Mathematical Modelling of Impact of Education on Average Age at the First Marriage in Tanzania

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1 Pue and Alied Mathematics Jounal 17; 6(1): htt:// doi: /j.amj ISSN: (Pint); ISSN: (Online) Mathematical Modelling of Imact of Education on Aveage Age at the Fist Maiage in Tanzania Fatma S. Seif 1, *, Saa A. Khamis 1, John K. Mduma, Estomih S. Massawe 3 1 Deatment of Science, The State Univesity of Zanziba, Zanziba, Tanzania Deatment of Economics, Univesity of Da es Salaam, Da es Salaam, Tanzania 3 Deatment of Mathematics, Univesity of Da es Salaam, Da es Salaam, Tanzania addess: fasasemo@hotmail.com (F. S. Seif), sakhamis3@gmail.com (S. A. Khamis) * Coesonding autho To cite this aticle: Fatma S. Seif, Saa A. Khamis, John K. Mduma, Estomih S. Massawe. Mathematical Modelling of Imact of Education on Aveage Age at the Fist Maiage in Tanzania. Pue and Alied Mathematics Jounal. Vol. 6, No. 1, 17, doi: /j.amj Received: Decembe 14, 16; Acceted: Januay 6, 17; Published: Febuay 3, 17 Abstact: This ae examines the imact of education on the aveage age at the fist maiage to women in Tanzania. A mathematical model fo the distibution of aveage age at the fist maiage in Tanzania is established and the effect of education on the yea at fist maiage is estimated fo the yeas 1999 and 3/4. Both descitive analysis and model estimate show that education level and education attainment have an imact on the aveage age at the fist maiage. It is established that as one attains futhe studies, he age at fist maiage is delayed. The model validation was esented gahically to illustate the validity of the model whee the ootion of eve maied (a,x 1,x ) fitted vey closely to the emiical data. Keywods: Imact of Education, Aveage Age, Fist Maiage 1. Intoduction Diffeent inteetations exist on what maiage is. Accoding to [1], maiage is defined as the stage at which a man woman ae socially o legally emitted to live togethe. Accoding to [], Maiage is the only institution that allows two eole to establish a vey stong and enduing elationshi that is fully backed by the law and society as a whole. Accoding to these definitions a maiage cannot be acceted without being suoted by law and society i.e. thee ae ules, noms and values which guide this institution. Maiage shall be enteed into only with the fee and full consent of the intending souses, that is, a maiage can be established afte the consent of the elated coules only [3]. Howeve in many cultues the consent of the coule only is not enough. The male aent of the bide must also give his consent. Ou societies have been chaacteized by low age at fist maiage. Accoding to [4], ealy maiage is the maiage of childen dolescent below the age of 18. The actice of ealy maiage is most common in Sub-Sahaan Afica and Southen Asia. A study conducted by [5] established that geneally Tanzanian women ente the maiage institution at the vey low age. The study emhasized that the minimum age is as low as six yeas. Many studies have shown that low age at fist maiage esult into numbe of oblems. Ealy maiage contibute to seies of negative consequences both fo young gils and society in which they live. agues by [6]. Low ages at fist maiage may also be associated with educed education among gils, although it is difficult to ascetain the causality, this was naated by [7]. A young bide may be less able to asset owe uthoity in he maiage esecially given that women get maied to men who ae on aveage seveal yeas olde. Maiage without consent is widely thought to be majo cause fo most of the divoces, and many of ealy maiages ae held without the consent of the gils, and this is the cause of instabilities of most maiages [4]. Instability is vey common in ealy maiages, since the women ente the union by foce and lack faithfulness and love to maintain the maiage so they un away back to thei aents o to towns

2 4 Fatma S. Seif et al.: Mathematical Modelling of Imact of Education on Aveage Age at the Fist Maiage in Tanzania to seach fo a bette living, getting emloyed as housemaids o becoming ostitutes. [4] agued that due to hysiological immatuity of sexual ogans, some of the young maied women faced health elated oblems like hysical ain duing intecouse, obstetic fistula due to ealy delivey, and othe comlications due to egnancy. [4] also believes that due to age diffeence, ealy maied women have and/o execise lowe sexual and eoductive ights than those who get maied at aoiate ages and due to lack of education they have less ability to make decisions on mattes elated to eoductive health, such as the use of contacetives and ights ove sexuality. Accoding to [8], ealy maiage cetainly denies childen of school age thei ight to the education they need fo thei esonal develoment, eaation fo adulthood, and effective contibution to the futue wellbeing of thei family and society. Thee have been many contibutions to educe the oblem of low age at fist maiage such as education. Studies have indicated that as the level of education inceases, the yea at the fist maiage is also affected. Thee is ositive coelation between the numbe of yeas of education one gets and the aveage age at fist maiage. Accoding to [9], women with at least some seconday school education ae less likely to get maied at younge ages than women with less education. The study elated to ealy maiage and female schooling attainment in Bangladesh was done by [1]. Thei esults indicate that each additional yea that maiage is delayed is associated with. additional yea of schooling and 5.6 ecent of highe liteacy. It is found by [1] that, as educational levels incease, ages at maiage also inceases. In this ae it is intended to develo nalyze a model on the imact of education on the aveage age at the fist maiage in Tanzania. A model fo fitting the age atten of fist maiage called PICRATE model is develoed by [11]. PICRATE meaning Phased-In-Constant Rate. In this ae, it is intended to modify the wok by [11] by incooating the imact of education at fist maiage fo women.. Model Fomulation A model fo the imact of education on aveage age at the fist maiage fo women is fomulated nalyzed. Accoding to [11], the PICRATE model is given by the fomula: ( a) = {1 ex[ ( a a) + I( ( a a ))]} (1) whee a is age (continuous, measued in yeas), a is the age at which maiage begin, ( a ) is the ootion of eve maied at age, is the ootion that will eventually may (imum value of ), is the imum age secific fist maiage ecuitment ate. x ex π C 1 1 I x = u du = N x () and Nc( x ) is the cumulative nomal distibution function with mean and standad deviation 1. The cuve fo ( a ) descibes the histoy of fist maiages in the cohot, but fo a stable oulation it also descibes the age-secific eve-maied ootion fo a oulation. Note that the cuve is detemined by the thee aametes a,, and [11]. In develoing the model of the imact of education on aveage age at the fist maiage, all aametes in PICRATE model ae etained excet a which is teated as a function of social economic vaiable (linea function of education). In fomulating the model, the following assumtions ae taken into consideation: The model is coect. Nonlinea egession adjusts the vaiables in the equation model chosen to minimize the sum-of-squaes. It does not attemt to find a bette equation. The vaiability of values aound the cuve follows aoximately a Gaussian distibution. The Standad Deviation of the vaiability is the same eveywhee, egadless of the value of indeendent vaiables. If the Standad Deviation is not constant but athe is ootional to the value of deendent vaiable, the data ae weighed to minimize the sum-of-squaes of the elative distances. The model assumes that indeendent vaiables ae known exactly. This is ae but it is sufficient to assume that any imecision in measuing the indeendent vaiable is vey small comaed to the vaiability in deendent vaiable. Lastly, the eos ae indeendent. The deviation of each value fom the cuve should be andom, and should not be coelated with the deviation of the evious o next value. The assumtion is violated if thee is any cayove fom one samle to the next [1]. The esults of nonlinea egession ae meaningful only if these assumtions ae tue (o nealy tue). Assuming that the social-economic factos affect the age at the fist maiage linealy, then a, the age at which maiage begin is assumed fo simlicity to be a linea function of education which can be exessed as: a = b + bx + bx (3) 1 1 whee, a as defined ealie, is the age at which maiage begin, b, b and b ae aametes to be estimated, 1 x1 and x ae dummy vaiables, whee 1 x eesents education at imay level, while x eesents education at seconday level. The non education gous fom the efeence gou.

3 Pue and Alied Mathematics Jounal 17; 6(1): Then the modified model will be: ( ) ( ( 1 1 )) a b + bx bx ( a, x1, x ) = 1 ex + I a b + bx + bx (4) 1 a ( a a ) = ( a + a ) + ( a a ) e ( ) ex a a + I a a 3. Estimation of the Model Paametes We conside the nonlinea equation ( 1,,..., k, 1,,..., ) Y = f X X X β β β + ε (5) Any nonlinea function can be exessed as a Taylo seies exansion. The Taylo seies exansion to a set of initial values β (,, ) fo the coefficients 1.,..., β.. a,..., (,, a) β is given by i β f Y = f( X..., X, β,..., β ) + ( β β ) 1, k 1.. i i. i= 1 βi 1 f + ( βi βi. )( βj βj. ) ε i= 1 j= 1 βi β j The subscit on the atial deivatives denotes that these deivatives ae evaluated at β1 = β1.,..., β = β.. A linea aoximation to the nonlinea function (5) is given by the fist two tems in the Taylo seies exansion (6) i.e. ( 1,..., k, β1.,..., β. ) Y f X X f f + β = β + ε i. i i= 1 βi i= 1 βi Fom the basic model (1), we have { } ' ' 1 ex o ( a a ) + I ( ( a a )) a ( a) a ( a) a a = ' a ' a ( a) a + + a + ε a ' a whee is the initial guess value of, is the initial guess value of, a as defined ealie, is the age at which maiage begin. The atial deivatives in (8) ae given by ( a) = 1 ex ( a a ) + I ( ( a a )) (6) (7) (8) 1 a a a = e a ( ) ex a a + I a a We substitute the exessions fo, a a a, a and in equation (8), to get aamete estimates fo,,. The estimated coefficients values fo,, which ae denoted by,, ae used as a new set of initial estimates, and the nonlinea equation is lineaized aound these values. The esult is a new linea egession equation. Fom equation (8) we then have ' 1 ( a) 1 ex ( a a ) + π Nc ( x) ' ( 1 ex π c 1 + a a + N x 1 ( a + a ) + ( a a ) ex ( a a ) ' ' ' 1 ex ( a a ) + π Nc ( x) + ' 1 a 1 ex ( a a) ' 1 ex ( a a ) + π Nc ( x) = 1 P 1 ex o( a a) + π( NC( x) 1 Po ( a a ) + ( a ao ) ex ( a a) 1 ex o ( a a ) + π Nc( x) 1 + a 1 ex ( a a) 1 ex o ( a a ) + π Nc( x) + ε Afte simlifying this equation, we let = 1, =.88 = 15 be a set of initial estimates. Substitute these values in equation (9), and then aly odinay least squae to this equation to get a new set of coefficient estimates (9)

4 4 Fatma S. Seif et al.: Mathematical Modelling of Imact of Education on Aveage Age at the Fist Maiage in Tanzania,. The ocess of elineaization is eeated until convegences is attained t this stage the values of aametes, will be attained. SPSS Softwae is used to estimate these aametes. At the second stage we use the modified model to estimate aametes which include aametes of the function of education, that is, we estimate aametes,, b, b1 and b by using the same ocedue of least squae method as it was used to estimate aametes of PICRATE model. On estimating the modified model aametes, STATA Softwae is used. Two datasets of the Tanzania Demogahic and Health suveys of 1999 and 3-4 ae used. 4. Analysis of the Model To assess the effect of education on the age at the fist maiage, a seies of vaiables (dummy o indicatos) fo the level of education comleted by esondents was ceated. The vaiables wee ceated accoding to the following categoies of education levels: no education, some o comleted imay education, some o comleted seconday education, highe education. Fo the uose of the statistical analysis the no education categoy was set as a efeence categoy. Estimation of the aametes fo the modified PICRATE model was done by using the STATA comute ackage fo the data of the yeas 1999 and 3/4 ovided by the Demogahic and Health Suveys (DHS). The estimation yielded the esults which ae the same fo the two yeas. The otimal esults wee obtained afte 1 iteations fo the data of the yea 1999 and 1 iteations fo the data of the yea 3/4. This was due to the natue of non lineaity of the function that was estimated Descitive Analysis of Education Level and Age at Fist Maiage The esults fo the education covaiates ae shown below. Table 1. Distibution of age at the fist maiage by education level. YEAR Education level Mean Median Mode 1999 No education Pimay Seconday Highe Education 5 5 3/4 No education Pimay Seconday 1 1 Highe Education Table 1 summaizes the findings fom the descitive analysis of the data. Fo both suvey yeas, the age at the fist maiage was lowe fo esondents with no education comaed to those with education. In both yeas, the mean age at fist maiage inceases gadually as the level of education inceases as seen in Figue 1 and Figue below. Figue 1. Vaiation in the mean age at the fist maiage by Levels of education in Tanzania fo the data of Figue. Vaiation in the Mean Age at Fist Maiage by Levels of Education in Tanzania fo the data of 3/4. Table. Inte-Yea Comaison of Mean Age at Fist Maiage /4 No education Pimay Seconday 19 Highe 5 4 Yea Table 3. Distibution of age at fist maiage by education attainment. Educational attainment Mean Median Mode Std Dev 1999 No education Incomlete imay Comlete imay Incomlete seconday Comlete seconday 3 3 Highe /4 No education Incomlete imay Comlete imay Incomlete seconday Comlete seconday Highe

5 Pue and Alied Mathematics Jounal 17; 6(1): Fom these obsevations, age at fist maiage can be egaded as inceasing function of education level o education attainment. 4.. Results of Estimating Paametes This section esents esults of the emiical models of age at fist maiage. The model is a modified PICRATE to the case of Tanzania. Fitting the model was done by non linea least squae technique using the STATA Softwae. Table 4 and table 5 below show esults of modified PICRATE model in Tanzania fom TDHS data in Dislayed in the table ae model summay and aamete estimates (coefficients) with thei significance/ -values. Table 4. Model Summay (1999). Numbe of obsevations 49 F(5,44) Pob>F. R-squaed.9991 Adj R-squaed.9991 Root MSE.46 Res. dev Table 5. Paamete Estimates (1999). Paamete Coef. Std. E P > t b n b b The F-value is and its obability (-value) is. imlying that the model is significant at 1% level since -value <.5. The coefficient of detemination, R-squaed, is.9991 imlying that the vaiation (99.91%) in the eve maied ootion model is exlained by the covaiates, namely education levels, in the fom of a escibed model (PICRATE model). This linea function of education is eesented by thee aametes which ae b, b1 and b. Out of the five aametes estimated in the model, fou of them ae statistically significant. These ae (-value =.<.5), (-value =.<.5), b (-value =.<.5) and b (-value =.<.5). bis 1 to be statistically insignificant (-value =.15>.5). The values of the significant aamete estimates fo the 1999 data ae theefoe = 1.43, =.5, b = 3.98, b = Table 6 and Table 7 below show the esults of the modified PICRATE model in the yea 3/4. Table 6. Model Summay (3/4). Numbe of obsevations 6863 F(5,6858) Pob>F. R-squaed.9984 Numbe of obsevations 6863 Adj R-squaed.9984 Root MSE.36 Res. dev Table 7. Paamete Estimates (3/4). Paamete Coef. Std. E P > t b n b b The F-value is and its obability (-value) is. imlying that the model is significant at 5% level since -value <.5. The coefficient of detemination, R-squaed, is.9984 imlying that the vaiation (99.8%) in the eve maied ootion model is exlained by the by covaiates, namely education levels, in the fom of a escibed model (PICRATE model). Again this linea function of education is eesented by thee aametes which ae b, b1 and b. All five aametes in the model ae statistically significant with <.5 fo each. The values of these aamete estimates fo 3/4 data ae =.9987( value =. <.5) =.17( value =. <.5) b = 3.48( value =. <.5) b1 =.477( value =. <.5) and b =.779( value =. <.5). 5. Emiical Validation of the Modified Picate Model Figue 3 and Figue 4 show the comaisons of the emiical data with the data comuted by the modified PICRATE model fo the yeas 1999 and 3/4 esectively. Eve maied ootion Cuent age of esondent Emiical data Modified PICRATE Model Figue 3. Fitting emiical data with modified PICRATE model (1999).

6 44 Fatma S. Seif et al.: Mathematical Modelling of Imact of Education on Aveage Age at the Fist Maiage in Tanzania Eve maied ootion Cuent age of esondent Figue 4. Fitting emiical data with modified PICTRATE Model (3/4)]. Fom the Figue 3 and Figue 4, the modified PICRATE model agees vey closely to the emiical data. These esults show that education level and education attainment has an imact on the aveage age at the fist maiage in Tanzania. Also these findings do not imly on the quality of education in Tanzania, they show that as gils attains futhe studies, he age at fist maiage is delayed. Delayed maiages may have esult in eduction in fetility ate and contibute in the eduction in oulation gowth ate. 6. Conclusions Emiical data Modified PICRATE Model In this ae, a model of the imact of education on the aveage age at the fist maiage was develoed and analyzed. At the fist stage PICRATE function ˆ( x ) (cumulative obability distibution function) and f ( x ) (obability distibution function) wee consideed. As an extension of the standad PICRATE model, it was assumed that the social economic vaiable (education) affect the age at the fist maiage linealy. Consequently, the age at which maiage begin in the PICRATE model was assumed fo simlicity to be a linea function of education. By doing that we obtained a modified PICRATE model. The aametes of modified PICRATE model wee then estimated by STATA Softwae. Otimal solution was obtained afte 1 iteations fo the data of 1999 fte 7 iteations fo the data of 3/4. In both cases almost all estimated aametes wee found to be statistically significant excet b 1 which is one of the aamete of linea function of education which estimates the effect of imay education on the mean of the fist yea of maiage. This aamete was found to be significant in 3/4 and insignificant in 1999 based on the eoted -values and selected level of significance. Geneally, descitive analysis of education level / educational attainment ge at fist maiage shows that education level and educational attainment have significant imact on the aveage age at the fist maiage. As one gets futhe education, he age at the fist maiage is delayed significantly. It is seen that high education level has a vey high imact on the delay of ealy maiage. It is aoiate fo the govenment to fomulate olicies which will encouage gils to get futhe studies. This will otect them fom ealy maiages. The education olicy needs to be eviewed. Fo examle the level of minimum education should be inceased fom standad seven to fom fou. This will incease the numbe of yeas in which the gils ae in school and hence delay ealy maiages. Refeences [1] Akte S, Rahman MM (9), Diect and Indiect Effects of Socioeconomic Factos on Age at Fist Maiage in Slum Aeas in Chinese, Jounal of Poulation, Resouces and Enviomental Bangladesh, Univesity of Rajshahi, Vol 7, No. 3 (. 79-8). [] Walsh J (8) Imotance of Maiage, Retieved on 7/9/1 fom htt:// Maiage-to-Society/Page1.html. [3] Univesal Declaation of Human Rights (1948) United Nation (UN), Retieved on 5/1/1 fom Aticle16No.htt:// on.of.human.ights.1948/otait.a4.df. [4] Pathfinde Intenational (6) Reot on causes and consequences of ealy maiage in Amhaa egion, Retieved on 1/1/1fom htt:// Ethioia. [5] Ngalinda, I (1998) Age at Fist Bith, Fetility, and Contacetion in Tanzania, Unlished PHD thesis, Belin. [6] Bayisenge, J (1) Ealy Maiage as a Baie to Gil s Education, Retieved on 1/1/1 fomwww.ifuw.og/fuwa/docs/ealy-maiage.df. [7] Beegle K, Kutikova S (8) Adult motality and Childen s tansition into maiage in Demogahic Reseach, Jounal of Pee Reviewed Reseach and Commentay in the Poulation Science, Max Planck Institute fo Demogahic Reseach Konad-Zuse St. 1, D-1857 Rostock, Gemany,Vol19, At 4, [8] UNICEF (1) The State of the Wold s Childen. [9] Singh S, Samaa R (1996) Ealy Maiage Among Women in Develoing Counties, Jounal of Intenational Family Planning Pesectives, Guttmache Institute, New Yok, Vol, No & 175. [1] Eica F, Attila A (8)Ealy Maiage, Age of Menache, and Female School Attainment in Bangladesh, Jounal of Political Economy, The Univesity of Chicago, Vol 116, No. 5, [11] Matthew AP, Leclec PM, Gaenne ML (9) The Picate Model Fo Fitting The Age Patten Of Fist Maiage, Jounal of Math. Sci. hum / Mathematics and Social Sciences (47e année, n 186, 9 (),. 17-8). [1] Motlusky H (1995) The Gah Pad Guide to Nonlinea Regession, Retieved on /3/11 fom htt://