Parameter estimation of beta-geometric model with application to human fecundability data

Transcription

1 Parameter estmaton of beta-geometrc model wth applcaton to human fecundablty data B. P. Sngh, P. S. Pudr and 3 Sonam Maheshwar 3 Faculty of Commerce, DST-CIMS, Banaras Hndu Unversty, Varanas Department of Statstcs, Unv. of Allahabad, Allahabad& DST-CIMS, BHU, Varanas 3*Department of Statstcs, Banaras Hndu Unversty, Varanas, *Corr Author: maheshwarsonam@gmal.com Abstract The present study deals wth the estmaton of the mean value of fecundablty by fttng a theoretcal dstrbuton from the observed month of frst concepton of the marred women who dd not use any contraceptve method before ther frst concepton. It s assumed that fecundablty s fxed for a gven couple, but across couples t vares accordng to a specfed dstrbuton. Under the classcal approach, methods of moment and maxmum lkelhood s used whle for Bayesan approach we use the above two estmates as pror for fecundablty parameter. A real data analyss from the thrd Natonal Famly Health Survey (NFHS-III) s analyzed as an applcaton of model. Fnally, a smulaton study s performed to access the performance of the several of methods used n ths paper Key words: Beta-geometrc dstrbuton, fecundablty, NFHS-III, pror dstrbuton, R-Envronment, VGAM Package. AMS Subject Classfcatons: 6D05; 6F0; 6F5.. Introducton The sze and composton of populaton s hghly related wth fertlty rate and growth rate of the populaton and fertlty and growth rate of the populaton s governed and related by the terms fecundty and fecundablty. Fecundty s related to the bologcal capacty to conceve of woman and fecundablty s the probablty of concepton durng a gven menstrual cycle of those women who dd not use any famly plannng method before ther frst concepton and are sexually actve. Fecundablty has an opposte relatonshp to the tme nterval requred to conceve from marrage to frst brth nterval. Ths tme nterval s

2 also known as concepton delay, concepton nterval and concepton wat. Concepton nterval and fecundablty are the two mportant and nter related fertlty parameters and these are regarded as the most drect measures of fertlty of a populaton. Thus the concept of fecundablty s one of the prncpal determnates of fertlty and n human reproductve behavor. Although, the theoretcal mportance of fecundablty s beyond queston, there are several dffcultes n estmatng t from drect observaton. Fecundablty s frequently estmated from the dstrbuton of watng tme to concepton. Gn (94) frst consdered brth ntervals as watng tme problems dependent on fecundablty. Hs work Deals wth pregnances and brth of frst order under constant fecundablty. However, It s not realstc to assume constant fecundablty as t s to be governed by many soco economc and demographc Varables. In fact, there s enough evdence that couples vary n ther fecundablty. About 30% of sexually actve couples acheve pregnancy n ther frst non contracepton cycle, a smaller proporton of the remanng couples acheve pregnancy n the second, and wth each addtonal unsuccessful cycle, the concepton rate contnues to declne, as the rsk sets become further depleted of relatvely fecund couples [Wenberg and Gladen (986)]. The fecundablty parameter s assumed to follow certan dstrbuton as fecundablty vares from women to women. One can assume one of many contnuous dstrbutons for fecundablty les n the parameter space [0, ]. Even though number of possble (Kotz & Van Dorp, 004; Johnson, Kemp, & Kotz, 005) unvarate contnuous dstrbutons defned on the standard unt nterval [0,] are avalable as the mxng dstrbuton of the success probablty random varable, the Beta dstrbuton denoted as Beta (a, b), where a and b are the two shape parameters of the Beta dstrbuton, s the most commonly used mxng dstrbuton to model the random varable defned on the standard unt nterval [0,] due to ts ablty of accommodatng wde range of shapes. Thus the betageometrc (BG) dstrbuton, represented by Geo (theta) ^ Beta (a, b), s consdered as a very versatle dstrbuton n modelng human fecundablty data n lterature. Extensve lteratures exst on the study of beta-geometrc dstrbuton. The assumpton of a beta dstrbuton for the probablty of concepton was orgnally proposed by Henry (957). For the women of constant fecundablty; a geometrc dstrbuton s

3 used for the watng tme to marrage tll concepton whle for the women of heterogeneous fecundablty the resultng dstrbuton for the watng tme tll concepton s beta geometrc. The parameters of ths mxed dstrbuton have practcal nterpretaton. Accordng to Sheps (964), fecundablty affects fertlty through ts relatonshp wth the average tme requred for a concepton to occur and can also be consdered as the transton probablty for the passage from the susceptble state to pregnancy. In a homogeneous populaton, fecundablty s equal to the recprocal of ts mean concepton delay but for heterogeneous populatons, the mean fecundablty s usually modeled on two parameters [Jan (969)]. Wenberg and Gladen (986) consdered that the decrease n concepton probablty over tme s a sortng effect n a heterogeneous populaton, rather than a tme effect. They proposed a general model for heterogenety and studed n detal the partcular case where the dstrbuton for the number of cycles to concepton has a beta-geometrc dstrbuton. Pual (005) develop tests of heterogenety n the fecundablty data through goodness of ft of the geometrc model aganst the beta-geometrc model along wth a lkelhood rato statstc and a score test statstc. Islam et al (005) also made an attempt to compare the two methods of estmaton of the mean value of fecundablty usng the Bangladesh Demography and Health Survey ( ) data. In leu of above consderatons, the paper s organzed as follows. In secton, we descrbe the model by assumng that fecundablty s fxed for a gven couple, but across couples t vares accordng to beta dstrbuton and hence obtan the beta-geometrc model as the uncondtonal dstrbuton of the concepton delay dstrbuton. In secton 3., we obtan the moment estmate of fecundablty parameter. In secton 3., we obtan the maxmum lkelhood estmators (MLE) of the unknown parameter of beta-geometrc dstrbuton. It s observed that the MLE s not obtaned n closed form, so t s not possble to derve the exact dstrbuton of the MLE. Therefore, we propose to use the asymptotc dstrbuton of the MLE to construct the approxmate confdence nterval. Further, by assumng beta pror of the fecundablty parameter, Bayes estmate s obtaned n secton 3.3. It s observed that the posteror dstrbuton of fecundablty parameter also becomes beta-geometrc. In Secton 4, demonstrate the applcaton of the beta geometrc dstrbuton to the data obtaned from the Natonal Famly Health Survey (NFHS)-3. In secton 5, a

4 smulaton study s also carred out to check the performance of classcal and Bayesan methods of estmaton.. The Model After a couple decdes to have a chld, the number of months elapsed before the tme of concepton s denoted by X. If the fecundablty at each month, stays constant over tme for a gven couple then X has a geometrc dstrbuton E( x) x P X x ;0 ; x 0,,,3... () and mean fecundablty s E x Ths s known as the condtonal dstrbuton of concepton delay. Now f vares among. couples accordng to beta dstrbuton, then has the followng densty functon f B(, ) ( ) ( ) ;, 0 Where B(, ) s the beta functon, s the gamma functon defned as x x e dx 0 and (, ) are two unknown non-negatve parameters. The mean and the varance of beta random varable θ are and ( ) ( ) respectvely. And the uncondtonal dstrbuton of the concepton delay X s gven by B(, x ) P( X x) g( x) f ( x, ) d ( ) ( ) 0 P X x f d () 0 B(, ) Ths dstrbuton s known as beta-geometrc dstrbuton. In the human reproducton lterature, P(X=x) s the probablty that concepton occurs at x for a randomly selected couple. Wenberg and Gladen (986) wrtten the beta-geometrc dstrbuton n terms of the parameter and / parameter and θ as the shape parameter, and s gven by, where p s nterpreted as the mean

5 y ( ) } 0 P( Y y n) y The mean and varance for the above defned form of beta-geometrc model s ( )( ) varance respectvely. ( ) ( ) 0 and 3. Estmaton of Fecundablty Parameter: In ths secton, we obtan the estmate of fecundablty parameter by usng the followng three methods of estmaton: 3. Moment Estmaton: Let us suppose that m and m are the two observed frst and second raw moment of the months requred for women to conceve for the frst tme after ther marrage. The correspondng populaton moment of X about orgn, condtonal on, as gven by the smple geometrc dstrbuton are x E x x x and E x x x0 x0 To obtan the uncondtonal moment of X, we have to put the value of to get r r r r x x x x x r... E p d ; where r r r r x x x x... x r ; Now, equaton m wth and m wth we get m m ˆ ; ˆ m ˆ m m m (3) From equaton (3), we can easly obtan the moment estmate of and, hence moments estmate of the fecundablty parameter. As the sample becomes very large, the

6 asymptotc varance-covarance matrx can be obtaned smlarly those obtaned by Islam et al (005) usng the method descrbed by Rao (95). 3. Maxmum Lkelhood Estmaton: Suppose that data are avalable on N ndvduals as x ;,,..., n. The lkelhood functon for data based on beta geometrc dstrbuton s gven as n B(, x ) L (4) B(, ) And the correspondng log-lkelhood L ;, s gven as log L L log B(, x ) n log B(, ) (5) The score functon U s defned as the gradent ofl, derved by takng the partal dervatves of L wth respect to and. The components of the score functon, U U U T are gven below L U N N x N (6) L U x N x N (7) The maxmum lkelhood estmates and can be obtaned ether by drectly maxmzng the above log lkelhood functon wth respect to or by solvng the two smultaneous equatons obtaned by equatngu 0. From equatons (6) and (7) we can see that the MLEs of, cannot be obtaned n closed form. Therefore, we need some numercal teratve procedures such as Newton-Raphson method. One can also use the vglm functon of R-Envronment to obtan the MLE of, one can easly obtan the MLEs of fecundablty parameter. samplng dstrbuton of ' ˆ, ˆ s N 0, matrx consstng of the followng elements:. Usng the nvarance property of MLEs, Further, the asymptotc, where s the Fsher s nformaton

7 log L ˆ log L and ˆ log L ˆ, ˆ The asymptotc 00% confdence ntervals (C.I.) for, Here V ˆ s the varance of ˆ obtaned from and / s ˆ z / ˆ V. z s the upper th 00 / percentle of a standard normal dstrbuton. The respectve asymptotc dstrbuton of fecundablty parameter s ' ' N(0, ) where,,. 3.3 Bayesan Estmaton: In many practcal stuatons, t s observed that the behavor of the parameters representng the varous model characterstcs cannot be treated as fxed constant throughout the lfe perod. In the ntroductory secton, we have already dscussed that the fecundablty parameter should not be assumed constant as t s governed by varous socoeconomc and demographc varables. Therefore, t would be reasonable to assume the parameters nvolved n the model as random varables. Keepng n mnd ths fact, we have also conducted a Bayesan study by assumng the followng beta pror for fecundablty parameter, h Here also, the reason of choosng the beta pror for the fecundablty parameter s straghtforward as t s the most commonly used mxng dstrbuton to model the random varable defned on the standard unt nterval [0,] due to ts ablty of accommodatng wde range of shapes. Here the hyper parameters and are assumed to be known real numbers. Based on the above pror assumpton, the jont densty functon of the sample observatons and becomes (8) n x L x,, Thus, the posteror densty functon of, gven the data s gven by (9)

8 x 0 h, L x L x h, d Puttng the expresson of equaton (8) and (9) n equaton (0), we get the posteror densty of n x n, x n x n It s to be noted that one more plus pont of choosng beta pror for fecundablty parameter s that t s conjugate pror.e. the posteror dstrbuton also comes beta. For the squared error loss, the Bayes estmator s the posteror mean and the mean fecundablty s * n x n Wthout loss of generalty, one can assume the value of and obtaned by method of moment and maxmum lkelhood. 4. Applcaton of the Model Here we demonstrate the applcaton of the beta geometrc dstrbuton to the data obtaned from the Natonal Famly Health Survey (NFHS)-3. NFHS-3 was conducted under the stewardshp of the Mnstry of Health and Famly Welfare (MOHFW), Government of Inda, and s the result of the collaboratve efforts of a large number of organzatons. The Internatonal Insttute for Populaton Scences (IIPS), Mumba, was desgnated by MOHFW as the nodal agency for the project. Fundng for NFHS-3 was provded by the Unted States Agency for Internatonal Development (USAID), DFID, the Bll and Melnda Gates Foundaton, UNICEF, UNFPA, and MOHFW. Macro Internatonal, USA, provded techncal assstance at all stages of the NFHS-3 project. A unform sample desgn was adopted n NFHS-III. In ths Study, only women who had ever been marred n age group 5-49 were used. In order to estmate the fecundablty for women, we have extracted 3767 women out of 83 women who have had at least one recognzable concepton (regardless of outcome). We have excluded women who were pregnant before marrage. Snce our study s based on brth hstory data, we exclude those (9) ()

9 conceptons of women occurrng more than 5 years precedng the survey to avod memory lapse of the respondents. Fnally, we have also excluded those women who dd not conceve durng ther frst 5 years or 80 months of marrage, because women who fal to conceve wthn 5 years of ther marrage are consdered to be the prmarly sterle. So from the above data, we have the followng: x x 78; n= 3767 Here, E x = and hence the mean fecundablty s Now the estmated values of the parameters nvolved n the model obtaned by usng the dfferent method of estmaton are gven as follows: Method of Moment: ˆ ; ˆ = ˆ = Method of MLE: ˆ ; ˆ = ˆ = Bayes Estmaton: ˆ= Usng Moment Estmates Pror ˆ= (Usng ML Estmate as Pror) All the methods of estmaton are near about the theoretcal value of fecundablty. Although, estmated mean value of fecundablty for Bayes estmate (Usng ML estmate as pror) s much closer to the true value, and hence, we can say that Bayes procedure s best for the above data set. Bayes estmate of fecundablty usng ML estmate as pror s near to true value than those obtaned by usng moment estmate as pror. Among method of moment and maxmum lkelhood, MLE perform better than MME n terms of true value. 5. A Smulaton Study: Here, we assess the performance of the moment estmate, maxmum-lkelhood estmate and Bayes estmate of mean tme to fecundablty wth respect to sample sze n. The measures that are employed for the comparatve study of estmaton methods for the model parameters are- MMEs, MLEs and Bayes estmates. Bases and Mean Square Errors (MSEs)

10 For each of the followng optons, we smulated sx sets of data wth samples of szes 00, 00, 400, 500, 750 and,000 respectvely, and based on each set of data we computed the above mentoned measures. 4, , 0.5 4, , 0.67 The above assessment s based on followng algorthm: () Generate 5,000 samples of sze n from beta-geometrc dstrbuton. The nverson method cannot be used to generate random sample from the beta-geometrc dstrbuton as the cumulatve dstrbuton functon s an ncomplete beta (.e. ncomplete gamma) functon. To overcome ths problem, we use VGAM package of R-envronment. () Compute the moment estmate, maxmum lkelhood estmate and Bayes estmate for the 5, 000 samples, say ˆ for =,,., (3) Compute the Average Estmates (AE), bases and mean-squared errors gven by And 5000 ˆ Bas 5000 ˆ 5000 MSE 5000 (4) We repeat these steps for n =00,00..., 000 wth θ = 0.67, 0.5, 0.5 and 0.0 hence computng AE, bas and MSE for n = 00, 00,..., 000. Form the fgure and tables -4, the followng observaton can be made For all the four choces of the fecundablty parameter, The magntude of the Bas and MSE decreases as the sample sze n ncreases thereby leadng to ncreased precson n the estmaton of the fecundablty parameter. Though all the consdered methods of estmaton are precsely estmatng the parameters, Bayes estmate of the fecundablty wth MME pror are consstently

11 performng better than the other methods studed. Ths s true for all the four optons of the fecundablty parameter. Bayes estmate of the fecundablty wth MME pror s more or less concde wth the Bayes estmate but the precson n estmaton s more n Bayesan estmaton wth MME pror both n terms of Bas and the MSE. The bases are negatve for the method of moment and maxmum lkelhood whle t s postve for Bayesan method of estmaton. The bases appear largest for 0.5 whle t decreases for other values. References. Gn, C. Premeres recherches sur la fecondablte de la femme. In Proceedngs of the Internatonal Mathematcs Congress, Toronto (94).. Wenberg, P. & Gladen, B.C. (986). The Beta-geometrc dstrbuton appled to comparatve fecundablty studes. Bometrcs, 4, Kotz, S., & Van Dorp, J. R. (004). Beyond beta: Other contnuous famles of dstrbutons wth bounded support and applcatons. New Jersey, USA: World Scentfc Publshng Company Incorporated. 4. Johnson, N. L., Kemp, A. W., & Kotz, S. (005). Unvarate dscrete dstrbutons (Vol. 444). Hoboken, NJ:Wley-Interscence. 5. Henry, L. (957). Fecondte et famlle-models mathematques, Populaton,, Sheps, M. C. (964). On the tme requred for concepton, Populaton Studes, 8, Jan, A. K. (969). Fecundablty and ts relaton to age n a sample Tawanese women. Populaton Studes, 3, Paul, S.R. (005). Testng goodness of ft of the geometrc dstrbuton: an applcaton of the human fecundablty data. Journal of Modern Appled Statstcal Methods, 4(), Islaam, S.M.S., Karm, M.A., Chowdhury, M.S. and Hossan, M.S. (005). Fttng and estmaton of type-i geometrc dstrbuton by usng C-language, Asan Journal of Informaton Technology, 4(), Rao, C.R. (95). Advanced Statstcal Methods n Bometrc Research, Newyork, Wlley.

12 Table: Average Estmate of wth ther Bas and MSE for alpha=4; beta=36; theta=0. and varyng sample sze n: n Average Estmate of Bas of MSE of MME MLE Bayes Bayes MME MLE Bayes Bayes MME MLE Bayes Bayes Table: Average Estmate of wth ther Bas and MSE for alpha=4; beta=; theta=0.5 and varyng sample sze n: n Average Estmate of Bas of MSE of MME MLE Bayes Bayes MME MLE Bayes Bayes MME MLE Bayes Bayes

13 Table 3: Average Estmate of wth ther Bas and MSE for alpha=4; beta=4; theta=0.5 and varyng sample sze n: n Average Estmate of Bas of MSE of MME MLE Bayes Bayes MME MLE Bayes Bayes MME MLE Bayes Bayes Table 4: Average Estmate of wth ther Bas and MSE for alpha=4; beta=; theta=0.67 and varyng sample sze n: n Average Estmate of Bas of MSE of MME MLE Bayes Bayes MME MLE Bayes Bayes MME MLE Bayes Bayes

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