987b). Statistics on Children in UNICEF Assisted Countries.
_ _ (1988a). The State of the World's Children, 1988. Oxford and
New York: Oxford University Press.
___(1988b). Statistics on Children in UNICEF Assisted Countries.
83
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