A Methodology for Studying Child Mortality Differentials in Populations with Limited Death Registration. Claire Noël-Miller Douglas Ewbank

Transcription

1 A Methodology for tudyng Chld Mortalty Dfferentals n Populatons wth Lmted Death Regstraton. Clare Noël-Mller Douglas Ewbank Indrect methods of estmaton reman an mportant tool for ganng an understandng of chld mortalty condtons n areas wth lmted, defcent or non-exstent vtal regstraton systems. The now commonly used Brass method estmates mortalty from reports of chldren ever born and chldren deceased organzed by duraton of ther exposure to the rsk of death, as approxmated for nstance by the age of the mother (Brass 1975; Brass and Coale 1968; Unted Natons 1983). Central to the Brass chld mortalty estmaton s the dea that the proportons of chldren dead to women aged 15-19, 20-24, 25-29, 30-34, 35-39, and approxmate the lfe table values of q(1), q(2), q(3), q(5), q(10), q(15) and q(20) respectvely. The basc Brass equaton: q( x) = k D (1.1) uses age group-specfc multplers (commonly known as k coeffcents) to translate data on the reported proporton deceased n a gven age group, D, nto lfe table estmates of the proportons deceased by varous ages, q(x). An mportant drawback of the Brass method les n ts nablty to allow for the formal test of dfferentals n mortalty across groups snce the mortalty schedule underlyng the calculaton of k s typcally consdered to be unversal. In an attempt to address ths lmtaton, Preston and Trussell ntroduced a multvarate approach to studyng mortalty dfferentals from data on chldren ever born and chldren survvng (Trussell and Preston 1984; Trussell and Preston 1982). Despte some mportant statstcal problems related to ts ntrnsc assumpton of lnearty between covarates of mortalty and mortalty outcomes (Hll 1989), ths method now consttutes the cornerstone of analyses of chld mortalty dfferentals for many populatons. The Preston-Trussell method usually reles on publshed tables of multplers (Brass 1975; ullvan 1972; Trussell 1975) themselves based on an ndex of the shape of the fertlty schedule and on a standard lfe table (Coale and Demeny 1983). In practce, publshed tables of multplers can be of lmted use when the actual age pattern of mortalty dffers substantally from the pattern underlyng the calculaton of the multplers. We propose an alternatve to the Preston-Trussell method for the multvarate analyss of assocatons between chld mortalty as measured by Brass questons and varous characterstcs of the envronment nto whch the chldren are born. We apply the method to data from the 1993 Gamban census and estmate the effect of varous covarates of mortalty va maxmum lkelhood. Partcular attenton s pad to the effects of educaton 1

2 and rural or urban locatons. Our model assumes a dchotomous dependent varable recordng falures (survvorshps n our case) and successes (deaths n our case) followng a gven number of trals (chldren ever born). Our approach s based on a skewed logstc regresson (also referred to as a scobt model), an elaboraton upon the standard logt model. The scobt model relaxes the assumpton that ndvduals wth ntal probablty P =.5 of experencng a success are most senstve to changes n the ndependent varable (Nagler 1994). The analyss offers a second nnovaton by relyng on multplers more carefully talored to local mortalty condtons. In leu of the North model varant of the regonal model lfe tables (Coale and Demeny 1983), typcally assumed for Afrcan countres, the calculaton of the k coeffcents s based on a lfe table more closely reflectng the mortalty experence of the Gamba. In addton, greater accuracy s obtaned by takng nto account possble varatons n the fertlty schedule across specfc sub-samples of the populaton. We examne how well the method performs n small samples relatve to the Preston-Trussell method and to the logt model. mlarly to the logt model, skewed logstc regresson belongs to a class of models known as generalzed lnear models (GLM), whch move beyond ordnary regresson models to accommodate non-normal response dstrbutons and modelng functons of the mean (Agrest 2002; Dobson 2001; Hardn and Hble 2001; McCulloch and earle 2001). The outcome varable,.e. the death or the survval of a chld, s assumed to result from a fxed number of trals (the number of chldren ever born) and to follow a bnomal dstrbuton 1. The lnk functon of the scobt model s the logt or the log odds of the probablty of experencng a success. The logt and the scobt models dffer n that the former assumes that ndvduals wth a probablty of success equal to 0.5 are most senstve to changes n the ndependent varables. Then, should women wth probablty of experencng a chld death dfferent than 0.5 be n realty most senstve to changes n the specfed covarates of mortalty, the logt model would be msspecfed. Rather than mposng an unrealstcally hgh value at whch ndvduals probablty of experencng a chld death s most senstve to stmulus, our model estmates an addtonal parameter for the maxmum pont of senstvty. 1 A Posson dstrbuton for the response varable has been proposed as an alternatve to the normal dstrbuton assumed n the Preston-Trussel method (see for nstance MacLeod 1999). The Posson dstrbuton s attractve because t s partcularly suted for non-negatve ntegers representng counts of a rare occurrence followng a number of events. However, the Posson dstrbuton assumes that the count data result from a small rsk n a large populaton (.e., number of trals). Ths s not the case when consderng the number of chldren ever born. Thus, the Posson dstrbuton s approprately used when there s no upper lmt to n, the number of trals and for counts of events that occur randomly ether n tme or n space. In some cases, when n s very large and when the probablty of success followng each tral,π, s very small, the Posson dstrbuton can be appled as an approxmaton for the bnomal. Ths s however not the case n our context. 2

3 The skewed logstc model presented n the paper s gven by: γ = β + β x + β x + + β x + γ (1.2) µ k k EPD, where γ represents the logt of µ, the probablty of experencng a chld death, µ β 0 s the constant term, the x represent covarates of mortalty, and γ EPD represents the logt of the expected proporton deceased (EPD). The values of the { EPD } used n equaton (1.2) were obtaned by turnng equaton (1.1) around so that: q ( A ) EPD = (1.3) k where, q ( X ) s a standard proporton of chldren who ded by exact age X and k s derved to more closely reflect age patterns of mortalty and fertlty n the Gamba. Fgures for the standard mortalty schedule (.e. the q ( X ) values n equaton (1.3)) were obtaned from a sngle year of age lfe table of the Mlomp area n énégal, coverng the perod for both males and females (Pson, Gabadnho, and Enel 2001). The Mlomp Demographc urvellance urvey (D) ste s a rural, mostly rce cultvatng area located n outhwest énégal, near the border between énégal and Gunea-Bssau (for more detals see INDEPTH 2002). Because of ts relatve proxmty to the Gamba, use of the Mlomp data nsures a smlar pattern of mortalty to that found n the Gamba. However, the level of mortalty n the Mlomp data was adjusted for that n the Gamba. The scobt model s also estmated wth values of the EPD derved a) by sngle year of age and b) for dfferent sub-groups of the populaton. The latter s done by assumng not one sngle fertlty schedule for the entre populaton, but rather dfferent fertlty schedules for dfferent sub-groups of the populaton defned accordng to the varous combnatons of lteracy levels and rural or urban locatons. The paper compares the performances of the skewed logstc regresson, the logt model and the Preston-Trussell method n estmatng covarates of mortalty based on small sample szes. We run Monte Carlo smulatons of the three methodologes appled to 1,000 samples of sze N=1,000 and 1,000 wth N=5,000 randomly drawn from the entre Gamba census. Results of the smulatons are compared to the estmates obtaned wth the scobt model appled to the entre census. In partcular, the Monte Carlo smulatons examne the senstvty of the estmated effects of educaton and rural or urban locatons. Prelmnary results suggest that n samples of 1,000 both the scobt and the logt coeffcents are based upwards, wth the logt coeffcents dsplayng a comparatvely smaller root mean squared error. In samples of N=5,000 the bas s reduced n the scobt model estmates and ther varance around the true estmate s very small. These results seem to ndcate that the logt model remans preferable n small samples whle scobt performs better n relatvely larger samples. 3

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