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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
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<t} = ~ f(u)du o and
11 Pr {t< - tea } . fit) ~ km aon 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<t+~ 1 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 .
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 ) .
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
~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
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~~ ~ ~
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.
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
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
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
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-
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.
