Keyfitz N., Caswell H. Applied Mathematical Demography

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16.2. Measurement of Fecundity 415

Suppose (Sheps, in Sheps and Ridley 1966) that the N representative

women with whom we start out include Nf(p) dp women of susceptibility

or fecundity between p and p + dp,wheredp is small. Among women of

fecundity p we will expect the fraction p to become pregnant in the ﬁrst

month, 1−p to go on to the second month, and, of these, (1−p)p to become

pregnant then, and so on. In other words, we will expect

Npf(p) dp

women to become pregnant in the ﬁrst month,

N(1 − p)pf(p) dp

in the second month, ..., and

N(1 − p)

m−1

pf(p) dp

in the mth month—all this for a given susceptibility p.

To ﬁnd the corresponding numbers for all women we add through the

several susceptibility groups, which in the limit is the same as integrating

with respect to p. The total expected conceptions for the ith month will be



(1 − p)

i−1

pf(p) dp. (16.2.2)

Expression 16.2.2 shows that for 0 <p<1 the number of conceptions

steadily decreases with time. Most of the decrease is due to women dropping

out through pregnancy, and there is no provision for bringing them back

in this way of measuring fertility.

The number of conceptions cannot but decline to zero in the course of

time as the entire group drops out through pregnancy.

But aside from this gross fact, which applies to a homogeneous group as

well, there is a selection factor that can be important if the variation in p,

expressed by f(p), is substantial. The pregnancy rate during the ith month

is (16.2.2) divided by the number of women still under observation



(1 − p)

i−1

pf(p) dp



(1 − p)

i−1

f(p) dp

, (16.2.3)

which can be regarded as the mean of p weighted by (1−p)

i−1

f(p). Because

1−p is less than unity, the weighting function (1−p)

i−1

f(p) shifts downward

(i.e., to the left on the usual form of chart) as i increases, so p

must decline

steadily with i. This is a selection arising from the removal by pregnancy

of the more fertile women.

The two processes are shown in Table 16.2, with a simple example in

whichhalfthewomenhavep =0.2 and half have p =0.3, and q is written

for 1 − p in (16.2.2) and (16.2.3). It shows in column 1 the chance of

416 16. Microdemography

Table 16.2. Example of removal of more fertile women in successive months, using

f(p)=0.5 for p =0.2 and f (p)=0.5 for p =0.3; that is, half the women are of

fecundability p =0.2 and half are of fecundability p =0.3

Probability of

Probability of Fraction of conceiving in

conceiving in women ith month for

ith month for remaining at women who have

all women beginning of not conceived by

entering month (i − 1)th month

Month



i−1

f (p) dp



i−1

f (p) dp



i−1

f (p) dp



i−1

f (p) dp

1 0.25 1.000 0.25

2 0.185 0.750 0.247

3 0.138 0.565 0.243

4 0.103 0.428 0.240

5 0.077 0.325 0.237

6 0.058 0.248 0.234

7 0.044 0.190 0.231

8 0.033 0.146 0.228

9 0.025 0.113 0.226

10 0.019 0.087 0.223

15 0.0054 0.025 0.213

20 0.0016 0.0078 0.207

30 0.00016 0.00079 0.2020

50 0.0000018 0.0000089 0.20014

becoming pregnant in the ithmonthasforecastforawomanwhenshe

comes under observation in month 1, and also (column 3) the chance of a

pregnancy in the ith month for a woman who has gone through the ﬁrst i−1

months without becoming pregnant. The rapid decline in the ﬁrst column

is no surprise, while the third column exhibits the more subtle selection

eﬀect, which is appreciable even with the small variation of our example.

The mean fecundability started at p =0.25, the mean of 0.2 and 0.3, and

by the tenth month had fallen to 0.223. The drop is more than halfway

to 0.2, to which it would ultimately tend as i became large. The greater

the variation, the more rapid is the decline in average fertility among the

women who remain.

To express (16.2.3) in terms of variation among women, expand (1 −

i−1

p of the numerator and (1 − p)

i−1

of the denominator in a Taylor

series around ¯p,themean



pf(p) dp. The constant terms are (1 − ¯p)

i−1

¯p

in the numerator and (1 − ¯p)

i−1

in the denominator, and the linear terms

vanish, with the result

≈ ¯p −

(i − 1)σ

1 − ¯p

, (16.2.4)

on using the fact that 1/(1 + α) ≈ 1 − α for α small.

16.2. Measurement of Fecundity 417

In application to the distribution of Table 16.2, consisting of two spikes,

¯p =0.25. The general rule is that the standard deviation of two numbers is

half the interval between them; if the numbers are a and b,theirvariance

+ b

−



a + b





a − b



In this case a =0.2andb =0.3, so σ

=0.0025. Then from (16.2.4) we

have for p

≈ ¯p −

(i − 1)σ

1 − ¯p

=0.25 −

(9)(0.0025)

0.75

=0.220,

as opposed to the 0.223 shown in Table 16.2. The diﬀerence is due to neglect

of higher moments and other approximations, the eﬀect of which becomes

more serious as i increases.

16.2.3 The Pearl Index Is the Harmonic Mean of the

Distribution

The Pearl index ˆp

of (16.2.1) is shown to estimate the harmonic mean of

the several p. For the expected value of N

is the initial number N times p,

of N

is Nqp,ofN

is Nq

p, and so on. The denominator of (16.2.1) can

be taken as the numbers of women nonpregnant at the beginning, after 1

month after 2 months, and so on. Entering these for any ﬁxed p and then

integrating over the range of p [overlooking that E(X/Y ) = EX/EY ]:



pf(p) dp +



(1 − p)pf(p) dp + ···



f(p) dp +



(1 − p)f(p) dp + ···

where we suppose that the survey continues until the last woman becomes

pregnant, or, more reasonable, that nonpregnant women left at the end of

the survey are excluded from all of its records. In the numerator again we

change p to 1 − q and assemble the integrals to obtain



[(1 − q)+q(1 − q)+q

(1 − q)+···]f(p) dp



(1 + q + q

+ ···)f (p) dp



f(p) dp



(1 + q + q

+ ···)f (p) dp

418 16. Microdemography



[1/(1 − q)]f(p) dp



(1/p)f(p) dp

. (16.2.5)

The quantity p

,ofwhichˆp

given by (16.2.1) is an estimate, is thus

shown to be the harmonic mean of the p’s, a statement that is meaningful

once all infecund women have been removed from the records.

16.2.4 The Gini Fertility Measure

Gini’s (1924) way of providing a measure of fertility that is unbiased in

the face of variation among women is eﬀectively to conﬁne the index to the

pregnancies and exposure of the ﬁrst month, bound to be unselected for

susceptibility if the sample is unselected. His index is

ˆp

− N

m+1

+ N

+ ···+ N

, (16.2.6)

where the denominator totals the women of proven fertility, and the nu-

merator has the relatively unimportant subtraction, if m is large, of those

who become pregnant in the (m + 1)th month.

If one wants to see formally what is happening in ˆp

when those not

pregnant by the mth month cannot be neglected, the algebra is only slightly

more involved:

ˆp

− N

m+1

+ N

+ ···+ N

is an estimate of



pf(p) dp − N



f(p) dp



(p + pq + pq

+ ···+ pq

m−1

)f(p) dp



p(1 − q

)f(p) dp



(1 − q

)f(p) dp

(16.2.7)

This is the arithmetic mean value of p, not quite in the original dis-

tribution f(p), but in the slightly diﬀerent distribution proportional to

(1 − q

)f(p) dp

Comparison of Pearl and Gini Estimates.Ifp does not vary among fer-

tile women, or does not vary greatly, we can be indiﬀerent as to whether

their arithmetic or harmonic mean is estimated. Thus in the homogeneous

case ˆp

is the best estimate of the common p in that it has the smallest

sampling error. If, however, p does vary considerably, the harmonic mean

is no substitute for the arithmetic, being always below it, and to use ˆp

to minimize sampling error at the cost of substantial bias. Take the arith-

metic example of Table 16.2, in which a group of women is equally divided

among those with probability of conceiving 0.2 and those with probability

0.3; then the arithmetic mean of their fecundity is 0.25 and the harmonic

16.2. Measurement of Fecundity 419

mean is 0.24, a small diﬀerence. For a group equally divided among those

with probabilities of conceiving of

,and

, the arithmetic mean is

0.917

=0.306.

Theharmonicmeanis

+1/

6+4+2

=0.25,

or about 18 percent low.

Thus mean fecundity is understated by ˆp

whenever variation among

women is considerable. The Pearl index, given as pregnancies per 100

woman-years of exposure, if unadjusted for selection will underestimate

the average probability of conception per month for all women in the un-

selected group. In commonsense terms this is so because ˆp

puts weight

on later months when the less fecund women are disproportionately repre-

sented. This defect of ˆp

is unimportant if couples are followed over a short

period. It is not a defect at all if we are interested in the mean fecundity

overaperiodoftimeofagroupofinitiallyfecundwomen,inwhichthose

who become pregnant drop out and are not replaced. Moreover, the mean

waiting time to pregnancy in a heterogeneous group of women will be ap-

proximated by 1/ˆp

, so that, if waiting time is the subject of interest, the

harmonic mean implicit in ˆp

is the one wanted.

16.2.5 Excursus on Averages

In the present section we have a type of selection to produce p

that lowers

the average below the unselected p

. The opposite occurred in Section 15.3,

where it appeared that choosing a woman at random gave the expected

number of her daughters as the arithmetic mean



,wheref

is the

fraction of women who have i daughters; choosing a girl at random and

asking how many daughters her mother had gave a larger average. It is

worth contrasting the four main kinds of average that enter demographic

work.

If families are chosen with probability proportional to number of daugh-

ters, the expected value is found by weighting by i and dividing by the

total number of daughters (



= 1 but



= 1):



(i)(if

)



This has the old name of contraharmonic mean (according to the Oxford

English Dictionary). If it is designated as C and the arithmetic mean as

420 16. Microdemography

A, the diﬀerence is (Section 15.3)

C − A =



−





− (



)



where σ

is the variance of the distribution. Thus C is always greater than

A, except in the trivial case where the numbers being averaged are the

same.

A similar aspect of selection, but over time, appears in Feller (1971, p.

12), who explains why we have to wait so long at a bus stop. Suppose,

for example, that the buses arrive independently at random according to

a Poisson process with constant α, so that the mean time between buses

is 1/α. When a would-be rider shows up at the stop, he ought on the

average to be midway between two buses and have to wait only

(1/α).

That is one way of calculating. The second is to say that, since the Poisson

process has no memory, and one moment of arrival has the same waiting

prospect as another, the time to the next bus must on the average be 1/α.

Unfortunately the second answer is the correct one. A selection occurs by

which one more often arrives in a long interbus period than in a short one.

A diﬀerent aspect of waiting times is evident when women are subject to

diﬀerent chances of having a child. What is the mean interchild period in a

group of women if the probability of pregnancy is p

for the ith woman, say

in a given month? The expected period for the ith woman is 1/p

, and the

mean time to conception (1/n)



(1/p

) months. This is the reciprocal of

the harmonic mean of the probabilities of conception; it is not n/



,the

reciprocal of the arithmetic mean. The Pearl index p

is another example

of the same harmonic mean.

The geometric mean turns up in applications of the life table. The chance

of a person surviving from age 30 to age 40, say, is the geometric average

chance of surviving over the 10 years taken to the tenth power. It has to

be a geometric average because the chance of survival is

⎡

⎢

⎣







···





⎤

⎥

⎦

Alternatively, if the chance of dying in the ith of n years is µ

,withan

arithmetic average ¯µ, the chance of surviving over the period is (e

−¯µ

)

and this is the geometric mean G of the chances of surviving the individual

years.

Thus we have instances of the arithmetic (A), geometric (G), harmonic

(H), and contraharmonic (C) means. For integral variables i with weights

16.2. Measurement of Fecundity 421

such that



= 1, these may be written, respectively, as

AG H C





/i)



For N unweighted values X

, integral or not, these means are



(1/X

)



Diﬀerences are not triﬂing. For a population with two values, X

and X

= 4, we have for the four averages

2.5, 2, 1.6, 3.4.

[Show that the contraharmonic mean is as much greater than the arithmetic

as the arithmetic is greater than the harmonic; that is, C − A = A − H.

Show also that G =

√

AH. Under what conditions do they hold?]

16.2.6 Graduation Uses Information Eﬃciently

Given that in practice samples used for fertility studies are small, one would

like to avoid the sampling instability of the Gini index, as well as what is,

from one point of view, the bias of the Pearl index. One would also like

to know something of the variation in fecundity among women that is not

revealed by ˆp

or by ˆp

. Variance could be inferred from the diﬀerence by

solving (16.2.4) for σ

, but this would have substantial error. Fortunately a

graduation due to Potter and Parker (1964) provides variance, along with

low sampling error and absence of bias in the mean.

16.2.7 Mean and Variance Simultaneously Estimated by

Graduation

Potter and Parker (1964) ﬁtted a beta distribution proportional to

a−1

(1 − p)

b−1

whose mean fecundity ¯p and variance σ

areasfollows:

¯p =



pf(p) dp =

a + b



(p − ¯p)

f(p) dp =

(a + b + 1)(a + b)

(16.2.8)

The a and b are not directly known, but we do know the mean and variance

of the waiting times of those becoming pregnant in terms of the same

422 16. Microdemography

constants a and b:

¯w =

a + b − 1

a − 1

,a>1; σ

(ab)(a + b − 1)

(a − 1)

(a − 2)

,a>2.

All that is needed is to solve for a and b in terms of the known ¯w and σ

After extended but straightforward algebra the estimates

a =

2σ

− ¯w

+¯w

,b=(¯w − 1)(a − 1) (16.2.9)

are reached. Substituting for a and b in (16.2.8) would give the mean and

variance of the probabilities of conceiving (not directly observable) in terms

of the mean and variance of waiting times as observed. In the example used

by Potter and Parker, ¯w =5.47 months and σ

=89.98, and from (16.2.9)

these provide a =2.746 and b =7.806; therefore mean fecundity ¯p is 0.260

and the standard deviation of p among women is σ =0.129. (A more

detailed secondary account is found in Keyﬁtz 1968, p. 386.)

Improvements on the moments ﬁtting sketched above have been pub-

lished by Majumdar and Sheps (1970). They develop maximum likelihood

estimators that make more eﬀective use of the data, these data still

consisting of waiting times for the whole group of women followed.

There is a certain arbitrariness in picking a family of probability distri-

butions to represent fecundity on the basis of mathematical convenience

rather than empirical evidence, and then drawing conclusions without at

least trying alternative distributions. This problem arises generally in stud-

ies of heterogeneity (Chapter 19). Recent discussions can be found in Wood

and Weinstein (1990), Wood (1994), and Weinberg and Dunson (2000). Un-

derstanding patterns of heterogeneous fecundity among women and how

they vary among diﬀerent populations is important in studies of the ef-

fects of heterogeneous fertility on variance in completed family size, or the

relative eﬀects of male and female fecundity within marriage.

The variation among couples is especially great for a contraceptive

method that is ineﬀective for some users. The ineﬀective users are selected

out by pregnancy, leaving a residue of eﬀective users whose probability of

conception is very low. For this reason and others, surveys of groups of

women practicing contraception are often tabulated by periods—ﬁrst year,

second year, and so on, so that the decline in the conception rate can be

traced. Tabulating in periods has the advantage that cases subsequently

lost to follow-up can be included as long as they are observed. The life

table method described below rescues the incomplete records.

16.2.8 Life Table Methods for Fertility

The theory so far has dealt with two cases: homogeneous, where all women

have the same probability of conception p, and heterogeneous, where their

several p values are distributed according to an (unknown) probability dis-

16.2. Measurement of Fecundity 423

tribution. In both we supposed that any given woman has an unchanging

probability of conception. If the sample is to be followed for a long pe-

riod, however, we need to allow also for change in individual women, either

because of a decline in fecundity with age or because of increased moti-

vation and skill in using contraception. The life table method in a fashion

embraces both factors. It also allows for the selection eﬀect of pregnancy.

To make the table we ﬁrst calculate the probability of conception month

by month. Like any life table, that for conception is based on two kinds

of data: number of events, and numbers exposed to risk. In the present

case the events are pregnancies, and the exposed are the women under

observation, for each month. If P

women are under observation through

the ith month, and A

(standing for “accident”) conceptions occur among

them, the conception rate for the group in that month is p

= A

,and

the probability of not conceiving is 1 − p

(Potter 1967).

We can multiply together the 1 −p

for successive months, and obtain a

column analogous to the life table l

that represents the chance of a child

just born surviving to exact age i. The technique is identical to that for

mortality, discussed at length in Chapter 2.

The life table model in which allowance is made for the single decrement

of pregnancy can be extended to provide for other risks, including the

death of the person, divorce of the couple, discontinuance of contraception,

and other contingencies. Among these the possibility that the couple will

drop the contraceptive is of the greatest interest for our analysis. Tietze

(1962) tells us that pregnancy rates for various IUDs during the 2 years

after insertion were considerably lower than discontinuance rates. This is

a standard problem in competing risks, of the kind dealt with in Section

2.6, and any of the methods there used would serve in this case too. But

the reﬁnements useful for mortality are not necessary for conception, where

small samples and biased data are general.

See Weinberg and Dunson (2000) for a discussion of some recent devel-

opments along these same lines, and Chapter 19 for discussion of individual

heterogeneity in general. Life table methods can be generalized to methods

based on hazard functions; these are reviewed in the context of fertility by

Wood et al. (1992) and Wood (1994).

16.2.9 Relation of Micro to Population Replacement

We saw that the replacement of a population, the ratio of girl children in

one generation to girl children in the preceding generation, is given by



l(a)m(a) da.

We can factor m(a)intov(a)f(a), where v(a) is the fraction married at

age a,andf(a) is the marital fertility rate. We can also go one step further

424 16. Microdemography

and say of married fertility that

f(a)=

1/{p(a)[1 − e(a)]} + s

where p(a) is natural monthly probability of conceiving at age (a), and e(a)

is the eﬃciency of contraception, both in a particular population. Then we

have for the net reproduction rate



l(a)v(a)

p(a)[1− e(a)]

+ s

da, (16.2.10)

but this remains purely formal without some source of information on p(a)

and e(a).

16.2.10 How Surer Contraception Reduces the Interval

Between Births

At one time “child-spacing” was a euphemism for contraception, and much

was said about the beneﬁts to the mother’s health if she spaced out her

children. This may have been good public relations at a time when authori-

ties frowned on contraception intended simply to reduce family size. While

the supposed healthful eﬀects of spacing are still referred to in some coun-

tries, yet in fact with the spread of safe and certain contraception women

often reduce the interval between such births as they decide to have. For

the mother who must give up a job in order to look after the family, two or

three children of about the same ages cost less in lost earnings than would

the same number of children spread over her reproductive life.

Whatever the motivation to compress childraising into a small time in-

terval, the couple could not aﬀord to yield to it when contraception was

uncertain. Compromise was necessary; the number of planned children

would be held below what was desired in order to allow for accidents. Even

if the chance of conception in any one month is as low as 0.01, and the

couple have 240 fertile months ahead of them, their expected prospective

children are 2.4, whether they want them or not. If they want 2 or 3 chil-

dren, they cannot aﬀord to have any deliberately. Even if they can achieve

a probability of conceiving in any month as low as 0.003, the chance of an

accidental pregnancy during 240 months is over one half:

1 − (0.997)

240

=0.514.

Thus only perfect contraception, or its equivalent in the form of easy legal

abortion, permits thoroughgoing family planning, and when such planning

is aimed at saving the time of the mother it will space the children as closely

as possible.

The idea of studying fertility in terms of the time required for the dif-

ferent processes involved in conception, pregnancy, lactation, and so on