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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
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 CONTINUOUSTIME 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
24 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) .
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
26 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
27 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
28 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.
29 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
30 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 .
