International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS)

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1 Internatonal Assocaton of Scentfc Innovaton and Research (IASIR (An Assocaton Unfyng the Scences, Engneerng, and Appled Research Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences (IJETCAS ISSN (Prnt: 4 ISSN (Onlne: 55 Performance of Generalzed Posson Regresson Model and Negatve Bnomal Regresson Model n case of Overdsperson Count Data Farhana Sada Lecturer. Insttute of Statstcal Research and Tranng (ISRT Unversty of Dhaka Dhaka, BANGLADESH Abstract: Ths paper represents the comparson between Negatve Bnomal Regresson model and Generalzed Posson Regresson model for overdsperson count data. For ths comparson, we used BDHS data n where the response varable s the total chldren ever born whch s a count data. When the response varable s count, then Posson Regresson Model as a Generalzed Lnear Model s wdely and popularly used to analyze such type of response varable and Posson Regresson model gves better result than the usual regresson model for analyzng count data. In ths paper, the descrptve statstcs of the total chldren ever born data exhbt the presence of overdsperson n the data set. Snce the total chldren ever born data used n ths study exhbt overdsperson, we can use Negatve Bnomal Regresson Model and Generalzed Posson Regresson Model. These two models have statstcal advantages over standard Posson regresson model and are sutable for analyss of count data that exhbt ether overdsperson or underdsperson. Keywords: Negatve Bnomal Regresson (NBR; Generalzed Posson Regresson (GPR; Overdsperson; Count data; Chldren ever born. I. Introducton Bangladesh s a small progressve country but ts populaton s the th largest n the world. It s most densely populated country n the world. Populaton growth refers to change n the sze of a populaton whch can be ether postve or negatve over tme dependng on the balance of brths and deaths. In Bangladesh, every year a large number of chldren born for varous causes or factors. To fnd out the nature of the determnants of populaton growth or fertlty, very lttle sophstcated and statstcally sound methods have been performed. In recent years, Posson type regresson models have been used to model count response varable affected by one or more covarates. Kng ( and Wnkelmann and Zmmermann (4 developed the generalzed even count models based on the Posson, negatve bnomal, and the bnomal dstrbutons. To understand the nature of socoeconomc and demographc factors related to populaton growth, a generalzed lnear modelng approach has been used n ths current study [5]. But there s more varablty around the model s ftted values than s consstent wth a Posson formulaton, that s, Over dsperson. To correct ths problem, we have used Negatve Bnomal Regresson and Generalzed Posson Regresson whch are also the specal case of Generalzed Lnear Model. But the man concern of ths study s to compare Negatve Bnomal Regresson and Generalzed Posson Regresson for explanng over dsperson. That means ths study wants to explan that whch model s better among Negatve Bnomal Regresson and Generalzed Posson Regresson for descrbng over dsperson of count data. Femoye, Wulu and Sngh ( noted that the Posson regresson model s not approprate when a data set exhbt overdsperson, a condton where the varance s more than the mean []. Bangladesh demographc and health survey data (BDHS have been used for ths study []. A number of canddate factors have been consdered. The number of chldren ever born s taken as the response varable The selected demographc and socoeconomc dfferentals lke mother's educatonal level, types of place of resdence, age at frst marrage, current workng status, husband's occupaton, dvson, havng rado etc. are taken as predctor varables for ths study. II. Methods A random varable Y s sad to have a Posson dstrbuton wth parameter θ f t takes nteger values IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 55

2 F. Sada, Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences, 4(, MarchMay, 3, pp.5553 z =,,. wth probablty [] e Pr{ Z z} z z!, z,,,... The parameter on the Posson regresson model may be wrtten as a loglnear model: Log x The Posson Regresson s to be ftted to the mean number of chldren, θ can be expressed as below for our ten ndependent varables (consderng = types of place of resdence, = has rado, 3 = has televson, 4 = relgon, 5 = educatonal attanment, = wealth ndex, = use of contraceptve method, = age at frst marrage, = partner s educatonal level, = current workng status. Log ( The parameters can be estmated by the maxmum lkelhood estmated method. Under the Posson, the mean, s assumed to be constant or homogeneous wthn the classes. However, by defnng a specfc dstrbuton for, heterogenety wthn the classes s now allowed. For example, by assumng to be a gamma wth mean E( and varance Var ( v and to be a Posson wth condtonal mean E(, t can be shown that the margnal dstrbuton of dstrbuton wth probablty densty functon [3] Pr[ Z z ] Pr( Z z ( z f ( d ( ( z ( Where the mean s E( Z and the varance s Negatve Bnomal regresson model may be wrtten as: z (, { log( r} z r log( log( z! z Therefore, the maxmum lkelhood estmates, ( ˆ, ˆ respect to and. Var ( Z follows a Negatve bnomal log( ( z ( z The lkelhood for log(, may be obtaned by maxmzng (, wth Generalzed posson regresson model uses response varable Z whch s count. Here, Z s defned as the total number of chldren ever born. The probablty functon of Z s gven by [] f ( z,, ( z ( z z! z And ( x exp( x ( z exp(, z,,... Where, x s a (k dmensonal vector of explanatory varables and β s a kdmensonal vector of regresson parameters. The mean and varance of z are gven by [] E( and V ( ( To estmate (β, α n the GPR model, we take the partal dervatves of the loglkelhood functon of the GPR model []. LogL(, ; z { z log( ( z ( z log( z log( z!} IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 55

3 F. Sada, Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences, 4(, MarchMay, 3, pp.5553 III. Results Table: Posson regresson verses Negatve bnomal regresson Posson Regresson Negatve Bnomal Regresson Varable Estmate pvalue Estmate pvalue Intercept Resdence Has Rado Has Televson Relgon Wealth Index Age At Frst Marrage Partner s Educaton Workng Status Educatonal level Current Contraceptve Method Table: Goodnessofft tests between Posson regresson and Negatve bnomal regresson Goodnessofft Test Measures Posson Regresson Negatve Bnomal Regresson Devance AIC. 335 Table3: Posson regresson verses Generalzed posson regresson Posson Regresson Varable Estmate Intercept.544. Resdence Has Rado..44 Has.4.54 Televson Relgon.33. Wealth Index.5. Age At Frst Marrage Partner s Educaton Workng.35. Status Educatonal.35.3 level Current. Contraceptve.4345 Method Generalzed Posson Regresson p p Estmate value value Table4: Goodnessofft tests between Posson regresson and Generalzed posson regresson Goodnessofft Test Measures Posson Regresson Generalzed Posson Regresson.4 Devance AIC..5 IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 5

4 F. Sada, Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences, 4(, MarchMay, 3, pp.5553 Table5: Negatve bnomal regresson Vs. Generalzed posson regresson Varable Negatve Bnomal Regresson Estmate pvalue Generalzed Posson Regresson Estmate pvalue Intercept Resdence Has Rado Has Televson Relgon Wealth Index Age At Frst Marrage Partner s Educaton Workng Status Educatonal level Current Contraceptve Method Table: Goodnessofft tests between Negatve bnomal regresson and Generalzed posson regresson Negatve Bnomal Generalzed Posson Goodnessofft Test Measures Regresson Regresson.4 Devance 3 4. AIC IV. Dscusson The comparson between Negatve Bnomal Regresson and the Posson Regresson models have been presented n Table. The regresson parameters estmates for these two models are somewhat smlar. Ths s expected, as estmates from these models are consstent. Of the explanatory varables, the coeffcent for the educatonal level of mother s negatve and sgnfcant whch ndcates that educated women have fewer chldren than uneducated women. As expected, the coeffcent for the dummy varable resdence s negatve ndcatng that urban famles have fewer chldren than rural famles. However, ths effect s statstcally sgnfcant. In summary, the estmaton results support the neoclasscal theory of fertlty. The standard errors of Negatve Bnomal Regresson model are slghtly larger than the standard errors of the Posson model. The devance for the Posson regresson model (Table s relatvely larger than the devance of the Negatve Bnomal Regresson model and thus ndcatng possble exstence for overdsperson. To test for overdsperson, AIC of Posson aganst Negatve Bnomal Regresson model s mplemented. Based on the AIC, the Negatve Bnomal Regresson model s not a better model for explanng overdsperson n our data set. Now we wll compare Generalzed Posson Regresson model aganst Posson model whch s the alternatve model of the Negatve Bnomal Regresson model for explanng overdsperson. IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 5

5 F. Sada, Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences, 4(, MarchMay, 3, pp.5553 The comparson between Posson model and Generalzed Posson Regresson s shown n Table 3. The parameter estmates from the both models (PR and GPR are somewhat smlar. Ths s expected, as estmated from these models are consstent. Posson regresson model founds three nsgnfcant factors whereas the Generalzed posson regresson model founds only one nsgnfcant factor ndcatng better explanaton of fertlty. Tabular results ndcate that not accountng for the overdsperson; the standard errors from the Posson Regresson model are under estmated. Consequently, the tstatstc for testng the sgnfcance of each regresson parameter s generally upward based for the PR model. Table 4 gves the estmates of dsperson parameter usng GPR model s postve ndcatng overdsperson. From Table 4, the GPR model s preferred to the PR model based on two goodnessofft measures: devance and AIC. For example, the GPR model has a smaller devance than the PR model. The estmated AIC for GPR model s.5 whereas t s. for the PR model ndcatng better fts usng GPR models. Table 5 presents the comparson between Negatve Bnomal Regresson models and the Generalzed Posson Regresson models. The parameter estmates for these two models are smlar. The standard errors of the GPR model are smaller than the standard errors of the Negatve Bnomal Regresson models. Table represents that the GPR model s preferred to the Negatve Bnomal regresson model based on two goodnessofft measures: devance and AIC. In ths table, the GPR model has a smaller devance than the PR model. The estmated AIC for GPR model s.5 whereas t s. for the PR model ndcatng better fts usng GPR models. V. Concluson Ths study demonstrates the performance of better model among Negatve bnomal regresson and the Generalzed posson regresson for explanng over dsperson count data. The emprcal results presented n ths study support the neoclasscal theory of fertlty. The ftted models suggest that the educatonal level of mother s a determnant of more chldren. The educated women are more lkely to bear fewer chldren than the women who have no educaton. Resdence s found to be the key factor of more chld bearng n Bangladesh. In rural areas, women desre more chldren than women n urban areas. As a result, the famly sze ncreases n rural areas than urban areas. Relgon has an oblgng effect on chldbearng. The populatons of Bangladesh are almost pous. But some of them are extremely conservatve. They thnk that the chldren are the gft of God. They don t care for the bad effects of over number of chldren. For ths reason, famly sze ncreases. Early age at frst marrage s also one of the man reasons of more chldbearng to the Bangladesh women. Partner s occupaton, havng rado, televsons, workng status, current contraceptve methods etc. are also appeared to be mportant as determnants of chldbearng. The am of our analyss s to obtan the smplest model that reasonably explans the varaton,.e., overdsperson n the data. For comparng Negatve Bnomal Regresson model and Generalzed Posson Regresson model, we use two measures of goodnessofft lke devance and AIC for overdsperson. These values ndcate that the Generalzed Posson Regresson model s more approprate for the present born data and leads to more effcent parameter estmates. Generalzed Posson Regresson model founds only one nsgnfcant factor but Negatve bnomal regresson model and the Posson regresson model found three nsgnfcant factors. That means, Generalzed Posson Regresson model exhbts better explanaton of chld bearng whch s a count data. Generalzed Posson Regresson model shows that the populaton s growng at a very frst rate n Bangladesh because of the effects of some typcal demographc and socoeconomc factors. Therefore, government should take approprate polcy and programs to reduce the level of chldbearng n consderaton to the exstng hgher rate of chldbearng among the women n Bangladesh. References [] F. Famoye, J.T. Wulu and K.P. Sngh, On the Generalzed Posson Regresson Model wth an applcaton to Accdent Data, Data Scence,, pp. 5. [] W. Wang and F. Famoye, Modelng Household Fertlty Decsons wth Generalzed Posson Regresson, Popul Econ,, pp. 33. [3] N. Ismal and A.A. Jeman, Handlng Overdsperson wth Negatve Bnomal and Generalzed Posson Regresson Models, Casualty Actuaral Socety Forum,, pp. 35. IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 5

6 F. Sada, Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences, 4(, MarchMay, 3, pp.5553 [4] Y. Cu, D. Y. Km and J. Zhu J, On the Generalzed Posson Regresson Mxture Model for Mappng Quanttatve Trat Loc Wth Count Data, Genetcs Socety of Amerca 4,, pp. 5. [5] M. Asaduzzaman and M. H. R. Khan, Factors Related to Chldbearng n Bangladesh: A Generalzed Lnear Modelllng approach, Brac Unversty Journal, vol. V, no.,, pp. 5. [] A.J. Dobson, Introducton to Generalzed Lnear Models,, Second Edton, London: Chapman and Hall/CRC. [] S. N. Mtra et al, Bangladesh Demographc and Health Survey, NIPORT, Dhaka, Bangladesh. (. IJETCAS 33; 3, IJETCAS All Rghts Reserved Page 53