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METHODS
.
This section of the report has five parts. First, the
duration data for fertility, breastfeeding, and con-
traception are described and evaluated using Eaplan-Meier
estimates of the survival functions (Kalbfleisch and
Prentice, 1980). Second, the models of conception rates,
termination of breastfeeding, and contraceptive di~-
continuation are analyzed using estimated hazard
functions with time-varying cover iates. In the next two
parts, the sample and variables used are defined.
Finally, the equations for the probability of breas t-
feeding and the probability of using contraception are
estimated using logistic regressions.
A DESCRIPTIVE STATISTIC FOR DURATION DATA:
THE KAPLAN~MEI} :R ESTIMATOR OF THE SURVIVOR E UNCTION
The survivor function was defined in the previous
section: '(t) is the probability that the event of
interest occurs sometime after the time, t, since the
preceding event. The events are births (in the case of
fertility), termination of breastfeeding, and Contras
ceptive discontinuation. An interval is said to be
censored if the respondent was interviewed before it was
closed by the next event. Since the information used is
from the last closed and the open birth interval, there
are at most two intervals for each woman, one of which
may be censored. All intervals (even those for the same
woman) are treated as if they were independent. The
Kaplan-Meier estimator is the nonparametric maximum
likelihood estimator of F. It is defined as follows,
using the example of birth intervals (from Kalbfleisch and
Prentice, 1980:11-16). Suppose birth intervals of length
22

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23
tl < t2 < e.- < tk are observed in a sample size N from
a homogeneous population with survivor function F (t) .
Suppose further that there are d birth intervals of
length tj (jSl,...,k) and mj birch intervals are censored
b y the interview in the interval [ tj, tj +l ) . The number
of items at risk just prior to t is nj = (m + dj) +...+
(mk + dk)-the number of birth intervals that~have not yet
been closed by a birth or censored by the interview. The
Kaplan-Meier estimator is now 9 iven by
it_
F (t) = 1r
{_~ _~\
jl tj < t \ no J
It is a step function with jumps at the observed birth
interval lengths. It neglects calendar time in that i t
only uses information on the length of the interval, as
if all intervals had a conenon origin time, too. This is
also true for duration of breastfeeding and duration of
contraceptive use. These estimated survivor functions
are used to evaluate the quality of the duration data.
Preference in reporting certain interval lengths, par-
t icular ly troublesome for breas "feeding and for Contras
ception, will be revealed in large jumps in the estimated
s urvivor functions.
A CONTINUOUS—TIME ESThMATOR OF THE HAZARD E UNCTION
WITH TIME-VARYING COVARIA1~:S
me hazard function was defined above, where it was Charm
to be related to the survivor function by a simple trans-
formation and therefore mathematically equivalent to it.
For analysis of a woman's reproductive history, the hazard
f unction is preferred because it allows us to examine the
probability of an event, such as a birth conditional on
other events or behaviors in the birth interval, such as
breastfeeding or contraception. Only a simple specif ica-
tion of the hazard will be considered here. While more
elaborate models have been descr ibed in the literatur e
(see , for example , Singer and Beckman, 1982), f itting
such models can be exceedingly complicated.
The basis for the estimation procedure is to approxi-
mate the hazard by a step function, def ining subper iods
of time from the start of the interval and assuming that
the hazard is constant within those ~ubperiod-, but
shifted proportionately by the covariates. As will be

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seen in the next section, the estimated survivor
functions, which contain the name information as.the
hazard functions, are quite smooth and change relatively
slowly. This implies that the hazard functions will also
tend to be smooth, so that the step function approxima-
tion probably does not result in the loss of a great deal
of information. In its structure, the hazard model
resembles an analysis of covariance with interactions.
It is 9 iven by
hp = eXp(ax + ApZp), p = 1, .. .,P,
where p indexes subper iods from ache s tar t of the interval;
X is a vector of covariates which are fixed with
respect to the interval;
Zp is a vector of possibly ti~varying covariates;
a,Ap are vectors of parameters to be estimated; and
hp is the level of the hazard in subperiod p.
The X variables are generally a net of background vari-
ables that describe the social and economic character-
istics of the woman~and her husbands These variables are
unchanged over the course of the interval and have the
same (proportionate) effect on the hazard at all dura~
tions. me Z variables represent the more dynamic aspect
o f the model and incorporate events or behaviors occur-
r ing dur ing the interval of interest that can be expected
to modify the fundamental shape of the hazard. Either
these var tables themselves change over the course of the
interval--for example, breastfeeding when births are the
dependent var. iable--or their impact on the depended t
variable changes--that is, they are interacted with
subperiod. This model is estimated by maximum likelihood
using the program RATE (Tuma and Pasta, 1980). In
addition to assuming a constant hazard within subperiods
which is shifted proportionately by the covariates, the
estimation procedure also assumes that all heterogeneity
is measured by the co~rariates X and Zp, and that all
intervals are mutually independent, even those for the
same woman. In other words, individuals with the same
value of all cavariates in subperiod p have exactly the
same hazard rate. It should be noted that this is only
one of a number of pass Able specif ications; a s imilar
model could have been estimated using a standard package
for the analysis of contingency tables (see Laird and
Oliver, 1981, or Allison, 1982) .

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25
SAMPLE DEFINITION
The analysis here is restricted to women who have been
stably married since before the birth of their third-to-
last child for the following reasons. Detailed informa-
tion is restricted to the last closed and the open birth
intervals. This means that two intervals are sampled
from a woman's reproductive history; the date of interview
determines which two are selected. The estimation pro-
cedures used here require pooling these two intervals and
neglecting their order and calendar time; that is, they
only use information on the length of the interval, as if
all intervals had a common origin time. Intervals for
the same woman are not linked in any way, but are treated
as independent observations. This means that the process
must be stationary over the intervals sampled. Stably
married women are selected to permit treating information
on social and economic characteristics of the woman and
her husband, collected at the time of the survey, as
fixed with respect to both intervals used. Women of
parities three and higher were chosen to eliminate
intervals between marriage and first birth and to permit
the option of using information on infant mortality from
the next-to-last closed interval. The sample selection
is diagrammed below.
h marriage (k_l)th birth k b
|. (k-1 ) th intermural ~
rth (k+l)th
~kth int e real |
birth interview
~ (k+1 ) th interval+
+-last closed ~ +~ opens +
--------------durat ion of ith marriage---------------------+
DEFINITION OF VARIABLES USED IN THIS: HAZARD MODEr~-C
Three dependent variables are considered at each stage of
the analysis: duration of the interval from a live birth
to the conception of the next live birth, duration of
breastfeeding, and duration of contraceptive use. The
first variable is so defined because it is assumed that
the length of gestation (when the risk of conception is
zero) is fixed at 9 months; it is measured by subtracting

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9 months from all closed intervals and def ining currently
pregnant women as having an open interval of length zero
(no exposure). This means that the period of gestation
has been eliminated from the analysis, thereby defining
the hazard as zero duration gestation. Spells of
breastfeeding and of contraceptive use in the closed
birth interval are always closed; spell from the open
birth interval are open only if the woman is still
breastfeeding or still contracepting at the time of
interview' otherwise they are closed.
The analysis of the hazard models ho three stages.
The first is descriptive. A model with no covariates is
estimated: h ~ A , p ~ 1, . .. ,8. This model contains
much of the Muse information as the estimates of the
survivor function. Nile eight subperiods are 0-2, 3-5,
6-11, 12-17, 18-23, 24-35, 36-47, and 48+ months. Shorter
subperiods are used at "e start of the interval because
this is where the hazard typically changes most rapidly.
These estimates serve as a baseline for comparing the
impact of the co~rariate';. For models with covariates,
the number of subperiods is reduced to six, where the
first five subperiods are the same as before and the
sixth subperiod is 24+ months. This is in part to reduce
the number of parameters, and in part because there are
few observations at the longer durations. Men models
winch eight subperiods were tested, the likelihood ratio
Statistics for the added parameters were generally not
s tat); tically ~ ign if icant.
Table 1 describes the variables used in each of the
three analyses. me set of background variables is the
ame in all three equations ~ It includes indicators of
female labor force participation, urban residence, and
years of schooling for the woman and her husband. As
already noticed, these variables can be expected to
influence primarily duration of breastfeeding and
duration of contraceptive use, rather than the time to
next conception. hey are included in the fertility
equation so that their direct impact on the time to next
live~birth conception apart from their indirect effect
through breastfeeding and contraception can be ascer-
ta ined . Extensive investigation of the functional
specification of these variables was not performed here
or elsewhere in ache analysis, in part because these
var tables did not per form as well as expected, perhaps
due to measurement error. rhe woman's parity at the
start of the birth interval is included in all three
equationes it is included in the fertility equation to

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capture parity-related changes in pregnancy wastage and
fecundity not measured elsewhere. The fertility equation
also includes indicators of contraceptive method (coitus-
dependent or coitus-independent); these are treated as
shift factors that are fixed for the interval. Duration
of use is a time-varying component that modifies the
shape of the hazard. The other time-varying covariates
are breastfeeding and child survival (survival of the
child whose birth began the interval). All three are
treated as indicator variables that are specific to
subperiods.
The variables indicating duration of contraceptive use
require some special comment. The WFS for Colombia and
Costa Rica provides information on length of use in the
closed and the open interval and on current status, but
do not give start and stop dates. In order to use this
information, one must make some assumptions. It is
assumed that women who do not breastfeed begin contracep-
ting immediately postpartum; it is also assumed that
women who breastfeed begin contracepting as soon as they
stop breastfeeding unless the sum of the duration of
breastfeeding and the duration of contraception is greater
than the length of the interval from last birth to next
conception (or to interview for the open interval); in
which case a period of overlap is allowed. However,
apparently very few women simultaneously breastfeed and
contracept (Pebley et al., 1981). When alternate assign-
ment strategies allowing a delay of 1-3 months between
the last birth and the beginning of contraception and
between termination of breastfeeding and initiation of
contraception were tried, results were not significantly
altered. As will be seen when the findings are described,
the principal impact of contraception on fertility is
through the two variables which indicate use of particular
contraceptive methods at any time in the interval. These
drastically shift the hazard downward. By comparison,
the time-varying covariates produce relatively small
rearrangements of the hazard. Poor quality of the data
on when contraception was actually used within the
interval may account for these findings.
The breastfeeding equation includes the same set of
background characteristics and parity. In addition, a
dummy variable indicating whether the child in question
survived past its second birthday is interacted with
subperiod (that is, its estimated coefficient is allowed
to vary with duration of breastfeeding and thus modify
the shape of the hazard and not just its level) to show

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TABLE 1 Variable Def inition for the Hazard Models
I~RTILl" 1!QlJ&TIOII
Dependent Variable:
Fised covariates (a)
Background Characteristics:
work ~ ince marriage:
marriage .
work away from home:
home .
woman's education: years of schooling.
husband ~ ~ educat ion: years of school ing .
urban residence: du~y-1 if the woman currently lives in an urban
area.
length of interval from birth to next concept ion .
dunany-1 if the woman has worked since
duty- 1 i f the woman has worked out ~ ide the
Other Demograph ic Charac t er i 8 t ic 8
contraceptive method: set of two duty variables.
coitus-intepcodent contracept ion: dus - ,r1 if the wean used the
pill, IUD or injections in the birth intermural in question.
coitus-dependent contraception: d~y-1 if the won used a
diaphragm, foam, condom or other coitus-dependent method in
the inverval in ques t ion .
parity: a counter indicating parity at the start of the interval.
Tiae-Varying Co~aristes (Zp)
breast feeding: a set of six dusmies-1 if the woman breast fed her
child in sub~eriod p and previous subperiode.
contraception: a set of disc du~ies-1 if the woman uset
contraception in subperiod p.
child survival: 8 set of disc du~ies-1 if the child born at the
start of the interval survives through aubperiod p.
liBl~l=EDIlIG ~QVATIOII
Dependent Variable:
Fixed Covariates (X)
Background Characteristics:
Other Demographic Q`arac teris t ice:
durat ion of breast feeding.
same as in fertility equation.
parity: s~ an in fertility equation.
Fixed Co,.riates that are ""erected with Bub-Period (Z)
child survival: dull if the child born at the start of the
birth interval Were breset feeding occurs survives longer than
two years.

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TABLE 1 (continued)
Dependent Variable:
Fised Coveriste. (A)
Background Characteristics:
Other Demographic Characteristics:
COlIl=C15PTION EQUATION
durat ion of contracept ive use .
Same as in fertility equation.
parity: sa~ as in fertility equation.
breastfeeding: du~y-1 if the woman breast fed in the birth
interval where cant racept ion occurs .
child Survival: du~y=1 if the child born at the start of the
birth interval where contracept ion occurs survives longer than
two years.
Fised Co~reristes thee are Interacted With Sub-Periot (Z)
desire for an additional child: dummy-1 if the woman stated that
she wanted an additional child after the child born at the
start of the birth interval where contraception occurs.
contraceptive method: dun~yzl if the woman if the woman used a
coitus dependent method (see fertility equation for a more
detailed descript ion ) .
SUBS~LES
The equations above have been reestimated for subsamples defined by
period to test for non-linearities and to attempt to uncover biases.
These results are included in the appendix.
Periods: birth intervals begun since 1960, since 1965, since 1970.
Data for all three equat ions are progressively restricted to
intervals from the more recent period.
in a descriptive way the shorter durations of breastfeed-
ing among women whose children die. Along with the set
of background characteristics and parity, the contracep-
tion equation also includes indicators for breastfeeding
and child survival that are treated as fixed with respect
to the interval of use. Variables indicating desire for
an additional child and contraceptive method are inter-
acted with subperiod. This specification was chosen
because it is not possible to determine the relative
timing of events in this portion of the data set.
Nevertheless, fertility desires and contraceptive method
are expected to influence not only the level of the
hazard, but also its shape. Breastfeeding and child
survival were retained in the equation since their effect
on duration of use is of some interest, but their impact
was not sufficient to warrant the estimation of
additional parameters required by an interaction model.
The models were estimated for several subgroups of the

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entire sample selected for this and ysis. In light o f
the concerns expressed earlier that intervals farther
away from the survey date are less representative, th e
period was restricted to the more and more recent past:
since 1960, since 1965, since 1970. All three equations
were estimated separately for each of these Upper ions .
LOGISTIC REGRESSION ANALYSIS OF THE: PROBABILITY OF
BREASTF':~'lING AND Am: PROBABILITY OF USING CONTRACEPTION
The decision to breastfeed or to contracept may be
d etermined by factors other than those related to the
duration of lactation and contraceptive use. As a f its t
step in understanding what determines the propensity to
breastfeed or to contracept, two dichotomous variables
were defined which indicate whether the woman ever breast-
fed and whether she ever used contraception in either the
last closed or the open interval. The predictor
var tables for both of these dependent var. tables are th e
same set of background covariates defined for the
fertility equation in Table 1. She estimation procedur e
u sed is log)'; tic r egress ion anal" is .