A Comparison of Bayesian and Classical Approach for Estimating Markov Based Logistic Model

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1 America Joural of Mathematics ad Statistics 5, 5(4): DOI:.593/j.ajms.554. A Compariso of Bayesia ad Classical Approach for Estimatig Markov Based Logistic Model Jaarda Mahata,*, Soma Chowdhury Biswas, Maidra Kumar Roy, M. Ataharul Islam 3 Departmet of Statistics, Uiversity of Chittagog Departmet of Statistics, Mawlaa Bhasai Sciece ad Techology Uiversity, Tagail 3 Departmet of Applied Statistics, East West Uiversity, Dhaka, Bagladesh Abstract Iferetial Statistics is the mai subject of statistics. Classical ad o-classical are the two types of approaches of estimatio. I Bayesia estimatio selectio of appropriate loss fuctio ad prior desity are very importat. Jeffery s o-iformative prior ad squared error loss fuctio were used. Lidley approximatio was used to solve the Bayesia itegral. The proposed procedure is applied to a logitudial data o pregacy complicatio i rural Bagladesh collected from the Bagladesh Istitute of Research for Promotio of Essetial ad Reproductive Health ad Techologies (BIRPERHT). I this study, we have coducted Markov model by maximum likelihood ad Bayesia approach ad compare them above the approaches. Bayesia approach of estimatio foud to be better tha the classical approach i this particular case. Keywords BSE, Credible iterval, MLE, T.K.. Itroductio Now-a-days Bayesia approach is widely used for decisio-makig. I this paper, we have applied Bayesia approach i Markov chai based logistic model. Markov chai models ca be used i aalyzig logitudial data. There is several discrete time Markov chai models proposed for aalyzig repeated categorical data over decades. To aalyze the biary sequece of presece or absece of diseases, Muez ad Rubistei [] itroduced a discrete time Markov chai for expressig the trasitio probabilities i terms of covariates. The techique suggests by them is applicable for first order Markov model. For aalyzig sequeces of ordial data from relapsig ad remittig of a disease, Albert [] developed a fiite Markov chai model. I additio, Albert ad Waclawiw [] developed a class of quasilikelihood models for a two state Markov chai with statioary trasitio probabilities for heterogeeous trasitioal data. Raftery [4], Raftery ad Tavare [3] proposed a higher order Markov chai model with depedece o cotributio of the past trasitios. Islam ad Chowdhury [8] applied a three state Markov model for aalyzig covariate depedece, also Islam ad Chowdhury [7] preseted a higher order versio of the covariate depedet Markov model. For aalyzig repeated observatios, there is a reewed iterest i the developmet * Correspodig author: johstatcu@gmail.com (Jaarda Mahata) Published olie at Copyright 5 Scietific & Academic Publishig. All Rights Reserved of multivariate models based o Markov chais. These models ca be employed for aalyzig data geerated from meteorology, epidemiology ad survival aalysis, reliability, ecoometric aalysis, biological cocers, etc. Muez ad Rubistei [] employed logistic regressio models to aalyze the trasitio probabilities from oe state to aother. The estimatio for first-order Markov models is quite straight forward, but still there is serious lack of geeralizatio i estimatio ad testig for models applicability for higher order Markov chais. Islam ad Chowdhury [7] provided a further geeralizatio for covariate depedet higher order models. Followig Azzalii [3], Heagerty ad Zeger [5] preseted a class of margialized trasitio models (MTM) ad Heagerty [4] proposed a class of geeralized MTMs to allow serial depedece of first or higher order. These models are computatioally tedious ad the form of serial depedece is quite restricted. Heagerty [4] provided derivatives for score ad iformatio computatios. Muez ad Rubistei [] itroduced a discrete time Markov chai for expressig the trasitio probabilities i terms of fuctio of covariates for a biary sequece of presece or absece of a disease. Islam, Chowdhury ad Sigh [6] suggest covariate depedece i a higher order Markov models is examied. The proposed model ad iferece procedures are simple ad the covariate depedece of the trasitio probabilities of ay order ca be examied without makig the uderlyig model complex. Aother advatage of the model lies i the fact that the estimatio ad test procedures for both the specific parameter of iterest ad the overall model remai easy for practical applicatios for ay logitudial

2 America Joural of Mathematics ad Statistics 5, 5(4): data. A simple alterative is also proposed. Applicatios are illustrated usig materal morbidity durig pregacy. I this paper, a attempt is made to estimate the probability of beig pregacy complicacies of wome accordig to the characteristics of the wome ad estimate the parameters by Bayesia approach ad method of maximum likelihood approach. This characteristics or variables may ofte be related to each other. Higher difficulties occur to ay miscarriage, ecoomic status ad age at marriage i this study.. Model Muez ad Rubistei [] i 985 proposed a two state Markov chai to model a discrete time-biary sequece. The trasitio matrix of the chai is: p M = p where p deotes the trasitio probabilities from to ad p is the trasitio probability from to. At each time poit, a vector of legth two cotais the probability of outcome of iterest ad its complemet. The first such vector (row) is P() = (p (), p ()), with the first elemet equal to the probability of the complemet of the evet of iterest. The matrix M ad iitial vector P() suffice to characterize the Markov chai. After time poit, the vector of occupatio probabilities is: p p j ( ) ( ( ) ( )) P j = p j, p j = P() M ; j > Muez ad Rubistei [] proposed model the trasitio probabilities p ad p by logistic regressios. p ( β X ) ( β X ) exp = ad + exp f( β / = f ( β / X ) p exp = + exp ( α X ) ( α X ) The vector X cotais covariates ad for the qth perso i the study is equal to, Xq = (, Xq, Xqp ). There are two logistic regressios, oe havig parameter vector (, p ) vector (, p ) β β β = ad the other havig parameter α β α =. Large positive (egative) values of β X ad α X yield large (small) trasitio probabilities. Sice the trasitio matrix is a fuctio of covariates, P (j) relates the occupatio probabilities to the iitial state for each patter of covariates. For trasitio to For biary radom variable Y t with covariate joit desity fuctio is as [ i i f ( x / β ) { p } { p } ] where, = i ad i are the umber of trasitios. 3. Bayesia Approach Xt the I this study, we use the Bayesia paradigm to make ifereces about parameters of the model [] for pregacy complicatio data. 3.. Prior ad Posterior Distributio I Bayesia aalysis prior ad posterior distributio are importat. Sice the parameters of Markov model of β lies betwee to +, the accordig to Jeffrey s o-iformative prior [5] of β are g( β ) = I. Where, I represet uit vector. The the posterior desity of β for the give sample is i i [{ p} { p} ] i i [{ p } { p } ] d = X ) + + X ) + + β i i i i dβ

3 8 Jaarda Mahata et al.: A Compariso of Bayesia ad Classical Approach for Estimatig Markov Based Logistic Model 3.. Bayes Estimators uder Squared Error Loss Fuctio The squared error loss fuctio is defied as L ββ ; = β β For squared error loss fuctio, Bayes estimators are the mea of the posterior desity [] β i i BSE = i i β X ) + + X ) + + dβ dβ The two itegrals appear i the ratio caot be solved to have a closed form. For evaluatig them, we use the Lidley [] approximatio. We have, ( β ) = = I( E u( )/ X L( β) + p( β) u( β) e dβ e L ( β) + p( β) dβ where, L ( β ) is the log likelihood ad p( β ) is the logarithm prior distributio. Sice, g( β ) = I the p( β) = log g( β) or, p( β ) = log I = The accordig to Lidley, I( ca be approximately evaluated as I( = u( β) + u ( β) + u ( β) p ( β) + L ( β) u( β) σ σ where, u( β ) is the fuctioal form of the parameter β, which is expected to posterior desity ad β maximum likelihood [] estimator of β. is the u( β) u( β) u ( β) =, u ( β) = β β 3 L β 3 p( β ) p ( β) =, L ( ) = β ad δ = L ( β ) β ( β ) Now we kow that the likelihood fuctio of biary depedece Markov model is Therefore, L= ll Takig log o both sides L = i i i [{ p } { } ] { } { } p p p i [ ]

4 America Joural of Mathematics ad Statistics 5, 5(4): ( ) log L = logl + logl = where, L ad L are the likelihood fuctio of trasitio to ad to respectively. Now cosiderig L + L [ ilog ilog( )] L = p + p X ) = ilog + ilog + + { β } = iβ ( i + i)log + exp( ) Differetiatig both sides with respect to β, we have L ) X i = i ( i + i) β + ) L = ( i+ i) p p β 3 L 3 = 3 ( i+ i) p p ( p p) β σ = = L ( i+ i) p p Therefore, Bayes estimator uder squared error loss fuctio is 3 ( i i) p p p p + βbse = β+ β ( i + i) p p i = where β is the maximum likelihood [] estimate of β. If ( i + i) is large the Bayes estimator teds to maximum likelihood estimator. I additio, the same way we estimate the parameters of the trasitio to. 4. Bayesia Credible Iterval If f ( / θ is the posterior distributio give the sample, we may be iterested i fidig a iterval such that ( ) ( ) P θ ( θ, θ )/ x = f θ / X dθ = α θ θ This iterval is called ( α ) % Bayesia credible [9] iterval of θ. I Bayesia aalysis, credible iterval becomes the couterpart of the classical cofidece iterval, also credible iterval may be uique for all models. The Bayesia credible iterval, o the other had, has a direct probability iterpretatio P( θ ( θ, θ)/ x) α ad is completely determied from the curret observed data x ad the prior distributio. 5. Numerical Aalysis The covariate depedet Markov model proposed i this paper is applied to the pregacy complicatio data coducted from BIRPERHT data for the period November 99 to December 993. The data were collected usig both cross-sectioal ad prospective study desigs. A multistage samplig desig was used for collectig the data for this study. Districts were selected radomly i the first stage, oe district from each Divisio. The Thaas were selected radomly i the secod stage, oe Thaa from each of the selected Districts. At the third stage, two Uios were selected radomly from each selected Thaa. The subjects comprised of pregat wome with less tha 6 moths duratio i the selected Uios. All the selected pregat wome from the selected Uios were followed o regular basis (roughly at a iterval of moth) throughout the pregacy. Agai, the subjects were followed at the time of delivery for a full- term pregacy ad 9 days after delivery or 9 days after ay other pregacy outcome. A total of 59 pregat wome were iterviewed i the follow-up compoet of the study. The followig pregacy complicatios are cosidered uder the complicatios i this study: hemorrhage, fits, covulsio, edema, excessive vomitig, ad cough or fever for more

5 8 Jaarda Mahata et al.: A Compariso of Bayesia ad Classical Approach for Estimatig Markov Based Logistic Model tha three days. If hemorrhage, fits, covulsio, edema, excessive vomitig, ad cough or fever for more tha three days occurred to the respodets, we cosidered complicatios ad was coded as, if o complicatios the coded as. The explaatory variables are: age at marriage (5 years or lower, more tha 5 years), ecoomic status (lower or upper), ay miscarriage (yes, o). The umber of trasitios for the two-state Markov chai of first order is displayed i Table. Table. Trasitio couts of Markov chai for first orders of pregacy complicatio of differet flow-up Trasitio Couts States Total the Muez-Rubistei model usig pregacy complicatio data. I this study, we have used three covariates because of complexity to fit the model. Three highly sigificat covariates are used i our study. Bayesia approaches have bee applied for estimatig the parameter of the above model. 5.. Compariso betwee Credible Iterval ad Cofidece Iterval Cofidece itervals for maximum likelihood estimators ad credible itervals of Bayesia estimators uder squared error loss fuctio were calculated for Muez-Rubistei model ad two types of trasitio are preseted i the followig table. For to trasitio Table. Cofidece iterval for maximum likelihood estimators Poit Iterval (95%) Lower Upper Legth Costat Ay miscarriage Ecoomic Status Age at Marriage Table 3. Credible iterval for Bayesia estimator uder squared error loss fuctio Poit Credible Iterval (95%) Lower Upper Legth Costat Ay miscarriage Ecoomic Status Age at Marriage From the above results, we have foud that the legth of Bayesia credible is lower tha the legth of cofidece iterval for all covariates. Moreover, to trasitios Table 4. Cofidece iterval for maximum likelihood estimators Poit Iterval (95%) Lower Upper Legth Costat Ay miscarriage Ecoomic Status Age at Marriage Table 5. Credible iterval for Bayesia estimator uder squared error loss fuctio Poit Credible Iterval (95%) Lower Upper Legth Costat Ay miscarriage Ecoomic Status Age at Marriage Therefore, from the above tables it is also see that, Baysia credible itervals are smaller legth tha cofidece iterval i to trasitios. Therefore, we ca say that Bayesia approach provides better result tha usual method maximum approach i Muez-Rubistei model. I additio, ecoomic status is egatively associated, ay miscarriage ad age at marriage are positively associate with pregacy complicatio. All the umerical aalysis was performed by R-Laguage (Versio-..). 6. Coclusios Markov chai is very importat part i real world situatio. Its applicatio icreasig day by day, i this study we applied pregacy complicatio data ad oly three covariates were used because complexity of the model. Meawhile differet types of approach were used for estimatig the parameters i this model. I Bayesia approach uder squared error of loss fuctio were used ad compare with method of maximum likelihood ad we have foud that legth of Bayesia credible iterval is smaller tha legth of cofidece iterval for all covariates i two types of trasitios. From the above aalysis, we coclude that Bayes estimators uder squared error loss fuctio give better result i Muez Rubistei model. A applicatio is icluded i this paper to illustrate the use of the proposed model for real life problems. REFERENCES [] Albert, P. S. (994). A Markov model for sequece of ordial data from a relapsig remittig disease. Biometrics, 5, 5-6.

6 America Joural of Mathematics ad Statistics 5, 5(4): [] Albert, P. S., & Waclawiw, M. A. (998). A two state Markov chai for heterogeeous trasitioal data: A quasilikelihood approach. Statistics i Medicie, 7, [3] Azzalii, A. (994). Logistic regressio for autocorrelated data with applicatio to repeated measures. Biometrika, 8, [4] Heagerty, P. J. (). Margialized trasitio models ad likelihood iferece for logitudial categorical data. Biometrics, 58, [5] Heagerty, P. J., & Zeger, S. L. (). Margialized multi-level models ad likelihood iferece (with Discussio). Statistical Sciece, 5, -6. [6] Islam, M. A., Chowdhury, R.I., & Sigh, K. P. (8). Covariate Depedet Markov Models for Aalysis of Repeated Biary Outcomes. Joural of Moder Applied Statistical Methods, 6(), [7] Islam, M. A., & Chowdhury, R. I. (6). A higher-order Markov model for aalyzig covariate depedece. Applied Mathematical Modellig, 3, [8] Islam, M. A., & Chowdhury, R. I. (4). A three state Markov model for aalyzig covariate depedece. Iteratioal Joural of Statistical Scieces, 3, [9] Kazmi, S. M., Aslam, M., & Ali, S. (). Preferece of prior for the class of life-tome distributios uder differet loss fuctios. Pak. J. Statist, 8(4), [] Muez, L. R., & Rubistei, L. V. (985). Markov models for covariate depedece of biary sequeces. Biometrics, 4, 9-. [] Podder, C. K., & Roy, M. K. (3). Bayesia estimatio of the parameter of Maxwell distributio uder MLINEX loss fuctio. Joural of Statistical Studies, 3, -6. [] Press, S. J. (989). Bayesia Statistics: Priciples, Models, ad Applicatios. New York: Joh Wiley & Sos. [3] Raftery, A., & Tavare, S. (994). Estimatig ad modelig repeated patters i higher order Markov chais with the mixture trasitio distributio model. Applied Statistics, 43(), [4] Raftery, A. E. (985). A model for higher order Markov chais. Joural of Royal Statistical Society B, 47, [5] Smith, G. P. (.d.). Expressig Prior Igorace of a Probability Parameter. Notes, Uiversity of Missouri o iformative priors.