ANNEXES
THE SINCULATE MEAN ACE AT MARRIAGE
A. BACKGROUND
OF METHOD
The singulate mean age at marriage. SMAM, is the mean age first
marriage among those who ever many (or, in practice, among those
who marry by some predefined age-limit). I t is computed from the
proportions who are single, that is, never married, in each age groupo
Since the most frequently considered age groups are five years in
length, the process of calculating the singulate mean age at marriage is
described for data classified by such age groups. It is assumed here
that no first marriages occur after age 50 or before age 15, though the
generalization of the pmedure described below to the use of other
age limits is straightforward.
Step 3: estimation of proportion wJlo ewr marry. The proportion
remaining single at age 50 is estimated as
where U(7) is the proportion single (never married) among those aged
4549; and U(8) is the equivalent proportion among those aged 50-54.
Once the proportion remaining single at age 50 is estimated. the proportion ever marrying by that age. RM,is clearly just its complement,
that is,
R M = 1.0-RN.
B. BAS~CCALCULATIONSTOOBTAIN
THE SINGULATE
MEAN AGE AT MARRIAGE
Sfep 4: calculation of number of person-years lived by the pmpomn nor
-ng.
Since RN is estimated to be the proportion who have not
married by age 50, the total time spent in the single state by this proportion is
1. &la wquircd
The following data are required for this procedure:
(a) The population aged 15-54 classified by age (five-year age
group) and by sex;
(6) The never-married population aged 15-54 classified by age
(five-year age group) and sex.
Step 5: calculation of singulate mean age at marriage. Lastly, the
value of SMAM is calculated as follows:
2. Computationalpmedwr
The steps of the computational procedure are given below.
Step I: calculation of pmportionr single for a giwn sex. Divide thc
number of single in each age group by the total population in the
same age group. The resulting proportion for the age group from
Si + 10 to 51 + 14 is denoted by U(i), with i usually ranging from 1 to
8.
Step 2: calculation of person-years lived in the single state. Add the
proportions single in each age group up to and including that for the
age group 4549 (i = 7) and multiply the sum by five. The resulting
quantity is denoted by RS1. Let RS2= RS I + 15.0. The quantity 15.0
is the number of person-years lived in the single state from birth to age
I5 by the hypothetical cohort of size one being considered. If the
lower limit at which marriage takes place is changed to some other age
x, the x should be substituted for IS and the calculation of RS I
should include all age groups from x to x +4 (when x is a multiple of
five) to 4549.
SMAM = (RS2-RS,)/RM.
3. DetaiIedexampIe
The data given in table 187 were collected during the National
"11
-.
TABLE
187. POPULATION
BY AGE GROUP. SEX AND MARITAL STATUS. PANAMA
1976
em-
nvrlad
~MIIOWI
(3)
(4)
(B.4)
That is, the value of SMAM as calculated here is the average number
of years spent in the single state by those who marry before age 50. It
should be noted that although, strictly, this value refers only to the
persons who many by age 50, in most applications it is handled as if it
referred to the totality of the ever-married population. Two facts justify this practice: first, in most populations the incidence of first
marriages after age 50 is very small; and secondly, in populations with
deficient data it is not uncommon to find that aner age 50 (and sometimes even after age 45) the proportions single increase as age
increases, implying that older cohorts were subject to lower firstmarriage rates than were younger group. Because such a trend is
fairly unlikely, the observed increases are usually attributed to =porting erron. Under such conditions, it would be unwise to incorporate
the data for older cohorts in the calculation of SMAM.
John Hajnal, "A e at marriage and ro rtions marrying". Popuhion Sudies, vol.
No. 2 (~ovember!9g pp. 111-136.
?P
(B.2)
T&
(5)
em-
nrNl
(71
509
1251
1 503
1414
1 223
1 043
832
660
Demographic Survey in Panama in 1976. They illustrate some of the
problems encountered when dealing with real data.
A quick examination of table 187 reveals that, especially at older
ages, the marital status of a substantial proportion of the persons
interviewed is unknown. It is difficult to imagine a reason for this
omission since a priori it would seem easy for a person to remember
whether he or she had been married.
A possible s o u ~ eof misunderstanding may be that the term "ever
married" was used in a broad sense to mean "having belonged to a
steble union". Because in many countries, unions not legalized by
marriage are socially unacceptable, a respondent in such a situation
might tend to avoid acknowledging his or her "marital status". It then
seems plausible to assume that most of the "unknowns" were. in fact,
"ever married". The validity of this assumption is, however, impossible to establish without recourse to further information. But if the
data available are to be used in computing singulate mean ages at
marriage for each sex, it is necessary to make some assumption about
their true meaning. As suggested, one extreme assumption is to suppose that all the "unknowns" are ever married. Another, less extreme
possibility is to assume that both the single and the ever-married persons within each age group have an qua1 probability of being
classified as "unknown" (this assumption would be plausible if most of
the unknowns were the result of random errors produced while processing the data). Under this assumption. the proportion single in
each age group would remain the same if the unknowns were ignored
as if they had declared their marital status properly.
Each of these assumptions leads to a dibrent way of computing the
proportion single in each age group and to somewhat different estimates of the desired singulate mean ages at marriage. The computational procedures and results are shown below.
(a) Wcutation of singulate mean age at marriage ignoring those of unknow marital st&
In the first instance, the pmportions single are calculated by ignoring the unknowns. The resulting proportions are shown in table 188
Then, using these pmportions single, the singulate mean ages at marriage are computed as usual.
of this type are often due to changing marriage patterns. but the fact
that the proportion single declines at ages 50-54 suggests that the relatively high value observed at ages 45-49 may be due, at least in part, to
misnporting of status rather than to true changes in the maniage patterns of the past.
Therefore, to calculate the singulate mean age of marriage in this
case, the reported proportions single were adjusted by replacing the
value of 0.0470 for age group 45-49 by the average of the two values
adjacent to it, which happen to be also the two smallest proportions
single observed. Thus. it was assumed that the proportion single at
ages 45-49 was 0.0324. Under this assumption, the singulate mean age
at maniage is computed in the usual way. The results of the main
steps are shown below:
SMAM =(RS2-RS3)/RM= 22.01.
It is interesting to note that the value of SMAM would have been
21.88 if the observed data. without adjustment. had been used. a value
that is very similar to that obtained above.
(b) Wcu&tion of singulcte mean age at mammageawm'ng that thaw of
unknow mon'tal status am ewr martied
The values of SMAM are computed next by assuming that the
"unknowns" belong, in fact, to the ever-married category. The proportions single obtained under this assumption are shown in table 189.
The computations needed to arrive at the final value of SMAM arc
summarized in cases 3 and 4.
(i) CPW 3: d e s (unknow cmi&red to be ever nmniedj
The results for case 3 are shown below:
TABLE188. PROPORTIONS
SINGLE IGNORING THOSE OF UNKNOWN
~ l u r ~ rSTATUS,
u . PANAMA,
1976
SMAM = (m2-RS3)/0.9337= 25.45.
(ii) CPW 4:females (unknownrr comidered to be ewr married)
Once more, the inconsistency apparent in age group 45-49 was
corrected by replacing the reported proportion single, U(7) (0.0441).
by the avenge of those adjaant to it (0.0305). An outline of the calculations follows:
(i) Gase I: moles (unknowm ignored)
The results of the main steps for case 1 are given below:
SMAM = (RSz-RS3)/0.9698 = 21.95.
RS 1 = 5.0(2.4762)= 12.381
RS2=RSI+15.0= 27.381
TABLE
189. PROPORTIONS
SINGLE ASSUMING THATTHOSE OF UNKNOWN
MARITAL =ATUS ARE EVER MARRIED. PANAMA,
1976
RN = (0.0893+0.0626)/2.0= 0.076
RU = 0.924
RS3= SO.0RN = 3.798
SMAM = (RS2-RS3)/RM = 25.52.
(ii) Caw 2: &males (unknowlls ignored)
Note that the proportions single among females do not. as expected.
decrease consistently as age increases The proportion single at ages
4549, for example, is greater than that at ages 40-44. Inconsistencies
35-39 ....................
40-44 ....................
45-49 ....................
50.54 ....................
0.1 165
0.0853
0.0783
0.0542
0.0602
0.03 10
0.044 1
0.0300
(0.0305)
(c) Comments on the &tailed em+
The SMAM estimates obtained by assuming that all those of unknown marital status were, in fact, ever married, are very similar to
those obtained earlier by ignoring the respondents with unknown
status. This similarity is due to the fact that the numbers of respondents of unknown marital status are a relatively small proportion both
of the total population interviewed and of the total population whose
marital status was reported. Yet, even though the estimates yielded
m similar. the question arises as to which set should be used i f further
analysis were to be undertaken. I f no more information were available about the survey in question. it would be advisable not to make
extreme assumptions, so that the SMAM estimates computed by
ignoring the unknowns would be accepted as representative.
I n this car, however, the extreme assumption turns out to be that
nearer to the truth. Indeed, since the question in the survey was "On
what date did you marry or form a union for the first time?", and i t
was used to establish both marital status and the time elapsed since
first union. all those who were unable to state the date of which their
first marriage or union began were classified as "unknown", a term
that in this context means "of unknown marital duration" rather than
"of unknown marital status".
This example illustrates very clearly the following point: tabulations
very often do not provide enough information about the data they
display. In order to assess the true meaning of those data, i t is important to know how they were obtained and how they were processed
before they become part of a given tabulation. Without this additional information. errors in the interpretation of the data are very
likely.
The singulate mean age at marriage is a measure that logically
should refer to the nuptiality experience ofa birth cohort. Yet. in practice, data on nuptiality for birth cohorts are very rarely available. Usually. the data at hand refer to the proportions single observed in a population at one or two points in time. Because such proportions refer to
a cross-section of the population. it is conceivable that i f nuptiality
patterns have been changing. the proportion single at age 30. say, may
be smaller than observed at age 40, a situation that would never arise
among members of the same birth cohort. Therefore, when there exist
rapidly changing marriage patterns, the singulate mean age at marriage should not be calculated directly from the distribution by marital
status and by age that is observed at a single census or survey. The
proportions single in a hypothetical cohort exposed to marriage rates
between two surveys or censuses need to be calculated before the
value of SMAM is computed.
I. CMputationalp~~~cdun
The basic steps in the calculations are the same as those described
before, except that the distribution of a hypotheticalcohort by marital
status has to be constructed before calculations of penon-years lived
are undertaken. Only those steps which are different from the steps
described in subsection B.2 are described below.
Step 1: calculation of proportions singlefor two points in time scpomted
by@ or IOyars. Proportionssingle are calculated for each age group
i and time-point j by dividing the single population by the total population in the age group, adjusted, i f necessary. for non-response. The
resulting proportions are denoted by U(i. j ).
Step 2: coleulorionoJ proportions single in a hypotheticalcohort exposed
to intersmy flsr-marriage rates. Let the proportion single for age
group i of the hypothetical cohort be denoted by U(i, s). Equations
(C.1) and (C.2) are used to calculate the values of U(i.s) when the
intersurvey interval i s five years in length, whereas equations (C.3).
(C.4) and (C.5) are used when the intersurvey interval is 10 years in
length. For age groups 20-24 and above, i= 2,. .. - 8 .
For the age group 15-19. for which i= 1.
Note that in equation (C.I)the values U(i. 2) and U(i - I. I)refer
to the same birth cohort. so that their quotient is a form of "survival
rate" in the single state for that cohort. When this survival rate is
applied to the previous hypothetical value. U ( i - I,s). i t transforms i t
in accordance with the ohserved change in the proportion single during the intersurvey period.
I f the intersurvey period is 10 years in length instead of five. i t is still
possible to calculate the proportions single in a hypothetical cohort
Now
for values of i ranging from 3 (age group 25-29) to 8 (age group 50-54).
For i equal to Iand 2. that is. for age groups 15-19 and 20-24, U(i. s )
is calculated as the average proprtion single at the time of the two
censuses or surveys:
and
Steps 3-6: calculation of singulate mean age at marriage/rom the proportions single in a hypothetical cohort. Once the proportions single.
U(i,s), Cave been calculated, the procedure to be followed in calculating the value of SMAM is exactly the same as that described in
steps"2-5 of subsection B.2.
2. First &ailed example
Table 190 shows data for the female population of Japan
enumerated in 1955 (i = I ) and 1960 (i = 2). I t illustrates how the
data are used to estimate proportions single for a hypothetical cohort.
Column (6) shows the resulting set of U(i .s ) values. In the steps out-
PROPORTIONSSINGLE. 1955 AND 1960. AND ESTIMATED PROPORTIONSSINGLE
TABLE190. FEMALE
IN A HYPOTHETICAL INTERSURVEY COHORT. JAPAN
I b i o o/
Jw
llOPmpm~
AF
lrCr
C;;r
/ZJ
1
Assumed value.
1935
/3J
IW
M
r""'nmyl
Wi, 2)lWi
/J)
- 1, 1)
Pmpw11..
single in
hvpork~~cd
do11
U i . 3)
161
lined below, these values are used to estimate an average value of
S U M for che period 1955-1960.
Stq I: cuhkatiion of pmprtions sing&for mch mmrr. Since in this
ase the values of U(i, j) are already in the appropriate form in
columns (3) and (4) of table 190, this step is unnecessary.
&p 2: cakuhtion ofpnywrtiou singkin a hpotktick cohort exposed
to Wrmary @t-mawiaemtu. In this example, the intersurvey
period is five yeus. so equations (C.1) and (C.2) are used in calculating U(i, s). For age p u p 15-19, for which i = 1,
For subsequent age p u p s , each value of U(i - 1, s ) is multiplied by
the ratios U(i,2)/U(i -1, I). Column (5) of table 190 shows the
values of these ratios for each value of i . For example, for i = 2,
U(2,s)= U(I, s)U(2,2)/U(l. 1)
= U(l, sX0.648/0.983)
= (0.987)(0.659)= 0.650.
Full results are shown in column (6) of table 190.
Step 3: c a l ~ i o onf pcrsn-ycon &miin the single state. The sum of
the first seven values of U(i, s ) is 2.11 1. so that
RS2= 5.q2.11 I)= 10.555.
Note that the value of U(8, s ) for ages 50-54 was not included in the
sum. Then, the number of person-years desired is
RS3= 15.0+RS2= 25.555.
It is of interest to compare the value of SMAM estimated for the
intemnsal period (24.40 years) with the estimates that would have
been obtained by considering each census separately. According to
the proportions single presented in columns (3) and (4) of table 190.
the value of SMAM in 1955 was 24.74 years. while that in 1960 was
24.79 years. Therefore, although the difference is relatively small, the
intercensal estimate of SMAM is lower than those obtained at the
end-points of the period. This outcome implies, in general, that
according to the first-marriage rates prevalent during the intercensal
period, women would marry at slightly younger ages than they would
according to the mixed cohort-period experience observed at a given
point in time. However, the difference observed is so small that, at
least in this case, what ought to be stressed is the similarity of the
different estimates and the fact that, according to them, the singulate
mean age at mamage in Japan during the period 1955-1960 remained
almost constant.
3. Second &tailed eyunpIe
Table 191 shows data for the female population of Tunisia as
enumerated in 1966 (j = 1) and as estimated for 1976 on the basis of
the 1975 census (j = 2). It indicates how the data are used to estimate
proportions single for a hypothetical cohort. Column (6) shows the
resulting set of U(i. s ) values. The steps outlined below illustrate how
these data are used to estimate an average value of SMAM for the
period 1966-1976:
Step I: rolevloion of p'oponiow single. In this case, the values of
U(i, j) are already in the appropriate form in columns (3) and (4) of
iabie 191. Therefore. this step is not necessary.
Step 2: calcukuion ofpmportions single in a Lpotk~ical
cohort exposed
to i n t e r s w w y j r s t ~ g mles.
e
Although the true intercensal period
in Tunisia was nine years in length, the basic data have been adjusted
to r e p a n t a 10-year period. Hence, equations (C.3). (C.4) and (C.5)
can be used to calculate U(i. s). Thus. for age gmup 15-19. for which
i = 1, for example,
Step 4: colnJorion of p'oportion ever manying. The proportion
remaining single is
RN = (0.045 +0.045)/2.0= 0.045.
Hence, the pmportion ever mariying equals
RM = 1.0-0.045 = 0.955.
*p 5: f ~ ~ t i i of
o np r m n - m l i d by those eMning single.
These penon-years are calculated as
and for age p u p 20-24, for which i = 2.
U(2, s ) = O.S(U(2, I) +U(2,2))= 0.5(0.270+0.473)= 0.372.
For subsequent age groups each value of U(i -2. s ) is multiplied by
the ratio U(i; 2)/U(i -2, 1). Column (5) of table 191 shows the value
of these ratios for each value of i. For example, for i = 3.
Step 6: rolevloion of s i n p h e maw age at d g c . The value of
SMAM is calculated as
SMAM = (RS2-RS3)/RM = (25.555-2.25)/0.955 = 24.40.
Full results arc shown in column (6) of table 191.
PROWRNONSSINOLE. 1966AND 1976,AND WIMATED PROPORTIONSSINOLE
TABLE191. FEMALE
IN A HY#rmEllCALINTEIISURVEY COHORT.TUNISIA
,Sowns: For ro rtions single in 1966.
ulation census of 1966; for roportions sinale in 1976. estimates fmm the (bl~popuIatbncensus by m a r a k e t . EwIution r i m e la n lYIif4 n de IajkonditC
en W e (Tunis, Olllcc national du planning familial et de la population. Hay 19%)
Assumed equal to value for ages 4549.
2
Step 3: calculation 4pemn-yws l i d in the single slate. The surn of
the first even valuer of U ( i . s ) is 1.568. so that
Note that the value of U(8,s) for ages 50-54 was not included in the
sum. Then the number of person-years desired is
Step 5: mahhion ojpmon-)reon l i d by t
SO. This value is calculated u
h W n i n g sing& at age
Slcp 6: Elrleuhrion ofsinguh~emarrn age at manbp. The value of
SMAM is
SMAM = (M2
-RS,)/RM = (22.840- 1.475)/0.9705 = 22.01.
Step 4: caleulotion ofptvporiion ever -ng.
proportion remaining single at age 50 is
An estimate of the
Hence, the proportion ever marrying by age 50 equals
RM = 1.O -0.0295 = 0.9705.
If this vdue d S W M is cornpad with the atimates that would be
obtained by win8 the data u reamled at the end-poinu of che period
being a n u i d e d (20.88 yun in 1966 md 22.71 p n in 1976), it
appeur that the inkrrenrri vdue d SWM b higher chm tbe mun
of the eccimaler at the end-point& ThL outcome ruthat Intmarriage mler declined rubtrntirlly and Ridy rapidly in T u n h during the period 1966-1976.
If
Annex
THE ELBADRY CORRECTION FOR DATA ON CHILDREN EVER BORN
A. BACKGROUND
OF M ~ O D
During fie colle&n or the p m s s i n g of information on children
ever born, an error consisting of the misclmification of women of zero
parity as women whose parity is not stated sometimes arises. This
error is often due to the ambiguous entries made by interviewers at the
time of the survey. For example, ambiguity is clearly present when a
dash (-) is used to indicate both that parity information about a given
woman was not obtained or that she declared herself as childless.
When errors of this type occur on a sufficiently large scale, the quality
of the entire set of information on children ever born is compromised.
especially if this type of information is to be used to estimate fertility
indirectlv bv anv of the methods described in chapter 11. Whenever a
of childless women are d k i f i e d incorrectly in
consideAbl;
the category "parity not stated", the exclusion of these women from
the denominator when calculating average parities by age would lead
to overestimates of average parity, especially at younger ages. If, on
the other hand, the women whose parity was not stated are not
excluded, the denominator wil! be too large, producing underestimates
of the true average parities.
In 1961, El-BadryO proposed a procedure to estimate the proportion
of women belonging to the category "parity not stated" who should
have been classified as childkss. This procedure is based on the
observed high cornlation existing between the proportions childless
and the proportions in the category "parity not stated", especially
when only the fint four or five five-year age groups are considered.
El-Badry's method is based on a very simple model. Let Zo(i) be
the true proportion childless in age group i and NS(i) be the reported
proportion of women classified in the category of parity not stated for
the same age group. Then, it is plausible to assume that the following
relation holds:
where a is the proportion of women who are truly childless and who
were erroneously classified as of "parity not stated"; and & is the constant. true level not stated. (It is expected that, in every survey, even
in the absence of the coding error being discussed, some women will
be classified as not having stated their parity. Usually, these are
women who were not present at the time of the interview and for
whom the selected informant did not know the true number of children ever born). Since the proportion aZo(i) of childless women has
been misclassified, the reported proportion in age group i , Z(i), must
be equal to
or
the true value of Z"(i): fit a line to the reported points [Z(i), NS(i)]
and estimate y and 8. Then the true proportion childless is
Z"(i)= Z(i)+(NS(i)-8).
and an estlmate of the zero-parity error, a , can be obtained by
a = y/(l.O+y).
to
(A.6)
(A.7)
Note that even though algebraically equation (A.6) is equivalent
Z"(i)=(I.O+y)Z(i),
(A.8)
equations (A.8) and (A.6) may (and usually do) produce slightly
different results in practice, since the fit obtained is very rarely exact
For the purpose of adjustment of the zero category, equation (A.6) is
recommended.
In most cases, when the required final product is a best estimate of
average parities by age group of women, an adjustment of the zeroparity category is not of immediate use, but the true level of nonresponse is. Once &, an estimate of this level, has been calculated, the
female population to be used as denominator in computing average
parities by age group can be obtained by multiplying the reported
female population in each age group by the factor(l.0-8). Thus. the
required output from the El-Badry correction procedure will very
onen be only &, obtained by fitting equation (A.5).
To conclude. it must be noted that slightly different computational
procedures have to be followed according to whether the data on children ever born refer to all women or only to ever-married women.
Therefore, two detailed descriptions of the computational procedure
are presented in the following sections. It is also useful to note now
that in the following discussions, index i is used to denote the age
group from Si + 10 to Si + 14.
The following data are required for this method:
(a) The number of women in the zero-parity category (childless
women) classified by five-year age group;
(6) The number of women in the category "parity not stated
classified by five-year age group;
(c) The total number of women classified by five-year age group.
However, when the question on parity was asked only of ever-married
women, the data required include both the total number of women
and the total number of ever-married women in each five-year age
group (15-49).
'
C. DATAON CHILDREN EVER BORN REFERRING
Z*(i)= Z(i)I(I.O-a).
(A.3)
Substituting
(A.3) in equation (A. I), one finds that
- equation
NS(i)= [a/(l.O-a)]Z(i)+&
64.4)
NS(i)= yZ(i)+&
(A.3
or
where y = al(l.0-a).
Equation (A.5) suggests a method of estimating
M. A. El-Badry, "Failure of enurneraton to make entries of zero
ulation censuses", Journal o
errors in recording childless qses in
the Amen'can St~listicdAssociatiot&v0°56. No. 2% (December 1%1{
pp. 909-924.
TO ALL WOMEN
I. Computationalprocedure
The steps of the computational procedure are given below.
Step 1: calculation of proporriots ntthparity not stated. If one denotes
by FP(i) the total number of women in age group i and by FNS(i)
the number ofwomen whose parity was not stated and who belong to
age group i , then the proportion whose parity was not stated is
NS(i)= FNS(i)/FP(i).
(c. 1)
Step 2: calculation of proportions childless. Letting n ( i ) be the
childless
in age group and Fp(i) be as
=ported
defined above*lhen the P
~ childless~ is
P
~
~
Z(i)= FZ(i)/FP(i).
(c.2)
~
Stq 3: atbmtio11oflinearpammeters 7 and 8. Once the proportions
of wanen whose parity was not stated and those of childless women
have k e n ~1Iculated.the points [Z(i).NS(i)] should be plotted. If
they follow approximately a stmight line, the use of this method of
adjustment is warranted. Otherwise. no adjustment is possible and the
user is advised to include all those women classified as "parity not
stated" in the denominator when computing average parities. Of
course, if the proportion of women with unknown parity in each age
gmup is small. their inclusion or exclusion from the denominator is not
likely to introduce serious biases in the average-parity values.
&en the plotted points do fall roughly o i a itraight line. the values
of its parameters have to be com~uted.1t is recommended that only
the points corresponding to the first four or five age groups be considered. To estimate the parameters of the straight line approximating
them, divide the selected points into two group of equal size according to their abscissa values (repeat the middle point if necessary) and
calculate the means of the abscissae and of the ordinates in each
group. The parameters of the line passing through these mean points
are estimates of the desired y and B values. The application of this
procedure is illustrated in the detailed examples that follow.
When the points are fairly well aligned. the use of the least-squares
method to estimate y and j3 is, of coune. also possible and may yield
the best results. See chapter V, subsection C.4 for a description of this
method of line-fitting.
Slcp 4: cal&tion of t n e proportions childlnu. As equation (A.6)
indicates, the m e proportion childless, Z*(i), is equal to the sum of
the reported proportion childks, Z(i). and the deviation of the
reponed ploportion whose parity was not stated from the estimated
constant kvel, that is, (NS(i)-8). Therefore, once /3 has been canputed, the calculation of Z*(i) is stmightfornard. Once the estimated
Z*(i) values arc available, the m e number of childkg women.
n*(i),
can be obtained as the product of each Z * ( i )and the reported
female population of age group i. FP(i). The values of J2*(i)
obtained in this way should be used when fertility is to be estimated
by using data on first births (see chapter 11, subsection B.3).
- Step 5: ccJfulcuion of &nmninafok
for estimation of awmge pities.
Let FP*(i) denote the number of women whose parity is usumed to
be known afier the true proportion childless has been estimated. Then
since. according to the model presented so far. the true level of nonmponse is the estimated value of @.When data on children ever born
are being used to estimate fertility indirectly by'the methods described
in chapter 11 or to estimate child mortality by those described in
chapter Ill, the average parity of women of age group i should be calculated by using FP*(i) as the denominator instead of the reported
number FP(i ).
2. LktaiIede"nple
The census carried out in Peru in July 1972 collected information
from all women on the number of children they had ever borne, and
the necessary information to apply the El-Badry adjustment procedure
was published. It is shown in table 192.
TABLE192. F E W POWLATlON ACCORDING TO DIFFERENT RESPONSES CONCERNING
CHILDREN EVER BORN. PERU. 1972
(Popahtion in thouaondr)
A quick examination of the published numbers (those shown in
columns (3). (4) and (5)) nveals that a very significant proportion of all
women in each age group was classified in the category of "parity not
stated". Especially striking is the fact that in age p u p 15-19 them arc
more women in that category than in the childless p u p . All these
observations appear to be consistent with the existence of a zero-parity
error of the type described in section A. Adjustment by means of ElBadry's procedure should be attempted.
SIcp 1: cohwiuion of proportion uith p ' t y not stafed. According to
equation (C.1). the proportion of women whose parity was not stated
in age group i, denoted by NS(I), isjust the quotient of the number of
women in that category in that age p u p and the total female population of the same age p u p . Therefom, NS(i) is obtained by dividing
each of the entries in column (5) (women with parity not stated) by the
comsponding entry in column (3) (total female population) of table
192. For example, for age group 20-24,
The complete set of NS(i) values is shown in column (7) of table
192.
SIcp 2: d&tion of pqonions chihhk. According to equation
(C.2). the proportion childless in each age p u p . Z(i). is obtained by
dividing the reported number of childkss women (column (4) of table
t92) by the total number of women in each age p u p (column (3) of
the same tabk). For instance.
Z(3)=51.1/471.5=0.10SJ.
The full set of Z(i) values is shown in column (6) of table 192.
Step 3: estimation of linear pamneten, @ and y. Figure 24 shows a
plot of the first five points [Z(i), NS(i)] calculated in the previous
steps. The Z(i) values am plotted on the x-axis. while those for NS(i)
appear on they -axis. Even in the largely expanded scale used in plotting these p i n & they display a fairly linear trend. The point farthest
right (corresponding to age gmup 15-19) is the one that deviates most
markedly from the trend defined by the others. However. given its
importance (the proportion childless is highest for that age group), it is
not eliminated from the calculations leading to the selection o_fabest
fitting straight line.
Two methods of fitting are illustrated. The first method is based on
group means, according to which the points [Z(i). NS(i )] are divided
into two g m u p each containing t h m points. Tablc I93 shows this
division and the means in each group. Using these mean values 7 is
estimated as follows:
= (0.1459 -0.3645)/(0.0800 -0.2 193)= 1.5693.
The value of 8is then calculated as
Bl = 0. I459 -( 1.5693)(0.0800) = 0.0204.
The line whose parameters are yl and
line in figure 24.
is represented us the mean
TABLE
193. F ~ NOFOA SllWOHT LINE BY USING GROUP MEANS.
PERU.1972
FkportdpIoportb
whh prrlty not
s t r t d NW)
O*
r
=
zz3)
d
15-19....................
20-24 ....................
25-29 ....................
Mean ....................
25-29 ....................
....................
35-39 ....................
Mean....................
30.34
Apoc*l
NW)
(3)
(a) R n t g r o ' ~ ~
0.3469
0.2027
0.1084
z1=0.2193
(b) hndmv
0.1084
0.0708
0.0608
0.0800
0.5524
0.3427
0.1985
I = 0.3645
0.1985
0.1321
0.1072
m2=0.1459
z2=
so that
y2= 0.0880/0.0572 = 1.5385
and
/I2= 0.2666-(1.5385)(0.1579)=
Reported
0.0236.
The least-squares line defined by these parameters is also shown in
figure 24. Note that the two fitted lines almost coincide. Therefore
either of the two can be chosen as representative of the reported
points. In canying out the next steps, the least-squares line is used as
representative.
Step 4: dcuhtion of ttuepmpo~ionschiIdless. Equation (A.6) is used
to calculate the tlue proportion childless, Z*(i). from the reported
Z(i), NS(i) and the estimated B. Z0(3), for example, is
Mean Ilne
Least-squares line
RaportuI proportbn childless Yt)
Sow:Based on data given in table 192.
For the sake of illustration. another tine is fitted using the method of
krst quuer Accardig to this method, the values of y and /I are
o b b t d wing the f d w i n g equafions:
are the means of the reported proportions of childwhere Z and
k m women md ofthose whose parity was not stated. respectively.
The crlfuktions necessary to estimate y2 and &are not camed out
in &tail. Only the most impoNnt inteimediate reiults are given.
Note that only the fimt five age p u p s are used in this case:
z =0.1579 and
= 0.2666
The complete set of Z*(i) values is shown in column (3) of table 194.
Note that according to these estimates of the t ~ proportions
e
childless.
approximately 14 per cent of all women in age group 4049 had zero
parity in 1972. This value appears rather high for a country like Peru,.
where mamage is probably nearly universal and where there is no reason to expect a higher-than-normal incidence of sterility. These high
values of Z*(i) at older ages may indicate that this type of adjustment
is not adequate for the data at hand. Hence, before adopting these
mults for future use, it would be advisable to consider the observed
proportions single at each age so as to assess the plausibility of the
estimated proportions childless. The search for any other rewons that
might explain the high proportions of women with parity not stated
yielded by the census is also recommended.
When El-Badry's adjustment procedure is applied to data that will
later serve to estimate fertility by using information on first births (see
chapter 11, subsection B.3), the actual number of ever-fertile women
(not childless) is required. This number can be estimated by subtracting the estimated proportions childless from one and then multiplying
the resulting number by the total fefnale population. For exampk.
the ever-fertile women in age group 20-24, FF(2). would be
or equivalently.
Step 5: ccrkukation of drnominolon for estimation 4 owmge parilies.
The value of FP* is calculated according to equation (C.3). Thus.
Other values of FP*(i) are shown in column (6)of table 194.
Note that even though in the first three steps of this procedure
attention is focused mainly on the first five age groups. all seven
PROFORTIONSCHILDLESS AND ESTIMATED TOTAL FEMALE POWLATION
TABLE
1%. ESIIMATED
WITH KNOWN PARITY. PERU,1972
(Popukltion in thousands)
,iktk.d
alu
AJF
Index
(4)
P'P
-
(1)
15-19....................
20-24 ....................
25-29 ....................
30-34 ....................
35-39 ....................
40-44....................
45-49 ....................
I
2
3
4
5
6
7
ZG2
Z*(i)
mu
=d
*~I(M
n*(i)
Nunbrr #
CCCI-
Wik
*~I(M
FFO
(3)
(4)
(5)
0.8757
0.5218
0.2833
0.1793
0.1444
0. I393
0.1413
611.4
301.9
133.6
68.4
53.9
41.5
34.8
86.8
276.7
337.9
313.0
319.4
256.3
21 1.4
groups a n required for the purpoje of fertility estimation. Table 194
refers therefore to all of them.
As pointed out before, if one is satisfied with the consistency of the
estimated values obtained in this and in previous steps, future fertility
or child mortality estimation should make u r of the estimated rather
than the reported values. Therefore, the FPe(i) estimates should be
uscd as denominators when calculating average parities.
D. DATAON CHILDREN EVER BORN REFERRING
EsrUMIcd
*A
";p"*ry
16)
68 1.7
564.9
460.4
372.4
364.5
290.8
290.4
However, this number refers only to zero-parity women among
those who have ever been mamed. The total number of childless
women in age group i, denoted by FZe(i), can be obtained only by
making some assumption about the fertility of single women. If one
accepts that all single women are childless, then
n * ( i )= lZoe(i)+(FP(i)-EMP(i))
(D.5)
In many countries, questions on fertility are asked only of evermarried women, on the assumption that all single women are childless.
To the extent to which this assumption is true, fertility for the total
female population can be estimated from data referring exclusively to
ever-married women; but, especially when El-Badry's adjustment procedure is applied, due allowance must be made for the fact that the
parity of single women is not =ported. For this reason. the computational procedure one must follow when fertility information is avail-.
a& only for ever-married women is described in some detail.
where FP(i) is the total number of women in age group i . Since only
very rarely does one have some basis for making other assumptions
about the childbearing experience of never-married women, it is
recommended that equation (D.5) be used to estimate the number of
childless women among all women in age group i. This number
FZe(i), should be used if fertility estimation is to be attempted by
using the first-birth method described in chapter 11, subsection BS.
Step 5: calcukltion of denominatorsjbr estimation of average parities.
Let FPe(i) denote the number of women whose parity can k
assumed to be known after the true number of childless women has
been estimated. Then
1. Computationalprocedure
The steps of the computatidnal pmedure are given below.
Sfep I: d ~ i o ofproporlions
n
nirhponponty
nor stated. If one denotes
by EMP(i) the number of ever-married women in the age group from
5i + 10 to 5i + 14. and by FNSdi) the number of ever-married women
of the same age group whose parity was not stated, then the proportion with parity not stated is
that is, the number of women of known parity equals the total number
of women. FP(i), minus those among ever-married women whose parity is unknown. The value of FPe(i) defined by equation (D.6) should
be used as a denominator whenever avenge parities are nquind to
estimate fertility or child mortality.
TO EVER-MARRIED WOMEN
,
2 m i O nofproportions chi,d~e551.noting by m d i ) (he
ofchildless
among
ever.ma*ed in age group
lettingEMP(~)be as defined
the proponion childless is
Slrp 3: estimation of linearpnmmeters, y and 4. As in the c a r where
data for all women are available, a graph of the points [Zdi), NSdi)]
should be made. If they fall approximately in a straight line, its
parameters, y and 4, should be estimated by using either group means
or the kast-squares method ( r e chapter V, subsection C.4). In general, only the first four or five age groups should be considered when
fittingthis line.
Sep 4: calcuhtion of tnw pporfions chiMeu among ever-manied
mmn. The true proportion childless among ever-married women of
age group i. Zoe(i), is obtained by adding to the reported proportion
Zdi) the deviation of the proportion whose parity is not stated from
its estimated constant kvel, (NSdi)-8); algebraically,
Then. by multiplying this estimated true proportion childless by the
corresponding number of ever-mamed women, an estimated number
of childless ever-married women. n o e , can be obtained as
The case of Thailand in 1970 is inrkuing because during the
census carried out in that year the question on children ever born was
asked only of ever-married women. Table 195 shows the basic data
collected during the 1970census.
Sfeps I and 2: calculation ofPkportions w'h parity not stated and proportions childless. Column (4) of table 1% shows the pmportions of
ever-mamed women who did not state their parity, NSdi); and
column (3) gives the proportions childless, Zdi), abo among evermarried women. These pmportions were computed using equations
(D.1) and (D.2). As an example. NSd3) and Zd3) an calcuhted
here:
NSd3)= 56.5/963.6= 0.0586
Sttp 3: estimation of linear pmumeters y and 8. Figure 25 shows a
plot of the points (Zdi). NSdi)) listed in table 1%. Note that only
four points were plotted (those with i ranging from 2 to 5). The first
point. corresponding to age group 15-19. falls outside the graph and is
not aligned with the others. Therefore, for the purpose of estimating y
and B. this point is ignored. Using the least-squares method in fitting a
line to the other four points, the following y and B estimates are
obtained:
y= 1.4797
and
@=
0.0147.
TABLE
195. FEMALE
~PULATION XCORD~NG TO DIFFEREM RES~ONSESCONCERNING
CHILDREN EVER BORN. THAILAND.
I970
T '
I&
15-19... .................
2&24 ....................
25-29 ....................
30-34
35-39 ....................
....................
i
fII
I
2
3
4
5
-7
mi)
(3)
1 885.4
1361.7
1 143.4
1 077.1
957.6
Slcp 4: cakdation of tnw pponiiavu clu'Idleca. Column (3) of table
197 l o w s the proportion childless estimated according to equation
(D.3). For example, for age group 20.24,
h-
Worn
unW
Mlh m a
357.6
843.6
963.6
989.6
906.6
148.7
59.2
25.7
15.8
11.6
w
T)
(4)
P"
f
W(nn Wuh
ri'Y
n#urnl
mso
(6)
23.6
99.2
56.5
37.0
28.4
The complete set of FP* values is shown in column (6) of tabk 197.
These values should be used as denominaton when calculating aver-
age parities.
Once the estimated true ptoportions childless am computed, the
estimated number of childless women among the ever-married
noe(i). cm be obtained by using equation (D.4). Thus, for age
group 20.24.
Then. the estimated number of childless women among all of the
women is computed by adding the number of single women to no*.
AU n * values am shown in d u m n (5) of table 197. These values
should be used in estimating the true proportion childless among all
women wben the first-birth method of ferlility estimation is used (see
chapter 11, r u b e a i n B.3).
Sq E a&&litm ofihomimtors for atimtion of swag @ties.
Using equation (D.6), FP* is estimated from the total number of
women md Ihe ever-maied women of age p p i . For example. for
ry p u p 20.24.
TABLE 1%.
R e m R m FROKHTWNSCHILDLWS AND OF PARITY NOT
~ v e r - m WOUEN.T
~ ~ e ~HAILAND. 1970
STATEDMONO
TABLE
197.
ADJUSTW DATA USING FROCORTIONS A M N O EVER-MARRIED
WOUEN.THAILAND.
1970
3. C o r r r m Y s a n d r t c u ' l a d e ~
To candude, it should k pointed out that it is excnmely important
to know how the data on feftility were collected in order to choose a
suitable adjustment procedure. Lack of information on the sources of
the da@k i n g used may cause serious estimation enurs. An exampk
of one possibk error is illustrated in table 198. which shows the
a d j W proportionsofchildless women, their actual numbers and the
adjusted number of women of known parity for Thailand in 1970.
lhew values were estimated by assuming, inwmctly. that the fertility
information r e f e d to all women. Under this assumption. the wmputationrl p m d u n for data on rll women (section C) was used to
anive at the estimates given in tabk 198. A line was fitted by using
kuc squares to the four points ranging from age groups 2 to 5. The
panme- y and /?estimated in this way have the values:
n*
A w m p r r i m between the estimates of
and FP* shown in
table 198ind the cowed values given in table 197 reveals that the two
FP* values are relatively smilrr. although the differences between the
w m c t value and that from "all womerS" arc not trivial. However, a
great difference i s evident between the two types of #2 estimates (the
w e number childless). This difference is easy to explain: the &Z*
values obtained when proportions calculated with respect to evermarried women arr used as input incorporate the single women, whik
those obtained using proportionsamong all women do not inwrponte
them. Of course, if one knows that single women were assumed to be
childkss, the appropriate adjustment can be made. But knowkdge of
this assumption implies knowkdge of the data sources, and such
knowkdge should allow the investigator to choose the appropriate
adjustment proadure in the first plag so that no such errors would
arise.
and
p= 0.01 52.
TABLE198.
ADJUSTEDDATA USINO PROPORTIONS AMONO ALL WOMEN, BASED ON AN
IN COB^ A&~UMP~~ON.THAII.AND,
1970
= * =
(Populorion in t h o d )
Eu&md
4
77
.....................
....................
....................
....................
35-39 ....................
15-19
20-24
25-29
30-34
lrlr
i
(2)
I
2
3
4
5
?zd
%
rrrrvJ
NST)
(4)
z.(I)
(3)
lZ'(i)
(6)
Z)
0.0789
0.0435
0.0225
0.0147
0.0121
0.0125
0.0729
0.0494
0.0344
0.0297
0.0762
0.1012
0.0567
0.0339
0.0266
143.7
137.8
64.8
36.5
25.5
1 856.7
1341.0
1 126.0
1 060.7
943.0
JIL
(7)
THE MEAN ACE OF CHILDBEARING
A. DIRECTMETHOD FOR CALCULATION OF MEAN AGE
OF CHILDBEARING
I. &is of method and its mionale
The mean age of childbearing is the average age at which women
bear their children, in the absence of mortality and age-distribution
effects. It is a measure of the timing of childbearing for a mortalityfree cohort of women passing through their reproductive life, or, alternatively, for a x t of period age-specific fertility rates.
If one denotes by f (i) the fertility rate experienced by women in
ihe age group from 5i + 10 to 5i + 14, the mean age of childbearing. p
is defined by
where a(i) is the central age-point in the ith age interval, and I is the
age group containing the upper age-limit of the childbearing span.
2. Datamqucqucd
As equation (A.l) indicates, knowidge of the fertility schedule to
which the population of interest is subject is sufficient to provide an
estimate of p. Since fertility rates are usually obtained by dividing the
number of births occurring during a given time period by the personyean lived by the population subject to the risk of childbearing during
that period, it follows that the data required are:
(a) The total number of births occurring during a given year,
classified by age of mother;
(b) The number of women at the mid-point of the year in question,
classified by age.
Knowledge of the= data would permit the calculation of period fertility rates from which a period mean age of childbearing can be calculated making use of equation (A.1).
When cohort fertiliiy rate; specific by age are available, a cohort
mean age of childbearing can be computed. In some instances, the
fertility rates available wmspond neither to a welldefined time
period nor to a single cohort. In such cases, even though a mean age of
childbearing can stiU be computed. its use as a parameter indicating
the timing of childbearing is valid only under the assumption of
unchanging fertility. Bearing this in mind, the way in which p is wmputed is discussed in the next section.
'
3. Comptatiodpnmdure
The steps of the computational procedure are given below.
Step 1: calculation of fertility mtes. The fertility rate for group i.
denoted by f(i), is calculated by dividing the number of births occurring to women in age group i by the total number of women in that
age P U P .
SIcp 2: cal&ion of mean age of childbearing. The denominator of
equation (A.l) is calculated by adding the fertility rates obtained in
the pnvious step. Usually, the age groups considered range from that
beginning at age 15 to that ending at age 49. Then the central age for
each of the age groups considered is calculated and multiplied by the
comsponding fertility rate. Addition of all these products (again using
as limits of summation the age groups wntaining ages I5 and 49)
yields the numerator of equation (A.l). Therefore, its division by the
sum of the f (i) values yields the value of p.
OF CHILDBEARING
1. Necessity of these methodt
Frequently, in countries lacking adequate data on fertility or mor-
tality, the data required for the calculation of p are either not available
or a n seriously flawed. Two methods for estimating the mean age of
childbearing, p have been proposed for such cases. It is worth noting
here that while equation (A.l) is exact (except for grouping erron) in
the sense that it defines p, the methods and equations presented below
are only estimators of
2. Ertimation method 1'
Let P(2) be the average parity per woman (number of children ever
born divided by the number of women) reported by women aged 2024 and P(3) be the average parity of women aged 25-29. Then. by
using linear regression. it was established that a good predictor of p for
populations where then is little practice of birth control is themtio of
P(3) over P(2) in accordance with the following equation:
This estimate of fi is relatively accurate and very simple to obtain.
Besides, it requires as input data only the values of P(2) and P(3).
which are vew frequently available.
3. Estimation method 2
When only the proportions of women married in each age group.
m(i), are known, a mean age of childbearing can be calculated by
assuming that marital fertility has the same shape as natural fertility.
This assumption is valid only kr populations where there is very little
practice of birth control within marriage. Furthermore, for the
estimated mean age of childbearing to be npnscntative of the entire
population, illegitimacy must be rare.
The standard natural fertility schedule for married women is shown
in table 199. Note that the value of the marital fertility rate at ages
15-19 depends upon the proportion of women married during those
ages, m(l). All other rates are fixed.
TABLE199. STANDARD
NATURAL FERTILITYSCHEDULE FOR
MARRIED WOMEN
I&x
7
r(YI
rlr)
W
15-19..................................
20-24 ....................................
l
m u l ~ i u ~ y
i
I
2
(3)
1.2
- 0.7rn(l)l
1.000
Proportion married in age group 15- 19.
If one denotes these model fertility rates by y(i), the rates for the
population in question would be computed as the product of the proportions married and the marital fertility rates, that is, as
and once f is obtained, the calculation of p is carricd out as described
above in subsection A.3.
All these methods of estimation are used in the following example.
' Based on Mama1 IV:
M e t e of Estiiwing +cC Drmo rqphic
111~ompIete
Dare (Unlud Natans yubluatm. ~ a f NO.
.
OF WOMEN BY MARITAL STATUS. NUMBER OF WOMEN
TABLE
200. PROFORTIONS
ANDCHILDREN BORN IN 1961
C. A DETAILEDEXAMPLE
The data shown in table 200 comspond to a population
enumerated in IWI. The mean age
- of childbearing- is to be estimated
for this population.
2. Dim1 calcukltion of mean age of childbewing
Slrp I: ealeuhion ojfrrrili~ymfes. Table 200 shows the number of
women enumerated in 1961 and the number of children born in that
year (obtained from vital registration data) clasifled by age of mother.
The fertility rates, f (i). for each age group can therefore be calculated
simply by dividing the number of births in 1961 by the comsponding
female population (this population was enumerated near the middle
of the year, so that adjustment is not necessary). Values off(i) are
listed in column (3) of table 201.' For i = 3. f (i) is calculated explicitly
here:
TABLE
201. CALCUWTlONOF MEAN AGE OF CHILDBEARING
BY THE DIRECT METHOD
c.rarJ
I
Av"'
15-19 ..............
~
%
X
i
W
I
*
(4)
0.2050
17.5
3.59
S/ep 2: calculation of mean age of chiidbearing. Column (4) of table
201 shows the central age values corresponding to each of the age
group considered and column (5) shows the products of these central
ages by the observed f ( i ) values. The sums of the latter and of the
products just cakulated are shown in the line marked "Total", that is,
and
So that, by equation (A. I),
It is important to point out that only when the assumption of
constant fertility holds true is this value of p equivalent to that of a
cohort. If the fertility scheduks fdlowed by differen1 cohons in the
w~ulationare dilferent, the value of fi cakulated above is that of a
&;nposite parameter that does not refer to any cohort, but rather to
the experience observed during a certain period (1961 in this m e ) . In
the latter case, inferences drawn from it must be treated with caution.
Furthermore, note that the central ages, a(i). used in computing the
mean age of childbearing. p, are highly de ndent upon the way in
which information on births in a year is togaed. When the number
of births occuning in a given year is obtained by means of a r m p k
survey or a census and the age of mothers is rucemined as of the time
of the survey and not as of the date on which the most recent birth
took place, it is clear that most mothers were younger at the time of
this birth than at the time of the survey or census. If it can be assumed
that births are uniformly distributed in time, women were. on the
average, six months younger at the time their births in the past year
took place. Therefore. the true age group considered when using survey data are shifled by six months to the younger ages, becoming
14.5-19.5, 19.5-24.5 and so on, instead of the usual 15-19, 20-24 etc.,
and, in consequence, their "central agesn, a(i), are also six months
smaller (17.22. 27. 32. 37 etc.). In general. these centrrl ages should
be used when calculating p from survey data.
3. Ertimation of mean age of c h i M n g
ming methad I
Next to be estimated is the mean age of childbearing, p by means of
the equation (B.l). For the population under consideration, it is
known that PO)= 2.389 and PO)= 3.812. so that the calculation of p
is strai~htforward:
Once more, this value of p is equivalent to that of a cohort only if fertility has remained constant during the past 3C or 40 years. When fertility has been changing, p calculated as given above mprcsents, at
best, a rough "average mean" for younger cohons (those under age
30). In such a case, its value should be interpreted with caution.
Note that the estimate of p obtained hek is slightly greater than that
obtained by the direct method, probably because equation (B.1) was
obtained from data pertaining to populations that practise tittle birth
control, and it is therlfore likely to overestimate the mean age of
childbearing in populations where some timitation of family size is
common.
4. Estimation of mean age of e h i M n g
wing method2
According to this method, marital fertility is assumed to have the
same pattern as natural fertility (that is, it is assumed that no paritydependent birth control is practised by the population in question).
Under this assumption, a value of p can be obtained from knowledge
of the proportion of mamed women in each age group.
Thc proportion of mamed women in each age group is equal to one
minus the proportion of women who are single, widowed or divorced.
Thus
where m(i) is the proportion mamed in age group i ; s(i) is the proportion single; and w(i) is the proportion widowed or divorced in the
same age group.
Cdumn (3) of table 202 shows the values of m(i). Column (4)
-
N MEAN AGE OF CHILDBEAIIN0.p. BY USING METHOD 2
TABLE
202. E ~ M A N OOF
h**uI
4
77
15-19..............
20-24 ..............
25-29 ..............
B34 ..............
35-39 ..............
4044..............
4549 ..............
TOTAL
hlx
I
W
I
2
3
4
5
6
7
3
0.732
0.925
0.944
0.924
0.87 1
0.788
0.692
shows the values of y(i ), the standard natural fertility schedule, copied
fram tabk 199. Note that d l ) was obtained as
Cdumn (5) shows the values off (i) as estimated by multiplying
the entries form (i) in column (3) by Hi) in column (4). Using these
values of/(I) and the central ages. o ( i ) . listed in column (6). the products a(i)f ( i ) arc obtained. Then the value of p is computed by
d i i i n g the sum of these products (shown as "Total" under cdumn
(7)) by the sum of the f ( i ) values (shown u "Total" under cdumn
(5)):
M
ldlrl
T?
(4)
0.688
1.000
0.935
0.853
0.685
0.349
0.05 1
a&d
T?
$j
0.503
0.925
0.883
0.788
0.597
0.275
0.035
4.006
I
(6)
17.5
22.5
27.5
32.5
375
42.5
47.5
A5jy
8.80
20.8 1
24.28
25.6 1
22.39
11.69
1.65
1 15.24
As might be expected. this value of p is higher than either of the two
estimates obtained previously, because natural fertility, embodying
unnstricted fertility at older ages, represents an upper bound to the
fertility experience of a human population. In this sense, h e value of p
just calculated is probably also an upper limit for the true value of p.
- In conclusion. by using the t h m - s b l e methods of estimating the
mean age of childbearing. p, an qua1 number of different values for
this parameter has been obtained. The estimate derived directly from
observed data and the definition of p is preferred. but were those data
unavailabk. method I would produce a relatively good estimate of the
desired parameter. The vrlue of p obtained by assuming that marital
fertility b closely rppmximated by natural fertility is, on the ocher
h a d , unacceptable. In general, method 2 b to k avoided if at all p
ibk.
LINEAR INTERPOLATION
A. COMPUTA~ONAI.
?ROCEDURE
lnterpohtion is wed frequently in this M d , and because
simplicity is the main criterion when choosing the type of interpolation
that is to be performed, the linear type is very onen pnfemd.
It is well known that any two points define a line uniquely. Thercfore, if t h m points are to lie on a straight line, but only one of the c e
ordinates of the third is known. the other one is uniquely determined
and can be calculated. This calculation is described below.
Consider the line defined by the points (x 1.y I) and (xzy 3. A third
point (x, y ) would lie on this line only if the following relation holds:
The third point, that for which the ordinate is unknown, is (0.0169,
Y). Aawding to this identification,
(Y2-y t)/(x2-x I)= (Y -y I)/(x -1 1).
Note that in this example the abscissae of the points defining the
line wen ordered by increasing value, that is. xI<x2. Furthermore.
I<X <X b sothat the
of can be said equal
8=(0.0169-O.Ol50)/(0.02W-0.0l SO)
= 0.00 19/0.0050 = 0.38
and therefore
y = (0.38) (0.0304)+(0.62) (0.0264)~0.0279.
(A'i)
that is, only if the slope of the line defined by (x I,y I) and (xr y 2) is
exactly the same as the slope of the one defined by (x I. y ,) and (x,y ).
Now s u ~ ~ o sthat
e the value of x is known, but not that of y . Solvini fory i i iquation (A.I), the following expression is obtained:
y = V2-y IMX
-X
I)/(x~-xI)+Y
13
8=(central - smallest)/(largest - smallest).
Using this mnemonic notation, equation (A.4) can be transformed to
(A.2)
y = By(largcst)+( I .O-8)y(smallest)
which allows the calculation ofy if (x I,y (xr y 3 and x are known.
However. equation (A.2) is not the simplest equation that one can use
is obtained by reananging its
in calculating A better
terms in the follow in^ wav:
Y = (0-X I)/(x~-xt))y2+
8=(x -x t)/(x2-x 1)
(A.3)
(A.4)
(A.5)
Equations (A.4) and (A.5) suggest a way in which linear interpolabe perfomed in two simple
First, calculate the value of
tion
8. the "interpolation factor". ~ ~ k by
l y using the values of the
observed abscirsre (x-values). Then, use quation (A.4) to calculate
the desired interpolated ordinate, y.
Ther steps are simple enough, but in following them it is important
to match the indices properly. Note that although 8 is obtained by
pivoting on x (this value is the only one repeated in equation (A.S)),
its value (that of 8)is "applied to" or multiplied by y 2 in equation
(A.4). whik y I is multiplied by (1.0 8).
Some of the cases where linear interpolation arises in this Manual
are illustrated in the following examples.
(B.2)
where Y(smallest) docs not mean the smallest y value. but rather, the
Y value associated with the smallest x ; and similarly,y (largest) is the
value associated with the largest x value used.
These mnemonic expressions do not improve the clarity of equations (A.4) and (AS), but they are included here in the hope that they
may allow the user to perform linear interpolation correctly and in as
mechanical a way as posible. They are used in the next example.
" .
(I.O-(x -XI)/(X~-XI))~I.
an expression that can be rewritten as
y = 8y2+(1.0-8)yI
where
(B.1)
C. SECOND
EXAMPLE
Among the West model life tables, the q(2) value 0.0858 is associated with mortality kvel 17 for females. while at mortality level IS the
value of q(2) for females is 0.1 164. The question then is what the
female value of q(2) would be at level 15.36.
Since three diffemnt values for the level are cited. and only two are
given for q(2). the kvels arc taken as abscissae and the q(2) values as
dinates. Then, a d i n g to mnemonic equation (B. I).
8=(15.36-15)/(17-IS)=O.36/2=0.18.
and by equation (B.2)
y = (0.18) (0.0858)+(0.82) (0.1 164)=0.1 109.
-
Note that in the application of equation (8.2). 8 = 0.18 is multiplied
by the value of q(2) associated with the largest level used. namely 17,
and not by the largest q(2) value. Because in this case the largest level
is associated with the smallest value of q(2). the distinction is essential.
B. FIRST
EXAMPLE
The birth rate of a West model stable population at momlity level
19 and growth rate 0.0150 is 0.0264. If the growth rate were 0.0200,
the value of the birth rate would be 0.0304. The question then is what
the binh rate value is for a population with the same mortality level
and a growth rate of 0.0169.
To anSWerlhk question by using linear interpolation.the two points
defininga line
be identikd first. The ~o-ordina~~s
of each point
are the pair of (growth rate. birth rate) values; thus.
-
D. THIRD
EXAMPLE
examinesa
of extrapolation.rather than one of
This
of.rHdoes not ~
interpolation.E
~ takes place
~ when the
~
fall between the values x I and x~ but rather. is outside the interval
defined by them. When this is the case. 8 is either negative or greater
than one. Yet. equations (A.4) and (A.5) still hold truc and may be
used to calculate the missing value y .
Consider then the following probkm. According to the West model
life ubles at level 21 the expectation of life at birth for males is 66.02.
whik at level 23 it is 71.19. The question is what the value ofeowould
be at level 24.
~~
(x y ,)= (0.0150.0.0264)
and
(x2. y 3 = (0.0200.0.0304).
239
~
Once more, the kvcb arr taken to represent the abscissae values;
therrforr,
be greater than that at level 23. This common-sense check would
immediately permit one to discard the value
a vrluc, u expected, greater thm one. In the second step, y is calcul8tCduuRI.l:
that would be obtained if. by mistake the values 8 and 1.0- 8 were
not matched to the right indices.
As a final comment, it should be pointed out that linear extrapolation is not mommended in most demographic applications, since the
assumption of linearity outside the bounds of printed or otherwise
known quantities is difficult to justify. Therefore, in choosing the
values x 1 and xz that will be used as anchors for linear interpolation.
one should select. if at all possible. those which enclose the reported
value of x .
Note that since 8 is water than one, (1.0-8) is negative. Furthermon, because the valuer of life expectancy at binh increase as the
kvel of mortality iacrrwr (that is, the slope of the line associating
kvel with cg v d u e is positive), it is expected that e~ at level 24 would
Annex
V
SMOOTHING OF AN AGE DISTRIBUTION
A. BACKGROUND
OF METHODS
Throughout this Manual, stress has been laid on the fact that age
distributions are frequently distorted by age-misreporting. Although it
is not recommended that age distributions be smootheda or adjusted
prior to applying any of the analytical procedures described in the
main body of this Manual, on the grounds that their results may be distorted by the smoothing or adjustment process, an age distribution that
is smooth and as close to comct as possible is still useful, particularly
as a basis for population projections This annex describes and illustrates some fairly simple procedures for adiusting distorted age distributions; the teciniqub da~scribed
in section B can be applied-to a single age distribution. whereas the procedure described in section C
requires a series of two or more age distributions.
B.
SMOOTHINGAND AWUSTMENT OF A SINGLE
AGE DtSTRtBUTtON
I. Fining of a stabIepopulation
The most drastic form of adjustment for an age distribution is to fit a
stable population to the population being studied. following the procedures described in chapter VII, and to adopt the age distribution of
the stable population as a representation of the true age distribution.
The adopted age distribution will be free of all irngularities, whether
these a n true or the result of error, and will also be free of any feature5
that the actual age distribution does not share with the models. Thus,
the extent of adjustment may be more seven than is actually required.
but the result will at least be internally consistent.
This procedure is not described in detail here, nor is it illustrated.
since the methods for fitting a stable population are covered
thoroughly in chapter VII.
2. RdwtIon of effects ofage-heaping
(a) Basis of method and its m t i o ~ l e
smoothing and adjustment procedures are usually applied to cumulated age distributions (that is, to the number of persons or the proportion o f persons under given ages) since the process of cumulaiion
removes the effects of errors that do not result in a net transfer of people across each of the age boundaries used. If heaping tends to
transfer people whose true ages are both under and over an attractive
age to this age, the use of attractive ages as boundaries for cumulation
will not minimize the effects of heaping on the cumulated age distribution. In such circumstances, the use of age boundaries half-way
between attractive ages would probably minimize net transfers across
boundaries? Therefore, a simple way of smoothing age distributions
would be to fit a succession of polynomials to the reported population.
expressed either as numben of persons or as the proportions of persons
under ages a +3. a +8. a 13 and a + 18. where a is a multiple of five.
+
l The term "to smooth" is used in this Manual in its most general
sense to mean the elimination or minimization of irregularities oRen
present in reported data or in preliminary estimates obtained from
them. In t h ~ ssense. the set of possible "smoothing techniques"
encompasses a wide variety of rocedures. ranging from the fitting of
models to simple avera4ing. #he traditional smoothing techniques
applied to age distributions and to observed age-spec~fic mortality
rates are part of this set, but the do not exhaust it. The somewhat
m u p e r pmcedum d e bbed in !(is M+
are necessafy because the
basr data available a n both defictent and mcomplete.
Ken Hill. Hania Zlotnik and Jane Durch, Proccdumsfor Reducing
the ~ E Iof SAge EnwJ on Indimt Dcmopyhic Estimation T'hniques,
Laboratories for Population S1atisti.m Sc~entificRe rt Series No. 35
(Chapel Hill, North Carolma. Carol~naPopulat~onE n t e r , 1982).
and to compute the values of these polynomials over their central
ranges in order to obtain a smoothed single-year age distribution.
However, the problem of smoothing is somewhat complicated by
the fact that ages ending in zero are generally more attractive than
ages ending in five, so that the cumulated population under ages ending in eight will tend to be too small, whereas the cumulated population under ages ending in three will tend to be too large. This tendency can be allowed for by fitting two polynomials, one to cumulated
populations under ages ending in eight and one to cumulated populations under ages ending in three. The two polynomials would then be
evaluated over a common central range; and best estimates of the true'
cumulated population, relatively free of heaping effects, would be
found by averaging.
(b) &to mpired
The information needed consists of the population classified by sex
and single year of age, preferably up to age 85 or beyond.
(c) Computatio~lpmedure
The steps of tHe computational procedure are given below.
Step 1: rwnulation ofpopulation counts. For each sex, the population
under each age ending in three (3, 13, 23 and so on) and in eight (8,
18. 28 and so on) is required. The single-year age distribution is used
to obtain the required numbers; the population under 3 years is the
sum of persons aged 0. I and 2, the population under age 8 can be
found by adding the numbers aged 5,6 and 7 yean to those aged 0-4.
and so on. Note that there is no need to use proportions; it is simpler
to uw the actual numbers.
Step 2: firring of poly"0mr'als to the cumulated age dstnbution. A
thirddegree polynomial is fitted to the population under ages
a +3, a + 13. a +23 and a +33. where a is an age ending in zero. and
evaluated between a + 13 and a +23; such a polynomial will probably
overestimate slightly the population under each age because of the
heaping bias. A similar polynomial is fitted to the population under
ages a +8. a + 18. a +28 and a +38. and again evaluated between the
ages of a + 13 and a +23; such a polynomial will probably underestimate the population under each age. Therefore, to reduce those
biases, the estimated population under each age within the range
a +I3 and a +23 can be calculated as the average of the estimates
produced by each polynomia1;as long as the gain of each zero is a loss
of a neighbouring five the resulting cumulated age distribution should
be almost free of heaping effects and can be broken up into conventional five-year age groups. The process continues by increasing a by
10. Note that ages under 13. which fortunately do not oRen exhibit
the typical heaping patterns, require special treatment; and that, in
general, the technique can only be applied up to age 72 (assuming that
a single-year age distribution is available up to age 85).
TG Gpula60n under age 13 can be shootied by fitting a thirddegree polynomial to the estimated population under ages 0, 10, 20
and 30. The populations under ages 20 and 30 will already have been
obtained as described above. The population under age 10 can be
estimated from both of the first two polynomials, that beginning from
age 3 and that beginning from age 8, and an average used as an estimate. The population under age 0 is, of course, zem. The polynomial
fitted to these four estimated values can then be calculated between
ages 0 and 13. Note that this procedure is likely to produce some fairly
minor discontinuities in the smoothed single-year age distribution,
though the distribution by five-year age groups should be unaffected.
In order to obtain an age distribution in conventional five-year
groups adjusted for the effects of heaping bias. a short-cut can be used
that avoids calculating the coefliaents for every polynomial and
instead estimates directly the popu 'ion under iwo exact ages. If one
denotes by N(x -) the population under age x , the population under
agea+I2bgivcnby
fi((a
+ I2)-)=
+ lo)-)
-0.032N((a +30) -1,
-0.048N(a -)+0.864N((a
+0.2 16N((a +20)-)
(B.1)
a d the population under age a + 17 is given by
&(a
+ 17)-)=
-0.0455N(a -)+O.UISN((a +lo)-)
+0.7735N((a +M)-)-0.0595N((a
The estimate of the population under age 5 from the adjusted
population under ages 10.20 and W can be obtained using the equation
(8.2)
+30)-).
Thus, if a L equal to 3, the population under age IS will be given by
equation (B.1) and the population under 20 by equation (B.2). If a is
equal to 8, on the other hand, quation (B.1) will give the population
under age 20 and quation (B.2) the population under 25.
F a the apecia1cases at the beginning of the age range. the following
equation can be used to estimate the population under age 10 from
the populations under ages 3.13.23 and 33, and the population under
age IS from the populations under ages 8, 18.28 and 38:
$((a +7)-)=0.1495N(a -)+I.WSN((a +lo)-)
0.3)
-0.2415N((a +20)-)+0.0455N((a +30)-).
The a h a t e of the p o p u m n under age 10 from h e populatiom
under ages 8.18.28 and 38 can be obtained from
-0.224N(28 -)+0.048N(38-).
fi(s -I=
o.~~~~R(Io-)-o.~I~sA(zo-)+o.M~sR(~-).
(B.s)
Then. to educe even more the possible biases caused by ageheaping, "best estimates" of the population~ndereach age x , multipk of five. arc obtained by avenging the N(x -) estimates obtained
above whenever two such estimates arc available for a given x. Thus
the final estimates of the population under age x, N *(x -), arc computed as
where the subindices 3 and 8 i n d i t e that the intermediate A(x -)
wimates arc obtained from ages ending in three and eight, rcrpectively.
M y . for ages x at the beginning and at the end of the age nnge
(age 5. for example) for which only one fi(x -) estimate is availabk.
Nb(x -) b set equal to it.
(d) A &ailadcxa@e
me pmdurc dbcdbCd
ihStnlled bl ap@$n8 it
IhC
age diibution of the male population of Sri Lanka, as reported by
the fertility suwev in Sri Lanka in 197XCTable 203 shows the basic
The computational procedure is described below.
81..............
82.. ............
83..............
84..............
85..............
86..............
87..
............
88..............
89.. ............
90.. ............
91..............
92..............
93..............
94.. ............
95.
.............
............
97.. ;...........
98..............
99.. ............
Not
slated ...
%..
n..............
78..............
Strp I: amkth gptphlkm unuus. The numbers of males under
ages 3.8, 13, 18 and so on are found by cumulating the numbers in
d v e single-yeu age p u p up to, but not including the upper
botmhy age goup. Tbur, if NV) is the population of age y and
N(x T ) b the population under age x,
and
Ma-)= N(3-)+N(3)+N(4)+N(S)+N(6)+N(f)
= lP8+~+5%+637+600+643=4,706.
Full results arc shown in tabk 204; the calculations arc continued up
N(x -)= 5'N(y).
P O
to the population under age 88.
Ministry of Plan Impkmcntation. Depanment of Census and
Stahtk~,Worlrl FdUv -SH
krrkcr 1975. F k I Rqrorr (Cokmbo, 1978).
TABLE^.
Air
(1)
3
8
13
18
23
28
33
38
43
48
53
58
63
68
73'
78
83
88
ADIUSIMENT. FOR HfA?INO.OF THE MALE AOE DISTRIBWTlON. SRl LANKA.1975
s
&
b
Ikn
(1)
(3)
(4)
de*
0
5
10
IS
20
25
30
35
40
45
50
55
1 628
4 706
7 712
10 599
13 098
I5 158
16 733
18 039
19 227
20 220
21 185
21 960
22609
23 090
23 454
23 716
23 830
23 885
60
65
70
75
5 912
8 879
11 618
13 946
IS 790
17 304
18 563
19661
20 654
21 514
22 241
22 82 1
23 255
?$'
5 975
8 956
11 629
13 951
I5 837
17 293
18 523
19 624
20 609
2 1 494
22 23 1
22 809
23 254
23 578
s=
T"
+-'"'a
* r ~ + J p b M
(6)
Y'r
"'r'
(2 928)
5 944
8 918
11 624
I3 949
IS 814
17 299
18 543
19 643
20 632
21 504
22 236
22 815
23 255
23 578
2 928
3 016
2 974
2 706
2 325
1 865
1 485
1 244
1 100
989
872
732
579
440
323
341.
2 826
3 049
3 012
2 760
2 334
1810
1 402
1 326
1 028
1046
881
724
570
459
315
377'
.-
Population aged 75 and over.
Step 2: fiting of pIynmuynmuds
to the cuw~ulotedage chdriburion. Estimates of the population under ages 15.20 and so on. up to age 75 can
be obtained from polynomials fitted to the numbers under ages ending
in three or eight by the use of equations (8.1) and (B.2). Thus. the
population under ages IS and 20 can be found as follows using the
populations under ages ending in three:
Equations (B.1) and (B.2) are then applied to the populations under
ages 10 years older and the process is repeated up to age 75. The full
set of estimates based on ages ending in three is shown in column (4)
of table 204, while column (5) shows the estimates obtained from ages
ending in eight. Estimates of the population under age IS based on
populations under ages ending in eight and of the population under 10
based on populations under ages ending in three arc obtained by
applying quation (B.3):
'
= 8,955.5
and
and the populations under ages 20 and 25 can be estimated from the
populations under ages ending in eight:
= (-0.048)(4.706)+(0.864)(
( -0.032)( 18.039)
10.599)+(0.2 16)(15.1 58)+
The estimate of the population under a g 10 from populations
under ages 8. 18.28 and 38 is obtained by applying equation (8.4).
Once two estimates arc availabk for each of the populations under
ages ending in zero or five, from 10 to 70. a best estimate at each age is
obtained by averaging the two values corresponding to it. Thus, for
example,
Full results arc shown in column (6) of table 204.
The population under age 5 is then found by substituting best estimates of the populations under ages 10.20 and 30 into equation (B.5):
and
The cumulated population estimates can then he successively subf i d 2 ~ - ) = (-0.0455)(4,706)+(0.33I~nlo.~99)+(o.773~MI~~I~8)+
from each other to provide an age distribution in five-yearage
p u p s . the population aged from x to x +4 being ohtitined by sub( -0.0595)( 18,039)
tracting N *(x -) from N *((x +5)-). Thus, for aee pmup 20-24.
FuU results a n shown in column (7) of table 204 and may be compared with the reported age distribution shown in column (8).
3. Conywrson nirh a standordage distribution
(a) lhris of method and its mtiode
In some circumstances
when lhe quality of lhe age data is poor Or
when ages seem to be affected by systematic misreporting other than
heaping, some
procedure more drastic than that
in mbsectionB.2 may be necessary. One approach to such adjustment
age distribution with some
involves the
of the
standard age distribution, followed by the adjustment of the standard
age distribution to reflect the main features of the reported one; h e
result should be an age distribution that retains some of the broad
characteristics of the remrted distribution while at the same time
being free of obvious bia;.
In practice, a stable population derived from a suitable model life
tabk and having a reasonable rate of growth for the application in
hand can be used as the standard ape distribution. ~articularlvif there
is reawn to believe that the actual population is stible or quasi-stable.
The comparison of the reported and model distributions is made easier
by the use of a transformation that linearizes the relationship between
age and the cumulated proportion of the population under each age.
in growing populations, such as those of most developing countries, a
suitable transformation is
where C(x) is the proportion of the population under age x. From
here on, the transformation defined by equation (B.6) is referred to as
the Y-transformation. Values of Y(x) for the population in question
can be plotted against standard values, YS(x), for a suitable standard
stable pbpulation; and deviations from an approximately parabolic
relationship can be noted (in practice, when the data are not
extremely distorted, this relationship deviates only very slightly from
that represented by a straight line). An adjusted age distribution can
be obtained by fitting a curve to the points that are regarded as least
biased and then reversing the transformation process to obtain a
smooth cumulated age distribution. Tests with model populations
have suggested that adequate results can be obtained by fitting a
second-degree polynomial (a parabola) to the selected points, a process that is simplified because one can require the fitted parabola to
pass through the origin (since the population proportion under age
zero has to be zero) and through the means of both the first half and
the second half of the selected points.
The following data are required for this method:
(b) Lkta qdd
(a) The population classified by sex and five-year age group;
( b ) A standard age distribution by five-year age group. A stable
population subject to a growth rate and mortality risks similar to those
of the population being studied is usually a suitable choice.
(c) ~ t a f i o m l p m e d u w
The steps of the computational procedure are described below.
Step I : caiahtion ofproportiom wdcr ages encling in zetv or*.
The
proportion of the population under each age x , denoted by C(x), is
found by summing the populations of each five-year age group below
the age in question and dividing the resulting sum by the total population of known age, that is, excluding those of unstated age.
Step 2: calculation of Y-tmnrfonnation of the rpponed proportions
undwagesfivc)~rors+.
The Y-transformation of each of the C(x)
proportions calculated in the previous step is obtained by using equation (8.6). ,For the sake of completeness, this equation is repeated
here:
Y(x)= In[(l.O+C(x))/(I.O-C(x))].
(B.6)
Note that the Y-transformation of any proportion C(x) assumes
values ranging Rom zero to infinity.
Strp 3: dcction o/a a age distribution. The standard population age distribution. against which the reported population is to be
compared, should be selected from that of stable populations whose
mortality level and pattern approximate as closely as possible those
observed in the population being studied. Among these stable populations, the one having the same growth rate as the reported population is preferable.
S ~ r p4: the Y-tranrfomtlltion of the standonl age disrribution. Steps I
and 2 are repeated for the selected standard age distribution.
Step 5: relation of the Y-tmfonmtiom of the wported and standonl
age distriburions. A second-degree polynomial is assumed to represent
the relationship between the Y-transformations of the reported and
the standard age distributions. To simplify the fitting procedure, the
polynomial selected is that passing through the origin, the mean d a
first group of pinu
and lhe mean of a second group of points, where
Ihe groups
are
size but exclude any extreme values that
fall well away from a likely parabola. Therefore, the fitted equation
Y(x)= ~(YS(X))~+@YS(X),
where a and can be estimated as
and
where ( B 1 , TI) and ( E 2 . vI) are the mean points for the first and
the second groups, respectively. As usual. YS denotes the Ytransformation of the standard.
Step 6: calculation of esrimared age distribution. Having estimated a
and /I
equation
,
(B.7) can be used to obtain adjusted values of the Ytransformation. Yb(x ), which can. in turn, be inverted to produce estimates of an adjusted age distribution. For the sake of completeness.
equation (B. 10) indicates how the adjusted values Y *(x ) are obtained
and equation (B.l I) shows how an estimate of the adjusted Cb(x),the
estimated proportion under age x , is calculated:
Y*(x)= ~(YS(X))~+@YS(X);
(B. lo)
These Cb(x) proportions can be multiplied by the total reported
population to find the estimated number of persons under age x , and
estimates of the population in five-year age groups can then be
obtained by subtraction.
(d) A dcrh'ledexample
Once more, the age distribution of the male population of Sri
Lanka, as recorded by the fertility survey in 1975. is used to illustrate
this method (see table 203).
Step I : calculation of proportiom un&r ages ending in Zen, or fiw.
Column (2) of table 205 shows the proportions under each age x , multiple of five, denoted by C(x). also including in this case that under
age
2: calcvlarion of Y-tmformarion f'
wportted pmprtions
"der ages x fiw )porn apart. The proportions C(x ) calculated in the
previous step are transformed, using equation (8.6). into Y(x) values.
Thus, for age 15.
Y(IS)= In[(l.O+C(15))/(1.0-C(I5))j
= In[(I.O+0.3716)/(l.O-0.3716)j = 0.7806.
The full set of Y(x) values is shown in column (4) of table 205.
Step 3: selection of a standard age distribution. Procedures for fitting a
stable population to the one observed are presented in chapter V11
and need not be described again here. For Sri Lanka, a West model
stable population of mortality level 18, with a growth rate of 0.030.
was selected. The proportions under each age x for this population
are shown in column (3) of table 205. They are referred to below 8s
"standard proportions".
TABLE
205.
ESTIMATIONOF AN ADJUSTED MALE AOE DISTRIBUTIONBY USlNO Y-TRANSFORMATIONSAND THE
WEST MODEL STABLE )OWLATION AS STANDARD.SRI LANK& 1975
Toale-Demeny West model stable population. level 18. r ~0.03.
Figwe 16. Pbt d the Y-tmnsformrtioa of the pmporhm under age x in the rrpatcdpopulation
again8t the Y-tnnsfumution of the equivalent proportiomin the standud. Sri Lank& 1975
Fitted parabola:
~ ' ( x ) = -0.oooosje (Ys(x)jY
0.8775 YS(x)
St? 4: the Y-tmmjionnotionof the stonhd oge distribution. Step 2 is
applied to the standard proportions under age x , with the results
shown in column (5) of table 205.
Step 5: nlorion o f r k Y-tmrY,~rm~tiotts
of the nprted ond s t o d d
ogc &tn'htionr. The [YS(x). Y ( x ) ] points should be examined before
245
+
attempting to tit a parabola to them, so that outliers can be discarded.
Figure 26 diplays a plot of the points, which show no obvious outlien.
Themfore, a panbola passing Ulrough the origin, the mean of the first
nine points and the mean of the second nine points (the ninth point
being included in both groups) is selected as an acceptable fit. Thus
TI= (0.0478 +0.2375 + . - . +2.0620)/9.0=
1.0476.
Equations (B.8) and (B.9) can now be used to estimate the parameters a and /I in equation (B.lO):
and
and
S2=
(2.3033+2.6068+ . . +6.5347)/9.0= 3.9978.
Figure 27. A compdsoo af the results af different d/lrrtnnat proceduresfor
m l e age dbhbutkn, Sri hnlta, 1975
Proportion in
age group from
xtox
+4
r
-
Observed
--- Graduated
-
Reference stable
0 Adjusted for heaping only
b
Step 6: caIcuIation of estimated age distribution. Substituting the
values of a and /3 just obtained in equation (B.IO), we obtained
adjusted values of the Y-transformation. Thus, for age IS,
Column (6) of table 205 shaws all the YL(x) values. Then, the
estimated proportions under each age x are obtained by ?versing the
Y-transformation by means of equation (B.1 I). Again using age I5 as
an example,
Column (7) of table 205 shows all the estimated Ce(x) values.
Adjusted pmportions in each five-year age group can now be obtained
by subtracting from the proportion under age x the proportion under
each age x -5. Thus, the adjusted proportion in age group 15-19.
5C15,
is found as
Figure 27 shows a plot of the adjusted proportions in five-year age
g m u p the comsponding proportions in the observed population, the
standard pmportions and the proportions obtained by applying the
smoothing procedure described in subsection B.2. It is clear that the
male population of Sri Lanka has been drastically destabilized by declining fertility, so that the procedure described in this section cannot
be expected to work well at younger ages; above age 25, however, its
results seem acceptable, and they provide an obviously better approximation to the observed population than the original standard age distribution on which the fitted one is based. The age distribution
obtained when the effects of heaping are removed as described in subsection B.2 is much closer to the reported distribution, damping only
slifitly (and probably, at least, in the right direction) the effects of
destabilization at younger ages and retaining the trough in the
reported age distribution between ages 30 and 45, though smoothing
out its irregularities. In this particular application, where agereporting does not seem too poor and where the reported age distribution is clearly not stable, the smoothed age distribution obtained by
the procedure that merely reduces the effects of age-heaping (subsection B.2) is probably the best approximation available to the true age
distribution.
AND SHORTER
ADJU-MENT
C. DEMENY
I. Basis of method and its rationale
Demeny and ~ h o r t e t proposed a method based on intercensal
probabilities of survival to adjust the age distributions produced by
two cpnsecutive censuses. This method assumes that the probabilities
of intercensal survival are known, that both censuses achieved the
same level of coverage and that both suffered from similar proportionate age-reporting errors. that is, that the recorded population of
age, gmup x is equal to the true population of the age group multiplied by an adjustment factor typical only of the age group and not
changing from census to census. Given two censuses five years apart,
the population aged 0-4 at the time of the first census is assumed to be
w m c t and is projected forward to the second census. An adjustment
factor for age group 5-9 is then obtained as the ratio of the projected to
the reported population aged 5-9 at the second census, and this factor
is then used to adjust the reported population aged 5-9 at the first
census. The adjusted population aged 5-9 at the first census is then
pmjected forward to the second census, and an adjustment factor for
age group 10-14 is found as the ratio of the projected to the actual
population aged 10-14 at the second census. The chaining process
continues from age group to age group until the entire age range has
been covered. Two adjusted age distributions are then available, that
* Paul Demeny and Frederic C. Shorter. Estimating Turkish MortaliIstanbul University. 1968).
The dam required are listed below:
(a) The distribution of the population by age group and sex'from
two consecutive censuses. The method is most readily applied when
the intercensal interval is five years, but it can also be applied to a 10year interval. However, it is difficult and even inadvisable to use data
corresponding to intervals that are not multiples of five, since in such
cases the differential effects of heaping are likely to distort the results
obtained;
(b) A suitable set of survivorship probabilities referring to the intercensal period.
The procedure for this method is described below.
Step 1: synthesizing of the popularion corresponding to a fivr-par infer-
&. If the two censuses were held 10 years apart, the first at time r I
and the second at time 1 % the method is best applied by inventing an
intervening age distribution, thus creating two five-year intervals, and
applying the method twice. The age distribution for the mid-point of
the 10-year interval can be created by finding the annual growth rate
of each age group from x to x + 4. as
and then applying the growth rate for five years to the initial population:
TECHNIQUE
FOR TWO AGE DISTRIBUTIONS
ly, Fe#ilityMdAge Stnetwe,
for the first census being made up of the adjusted populations of each
age group and that for the second census consisting of the projections
of the adjusted population from the first census. The final step is to
scale the two adjusted age distributions to agree with the reported total
population.
The method depends upon the availability of a suitable mortality
schedule in order to carry out the projection, and a mistake in the
specification of mortality will introduce a systematic bias in the results.
The other two assumptions involved, that of no change in enumeration completeness and that of a constant proportional error for a given
age group at both censuses, may also be frequently untenable. A
change in enumeration completeness may result in large distortions of
the entire estimated age distribution if this method is applied, though
such large errors are likely to be obvious. Because of these potential
problems, the estimates yielded by this procedure, though useful, will
have to be carefully scrutinized before being adopted for future use.
Here. @Ix and fl2, denote, respectively. the reported populations
in the age gmup fmm x to x + 4 in the first and second censuses.
Step 2: projection of initid age distribution. The population of age
group 0-4 at the first census is projected forward to age group 5-9 at
the second census, and an adjustment factor for age group 5-9 is
obtained as the ratio of the projected to the reported population aged
5-9 at the time of the second census. Thus. if k ( s ) is the adjustment
factor for the age group from x to x +4,
The projected population aged 5-9 is taken as the adjusted value.
flZ5. at the second census. Thus.
The reported population aged 5-9 at the first census is then adjusted
by k ( 5 ) to obtain the adjusted population aged 5-9 at the first census.
fl Is, and this population is then projected forward to age gmup 10-14
at the second census; thus,
and
Statistics Institute Paper No. 2 (Istanbul.
k(10)=(#la&10/&3/#21,
where the fint two terns m simply the adjusted population aged 1014 81 the accond census, that is,
The procedure continues until the open-ended interval (which
iequires a slightly modified treatment) is reached. Since it is assumed
that the adjustment factor required is the same for both censuses, that
for the open-ended interval, k(A +), is equal to the ratio of the
adjusted to the reported population aged A and over for both age distributions. That is,
However, the adjusted population aged A and over at the second
census is also equal to the sum of two elements, the first being the projection to the time of the second census of the adjusted population
aged from A -5 to A 1 at the first census, and the second being the
survivors to the second census of the adjusted population (not known)
aged A and over at the first census. Thus,
-
where TA+5/TAis the probability that the population aged A and
over has of surviving five years. By combining equations (C.4) and
(C.5). and rearranging terms an expression for k(A +) is obtained:
A part of the example given by &!meny and Shorcer is presented
here to illustrate the calculations. The female population of Turkey as
recorded by the 1955 and 1960 censuses is adjusted below. The basic
data are shown in columns (2) and (3) of table 206 (the 1955 population data have been adjusted in order to allow for the eficts of migration). Column (4) shows survivorship ratios taken from a South model
life table of level 16.3. with an expectation of life at birth of 58.25
years.
Step I: synthesizing of the population for a @-par interval. The
actual intercensal interval is five years, so this step is not required.
Step 2: pjection of initial agckstriktion. he adjusted-population
aged 5-9 in 1960 is taken to be the 1955 population aged 0-4 projected
five ycars. Thus, if one denotes by $3, the ratio &, +5/&x,
The adjustment factor for age group 5-9, k(5). is then calculated as the
ratio of the adjusted to the reported population aged 5-9 in 1960.
Thus,
k(S)= #25/flz5= 1,813/1.928=0.940.
The adjustment factor, k(5). is then applied to the 1955 population
aged 5-9:
d l 5 = k(5) fl 15= (0.940)(1.570)= 1,476.
Then, the adjusted 1955 population aged 5-9 is projected forward to
1960 to obtain the adjusted 1960 population aged 10-14:
Since all the variables in equation (C.6) are known, the value of
&(A +) may be readily obtained.
Step 3: s d n g of @wedage distributions. The two age distributions
derived in step 2 are based on the provisional assumption that the
youngest age group is accurately recorded by both censuses. If this
assumption is not valid, as will frequently be the case, both the
adjusted age distributions will have been scaled according to the ratios
of the reported to the true populations aged 0-4. while being
unaffected in age pattern. The assumption that age group 0-4 is
comctly reported is not necessary if one scales the adjusted population
age 'stributions by the ratio of the total reported population to the
total adjusted population at each point in time. The scaling factors are
unlikely to be exactly the same for both age distributions, but they
should be very similar.
The procedure continues until one reaches the open interval, in this
ca-x 75 and over. The last normal step completed is the calculation of
film The adjustment factor for the population aged 75 and over is
found by applying equation (C.6):
TWO CONSECUTIVE ENUMERATIONS OF THE FEMALE POPULATION USING COHORT SURVIVAL. TURKEY.
1955 AND I%O
TABLE
206. ADJUS~MENTOF
(Population in rhommk)
04....,................................
5-9.....................................
10-14...................................
15-19...................................
20-24.. .................................
25-29...................................
30-34.. .................................
35-39...................................
4044...................................
45-49 ...................................
50-54.. ................................
55-59...................................
6064...................................
6569...................................
70-74.. .................................
75 + ....................................
TOTAL
1 889.0
1 570.0
1097.0
1 094.0
1 117.0
1 027.0
735.1
528.5
624.7
475.6
531.0
310.4
389.3
173.7
172.1
159.3
1 1 893.7'
2 079.0
1 928.0
1489.0
I 059.0
1 128.0
1 177.0
985.8
693.8
548.4
500.0
58 1.2
369.4
461.4
213.8
195.7
182.1
13 591.6'
'The slight differencesobserved are due to rounding.
0.9599
0.9903
0.9907
0.9873
0.9846
0.9830
0.98 13
0.9783
0.974 1
0.9660
0.9523
0.9272
0.8829
0.8085
0.6936
0.4414
(2 079.0)
1813.0
1 462.0
1 067.0
1 089.0
1061.0
910.7
666.5
496.9
55 1.3
506.8
440.9
343.6
256.1
168.3
167.2
13 079.3
0.940
0.982
1.008
0.965
0.902
0.924
0.96 1
0.906
1.103
0.872
1.194
0.745
1.198
0.860
0.9 18
(1 889.0)
1 476.0
1 077.0
1 103.0
1 078.0
926.4
679.2
507.9
566.0
524.6
463.0
370.6
290.0
208.1
148.0
146.2
1 1 453.0
1 962.0
1 533.0
I 118.0
1 145.0
1 119.0
%2.0
705.3
527.4
587.7
544.8
480.8
384.8
301.1
216.1
153.7
151.8
11 892.5'
2 160.0
1 884.0
1 519.0
1 109.0
1 132.0
I 102.0
946.3
692.6
516.3
572.9
526.6
458.1
357.0
266.1
174.9
173.6
13 590.4'
where fin+ is equal to Tso/T* The adjusted populations in the
open interval am then obtained by multiplying the recorded populations aged 75 by k (75 ), so
+
+
and
The first adjustment of the 1960 age distribution is shown in column
(5) in table 206, the adjustment factom am shown in column (6). and
the fin1 adjustment of the 1955 age distribution is shown in column
(7).
Step 3: scaling of Jusfed oge distributiom. I t will be seen that the
sum of column (7) in table 206, representing the total adjusted 1955
population, is not equal to the sum of column (2). repenting the
total mported 1955 population. This difference is an indication of the
actual error in the population aged 0-4 in 1975 in relation to the
overall level of enumeration. Therefore. the assumption concerning
the completeness of enumeration of the population aged 0-4 can be
dropped simply by multiplying the adjusted population of each age
group by the ratio of the total reported population to the total adjusted
population. Thus, for example, the final adjusted population aged 0-4
in 1955 is equal to the first adjusted value given in column (7) oftable
206 multiplied by 11.893.7/11.453.0. Columns (8) and (9) show the
final adjusted populations for 1955 and 1960. respectively.
ADULT SURVIVORSHIP RATIOS FOR COALE-DEMENY MODEL LIFE TABLES
FOR USE WITH ADULT MORTALITY ESTIMATES FROM ORPHANHOOD
.
.
.
.
TABLE
207. FEMALE
ADULT SURVIVORSHIP PROBABILII'IES FROM A G E ~ SI(N )/1(25)
FOR MATERNAL ORPHANHOOD ANALYSIS, NORTHMODEL
TABLE
208. FEMALE
ADULT SURVIVORSH~P fROBABILITIES PROM AGE 25 I(N)N(25)
FOR MATERNAL O W H A W O O D ANALYSIS, SOUTH MODEL
.
TABLE
209
FEMALE
ADULT SURVIVORSHIPPROBABILITIES FROM AGE 25. I(N)/1(25).
.
FOR MATERNAL ORPHANHOODANALYSIS EAST MODEL
.
TABLE~IO
FEMALE
ADULTSURVlVORSHlPPROBAILITIES
FROM ~ 0 ~ 2 5 . l ( N ) ! l ( 2 5 )
FOR MATERNAL ORPHANHOOD ANALYSI S. WESTMODEL
.
/(N
TABLE
2 1 1. MALEADULT SURVIVORSHIP PROBABILITIES FROM A G E ~ ~ . ~ , )/1(32.5).
FOR PATERNAL ORPHANHOOD ANALYSIS NORTHMODEL
.
TABLE
2 12. MALEADULT SURVIVORSHIP PROBABILITIES FROM AGE 32.5. I(N)/1(32.5).
FOR PATERNAL ORPHANHOOD ANALYSIS SOUTH
MODEL
.
TABLE
215
.
MALE
ADULT SURVIVORSHIPPROBABILITIES FROM AGE 37.5.
.
I(N)//(37.5).
FOR PATERNAL ORPHANHOODANALYSIS NORTH MODEL
TABLE
2 16. MALE
ADULT SURVIVORSHIP PROBABILlTlES FROM ~ G E 3 7 . 5 .I(N)/1(37.5).
.
POll PATERNAL ORPHANHOODANALYSIS SOUTH MODEL
TABLE
217
.
MALE
ADULT SURVIVORSHIP F+ROBABILI'TlES
FROM AGE 37.5, I(N)/1(37.5).
MODEL
.EAST
FOR PATERNAL ORPHANHOOD ANALYSIS
,
TABLE^^^. MALEADULT SURVIVORSHIP PROBABIL~IESFROM ~ G E 3 7 . 5 I(N)/1(37.5).
.WESTMODEL
FOR PATERNAL ORPHANHOOD ANALYSIS
ADULT SURVIVORSHIP RATIOS FROM COALE-DEMENY MODEL LIFE TABLES
FOR USE WITH ADULT MORTALITY ESTIMATES FROM WIDOWHOOD
.
.
.
.
ADULT SURVIVORSHIP PROBABILITIES FROM AGE 20 I(N ) / / ( 2 0 ) FOR ANALYSIS BASED ON
TABLE
2 19. FEMALE
SURVIVAL OF FIRST WIFE NORTHMODEL
.
TABLE
220 .
FEMALE ADULT SURVIVORSHIP PROBABILITIES FROM AGE 20 I(N ) / / ( 2 0 ) FOR ANALYSIS BASED ON
SURVIVAL OF FIRST WIFE SOUTH MODEL
.
TABLE
22 1
.
FEMALE
ADULT SURVIVORSHIP PROBABILITIES FROM AGE 20. I(N )/1(20).FOR ANALYSIS BASED ON
SURVIVAL OF FIRST WIFE EAST
MODEL
.
FEMALE
ADULT SURVIVORSHIP
TABLE
222
.
PROBABILITIES FROM AGE 20. I(N )N(2O).FOR ANALYSIS BASED ON
SURVIVAL OF FIRST WIFE WEST MODEL
.
TABLE 223
.
.
.
.
.
MALE
A M I L T SURVIVORSHIP PROBABILITIES FROM AGE 20 I(N )/1(20)FOR ANALYSIS BASEDON
SURVlVAL OF FIRS HUSBAND NORTH
MODEL
.
TABLE
2 2 4 . MALE
ADULT SURVIVORSHIP PROBABILITIES FROM AGE 20 I(N )//(H))
FOR ANALYSIS BASED ON
.
SURVIVAL OF FIRST HUSBAND SOUTH MODEL
.
.
TABLE^^. MALE
ADULT SURVIVORSHIP PROBABILITIES FROM AGE 2 0 I ( N )/1(20)
SURVIVASOF FIRST HUSBAND EAST
MODEL
.
.
.
FOR ANALYSIS BASED ON
T M L E ~ MALE
~ ~ ADULT SURVIVORSHIP PROBABILIT1ES FROM AGE^^, I ( N ) / 1 ( 2 0 ) FOR ANALYSIS BASED ON
SURVIVAL OF FIRST HUSBAND. WEST
MODEL
VALUES OF PROBABILITY OF SURVIVING FROM BIRTH. FROM COALE-DEMENY
MODEL LIFE TABLES FOR FEMALES, MALES AND BOTH SEXES COMBINED
T u ~ e 2 2.7 FEMALE HOBABILITYOFSURVIVINC
.N ~ T H
FROM BIRTH. I(x )
MODEL
TABLE^^^. MALEPROBABILITY O F SURVIVING FROM BIRTH. /(x ). NORTHMODEL
.
.
L ~SURVIVING FROM BIRTH ! ( x ). FOR BOTH SEXES COMBINED NORTH MODEL
TABLE
229. P R O B A B IOF
Note: Values of I(x) based on a sex ratio at birth of 105 males per 100 females.
T-230
.
F e w PROIMILI'W OF SURVIVJNOFROM IIRTH./(x
.
) SOU'TH MODEL
TABLE
'23 1 . MALEPROBABILITY OF SURVIVING FROM BIRTH. I ( x ) . SOUTH MODEL
hhhtticy o / r r r l h~ h~I(x)
1(1)
10)
1(3)
1(4)
r(5)
I(IO)
0.66445
0.68878
0.71075
0.73076
0.74911
0.76604
0.78172
0.79632
0.80996
0.82248
0.83358
0.84441
0.85498
0.86525
0.87522
0.88487
0.89419
0.90324
0.91382
0.92390
0.93395
0.94395
0.95383
0.96352
0.53955
0.57082
0.59976
0.62670
0.65187
0.67551
0.69776
0.71878
0.73868
0.75682
0.77468
0.79188
0.80845
0.82442
0.83978
0.85455
0.86875
0.88267
0.89888
0.91265
0.92580
0.93833
0.95020
0.96137
0.48194
0.51641
0.54857
0.57869
0.60702
0.63375
0.65903
0.6830I
0.70580
0.72654
0.74751
0.76765
0.78700
0.80558
0.82343
0.84056
0.85701
0.87307
0.89165
0.90700
0.92155
0.93528
0.94816
0.96012
0.45 107
0.48725
0.52113
0.55296
0.58298
0.61 137
0.63828
0.66384
0.68818
0.71030
0.73294
0.75466
0.77549
0.79548
0.81467
0.83307
0.85073
0.86788
0.88762
0.90377
0.91906
0.93345
0.94690
0.95933
0.43401
0.47115
0.50598
0.53876
0.56971
0.59901
0.62681
0.65326
0.67845
0.70134
0.72490
0.74749
0.76914
0.78991
0.80983
0.82893
0.84725
0.86498
0.88528
0.90183
0.91752
0.93230
0.94609
0.95882
0.39926
0.43683
0.47244
0.50626
0.53846
0.56917
0.59852
0.62661
0.65353
0.67848
0.70400
0.72855
0.75214
0.77482
0.79662
0.81758
0.83772
0.85725
0.87921
0.89709
0.913%
0.92976
0.94439
0.95777
~15)
0.38400
0.42157
0.45733
0.49143
0.52401
0.55520
0.58509
0.61378
0.64135
0.66701
0.69325
0.71853
0.74289
0.76636
0.78896
0.8 1072
0.83167
0.85202
0.87480
0.89345
0.91104
0.92749
0.94273
0.95664
I(#))
0.36199
0.39948
0.43539
0.46984
0.50293
0.53475
0.56540
0.59493
0.62341
0.64993
0.67724
0.70363
0.72914
0.75379
0.77758
0.80054
0.82270
0.84426
0.86836
0.88814
0.90680
0.92425
0.94038
0.95506
TABLE
232.
.
M
..............
1
2..............
3..............
l
y qlunlvlngfmr YrIC I(xJ
0.27377
024799
0.22244
0.19706
0.16869
0.31028
0.28348
0.25660
0.22953
0.19887
0.34631
0.31884
0.29097
0.26255
0.22993
4..............
0.38181
0.32542
0.35398
0.29595
0.26170
5..............
0.41675
0.35987
0.38883
0.32961
0.29404
6..............
0.4545111
0.42334
0.39421
0.36344
0.32684
7
0.48486
0.45746
0.42841
0.39734
0.35997
8..............
0.51800
0.49116
0.46238
0.43124
0.39337
9..............
0.55052
0.52442
0.49610
0.46508
0.42693
I0
0.58162
0.55625
0.52845
0.49764
0.45946
11 ..............
0.61363
0.58916
0.56195
0.53140
0.49310
12.
0.64514
0.62183
0.59548
0.56543
0.52123
13..............
0.67591
0.65389
0.62858
0.59923
0.56140
14..............
0.70593
0.68531
0.66118
0.63274
0.59552
I5..............
0.73519
0.71606
0.69325
0.66588
0.62950
16
0.76366
0.7461 1
0.72475
0.69859
0.66326
17
0.79136
0.77545
0.75563
0.73082
0.69674
18
0.81849
0.80429
0.78610
0.76278
0.73005
19
0.84684
0.83463
0.81845
0.79701
0.76608
20
0.871 16
0.86081
0.84666
0.82734
0.79876
21
0.89391
0.88538
0.87331
0.85628
0.83037
22
0.91498
0.90820
0.89824
0.88364
0.86068
23
0.93420
0.92906
0.92117
0.90910
0.88938
24
0.95138
0.94773
0.94183
0.93232
'0.91606
Note: Values of I(x) based on a sex ratio at birth of I05 males per 100 females.
..............
..............
.............
..............
..............
..............
..............
..............
..............
..............
..............
..............
.
PROBABILITY OF SURVIVING FROM BIRTH I(x
). FOR BOTH SEXES COMBINED SOUTH MODEL
TABLE
233. FEMALE
mOBABILIN OF SURVIVINQ FROM BIRTH.I(X). EASTMODEL
.
TMLE235
.........,...
..............
..............
4..............
5..............
6..............
7..............
8..............
9..............
10..............
11..............
12..............
13..............
I4..............
I5..............
16..............
17..............
18..............
19..............
20..............
21..............
22..............
23..............
24..............
1
2
3
.
.
PROBABILITYOF SURVlVlNO FROM BIRTH. /(x ) FOR BOTH SEXES COMBINED EASTMODEL
0.26680
0.21347
0.24029
0.18643
0.15734
0.30420
0.27637
0.24787
0.21871
0.18684
0.34124
0.31248
0.28268
0.25174
0.21742
037783
0.34848
0.28537
0.31773
0.24894
0.41395
0.38431
0.35293
0.31945
0.28123
0.44955
0.41989
0.38818
0.35389
0.31418
0.48460
0.45518
0.42341
0.38857
0.34768
0.51909
0.49014
0.45855
0.42343
0.38161
0.55302
0.52472
0.49354
0.45837
0.41590
0.58636
0.55891
0.52833
0.49334
0.45047
0.61956
0.59301
0.56312
0.52837
0.48515
0.65209
0.62653
0.59735
0.56282
0.51922
0.68404
0.65964
0.63140
0.59736
0.55368
0.71535
0.69229
0.66519
0.63189
0.58843
0.74597
0.72441
0.69865
0.66634
0.62338
0.77589
0.75594
0.73171
0.70059
0.65844
0.80508
0.78687
0.76431
0.73462
0.69353
0.83351
0.79638
0.76829
0.81712
0.72851
0.86115
0.84666
0.82784
0.80153
0.76329
0.88800
0.85867
0.83427
0.79777
0.87547
0.91277
0.90222
0.88758
0.86549
0.83138
0.93485
0.92648
0.91443
0.89536
0.86473
0.95444
0.94819
0.93879
0.92305
0.89650
0.97100
0.96673
0.95995
0.94776
0.92588
No&: Values of I(%)based on a sex ratio at birth of 105 males per 100 females.
0.12461
0.15043
0.17766
0.20615
0.23575
0.26633
0.29776
0.32995
0.36281
0.3%22
0.42986
0.46284
0.49655
0.53090
0.56581
0.60117
0.63691
0.67287
0.70893
0.74496
0.78103
0.81828
0.85493
0.89019
.
TABLE 236
FEMALE
PROBABILITY OF SURVIVING PROM BIRTH./(x),
WEST MODEL
. .
) WEST MODEL
TmLe 237. MALEPROBABILITY OF SURVIVING FROM BIRTH I(x
VALUES OF PROBABILITY OF SURVIVING FOR BOTH SEXES COMBINED. FOR AGES UP TO 15.
FROM COALE-DEMENY MODEL LIFE TABLES USING DIFFERENT SEX RATIOS AT BIRTH
TABLE
239.
. .
NORTH MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH [ ( x )
FOR AOES UP TO I 5 BOlll SEXES COMBINED: SEX RAT10 AT BIRTH OF 1.02
.
.
MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH.I(x )
TABLE
240. NORTH
.
FOR AGES UP TO I 5 BOlllSEXES COMBINED: SEX RATIO AT BIRTH OF 1.03
.
. .
TwLe 24 1 NonmMODEL VALUES FOR PROBABILITY OF SURVlVINO M O M BIRTH /(x )
FOR AGES UP TO I5 BOTH SEXES COMBJNED: SEX RATIO AT BIRTH OF 1 04
.
TABLE
242
.
.
NORTH
MODEL VALUES FOR PROBABILITYOF SURVlVlNO FROM IRTH./(x).
FOR AGES UPTO IS. BOTH SEXES COMBINED: SEX RATIO AT BIRTHOF 1.05
.
T~B1r243
.
NORTHMODEL VALUES FOR PROBABILITY OF SURVlVlNO FROM BIRTH.I(x )
.
FOR AGES UPTO I5 BUl'H SEXES COMBINED: SEX RATIO AT BIRTH OF 1.06
TABLE
244.
.
NORTH MODEL VALUES FOR PROBMILITY OF SURVIVING FROM BIRTH [(x ).
FOR AGES UP TO I 5 BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.07
.
.
.
TABLE
245 SOUTH MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH.~(x)
FOR AOES UP TO 15.BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.02
TA~L
246
E . SOUTH
.
.
MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH /(x )
FOR AGES UPTO I5 BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.03
.
.
TABLF.247. SOUl'H MODEL VALUES FOR PROBABILITYOF SURVlVlNO FROM BIRTH [ ( x ).
FOR AGES UPTO IS. BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.04
TML~248.~OU'l'HMODEL VALUES FOR PROBABILITY OF SURVlVlNO FROM BIRTH./(x).
POR AOES UPTO IS.
SEXES COMBINED: SEX RATIO AT BIRTH OF 1.05
.
.
. .
T m L e 249 SOUTH MODEL VALUES FOR PROBABILI7YOF SURVIVING FROM BIRTH /(x )
BOTli SEXESCOMBINED: SEX RATIO AT BIRTH OF 1.06
FOR AOES UPtO
.
TABLE
250. SOUTH MODEL VALUES FOR PROBbBILIW OF SURVIVING FROM BIRTH./(x )
FOR AGES UPTO 15. BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.07
. .
TABLE
251 . EAST MODEL VALUES FOR FUOBABILITYOF SURVIVING FROM BIRTH / ( x )
.
FOR AGES UPTO I5 KllN SEXESCOMBINED: SEX RATIO AT BIRTH OF 1.02
TABLE
252.
.
EAST MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH /(x ).
FOR AGES UPTO IS. BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.03
.
.
TABLEZS~ EAST' MODEL VALUES FOR PROBABILITYOF SURVIVING PROM BIRTH.& )
FOR AOES UPTO 15. DOTH SEXES COMBINED: SEX MTlO AT BIRTH OF I 04
.
Rdblunty tfanl*
~d
1 ..............
2..............
3..............
4..............
5..............
6..............
7..............
8..............
9..............
10..............
1 1 ..............
12..............
13..............
14..............
I5..............
16..............
17..............
18..............
I9..............
20..............
21..............
22..............
23..............
24..............
/(I)
10)
10)
0.53278
0.57040
0.60470
0.63618
0.66525
0.69221
0.71734
0.74085
0.76290
0.78366
0.80307
0.82100
0.83828
0.85491
0.87089
0.88621
0.90089
0.91491
0.92827
0.9410d
0.95371
0.96514
0.97541
0.98424
0.46330
0.50254
0.53907
0.57322
0.60530
0.63552
0.66409
0.69115
0.71685
0.74130
0.76508
0.78725
0.80857
0.82902
0.84865
0.86746
0.88558
0.90299
0.91%3
0.93538
0.94997
0.96282
0.97412
0.98363
0.43407
0.47400
0.51146
0.54675
0.58010
0.61169
0.64169
0.67026
0.69749
0.72350
0.7491 1
0.77307
0.79607
0.81814
0.83931
0.85957
0.87909
0.89783
0.91575
0.93276
0.94816
0.96165
0.97345
0.98330
~(r)
0.41519
0.45555
0.49363
0.52965
0.56381
0.59630
0.62724
0.65676
0.68498
0.71200
0.73881
0.76392
0.78801
0.81112
0.83327
0.85449
0.87488
0.89443
0.91312
0.93095
0.94687
0.96081
0.972%
0.98306
lyx)
45)
/(lo)
0.40176
0.44245
0.48095
0.51749
0.55224
0.58534
0.61695
0.64716
0.67609
0.70382
0.73146
0.75739
0.78227
0.80613
0.82898
0.85086
0.87184
0.89193
0.91 115
0.92955
0.94588
0.96015
0.97256
0.98286
0.37321
0.41372
0.45243
0.48946
0.52495
0.55900
0.59170
0.62316
0.65345
0.68264
0.71201
0.73964
0.76622
0.79178
0.81634
0.83992
0.86257
0.88431
0.90516
0.92517
0.94261
0.95792
0.97118
0.98209
. .
TABLE
254. EASTMODEL VALUES FOR PROBABILITYOF SURVIVINO FROM BIRTH / ( x )
FOR AOES UP TO I5 BOTH SWESCOMBINED: SEX RATIO AT BIRTH OF 1.05
.
0.35953
0.39983
0.43849
0.47564
0.51 135
0.54574
0.57887
0.61084
0.64168
0.67147
0.70144
0.72978
0.75711
0.78345
0.80879
0.83318
0.85665
0.87920
0.90087
0.92170
0.93991
0.95591
0.96978
0.98122
TABU!
255.
.
.
.
.
EAST MODEL VALUES FOR PROIMILIIY OF SURVIVINO FROM YRTH. /(x )
FOR AOES UPrO I5 BOTHSWSCQUBINED: SF% RATIO AT BIRTH OF 1.06
. .
TABLE 256 EAST MODEL VALUES FOR PROBABILITYOF SURVIVINO FROM BIRTH /(x )
FOR AOEd UP TO I 5 W l H SEXES COMBINED: SEX RATIO AT BIRTH OF 1.07
TABLE
257.
WEST MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH
FOR AGES UP IXI I5 BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.02.
.
TABLE
258
.
.[(x ).
WEST MODEL VALUES FOR PROBABILITYOF SURVIVING FROM BIRTH.I(x).
FOR AGES UPTO I5 mSEXESCOMBINED: SEX MITO AT BIRTH OF 1.03
.
. .
TABLE
259. WESTMODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH /(x )
FOR AGES
ur TO 15.m~ e x COMBINED
w
.SEX MTIO
AT BIRTH OF 1.04
f~ ~ ~. 2W ~6sMODEL
r0
VALUES FOR C l l O I M l L l l Y OF SURVIVING FROM BIRTH. / ( x ).
FORAGESUPTO I5 ~1)(SW(ESCOMUNED:SEXMTlOATBlltmoF1.05
.
.
.
.
.
TABLE
261. WEST MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH / ( x )
FOR AGES UPTO I S BOTH SEXES COMBINED: SEX RATIO AT BIRTH OF 1.06
.
.
TmLE262
WEST MODEL VALUES FOR PROBABILITY OF SURVIVING FROM BIRTH l ( x )
FOR AGES UP TO I 5 BOTH SEXESCOMBINED: SEX RATIO AT BIRTH OF 1.07
.
Annex X
FIVEYEAR AND TEN-YEAR SURVIVORSHIP PROBABILITIES
FROM COALE-DEMENY MODEL LIFE TABLES'
FIVE-YEAR SURVIVORSHIP PROBABILITIES fix
TABLE
263. FEMALE
Value listed for age 75 + is T(80)/T(75).
+,.NORTH
MODEL
.
FIVE-YEAR SURVIVORSHIP PROBABILITIES SSx. +4
TABLE
264. FEMALE
~
*f
0..............
5..............
10..............
IS..............
20..............
25..............
30..............
35..............
40..............
..............
..............
..............
..............
..............
..............
........
45
50
55
60
65
70
IS+'
-
~ x +S4 . x
/ w r. n v l r r r l :
J
6
7
8
0.73046
0.92985
0.94323
0.92649
0.9 1544
0.9 1064
0.90656
0.90368
0.90076
0.88425
0.84772
0.77684
0.66779
0.52979
0.36504
0.22 194
0.75838
0.93648
0.94836
0.93305
0.92289
0.91846
0.91469
0.91181
0.90874
0.893 16
0.859 10
0.79309
0.69096
0.55946
0.40003
0.24664
0.78287
0.94245
0.95297
0.93895
0.92%5
0.92555
0.92200
0.91915
0.91599
0.90122
0.86940
3.80780
0.71 1%
0.5865 1
0.43 189
0.26706
0.8046 1
0.94788
0.95720
0.94436
0.93580
0.93 199
0.92867
0.92583
0.92257
0.90856
0.87877
0.82124
0.73 1 16
0.61 131
0.46113
0.28474
0.82407
0.95284
0.96106
0.94930
0.94144
0.93791
0.93479
0.93 195
0.92860
0.91530
0.88740
0.83360
0.74886
0.63422
0.488 13
0.30058
0.84166
0.95741
0.96465
0.95386
0.94663
0.94336
0.94044
0.93760
0.93416
0.92 154
0.89535
0.84503
0.76523
0.65542
0.51319
0.31518
0.85765
0.96 164
0.96795
0.95810
0.95 145
0.94842
0.94567
0.94283
0.9393 1
0.92730
0.90274
0.85566
0.78045
0.675 14
0.53646
0.32887
0.87228
0.96556
0.97104
0.96204
0.95594
0.95312
0.950U
0.9477 1
0.94411
0.93266
0.90962
0.86558
0.79465
0.69358
0.55826
0.34197
11
R
I3
I4
IS
16
10
0.89862
0.97293
0.97648
0.9689 1
0.96379
0.96142
0.95918
0.95635
0.95265
0.94236
0.922 11
0.88367
0.82056
0.72684
0.59736
0.36690
17
18
..............
..............
..............
..............
..............
..............
30..............
35..............
40..............
45..............
50..............
55..............
0.96520
0.99 174
0.99195
0.98887
0.98644
0.98493
0.98323
0.98032
0.976 11
0.96834
0.95522
0.93 149
0.88924
0.8 1692
0.70368
0.45024
0.97244
0.99369
0.99362
0.99 103
0.98892
0.98756
0.98597
0.983 11
0.97894
0.97 148
0.95925
0.93729
0.89768
0.82858
0.7 1791
0.462%
'........
.
4
9
..............
..............
..............
y
3
0.88573
0.96922
0.97392
0.96572
0.960 13
0.95752
0.95508
0.95227
0.9486 1
0.93768
0.91604
0.87486
0.80796
0.7 1088
0.57870
0.35468
60
65
70
75+
f
2
0..............
5..............
10..............
IS..............
20
25
30
35
40
45
50
55
0
5
10
I5
20
25
d
SOUTH MODEL
I
.
.
.
..............
..............
..............
..............
..............
..............
..............
..............
60..............
65..............
70..............
75+ '........
~
.
'Value Sird for age 75 + is T(80)/T(75) .
FIVE-YEAR SURVIVORSHIP PROBABILITIES fix
. ++ EASTMODEL
TABLE
265. FEMALE
0.............
5.............
10.............
IS.............
20............
25............
30............
35............
40............
45...........
M...........
55.........
60.............
65..............
70.............
75
+'........
---.
0..............
5..............
10..............
Ik .............
20..............
25..............
30..............
35..............
40.............
45..............
50..............
55..............
60..............
65..............
70..............
75+' ........
0.91 146
0.973 15
0.97697
0.96935
0.962 19
0.95686
0.95223
0.94801
0.94262
0.92946
0.90237
0.85687
0.78778
0.69237
0.57025
0.35 131
I7
0.97458
0.99158
0.99185
0.98866
0.98587
0.98360
0.98082
0.977 16
0.97155
0.96144
0.94426
0.91506
0.86601
0.78840
0.67674
0.43321
0.92099
0.97587
0.979 16
0.97220
0.96570
0.96084
0.95653
0.95245
0.94707
0.93442
0.90889
0.86594
0.80003
0.70750
0.587 15
0.362 13
-
Id
0.98040
0.99325
0.99327
0.99054
0.98820
0.98626
0.98370
0.98018
0.97467
0.96499
0.94897
0.92 168
0.87506
0.79974
0.68958
0.44484
'Value listed for age 75 t IS T(80)/T(75).
-
T m e 2 6 7. MALE FIVE-YEAR SURVIVORSHIPPROBABILITIES.
++ NORTH MODEL
B msSx . + 4 . * d r l ~ :
A"
'Vllue listed for age 75+ is T(80)/T(75).
.
TABLE
268. MALE
FIVE-YEAR SURVIVORSHIP PROBABILITIES sSx.
d r y lcrcl:
s l m l ~ ~ U 1 5sx.
y . +*for
Ag
..............
0
5..............
10..............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70..............
75+' ........
.
+4 S O U T H MODEL
I
2
3
4
J
6
7
8
0.74480
0.93998
0.95242
0.92878
0.9 1397
0.91454
0.91 145
0.89969
0.88330
0.86009
0.82286
0.71024
0.66690
0.54325
0.39 159
0.24354
0.77054
0.94540
0.95649
0.9348 1
0.92128
0.92 167
0.91858
0.90751
0.89 199
0.86983
0.83437
0.77501
0.68640
0.56750
0.4 1948
0.26106
0.793 14
0.95028
0.960 15
0.94027
0.92790
0.92809
0.92499
0.9 1459
0.89984
0.87865
0.84480
0.78838
0.70406
0.58956
0.44475
0.27599
0.81319
0.9547 1
0.96350
0.94524
0.93392
0.93395
0.93086
0.92 103
0.90699
0.88669
0.8543 1
0.80060
0.72019
0.60969
0.46792
0.28932
0.831 17
0.95876
0.96656
0.9498 1
0.93945
0.9393 1
0.93623
0.92695
0.9 1354
0.89405
0.86305
0.8 1 185
0.73502
0.62826
0.4892 1
0.30131
0.84739
0.96250
0.96939
0.95402
0.94455
0.94427
0.94 1 18
0.93240
0.9 1959
0.90086
0.871 11
0.8222 1
0.74875
0.64544
0.50896
0.3 1253
0.862 16
0.96595
0.97202
0.95793
0.94926
0.94885
0.94578
0.93745
0.92520
0.907 16
0.87858
0.83 183
0.76151
0.66 145
0.52727
0.32304
0.87567
0.969 15
0.97446
0.96 I56
0.95366
0.95313
0.95005
0.942 16
0.93043
0.9 1304
0.88555
0.84080
0.77341
0.67637
0.54438
0.333 11
9
I0
I1
I2
I3
I4
I5
16
'Value listed for age 75 + is T(80)/T(75).
+*
TABLE
269. MALEFIVE-YEAR SURVIVORSHIP PROBABILITIE$ PxJ EASTMODEL
"f
-laVrm.
5SXd +4./-nrmYly**l:
I
2
3
4
J
6
7
8
0.78453
0.9485 I
0.95982
0.94076
0.92848
0.92330
0.91007
0.89056
0.86849
0.843 12
0.80825
0.7547 1
0.67179
0.55932
0.42603
0.26696
0.80717
0.95289
0.96295
0.94530
0.93399
0.92925
0.9 1703
0.89887
0.87806
0.85361
0.8 1970
0.76789
0.68800
0.578%
0.447%
0.27948
0.82685
0.95687
0.96581
0.94943
0.93902
0.93467
0.92339
0.90649
0.88676
0.86313
0.83015
0.78000
0.70285
0.5%93
0.46815
0.29070
0.84420
0.96051
0.96844
0.95324
0.94364
0.93%4
0.92922
0.9 1347
0.89476
0.87189
0.83977
0.791 10
0.7 1649
0.61348
0.48666
0.30090
0.85969
0.96387
0.97085
0.95676
0.94791
0.94424
0.93461
0.91992
0.902 16
0.87996
0.84862
0.80137
0.72913
0.62878
0.50378
0.31034
0.87364
0.96698
0.97310
0.%002
0.95 187
0.94851
0.93960
0.92590
0.90902
0.88749
0.85686
0.81091
0.74089
0.64300
0.5 1976
0.3 1932
0.88630
0.96987
0.97520
0.96306
0.95556
0.95248
0.94428
0.93149
0.91540
0.89448
0.86454
0.8 1980
0.75 186
0.65630
0.5347 1
0.32795
0.89788
0.97258
0.97715
0.96589
095QOI
0.95620
0.94864
0.93671
0.92139
0.90102
0.87171
0.82815
0.76216
0.66877
0.54871
0.33629
9
10
I1
I2
I3
I4
15
16
0..............
5..............
10..............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70..............
75 +.........
0.90852
0.9751 1
0.97899
0.96857
0.96224
0.95968
0.95273
0.94161
0.92702
0.9071 8
0.87846
0.835%
0.77181
0.68049
0.56188
0.34443
0.91835
0.97748
0.98072
0.97109
0.96530
0.96297
0.95659
0.94624
0.93230
0.9 12%
0.88481
0.84334
0.78093
0.69154
0.57430
0.35248
0.92840
0.97985
0.98221
0.97328
0.96800
0.96592
0.96012
0.95050
0.937 16
0.9 1822
0.89055
0.84992
0.78890
0.70102
0.58486
0.35989
0.93728
0.98 195
0.98367
0.97538
0.97054
0.96863
0.96328
0.95421
0.94127
0.92254
0.89514
0.85515
0.79532
0.70874
0.59344
0.36634
0.94541
0.98394
0.9851 1
0.97745
0.97302
0.97128
0.96637
0.95786
0.94535
0.92688
0.89978
0.86046
0.80187
0.71664
0.60226
0.37316
0.95288
0.98582
0.98649
0.97945
0.97543
0.97387
0.96939
0.96 I43
0.94938
0.93 122
0.90447
0.86586
0.80854
0.72472
0.6 1 130
0.38035
0.95976
0.98760
0.98783
0.98 140
0.97778
0.97638
0.97232
0.96493
0.95337
0.93553
0.909 18
0.87132
0.8 1530
0.73293
0.62054
0.38792
0.96612
0.98931
0.98913
0.98329
0.98006
0.97883
0.975 18
0.96834
0.95728
0.93983
0.91 389
0.87682
0.82214
0.74 124
0.62993
0.39584
I7
I8
19
20
21
22
23
24
0..............
5..............
10..............
15..............
20..............
25..............
0.97201
0.99091
0.99038
0.98512
0.98226
0.981 19
0.97794
0.97165
0.9611 1
0.94407
0.91861
0.88233
0.82901
0.74963
0.63942
0.40410
0.97780
0.99243
0.99158
0.98689
0.98438
0.98346
0.98061
0.97486
0.96484
0.94824
0.92326
0.88782
0.83589
0.75804
0.64898
0.41266
0.98343
0.99386
0.99273
0.98858
0.98642
0.98565
0.98318
0.97796
0.96846
0.95232
0.92786
0.89326
0.84271
0.76642
0.65852
0.42146
0.98856
0.99523
0.99383
0.99020
0.98837
0.98775
0.98563
0.98094
0.97197
0.95629
0.93236
0.89861
0.84945
0.7747 1
0.66801
0.43048
0.99205
0.99628
0.99498
0.99199
0.99045
0.98985
0.98802
0.98381
0.97552
0.96057
0.93744
0.90486
0.85756
0.78501
0.68010
0.44135
0.99491
0.99730
0.99608
0.99368
0.99246
0.99200
0.99053
0.98699
0.97%2
0.96577
0.94390
0.91299
0.86816
0.79851
0.69608
0.45533
0.99704
0.99818
0.9971 1
0.99527
0.99438
0.99404
0.99292
0.99006
0.98374
0.97123
0.95094
0.92205
0.88015
0.8 1398
0.71469
0.47 144
0.99851
0.99889
0.99801
0.99674
0.99613
0.99590
0.995 1 1
0.99294
0.98779
0.97693
0.95863
0.93224
0.89388
0.83 197
0.73674
0.19020
0..............
5..............
10..............
15..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70..............
75+ ........
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70..............
75 + ........
'Vdue listed for age 75 + is T(80)lT(75).
.
.
TABLE
270. MALE
FIVE-YEAR SURVIVORSHIP PROBABILITIES sSx
.(Lnha*/yolrlyYry. 5s,
A!"
..............
..............
..............
.............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70..............
75 +a ........
0
5
10
15.
I
2
3
4
0.78598
0.94170
0.94345
0.92 179
0.90238
0.88887
0.87 123
0.84849
0.82417
0.79284
0.75 188
0.69526
0.61357
0.5 1773
0.3962 1
0.253 10
0.80839
0.94685
0.94829
0.92836
0.9 1067
0.89838
0.882 19
0.861 18
0.83840
0.80869
0.76954
0.7 1503
0.63658
0.54303
0.42347
0.26885
0.8279 1
0.95 148
0.95266
0.93432
0.91815
0.90695
0.89209
0.8726 1
0.85 122
0.82297
0.78552
0.7329 1
0.65732
0.56594
0.44803
0.28214
0.845 14
0.95568
0.95662
0.9397 1
0.92495
0.91475
0.90 108
0.88302
0.86287
0.83595
0.80001
0.749 18
0.67622
0.58676
0.47062
0.29420
Value listed for age 75
l
+ is 7ls0)/7l75) .
.
.
+I WEST MODEL
+4.Jiv~ry*rl:
5
6
7
8
0.86050
0.95953
0.96024
0.94466
0.931 17
0.92 187
0.90932
0.89255
0.87355
0.84784
0.8 1329
0.764 10
0.69354
0.60585
0.49 123
0.30500
0.8743 1
0.96306
0.96359
0.9492 1
0.93690
0.92844
0.9 1690
0.90 132
0.88336
0.85880
0.82552
0.77785
0.7095 1
0.62347
0.51025
0.3 1498
0.88685
0.96633
0.96667
0.95343
0.94220
0.93452
0.9239 1
0.90944
0.89246
0.86894
0.83684
0.7906 1
0.7243 1
0.63983
0.52797
0.32449
0.89827
0.96937
0.96956
0.95735
0.947 12
0.94017
0.93044
0.9 1699
0.90090
0.87838
0.84736
0.80246
0.73809
0.65504
0.54445
0.33364
.
10-YEAR
SURVIVORSHIP PROBABILITIES
TABLE
27 1. FEMALE
0:.............
5..............
10..............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70+' ........
0..............
5..............
10..............
IS. .............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60
65..............
..............
70+ '........
I&
.
+4.
NORTH
MODEL
9
10
11
I2
I3
I4
IS
16
0.84440
0.92343
0.933 18
0.92466
0.91301
0.89977
0.88684
0.87580
0.86121
0.83 163
0.77801
0.69423
0.57705
0.43542
0.18157
0.8598 1
0.93075
0.93915
0.931 12
0.92042
0.90837
0.89639
0.88578
0.87 162
0.84342
0.79265
0.71258
0.59903
0.45922
0.19229
0.87426
0.93763
0.94475
0.937 19
0.92739
0.9 1646
0.90538
0.8952 1
0.88 143
0.85454
0.80654
0.73005
0.62000
0.482 12
0.20304
0.88783
0.94410
0.95002
0.94290
0.93395
0.92410
0.91386
0.90407
0.89068
0.86505
0.8 1972
0.74679
0.64037
0.50485
0.21417
0.90208
0.95046
0.95470
0.94776
0.93956
0.93072
0.92 122
0.91 165
0.89849
0.87399
0.83 102
0.76094
0.657 13
0.52289
0.22360
0.91578
0.95676
0.95963
0.95300
0.94551
0.93752
0.92859
0.9 1907
0.90586
0.88205
0.84086
0.7732 1
0.67 182
0.53887
0.23258
0.92840
0.96259
0.%433
0.95805
0.95 123
0.944 10
0.93579
0.92639
0.9 1324
0.89020
0.85088
0.78579
0.68700
0.55555
0.2422 1
0.93983
0.%799
0.96879
0.%289
0.95676
0.95046
0.94277
0.93357
0.92057
0.89839
0.86 102
0.79861
0.70265
0.57292
0.25246
17
18
19
20
21
22
23
24
0.95023
0.97299
0.97300
0.9675 1
0.96207
0.95657
0.9495 1
0.94059
0.92783
0.90657
0.87 123
0.8 1162
0.7 1865
0.59085
0.26332
0.95972
0.97761
0.97697
0.97 192
0.967 13
0.9624 1
0.95600
0.9474 1
0.93494
0.91470
0.88 144
0.82473
0.73490
0.60924
0.27473
0.9684 1
0.98 193
0.98073
0.97610
0.97 195
0.96798
0.96222
0.95401
0.94 194
0.9227 1
0.89157
0.83782
0.75 128
0.62796
0.28663
0.97644
0.98594
0.98425
0.98005
0.97653
0.97328
0.968 16
0.96036
0.9487 1
0.93055
0.90154
0.85080
0.76766
0.64685
0.29892
0.98390
0.9897 1
0.98757
0.98380
0.98086
0.97830
0.9738 1
0.96645
0.95525
0.938 18
0.9 1 129
0.86357
0.78390
0.66576
0.31 150
0.99104
0.99333
0.99077
0.98740
0.98505
0.983 17
0.97930
0.97235
0.96161
0.94544
0.92018
0.875 17
0.79892
0.6835 1
0.3238 1
0.99468
0.99583
0.99405
0.99 169
0.98986
0.98825
0.98490
0.9787 1
0.96884
0.95427
0.93 186
0.891 19
0.82055
0.71027
0.34077
0.99725
0.99761
0.99626
0.99457
0.9933 1
0.99221
0.98%7
0.98465
0.97625
0.96385
0.94486
0.90950
0.84602
0.74282
0.36064
Value listed for age 70+ is T(80)/T(70).
.
TABLE
272 FEMALE
~OYEAR
SURVIVORSHIP PROBABILITIES. I$x
'Vahe listed for age 70+ is T(80)/T(70)
.
. +*
SOUTH
MODEL
.
R
PROBABILITIES I$r.
TMLE273. FEMALE~ O Y E ASURVIVORSHIP
0..............
5..............
10..............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65..............
70+ ........
0..............
5..............
10..............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
++ EASTMODEL
0.88698
0.95074
0.94702
0.93270
0.92068
0.91 115
0.90273
0.89362
0.87613
0.83872
0.77321
0.67503
0.54544
0.39482
0.16435
0.89877
0.95553
0.95194
0.93885
0.92788
0.91908
0.91 104
0.90203
0.88496
0.84928
0.78704
0.69277
0.56602
0.41541
0.17357
0.91072
0.96003
0.95614
0.94404
0.9341 1
0.9261 1
0.91853
0.90959
0.89280
0.85864
0.79936
0.70857
0.58421
0.43355
0.18209
0.92197
0.96464
0.96085
0.94989
0.94084
0.93331
0.92578
0.91661
0.89987
0.86690
0.81006
0.72220
0.59985
0.44899
0.18974
0.93225
0.96894
0.96529
0.95543
0.94724
0.94019
0.93280
0.92349
0.90690
0.87517
3.82084
0.73602
0.61587
0.46496
0.19789
0.94170
0.97295
0.96946
0.96066
0.95331
0.94676
0.93957
0.93023
0.91386
0.88343
0.83167
0.75002
0.63222
0.48146
0.20658
0.95038
0.97669
0.97340
0.96561
0.95907
0.95304
0.94609
0.93678
0.92072
0.89 165
0.84251
0.76412
0.64887
0.49844
0.21578
0.95842
0.98020
0.97709
0.97027
0.96453
0.95903
0.95238
0.94316
0.92746
0.89980
0.85333
0.77829
0.66573
0.51582
0.22552
17
18
19
20
21
22
23
24
0.98057
0.98948
0.98699
0.98280
0.97926
0.97535
0.96975
0.961 12
0.94681
0.92346
0.88504
0.82043
0.71683
0.56956
0.25753
0.98683
0.99222
0.98992
0.98653
0.98368
0.98027
0.97504
0.96667
0.95287
0.93098
0.89522
0.8341 1
0.73368
0.58762
0.26896
0.991 14
0.99452
0.99286
0.99034
0.98797
0.98490
0.98005
0.97215
0.95924
0.93914
0.90644
0.84961
0.75357
0.60985
0.28273
0.99440
0.99629
0.99501
0.99317
0.99141
0.98896
0.98485
0.97786
0.96629
0.94852
0.9 1965
0.86831
0.77831
0.63853
0.30008
0.99681
0.99769
0.99679
0.99555
0.99433
0.99250
0.98922
0.98330
0.97331
0.95812
0.93344
0.88828
0.80557
0.67131
0.31980
0.99842
0.99875
0.99818
0.99743
0.99665
0.99540
0.99298
0.98832
0.98018
0.96783
0.94767
0.90950
0.83567
0.70915
0.34230
0.96637
0.97378
0.98349
0.98657
0.98059
0.98388
0.97469
0.97886
0.96971
0.97462
0.96474
0.97018
0.95842
0.96421
0.94936
0.95535
0.93409
0.94054
0.90785
0.91574
50..............
0.86405
0.87464
0.79245
0.80652
55..............
60..............
0.68276
0.69982
0.53354
0.55148
65..............
70+ ........
0.23576
0.24644
'Value listed for age 70+ is T(80)/T(70).
.
..............
..............
..............
..............
20..............
0
5
10
I5
0.65634
0.84223
0.85955
0.83 144
0.81099
0.79587
0.76937
0.73122
0.68660
0.63032
0.55070
0.44525
0.31 157
0.16835
0.06188
0.68748
0.85575
0.87082
0.84458
0.82567
0.81 169
0.787 15
0.75163
0.70947
0.65572
0.57934
0.47678
0.3445 1
0.19911
0.07761
0.7 1539
0.86793
0.88098
0.85648
0.83898
0.82606
0.80330
0.77015
0.73032
0.67902
0.60572
0.SM 1 1
0.37570
0.22903
0.09189
0.74062
0.87902
0.89026
0.86732
0.851 13
0.839 18
0.81806
0.78716
0.74950
0.70048
0.63017
0.53353
0.40524
0.25806
0.10513
0.76363
0.889 17
0.89877
0.87728
0.86229
0.85 126
0.83 168
0.80286
0.76723
0.72036
0.65292
0.55920
0.43331
0.28626
0.1 1745
0.78473
0.89850
0.9066 1
0.88649
0.8726 1
0.86242
0.84429
0.8 1742
0.78370
0.73890
0.6742 1
0.58337
0.45998
0.31350
0.12904
0.8W
0.79908
0.75624
0.69420
0.60620
0.48539
033982
0.14014
0.82224
0.9 1521
0.92066
0.90298
0.891 12
0.88249
0.86698
0.84367
0.8 1347
0.7725 1
0.71304
0.62778
0.50962
0.36526
0.15086
9
10
11
I2
13
I4
IJ
16
0..............
5..............
10- ............
I5..............
20..............
25..............
30..............
35..............
40..............
45..............
50..............
55..............
60..............
65
70+' ........
0.83904
0.92273
0.92700
0.91042
0.89947
0.89154
0.87724
0.85558
0.82699
0.7878 1
0.73080
0.64824
0.53280
038989
0.16134
0.85472
0.92977
0.93294
0.91741
0.90732
0.90005
0.88688
0.86678
0.83974
0.80228
0.74762
0.66769
0.55497
0.41366
0.17169
0.8694 1
0.93636
0.93852
0.92399
0.91471
0.90805
0.89597
0.87735
0.85 178
0.8 1594
0.76356
0.6862 1
0.57620
0.4366 1
0.181%
0.88319
0.94258
0.94376
0.93016
0.92166
0.91561
0.90454
0.88733
0.86320
0.82922
0.77941
0.70465
0.59734
0.45977
0.19267
0.8974 1
0.94849
0.94830
0.93532
0.92742
0.92 189
0.91180
0.89600
0.8732 1
0.84041
0.79216
0.71923
0.6 I392
0.47769
0.20150
0.91033
0.95403
0.95278
0.94058
0.93334
0.92823
0.91884
0.903%
0.88 1%
0.85006
0.8032 1
0.73195
0.62853
0.49363
0.20978
0.92232
0.95932
0.957 19
0.94578
0.9391 8
0.93449
0.92582
0.91 190
0.89076
0.85984
0.8 1444
0.74498
0.64379
0.51041
0.2 1872
0.93368
0.96436
0.96149
0.9H)90
0.94492
0.94064
0.93273
0.9 1978
0.89957
0.86974
0.82603
0.75853
0.65945
0.52781
0.22834
I?
I8
I9
10
21
22
23
24
..............
a94424
0.96913
0.96566
0.95589
0.95054
0.94671
0.93951
0.92757
0.90835
0.87968
0.83770
0.77227
0.67561
0.54591
0.23862
0.95404
0.97367
0.96970
0.96075
0.95599
0.95259
0.946 13
0.93522
0.9 1704
0.88960
0.84942
0.786 18
0.692 1 2
0.56455
0.24955
0.963 15
0.977%
0.97358
0.96543
0.961 26
0.95828
0.95255
0.94267
0.92557
0.89943
0.861 1 I
0.80015
0.70883
0.5836 1
0.26106
0.97163
0.98202
0.97730
0.96994
0.96634
0.96376
0.95876
0.94989
0.93390
0.9091 1
0.87268
0.81408
0.72563
0.60290
0.27308
0.97957
0.98586
0.98085
0.97425
0.971 18
0.96901
0.96472
0.95687
0.94 198
0.9 I856
0.88407
0.82785
0.74235
0.62228
0.28552
0.98706
0.98947
0.98428
0.97846
0.9759 1
0.97414
0.97052
0.96337
0.94906
0.92658
0.893 72
0.83984
0.75734
0.64009
0.29749
0.99158
0.99250
0.98840
0.9841 1
0.98215
0.98042
0.97703
0.97067
0.95786
0.93757
0.40767
0.85774
0.78031
0.66803
0.3 1507
0.9952 1
0.99509
0.99 175
0.98853
0.987 12
0.98581
0.98321
0.97813
0.96729
0.94976
0.923%
0.87867
0.80788
0.70242
0.33632
..............
35..............
2S
30..............
40..............
45..............
50..............
55..............
60..............
65..............
70+'........
..............
0
5..............
10..............
15..............
20..............
25..............
M .............
35..............
40..............
45..............
50."...........
55
60
65
fO+'........
..............
..............
..............
v d u e listed for r p m+ is T ( ~ ) I T. ~ )
0.80420
0.907 17
0.91388
0.89502
0.88219
0.87279
0.85602
.
TABLE
277
'Vdue listed for age 70+ is T(80)lT(70) .
MALEIO-YEAR SURVlVOllSHlPPROMIILITIE& ma$
.
A A.
EAST MODEL
+*WEST MODEL
T~O~a278
. MALE10-YEARSuRvtvorsHIrMOBA#Lm&S
I
I
4
,
I
i
..............
..............
..............
..............
..............
..............
..............
..............
..............
..............
...........
..............
..............
0
5
10
I5
20..............
25
30
35
40
45
50
55.60
65
70+'
........
Value S
17
16
I9
a
21
22
ZJ
24
0.96172
0.97853
0.97229
0.96354
0.95796
0.95199
0.94146
0.92465
0.89893
0.86Q68
0.805 19
0.72727
0.62279
0.48986
0.2 1487
0.96889
0.98 172
0.97606
0.96848
0.%384
0.95871
0.94921
0.93343
0.90866
0.87138
0.8 1694
0.74022
0.63647
0.50329
0.22288
0.97558
0.98479
0.97970
0.97324
0.96949
0.965 19
0.9567 1
0.94198
0.91826
0.88205
0.82877
0.75336
0.65043
0.5 1709
0.23137
0.98 182
0.9877 1
0.98320
0.97781
0.97490
0.97 140
0.96394
0.95029
0.92767
0.8926 1
0.84057
0.76656
0.66457
0.531 18
0.24033
0.987 13
0.99040
0.98652
0.98221
0.98000
0.97714
0.97057
0.95793
0.93647
0.90275
0.85234
0.78013
0.67942
0.54623
0.24997
0.99123
0.99288
0.98976
0.98648
0.98495
0.98280
0.97742
0.96645
0.94109
0.91578
0.868(#1
0.79885
0.70054
0.56833
0.26357
0.99460
0.995 14
0.99278
0.99049
0.98954
0.98805
0.98392
0.97488
0.958 13
0.92993
0.88584
0.82058
0.72566
0.59521
0.2801 1
0.9971 1
0.99705
0.99545
0.99403
0.99353
0.99262
0.98972
0.9828I
0.96922
0.94500
0.90569
0.84582
0.75582
0.62850
0.30041
.
i for age 70+ is T(80)/T(70)
LOGIT VALUES BY SINGLE YEAR OF ACE FOR COALEDEMENY
FEMALE LEVEL 16 MODEL LIFE TABLES
TABLE
279.
.
LOG1TTRANSFORMATIONOF M E COMPLEMENTOF THE
PROBABILITY OF SURVIVING 1 I(% ) NORTH
MODEL
.
-
.
TABLE
280
LOOIT TRANSFORMATION OF M E COMPLEMENT OF THE
~ O B A B I OF
L ~SURVlVINO. I
I(x
).SOUTH MODEL
-
TABLE
281. LOOITTMNSFORMAllONOF THE COMPLEMENT OF THE
PROMBILITYOF SURVIVINO 1 - /(x ) EAST
MODEL
.
.
Annex XII
GLOSSARY a
tendency for enumerators or respondents to report certoin ages instead of others; also known as age preference or "digit
preference". Preference for ages ending in zero or five is
widespread.
Agepattern offerrility: relative distribution of a set of "age-specific fertility rates". It expresses the relative contribution of each age group
to "total fertility".
Age mtio: ratio of the population in a given age group to the average
of the populations in the two neighbouring age groups, times 100.
Age-spccifiefdility rate: number of births occurring during a specified
period to women of a specified age or age group, divided by the
number of person-years lived during that period by women of that
age or age group. When an age-specific fertility rate is calculated for
a calendar year, the number of births to women of the specified age
is usually divided by the mid-year population of women that age.
Age-specific mortality mte: number of deaths occurring during a
specified period to persons (usually specified by sex) of a specified
age or age group, divided by the number of person-years lived during that period by the persons of that age or age group. When an
age-specific mortality rate is calculated for a calendar year, the
number of deaths to w m n s of the s~ecifiedaee is usuallv divided
by the mid-year pophation of persbns of thit age. ~ge-specific
mortality rates are generally denoted by .M,. the annual death rate
of persons aged from x to x + n.
Age s t a ~ z a t i o n a: procedure of adjustment of crude rates (birth.
death or other rates) designed to reduce the effect of differences in
age structure when comparing rates for different populations.
Birth history: report of the number and dates of all live births experienced by a particular woman: see also "Pregnancy history". The
sex of each child. the survival of each child to the date of the interview, and, where pertinent. the date of death are also generally
recorded.
Birth order: the ordinal number of a given live birth in relation to all
previous live births of the same woman (e.g., 5 is the birth order of
the fifth live birth occurring to the same woman).
Bir~hmte: see "Crude birth rate".
Chandmpekamn-Deming technique: procedure to estimate the coverage
of two independent systems collecting information about demographic or other events, based on the assumption that the probability of an event being recorded by one system is the same regardless
of whether the event is recorded by the other system. The events
from both systems are matched to establish M. the number of
events recorded by both systems; U I . the number recorded only by
system I; and U2. the number recorded only by system 2. The
Chandrasekaran-Demingformula then estimates total events. N. as
Age-+ng:
Chilhring ages: the span within which women arc capable of bear-
ing children, generally taken to be from age IS to age 49 or, sometimes, to age 44.
C h i h n ewr born@): number of children ever borne alive by a particular woman; synonymous with "parity". In demographic usage,
stillbirthsare specifically excluded.
'
Cohort:,groupof persons who experienced the same class of events in
the same period. 'Thus, an age cohort is a group of people born during a particular period. and a marriage cohort is a group of people
who married during a particular period. The effects of a given set of
mortality a fertility rates are often illustrated by applying them to
hypothetical cohorts.
Coho# fertiliiry: the fertility experienced over time by a group of
-
-
'~ermsin quotation marks are listed in the glossary.
women or men who form a birth or marriage cohort. The analysis
of cohort fertility is contrasted with that of "period fertility".
Cndr birth mte: number of births in a population during a specified
period divided by the number of person-years lived by the population during the same period. It is frequently expressed as births per
1,000 population. The crude birth rate for a single year is usually
calculated as the number of births during the year divided by the
mid-year population.
Cndr dcoth mte: number of deaths in a population during a specified
period divided by the number of person-years lived by the population during the same period. It is frequently expressed as deaths per
1.000 population. The crude death rate for a single year is usually
calculated as the number of deaths during the year divided by the
mid-year population.
C h e d f e r t i l i t y : an estimate of the average number of children ever
borne by women of some age x, obtained by cumulating "agespecific fertility rates" up to age x : also onen calculated for age
groups.
h t h mte: see "Crude death rate".
De facto popuhtion: population enumerated on the basis of those
present at a particular time. including temporary visitors and
excluding residents temporarily absent. See "fkjurppopulation".
De jure population: population enumerated on the basis of normal
residence. excluding temporary visitors and including residents temporarily absent. See "Ikf i t 0 population".
LXgitprrfm&: see "Age-heaping".
I)lcolrccordsysrrm: see "Chandrasekaran-Deming technique".
Expectation of life at b i ~ h average
:
number of years that a member of a
"cohort" of births would be expected to live if the cohort were subject to the mortality conditions expressed by a particular set of
"age-specific mortality rates". Denoted by the symbol eo in "life
table" notation.
Fertility history: either a "birth history" or a "pregnancy history".
Fornanlsurviml: pnredure for estimating the age distribution at some
later date by projecting forward an observed age distribution. The
procedure uses "survival ratios", often obtained from model :life
tables". The procedure is basically a form of population projection
without the introduction of new entrants (births) to the population.
Gned fertility mtc: ratio of the number of live births in a period to
the number of person-years lived by women of "childbearing ages"
during the period. The general fertility rate for a year is usually calculated as the number of births divided by the number of women of
childbearing age at mid-year.
Gnrss mpdwtion mle: average number of female children a woman
would have if she survived to the end of her childbearing years and
if, throughout, she were subject to a given set of "age-specific fertility rates" and a given "sex ratio at birth". This number provides a
measure of replacement fertility in the absence of mortality.
Growth mte: the increase or decrease of a population in a period
divided by the number of person-years lived by the population during the same period. The increase in a population is the result of a
surplus (or deficit) of births over deaths and of immigrants over emigrants. (The annual increase is onen expressed as a fraction of the
total population at the beginning of the year. but this convention
has the inconvenient characteristic of not being readily defined for a
five-year interval and of being unequal to the difference between
the birth rate and the death rate even in the absence of migration.)
See also "Rate of natural increase".
I+
~ a l i t nue:
y
number of deaths of children under one year of
age occurring in the same year; also used in a more rigorous sense to
mean the number of deaths that would occur under one year of age
m a "life tabk" with a "radix" of 1.000, in which sense it is denoted
by the symw 190
Life W :listing ofthe number of s
w
i
mat different ages (up to the
w
e
s
t age attained) in a hypothetical " d o r t " subject from birth
to a partidat s t of "age-speci8c mortality rates*'. The rates arc
u a d y thqc observed in a given population during a particular
period of time. The survivm of the "radix" to age x arc generally
denoted by J(x). The tabulatiom ~ ~ ~ l oaccompanying
n l y
a life
tabk include $her featurn of the cohort's experience: its expectation of life at each age x, denoted by ex; the probability of dying
from each age x to age x n , denoted by ,qX ; the person-years
lived by the hypothetical cohort as it ages from age x to age x n ,
denoted by, Lx ( a h equivalent to the population aged x , x n in
a "Jtationry population" experiencing a number of b i s each year
q u a l to the radix of the life tabk); and the person-years lived by
the hypotheticalcohort from age x onward, denoted by T(x).
Logit: The logit of a pmportion p is 45 In(p /(I -p)]. As a linearizing
tmnsfonnation, the logit has been proposcd as the basis of a model
life-tabk system in which the log~tof a probability of dying by age
xGqo) is related lineruly to the logit of a standard probability of
dying by clge xGq6) rn that
+
+
+
where a is a measure of mortality level relative to the standard and
B is a parameter that alters the shape of the standard mortality function.
~ t d j c r l i i t y any
:
measure of fertility in which the births (in the
numerator) are b i to manied women and in which the number
of person-years lived (m the denominator) also pertains to married
women. In some instance5 the category of married includes persons
incon#mualunioaa
Man qpc of & M n g : average age at which a mortality-free
"cohort" of women bear their children according to a sct of "agespecific fertility rates".
Akm uge 4 c h i M n g in thepq&Iion: average age of the mothers
of the children born in a population during a year. This measure
incoporates the effects of both mortality and the age distribution.
A M m the value associated with the central member of a set that is
a d e n d by size or some other characteristic expressed in numbers.
ion nue: number of migrants during a specified period divided
by the person-years lived of the population exposed to migration.
A h see "Population change due to migration".
M d l &fe Urbk: expression of typical mortality experience derived
from a group of 0bSe~Cd"life tables".
M o w onmp: the successive averaging of two or more adjacent
values of a series in order to remove sharp fluctuations.
Mjm indcx: an index of digit preference that essentially sums in turn
the population ending in each digit over some age range, oRen 1089, expressing the total as a percentage of the total population, and
which avoids the b i i introduced by the fact that the population is
not evenly dirtributcd among all ages by repeating the calculations
10 times, once for each beginning digit, and averaging the results.
The daerence between the average percentage for each digit and
the expected value of 10 per cent provides a measure of the preference for or avoidana ofthe digit over the age range considered.
Nufwd faH&ty: age pattern of "marital fertility" observed in noncontraceptive populations where reproductive behaviour is not
affected by the number of children alrtady born.
Net migmtion: the ditrerence between gross immigration and gross
emigration.
Nu -tion
m:the average number of female children born per
wormn in a "cohort" subject to a given sct of "age-specific fertility
rates". a given set of "age-specific mortality rates" and a given "sex
ratio at birth". This rate measives replacement fertility under given
conditions of fertility and mortality: it is the ratio of daughters to
mothers, assuming continuation of the specified conditions of fertility and m d t y .
h&&
Mhf:
h a refinement of the "revem-survival" procedure
for fertility estimation, whereby estimates of "age-specific fertility
mtes" for the ncent past arc obtained by relating mothers to their
own children, using information on relationship and other charact e r h b avdabk from a censusor survey.
Pmily: see "Children ever born".
PorHol binh mrr: the proportion ofthe population that enten (thrt is,
is born into) a &en age category in a year. The age categorieswed
are nonnally open-ended; thus, the partial birth rate x + designates
the proportion of the population becoming x y e m and older.
Pmicll &ah mte: the proportion of the population that kaves (that is,
dies out of) a given age category in a year. See "Partial birth rate".
Periodfdility: fertility experienced during a particular period of time
by women from all relevant b i or madage "cohorts"; ace
"Cohort fertility".
P/Fmtio nuthod: a consistency check for survey information on fenil-
ity. Information on m n t fertility k cumulated to obtain measures
that arc equivalent to average parities. Lifetime fertility in the form
of reported average parities by age group, P, can then be compared
for consistency with the parity equivalents, F, by calculating the
ratio P IF for successive age groups. If certain assumptions about
error patterns arc met, an improved estimate of fertility can sometimes be obtained by correcting the age pattern of cumnt fertility to
a p e with the level of lifetime fertility reported by younger women.
Populorion change due lo &gnation: the sum of in-migrants minus outmigrants during a specified period of time. The change may also be
expressed as a rate by dividing the change by person-years lived in
the population during the same period.
Pn?gmncy kirto'y: a report of the number and dates of occumnce of
all the pregnancies experienced by a particular woman. The
outcome of the pregnancy-live birth, stillbirth, foetal death-is also
rtcorded.
R d k : the hypothetical birth "cohort" of a "life table". Common
values arc 1,1,000 and 100,000.
Rate of n a t d i n e m : the differencebetween the births and deaths
occurring during a given period divided by the number of personyears lived by the population during the same period. This rate,
which specifically excludes changes resulting from migration, is the
difference between the "crude b i rate" and the "crude death
rate".
RcImqmtiw m y : survey that obtains information about dem*
graphic events that occumd in a given past period, generally terminating at the time of the survey.
RNCrsepmjmtion: see "Reverse survival".
R ~ Kswn'wl:
C
a technique to estimate an earlier population from an
observed population, allowing for those members of the population
who would have died aocording to observed or assumed m o d i t y
conditions. It is used as a method of estimating fertility by cakulating from the observed number of survivors of a given age x the
expected number of births that c a u m d x years earlier. (In situations for which both fertility and mortality are known or can be reliably estimated, revem survival can be used to estimate migration.)
Robwnus: a characteristicof estimates that are not greatly affected by
deviations from the assumptions on which the estimation procedum
is based.
Sex mtio at birth: number of male births for each female birth or male
births per 100 female births.
Singdare merin age at modage (SMAM): a measure of the mean age at
first marriage, derived from a set of proportions of people single at
different ages or in different age groups, usually calculated
separately for males and femaks.
Stablepopukation: a population exposed for a long time to constant fertility and mortality rates, and closed to migration, establishesa fixed
age distribution and constant growth rate characteristic of the vital
rates. Such a population, with a constant age s t ~ c t u r and
e constant
rate of growth, ii called a "stable population".
Stutiomry popu&tion: a "stable population" that has a zero growth
rate, with constant numbers of births and deaths per year. Its age
structure is determined by the mortality rates and is equivalent to
the person-years lived (,Lx) wlurnn of a conventional "life table".
Survival mtio: the probability of surviving between one age and
another; often computed for age groups, in which case the 18th
correspond to those of the person-years lived function, ,Lx, of a
"lifetable". Also called "survivorship probabilities".
~ ~ ~ m h i p ~ l i tsee
i "Survival
e s :
ratio".
W h e t i ~ ~ tthe
y :average parity calculated for a hypothetical coho17
exposed indefinitely to a set of period "age-specific fertility rates".
Totalfirtility M& (TFR): the average number of children that would
be born per woman if all women lived to the end of their childbear-
ing years and bore children according to a given set of "age-specific
fertility rates"; a h refemd to as total fertility. I t is frequently used
to compute the consequence of childbearing at the rates currently
observed.
UNld Nations obrlsrx arrvmcy indcx: an index of age-reporting accuracy that is bucd on deviations from the expected mgularity of
population size and sex ratio, age group by age group. The index is
calculated as the rum oT: (a) the mean absolute deviation from 100
of the age ratios for maks: ( b ) the mean absolute deviation from
100 of the are ratios for females: and I c ) three times the mean of
the absoluteYdiffe~ncein reported wx
from one age group to
the next. The United N a t i w defines ap/sex data as "accurate".
A&
"inaccurate". or "highly inaccurate". depending upon whether the
index is less than 20. from 20 to 40. or greater than 40.
indcx: a measure of the quality of age-reporting based on the
extent of preference for a particular target digit or digits. The index
essentially compares the reported population at ages ending in the
urge1digit or digits with the population expected on the assumption
that population is a linear function of age. For a particular age
range. onen 23-62, the population with ages ending in the target
digits is divided by one tenth of the total population: the result is
then multiplied by 100 and divided by the number of different target digits. A value of 100 indicates no preference for those digits.
whereas values over 100 indicate positive preference for them.
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