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OCR for page 9
TEE MODEL
In an effort to understand the dynamics of fertility as
regards its responsiveness to breastfeeding, contra-
ceptive use, and their durations at the birth interval
level, the reproductive history of a woman is treated as
a sequential process marked by events including births,
initiation and termination of breastfeeding, initiation
and termination of contraceptive use, and infant deaths.
Although decisions about breastfeeding or contraception
and about childbearing are simultaneously determined, and
child mortality both affects and is affected by fertility,
this highly endogenous system will be broken into a
multi-stage process.
The stages of the model are as follows. A set of
background social and economic variables is developed for
the purpose of predicting whether a woman breastfed or
whether she used contraception during the last closed or
the open interval. Durations of breastfeeding and of
contraceptive use are then predicted for waken with posi-
tive durations on each variable, respectively. Finally,
a model for the full interval between births is devised
and tested. Because the WFS collected information on
relatively few social and economic variables which might
be expected to influence the decision to breastfeed or to
contracept, and the duration of either, and because we
know relatively little about this decisionrmaking process,
the first portion of the model is not as fully developed
as it might otherwise be. In contrast, more information
is available about how the intermediate variables ought
to enter a model of the waiting time to the next birth.
Thus in our conceptual scheme, a set of social and
economic character istics influences whether and how long
a woman breastfeeds or contracepts, and these two
intermediate variables, along with child mortality,
9
OCR for page 10
10
influence the waiting time to the next birth. This Salem
can be thought of as a translation of the Davis and Blak e
(1956) framework to the individual level, with the first
portion involving behavioral modeling and the second bio-
metric modeling. It is shown in diagram form at the
beginning of the section on final results.
Th e model will be decor ibed from the inside out:
f irst, the biometric model of fertility; then determinant'
of the duration of breastfeeding and of contraception;
and f inally, determinants of the probability of breast-
feeding or contraception. To do this, we must define
some statistical terms and functions. The reproductive
history of a woman is taken to be a point process, that
is, a random collection of points along the time axis
(Cox and Isham, 1980) . Events such as a birth, initiatior
or termination of breastfeeding, initiation or termination
of contraception, or a child death define the point=; the
time between related events def ines intervals on the axis,
f or example, birth intervals or duration of breastfeeding.
Such a process can be specified in three ways, each
containing the same information expressed differently:
the interval specif ication , the counting specif ication,
and the conditional intensity specification. The interval
specification is bred on the joint distribution of
intervals. The counting specification i'; based on Me
dis tr ibution of the number of events in some f ixed
interval. The conditional intensity specification tells
the probability of an event in a small time interval,
conditional on the history of the process up to that
point in time tcox and I sham, 1980 ) . For example, i t
would tell the probability of a birth at a given time
instant conditional on the woman's reproductive history.
For our purposes, this is the most intuitive approach to
modeling the childbearing process.
The intensity specification is a generalization of the
well-known hazard function. The hazard approach requires
the existence of a density function for waiting times,
that is, times between events. The relationships among
the hazard, and the density and the distribution functions
for waiting times is given below. Let T be a random
variable denoting time between events. Suppose it has
probability density function f(t) and cumulative
distribution function F(t), then
F(t) = Pr{T
OCR for page 11
11
Pr {t< - tea }
.
fit) ~ km
a—on a
The survivor function is defined as
F(t) - 1 - F(t) ~ Pr{T>t}.
The hazard function is now given by
.
( ) i_0+ Pr{t t}
a l im Prt ~ ]?t tit }
~—0+
=
f(t)
il(t)
.
1
Pr{T> t}
The hazard function is related to the survivor and
dens ity functions by
£'(t)
in(t) = - =
-
_ log F(t)
F (t ) at
.
Us ing the initial condition that F (O) = 1 and integrating
one has
F(t) = exp[-J h (u)du]
o
from which it follows that
f(t) = h(t)exp[-l h(u)du]
o
.
Clearly information about the hazard function is equines
lent to information about the dens ity of waiting times or
the ~ urvivor function .
OCR for page 12
12
TO rERTILITY EQUATION:
CONCEPTION
MODELING THE TIME TO NEXT
our approach will be to model the conditional probability
of having a birth (or conception), given that it has not
already happened in the birth interval in question, and
then to consider how this hazard function (or conception
rate) is modified by breastfeeding or contraception, or
other characteristics such as a child death. Consider
the following schematic representation of the birth
interval:
Itch birth
concept ion leading
to (k+1 ) th birth
+~nonsuscept ible period-+ +-ouscept Able periods ~ -gestat ion-+
( k+ 1 ) ah bjr th
The discussion of the hazard function is based on thin
decomposition of the birth interval into components
influenced by different factors: a nonsusceptible period
following a birth composed of postpartum and possibly
lactational amenorrhea, a susceptible period where the
probability of conception may be modified by contracep-
tion, and a period of gestation. The goal is to derive
some notion of the shape of the hazard function and how
it behaves by consideration of these components. The
model will not explicitly include fetal mortality.
(Underreporting of pregnancy wastage is very severe in
the WAS; see Chidambaram et al., 1980). Spontaneous
abortions are implicitly included in the sense that they
reduce observed conception rates in the susceptible
period. In addition, the probability of conception for
any given woman is influenced by a variety of factors,
some of them genetic, which cannot be measured and which
will not be included in the model . Th is means that th e
estimated hazard function may appear to decline where the
true hazard does not. (Explicit modeling of this unmea-
sured heterogeneity is beyond the scope of the present
research; for a discuss ion of the biases involved see
Flinn and Heckman, 1982; Singer and Heckman, 1982;
Heckman and S. inger, 198 2 ) .
OCR for page 13
13
For convenience, instead of considering the intervals
between births, consider the intervals between conceptions
leading to live births, 80 that an interval begin. with a
period of gestation followed by postpartum amenorrhea.
For the purposes of modeling, we will assume that the
period of gestation is fixed at 9 months. This follows
standard biometric practice (see Leridon, 1977). During
this period, the probability of conception is zero. The
period of amenorrhea following a birth is often taken to
be fixed at 2-3 months. (Leridon cites a mean of 58 days
[1977:831; Bongsarts cites 1.5 to 2 months [19831.) Bow-
ever, this distribution is considerably more variable
than is that for the duration of gestation, and seems to
bear incorporating into the model. (Leridon suggests a
maximum of 11 months of amenorrhea in the absence of
lactation [1977:831.) The period of amenorrhea may be
followed by irregular cycles. In 90 percent of cases,
ovulation returns before the first menses or in the first
cycle (Leridon, 1977:84; Bong marts, 1983). However, the
first few cycles following the return of menses tend to
be more variable in length, with the proportion of anovu-
latory cycles falling from 10 percent to less than 5
percent in the first five cycles (Leridon, 1977:84).
After this period of irregular cycles, the risk of
conception is usually taken to be a constant, estimated
to be .20 or .25 near marriage for women in their
twenties, with a mean waiting time to conception of 5 to
8 months depending on the variability of fecundity
(Leridon, 1977:33-36). Specifically, the distribution of
waiting times in the susceptible period is assumed to be
exponential with a constant occurrence rate. Combining
this information yields the graphical display of the
conception rate shown at the top of page 14. Here the
hazard is constant and equal to zero for 12 months during
the nonsusceptible period (9 months gestation plus 3
months postpartum amenorrhea), and then gradually rises
to a second constant level as the woman enters the
susceptible period.
Now consider how this hazard may be modified by breast-
feeding and by contraception, respectively. Breastfeeding
lengthens the period of postpartum amenorrhea. The extent
of this impact depends on the duration of breastfeeding:
with up to 3 months of breastfeeding, the duration of
amenorrhea exceeds the duration of breastfeeding; after 3
months, each additional month of breastfeeding adds less
than 1 month of amenorrhea (Leridon, 1977:85; Bongaarts,
1983). The impact of breastfeeding on amenorrhea also
OCR for page 14
~4
DIAGRAM 1 Hazard Function for Conceptions for In To
do not Breastfeed and do not Contracept
concept ion
rate
~ 2
1\
v
10
+---gestat ion---+ 4-
1
amenorrhea
t+-nonsuscept ible period-+
kth concept ion
20
period of irregular cyc lee
+--suscept ible- ++
( k-1 ) th concept ion
months
depends on both the fr equency of nursing and whether
breastfeeding is full (no supplementation) or partial
(the infant receives other nourishment). Full breast-
feeding appears to have a much greater impact than
partial on amenorrhea, but neither has any apparent
effect after 18 months. In addition, little is known
about the return of ovulation, the propor tion of anovu-
latory cycles, or the regularity of menses among women
who have stopped lactating. Once menses return, it is
plaus ible that the same model of constant conception
rates is appropr late both for noncontracepting women wh o
are breastfeeding and for those who are not, although
there is some evidence that conception rates may be lace r
among women who continue breastfeeding after the menses
resume. The expected impact of breastfeeding is shown by
the dashed line in Diagram 2 on page 15. Contraception
reduces the conception rate dur ing the susceptible
period. Most biometric models assume that a woman
contracepts with an ef festiveness c, which is
method~specif ic and constant throughout the per iod of
contraceptive use . Dur ing this per iod, the conception
rate is proportionately reduced by the factor c. This
is shown by the dotted line in Diagram 2. Once contrace'
tion ceases, the conception rate shifts back up and is
the same as for women who do not breastfeed and do not
contracept. Thus, breastfeeding shifts the curve to the
r ight, whereas contraception shifts it down. it is beyond
OCR for page 15
15
DIAGRAK 2 Hazard Function for Conception:
concept ion
rate
2
Three Cases
10 20 months
"natural fertility: when who do not breastfeed Id to not
contracept .
case 1: women Who breastfeet and do not contracept.
case 2: wowed who contracept and do not breastfeed.
the scope of the present research to study interactions
of brea~tfeeding and contraception; moreover, only a
small fraction of women simultaneously breastfeed and
contracept in the countries we are studying (Pebley et
al., 1981).
Infant and child mortality pose a difficult problem in
this study of fertility. Short birth intervals and high
child mortality are often mutually reinforcing. First,
mortality rates of children born only a short time after
an older sibling are known to be higher than rates for
children born after long intervals (Wolfers and Scrimshaw,
1975). Second, a child death may truncate breastfeeding
or alter contraceptive practice.
This last effect should
be captured through the inclusion of breastfeeding and
contraception in the fertility equation. Nevertheless,
there may be an additional behavioral effect if couples
try to replace children who die, for example, by
increasing coital frequency. To determine whether there
is a residual impact of child mortality on fertility
apart from changes in breastfeeding and contraception;
therefore, child mortality is included in the model.
For the most part, social and demographic character-
istics of the woman or her husband that may affect
fertility are expected to work primarily through their
influence on breastfeeding and contraception. Two
possible exceptions to this are age and parity. Empirical
---.. "~ -..~- char intrauterine mortality and the
probability of stillbirth rise with parity (Leridon,
1977; Bongaarts, 1983). Since these pregnancy outcomes
are much less likely to be reported than live births
L =:31~1 ~= I—~~ ~ ~
OCR for page 16
16
(Chidambaram et al., 1980), this increasing risk may lead
to longer lie-birch intervals and conception rates the t
appear to be rawer at higher parities. Parity h" be0
included among the covariates to help alleviate this
problem. me particular specification for parity and the
other cavar fates will be deacr ibed following the dis-
cus`;ion of estimation procedures. In addition, result
will be presented separately for f ive-year color ts for
two reasons: fecundity decline. with age, and reproduc-
tive behavior may differ substantially for different
cohorts. Duration of marriage is omitted fram this model
because, although coital frequency may decline with
duration of marriage, recent research using WFS data
shows little or no effect of marriage duration on
fertility (Casterline and mberaft, 1981). Although
social and economic characteristics of the woman or her
h usband can be expected to influence fertility pr imar fly
through their influence on breastfeed~ng and contracep-
tion, these variables are included in the model to
determine whether there i'; any residual impact on
fertility. Therefore, ache model includes information on
the education of both spouses and whether they reside in
urban areas. Although it would be of some interest to
examine the joint decisions about female labor force
participation, fertility, breastfeeding, and oontracep~
Lion, to do so adequately is beyond the scope of the
present research; the model therefore includes only some
rudimentary information on the woman's work experience
since marriage.
THE BREASTl?~:~:nING EQUATION: MODELING TlIE: DURATION OF
BREASTFI4:~ ING
In formation on the duration of breastfeeding for the last
closed and the open birth interval it; available from the
WFS core questionnaire. We are therefore limited to
modeling 'the probability of breastfeeding in either the
last closed or the open interval and the duration of
breastfeeding, that is, discontinuation rates, for women
who breastfeed. Fore careful modeling of the probability
of initiating breastfeeding following any birth is not
possible with the simple estimation strategy used here.
A schematic representation of breastfeeding for the last
closed and the open birth intermural for a woman who breast-
feeds in both intervals and who has stopped breastfeeding
by the time of interview is given on page 17.
OCR for page 17
17
tth birth
kth integral—~ +
—breast feeding--- note +
(k+1 ) h birth
interim
(k. 1 ) th interve 1 -___+
+--breas t feedings not ~ +
Women may discontinue breastfeeding for two reasons:
some women may discontinue immediately because of medical
or physical problems; others will continue breastfeeding
until some later, perhaps socially prescribed, weaning
date. Those in the first group will have very short
durations of breastfeeding, shown by the function f1 in
Diagram 3 below; those in the second group will have
longer durations, grouped around a second modal value
shown by the function f2. As a consequence, the density
of waiting times for all women may be a mixture of these
two densities.
DIAGRAM 3 A Two-Fold Mixed Weibull Density Function
1.
~`fl(~)
-
/~(t)
— _ ~
I\
\\ f2(t)
,~
The role of child mortality in relation to breastfeed-
ing is a particularly difficult problem. For some
children, death truncates breastfeeding; for others,
short breastfeeding may precipitate death; for still
others, poor health may result in early termination of
breastfeeding, which can in turn result in worse health
OCR for page 18
18
and possibly death. This poses a severe endogeneity
problem. Ideally, breastfeeding and child mortality
should be modeled simultaneously as a multi-state
process, indicating whether breastfeeding terminated
before the child died. Unfortunately, the WFS has
grouped the data on date of death into quite large
intervals so that it is not possible to tell how long
before the child's death breastfeeding was terminated.
Even if estimation of such a model were possible, one
could not determine without additional information
whether breastfeeding was terminated because a child was
unhealthy or whether early termination of breastfeeding
resulted in a child death. Therefore, from these data,
it will be possible to determine only descriptively
whether children who survive are breast fed longer than
those who do not;
Social and demographic characteristics of the woman
and her husband can also be expected to influence
duration of breastfeeding. If breastfeeding is being
used for family limitation purposes, then its duration
should be longer at higher parities, although empirical
evidence for this is weak (Jain and Bongsarts, 1980; Butz
and DaVanzo, 1978). m e greater availability of com-
mercial supplements, as well as better opportunities for
women to participate in the labor force in urban areas,
may reduce duration of breastfeeding among urban resi-
dents. In addition, it is known that women's education
and breastfeeding are inversely related (Jain and
Bongaarts, 1980). Ideally, a model of breastfeeding
would include information on other opportunity costs of a
woman 's time; unfortunately, such information is not
available from the WES. However , some rudimentary
information on the woman's work experience since marriage
is included in the model to test its value as a proxy for
the opportunity costs of women 's time.
THE: CONTRACEPTION EQUATION: MOD=TNG DISCO=I=ATION
RATES
Th e family planning module of the WES questionnaire used
in Colombia and Costa Rica includes information on
duration of contraceptive use for the last closed and the
open interval . These are the only two coun tr ies for
wh ich such information is available . As in the case of
OCR for page 19
19
breastfeeding, both the probability of using contraception
in either interval and contraceptive discontinuation are
modeled. The diagram below gives a schematic represen-
tation of contraceptive use in the last closed and the
open interval for a woman who contracepts in both
intervals and is still contracepting at the time of
interview.
Ith birth
+ _________Icth interval
(k+1 ) eh birch
ineervie
note ~~ contracept ing- ++--not~ ~~ ^~~s^~ i "~___
A- eve rim e
Studies of contraceptive discontinuation abound in the
literature. One example using a similar methodology is
Potter and Phillips (1980). Diagram 4 below gives the
estimated hazard rate of discontinuing any method among
pill acceptors.
DIAGRAM 4 Observed and Predicted Monthly Probabilities
of Discontinuing Any Method Among Pill Acceptors
d i~cont inuat ion
rate .
\
24 36 48 60 months
The hazard function for contraceptive discontinuation
should be shaped differently for different methods:
users of coitus-dependent methods may be expected to use
OCR for page 20
20
for shorter per iod'; than users of more modern coitus-
independent methods. Women who want an additional child
are likely to contracept for shorter durations. In
addition, a child death may lead some women to discontinue
contraception earlier than they would have otherwise.
However, the model of contraceptive discontinuation will
not include reasons for discontinuation since these may
change with the length of use. It has also been argued
that breastfeeding and contraception are competing ways
of postponing the next birth. If this is the case, we
would expect women who brea~tfeed not to contracept or to
contracept for shorter durations. The concentration of
family planning services in urban areas, as well as
better availability of supplies and medical care, may
make urban women more likely to use and to continue to
use contraception. Research has also shown that more-
educated women continue contraception longer than do the
less-educated (Potter and Phillips, 1980). Participation
in the labor force may increase the incentive to contra-
cept for longer periods. Although detailed information
on labor force participation is not available, the
less-than-ideal data available are included in the model
to test their utility as a proxy for opportunities in the
labor force.
TlIE: PROB"ILITY OF BREASTFEF~ING "D THE PROB"ILITY OF
US ING CONTRACEE'TION
Th e pr evalence of both breastfeeding and contraception
dif fer s quite markedly between Colombia and Costa Rica .
For example, 19 percent of Colombian women and 7 percent
of Costa Rican women were breastfeeding at the time of
the survey; while 52 percent of Colombian women and 78
percent of Costa Rican women were using contraception.
Th erefor e, ther e is some interest in investigating the
determinants of the propensity to breastfeed and the
propensity to contracept in either of the last two birth
intervals for these two counts ies . Only the background
variables describing social and economic characteristics,
taken to be fixed over the last closed and the open
i nterval, are expected to influence these probabilities .
It is hypothesized that more-educated women and those
w ith more-educated husbands will be more likely to con-
OCR for page 21
21
tracept and less likely to breastfeed; women living in
urban areas will be more likely to have access to and
therefore use contraception, but may also be less likely
to breastfeed; women with experience in the labor force
since marriage and those who have worked outside the home
may have more ~modern. ideas and be more likely to
contracept and less likely to breastfeed than women who
have not had these experiences.
Representative terms from entire chapter:
child mortality
