Bayesian Survival Analysis without Covariates Using Optimization and Simulation Tools

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1 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) Bayesan Survval Analyss wthout Covarates Usng Optmzaton and Smulaton Tools Yasmn Khan 1, Athar Al Khan 2 Research Scholar, Department of Statstcs & Operatons Research, Algarh Muslm Unversty, Algarh, Uttar Pradesh, Inda 1 Professor, Department of Statstcs & Operatons Research, Algarh Muslm Unversty, Algarh, Uttar Pradesh, Inda 2 ABSTRACT: Webull and exponental dstrbuton s one of the most mportant and flexble dstrbutons n survval analyses. In ths paper, Bayesan regresson analyss wthout covarate wth censorng mechansm s carred out for ntrauterne devce (IUD) data problem. Throughout the Bayesan approach s mplemented usng R and approprate llustratons are made. KEYWORDS: Bayesan Inference, Rght censorng, LaplaceApproxmaton, Survval functon. I. INTRODUCTION The analyss of survval data s a major focus of the statstcs busness. The man theme of ths chapter s the analyss of survval data usng two parametrc models, namely Webull and exponental. These models are very flexble and have been found to provde a good descrpton of many types of tme-to-event data. These are the dstrbutons whch occupy a central role because of ther demonstrated usefulness n a wde range of stuatons. There are many potental lfe tme models but these models are used qute effectvely to analyze skewed data sets and gve best data ft. Webull dstrbuton has two parameter, shape and scale. Its densty, survval and hazard functons, respectvely are: a y f ( y) b b y exp b a1 a a y S( y) exp b Where a > 0 and b > 0 are the shape and scale parameters respectvely. Correspondng probablty densty, survval and hazard functon of exponental model are: 1 f ( t) exp b y S( t) exp b One mportant feature of survval data s the presence of censorng, whch create specal problems n the analyss of the survval data. Lfetme data are censored when the exact falure tme for a specfc tral s unknown. When analyzng censored data, Bayesan methods have an mportant advantage over classcal methods. From a classcal perspectve, confdence nterval and other nferental statements must be made wth respect to repeated samplng of the data. From Bayesan perspectve, only the observed censorng pattern s relevant. There are several categores of censorng but n ths chapter we wll dscuss only rght censorng mechansm. y b Copyrght to IJIRSET DOI: /IJIRSET

2 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology Lkelhood functon for rght censored data s, L (An ISO 3297: 2007 Certfed Organzaton) n n Pr( y, ) 1 1 [ f ( y )] [ S( y )] Where s an ndcator varable whch takes value 1 f observaton s censored and 0 otherwse. Secton III begns wth a bref dscusson of the Bayesan analyss of ntercept model based on the Webull and then exponental dstrbuton. There has been growth n the development and applcaton of the Bayesan nference. The Bayesan nference enables us to ft very complex model that cannot be ft by alternatve frequentst methods. To ft the Bayesan models, one needs a statstcal computng envronment. An envronment that meets these requrements s the R (R Development Core Team [1]) software. In ths paper, an attempt has been made to llustrate the Bayesan modellng by usng the R language. The package used n the paper to get the posteror summary s LaplacesDemon Statstcat [2] whch reveals the conceptual smplcty of the Bayesan approach for survval data analyss. Ths package has several optmzaton and smulaton algorthms. The default optmzaton algorthm s LBFGS (Broyden-Fletcher-Goldfarb- Shanno). Also a very popular algorthm,.e. the Nelder-Mead [3], s a dervatve-free, drect search method whch s effcent n small-dmensonal problems. The LaplacesDemon offers numerous MCMC algorthms for smulaton n Bayesan nference, and are, Random-Walk Metropols (RWM), Metropols-wthn-Gbbs (MWG), Delayed Rejecton Metropols (DRM), etc. Ths package provdes a complete Bayesan envronment to the user. Secton VII provdes a model compatblty study based on DIC and devance crteron so that the choce of these two models can be justfed for the data under consderaton. Fnally, n the last secton a bref dscusson and concluson s gven. 1 II. RELATED WORK A sgnfcant amount of work has been done for estmatng the lnear regresson model for censored data khan and khan[4], Khan and khan[5], Khan et al[6], Collet[7] and Puja et al. [8] dscuss the Bayesan survval analyss of head and neck cancer data and conclude that whch therapy has gve better performance for patents. In ths paper all analyses and computaton were undertaken usng LaplacesDemon package avalable n R software. III. BAYESIAN FITTING OF INTERCEPT MODEL WITH CENSORING Here we want to make nferences about the response and ntercept. In ths secton, we wll consder these two models, Webull and exponental models for the data whch s dscuss here are the tme to dscontnuaton of the use of an IUD. The data n Table 1 refer to the number of weeks from the commencement of use of a partcular type of ntrauterne devce (IUD), known as Mult load 250, untl dscontnuaton because of menstrual bleedng problems. Data are gven for 18 women, all of whom were aged between 18 and 35 years and who had experenced two prevous pregnances. Dscontnuaton tmes that are censored are labelled wth an astersk. Table I: Tme n weeks to dscontnuaton of the use of an IUD * 18* 19 23* * 54* 56* * * 107* The ntercept model s n the form of y e y ~ Webull( a, b) Copyrght to IJIRSET DOI: /IJIRSET

3 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology Where, a and b are shape and scale parameter and (An ISO 3297: 2007 Certfed Organzaton) X, log( b) X ~ N(0,1000) ~ halfcauchy(25) IV. FITTING OF WEIBULL MODEL Webull dstrbuton has two parameters, shape and scale. It can be denoted as thus, the lkelhood s p( y a, b) whch mples log-lkelhood as n 1 y ~ W ( a, b) a y b b a1 y exp b n a1 a a y y log p( y a, b) log exp 1 b b b Bayesan fttng of Webull model for ths data can be done n R by usng functon LaplaceApproxmaton and then wth LaplacesDemon. Its fttng ncludes codes for creaton of data and defnton of model. R codes to ft Webull model s beng descrbed below. A.CREATION OF DATA LaplaceApproxmaton functon requres data that s specfed n a lst. Though most R functons use data n the form of a data frame, Laplace's Demon uses one or more numerc matrces n a lst. It s much faster to process a numerc matrx than a data frame n teratve estmaton. For the above data of 18 patents wth dscontnuaton tme has gven the survval tmes n weeks as a response varable. Snce ntercept s the only term n the model, a vector of 1s s nserted nto desgn matrx X. Thus, X=1 ndcates only column of 1s n the desgn matrx. lbrary(laplacesdemon) y<-c(10,13,18,19,23,30,36,38,54,56,59,75,93,97,104,107,107,107) censor<-c(1,0,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,0) N<-18 J<-1 X<-matrx(1,nrow=length(y)) mon.names<-c("lp","shape") parm.names<-as.parm.names(lst(beta=rep(0,j),log.shape=0)) MyData<-lst(J=J,X=X,mon.names=mon.names,parm.names=parm.names,y=y) There are J=1 ndependent varables (response varable). The R codes defned above must have a name specfed for each parameter n the vector parm.names, and parameter names must be ncluded wth the data n a lst called as.parm.names. The user must specfy the number of observatons n the data as ether a scalar n or N. If these are not found by the LaplaceApproxmaton or LaplacesDemon functons, then t wll attempt to determne sample sze as the number of rows n y or Y. B. INITIAL VALUES Laplace's Demon requres a vector of ntal values for the parameters. Each ntal value s a startng pont for the estmaton of a parameter. When all ntal values are set to zero, Laplace's Demon wll optmze ntal values usng a ht-and-run algorthm wth adaptve length n the LaplaceApproxmaton functon. Intal.Values <- c(rep(0,j), log(1)) a Copyrght to IJIRSET DOI: /IJIRSET

4 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) C. MODEL SPECIFICATION Model<-functon(parm,Data) { beta<-parm[1:data$j] shape<-exp(parm[data$j+1]) beta.pror<-sum(dnorm(beta,0,1000,log=t)) shape.pror<-dhalfcauchy(shape,25,log=t) mu<-tcrossprod(beta,data$x) scale<-exp(mu) w1<-log(scale)+log(shape)+(shape-1)*log(y)-scale*y^shape s1<--scale*y^shape LL<-censor*w1+(1-censor)*s1 LL<-sum(LL) LP<-LL+beta.pror+shape.pror Modelout<-lst(LP=LP,Dev=-2*LL,Montor=c(LP,shape),yhat=mu,parm=parm) return(modelout) } D. FITTING OF MODEL WITH LAPLACE APPROXIMATION M1<-LaplaceApproxmaton(Model,Intal.Values,Data=MyData,Sample=1000,Iteratons=10000) The output summary s tabulated n Table 2 and Table 3. Table 2: Approxmated posteror summary of IUD data usng LaplaceApproxmaton functon wth posteror mode, posteror sd and ther quantles. Mode SD LB UB beta Log.shape Table 3: Smuated posteror summary of IUD data usng samplng mportance resamplng n LaplaceApproxmaton functon wth posteror mean, posteror sd and ther quantles. Mean SD LB Medan UB beta Log.shape V. MEDIAN SURVIVAL TIME Medan survval tme s the tme beyond whch 50% of the ndvduals n the populaton under study are expected to survve, and s gven by that value t(50) whch s such that S(t(50)) = 0.5. As we know Survval tme dstrbuton s always postvely skew, the medan s the preferred summary measure of the locaton of the dstrbuton. Once the survvor functon has been estmated, then t s easy to obtan an estmate of the medan survval tme. The estmated medan survval tme, {t(50)}, s defned to be the smallest observed survval tme for whch the value of the estmated survvor functon s less than 0.5. t ( 50) mn t S( t ) 0.5, Where t s the observed survval tme for the th ndvdual, 1,2,... n. snce the estmated survvor functon only changes at a death tme, ths s equvalent to the defnton where j t t ( 50) mn S( t ) 0.5, t s the jth ordered death tme, j 1,2,... r. j Copyrght to IJIRSET DOI: /IJIRSET

5 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) Fg 1. Medan survval tme and other percentle of IUD data. The correspondng nterval for the true medan dscontnuaton tme s (53.81,126.69), so that there s 95% chance that the nterval from 54 days to 127 days ncludes the true value of the medan dscontnuaton tme. VI. SIMULATION STUDY: INDEPENDENT METROPOLIS ALGORITHM For the purpose of smulaton from jont posteror dstrbuton, ndependent Metropols algorthm wll be performed. Now, n ths secton we have to explore IUD data usng functon LaplacesDemon. It maxmzes the logarthm of the unnormalzed jont posteror densty wth MCMC and provdes samples of the margnal posteror dstrbutons, devance, and other montored varables. In LaplacesDemon functon there s an argument called Algorthm, here the algorthm used for smulaton from jont posteror dstrbuton s ndependent-metropols algorthm. Multvarate normal has been treated as a proposal dstrbuton q ( ). Here, the proposal dstrbuton does not depend on the prevous state of the chan. The IM algorthm s effcent when the proposal s a good approxmaton of the target posteror dstrbuton. Good ndependent proposal denstes can be based on LaplaceApproxmaton (Terney and Kadane[9] Terney et al.[10] and Erkanl[11]). Thus, a generally successful proposal can be obtaned by a multvarate normal dstrbuton wth mean equal to the posteror mode and precson matrx 2 log p( y) H ( ) j that s, mnus the second dervatve matrx of the log-posteror densty Copyrght to IJIRSET DOI: /IJIRSET

6 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) log p( y) log p( y ) log p( ) evaluated at the posteror mode. Consequently, an effcent proposal s gven by, 1 q ( ) N, H ( ) The acceptance probablty, when proposng a transton from to, s gven by p( y) q( ) mn 1, p( y) q( ) Whch can be re-expressed as, w( ) mn 1,, w( ) Where, w( ) p( y)/ q( ) s the rato between the target and the proposal dstrbuton and s equvalent to the mportance weght used n mportance samplng Ntzroufras[12]. Ths theory s mplement n LaplacesDemon wth object name M2, Intal.Values<-as.ntal.values(M1) M2<-LaplacesDemon(Model,Data=MyData,Intal.Values,Covar=M1$Covar,Algorthm="IM", Iteratons=10000,Status=F,Specs=lst(mu=M1$Summary1[1:length(Intal.Values),1])) The output obtaned from M2 object s summarzed n Table 4. Table 4 contans posteror mean, posteror sd and respectve quantles. The margnal posteror densty plots of both parameter.e. scale and shape parameters from LaplaceApproxmaton and LaplacesDemon functons are reported s Fgure 2. Table 4: Smulated posteror summary usng LaplacesDemon functonwth posteror mean and quantles. Mean SD LB Medan UB beta Posteror densty of scale Posteror densty of shape Densty LaplaceApproxmaton LaplacesDemon Densty LaplaceApproxmaton LaplacesDemon N = 1000 Bandwdth = N = 1000 Bandwdth = 0.25 Log.shape Fg 2: Margnal posteror densty plots of scale (left) and shape (rght) parameter. Sold lne depcts the margnal posteror densty obtan by LaplaceApproxmaton method wth posteror mode and dotted lne s posteror densty plot obtan by smulaton Copyrght to IJIRSET DOI: /IJIRSET

7 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) usng LaplacesDemon functon wth posteror mean Smlarly the sold and dotted lne of shape parameter n rght, depcts the margnal posteror densty wth posteror mode 1.68 amd posteror mean 1.66, respectvely. VI. FITTING OF EXPONENTIAL MODEL R-codes for the fttng exponental dstrbuton s beng descrbe below, lbrary(laplacesdemon) y<-c(10,13,18,19,23,30,36,38,54,56,59,75,93,97,104,107,107,107) censor<-c(1,0,0,1,0,1,1,0,0,0,1,1,1,1,0,1,0,0) N<-18 J<-2 X<-matrx(1,nrow=length(y)) mon.names<-c("lp") parm.names<-as.parm.names(lst(beta=rep(0,j))) MyData<-lst(J=J,X=X,mon.names=mon.names,parm.names=parm.names,y=y) Intal.Values <- c(rep(0,j)) Model<-functon(parm,Data) { beta<-parm[1:data$j] beta.pror<-sum(dnorm(beta,0,1000,log=t)) mu<-tcrossprod(beta,data$x) scale<-exp(mu) w1<-log(scale)+log(1)+(1-1)*log(y)-scale*y^1 s1<--scale*y^1 LL<-censor*w1+(1-censor)*s1 LL<-sum(LL) LP<-LL+beta.pror Modelout<-lst(LP=LP,Dev=-2*LL,Montor=c(LP),yhat=mu,parm=parm) return(modelout) } M2<-LaplaceApproxmaton(Model,Intal.Values,Data=MyData,Sample=10000,Iteratons=10000) Table 5: Approxmated posteror summary of IUD data usng LaplaceApproxmaton functon wth posteror mode, posteror sd and ther quantles. Mode SD LB UB beta[1] beta[2] Table 6: Smuated posteror summary of IUD data usng LaplacesDemon functon wth posteror mean, posteror sd and ther quantles. Mean SD LB Medan UB beta[1] beta[2] Copyrght to IJIRSET DOI: /IJIRSET

8 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) Posteror densty of scale Densty LaplaceApproxmaton LaplacesDemon N = Bandwdth = Fg 3: Margnal posteror densty of scale parameter. Sold lne ( ) s the densty obtan as a result of LaplaceApproxmaton functon and dotted lne (-----) s the densty obtan by LaplacesDemon functon. The posteror mode s and posteror mean s also evdent from Table 5 and Table 6 after dong exp(beta[1]). A. MEDIAN SURVIVAL TIME OF EXPONENTIAL DISTRIBUTION. Medan of exponental dstrbuton s gven as, t med log b Densty plots of medan dscontnuaton survval tmes and also percentle s reported n Fgure 4. Fg 4. Medan survval tme and other percentle of IUD data under the assumpton of exponental dstrbuton. The 90th percentle of the dstrbuton of dscontnuaton tmes s 287 days. Ths means that on the assumpton that the rsk of dscontnuaton the use of an IUD s ndependent of tme and 90% of women wll have a dscontnuaton tme less than 287 days. Copyrght to IJIRSET DOI: /IJIRSET

9 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) VII. MODEL COMPARISON In ths secton, a goodness-of-ft crteron tests would be appled n order to verfy whch dstrbuton fts better for ths data. To compare the two models; namely exponental and Webull dstrbuton, the model selecton crteron preferred by the Bayesans and lkelhoodsts are devance and devance nformaton crteron (DIC, Spegelhalter et al.[13], s a model assessment tool). A smaller DIC and devance ndcates a better ft to the data set. Table 7: Model comparson of Webull and exponental model for IUD data. Both devance and DIC crteron support Webull dstrbuton s a better choce as compared to exponental dstrbuton. Model Devance DIC Webull Exponental VII. CONCLUSION In trals nvolvng contraceptves, preventon of pregnancy s an obvous crteron for acceptablty. The IUD data whch was dscussed n the whole chapter s the tme orgn corresponds to the frst day n whch woman uses IUD, and the end pont s the dscontnuaton because of bleedng problems. Some women n the study ceased usng IUD because of the desre of pregnancy, or because they had no further need for a contraceptve, whle others were smply lost to follow-up. The data was fully analyzed n Bayesan framework. Fg 5: Estmated survval curve for Webull and exponental dstrbuton showng ther medan survval tme. The IUD data had been analyzed both by both Webull and exponental models. The dscontnuaton tmes when modeled under an exponental dstrbuton, the medan dscontnuaton tme s days whch could also be seen n above Fgure, and an estmate of the 90 percentle of the dstrbuton of dscontnuaton tmes s 287 days as reported n Fgure 4. Ths means that on the assumpton that the rsk of dscontnuaton the use of an IUD s ndependent of tme and 90% of women wll have a dscontnuaton tme less than 287 days. On the other hand, under the assumpton of Copyrght to IJIRSET DOI: /IJIRSET

10 ISSN(Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology (An ISO 3297: 2007 Certfed Organzaton) Webull dstrbuton, the estmated medan dscontnuaton tme for IUD data s days and the 90 percentle s 162 days as reported n Fgure 1. So 90%of women wll have a dscontnuaton tme less than 162 days. Hence t s very clear that the medan s therefore estmated more precsely when the dscontnuaton tme are assumed to have a Webull dstrbuton. It would also be evdent from Table 7. Table 7 shows the devance and DIC of both models. Here the devance of Webull model s 100 whereas exponental model have 207, whch prove that Webull model s sutable for ths data and exponental model does not provde an acceptable ft to ths data. REFERENCES 1. R Development Core Team, R: A Language and Envronment for Statstcal Computng, R Foundaton for Statstcal Computng, Venna, Austra, ISBN , URL Statstcat, LLC, LaplacesDemon: Complete envronment for Bayesan Inference, R package verson , URL com, J. A. Nelder, and R. Mead, A smplex method for functon mnmzaton, The Computer Journal Vol. 7, , Y. Khan, and A. A. Khan, Bayesan regresson analyss of Webull model wth R and Bugs, Statstcal Methodologes and Applcatons, Vol. 2, , Y. Khan, and A. A. Khan, Bayesan analyss of Webull and log normal survval models wth censorng mechansm, Internatonal Journal of Appled Mathematcs, Vol. 26, , Y. Khan, M. T. Alhtar, R. Shehla, and A. A. Khan, Bayesan modellng of forestry data wth R usng optmzaton and smulaton tools, Journal of Appled Analyss and computaton, Vol. 5, 38-51, D. Collet, Modellng Survval Data n Medcal Research, London: Chapman & Hall, M. Puja, K. S. Puneet, R. S. Sngh, and S. K. Upadhyay, Bayesan Survval Analyss of Head and Neck Cancer Data Usng Lognormal model, Communcatons n Statstcs-Smulaton and computaton, L. Terney, and J. B. Kadane, Accurate approxmatons for posteror moments and margnal denstes", Journal of the Amercan Statstcal Assocaton, Vol. 81, 82-86, L. Terney, R. E. Kass, and J. B. Kadane, Fully exponental Laplace approxmatons to expectatons and varances of non-postve functons, Journal of the Amercan Statstcal Assocaton, Vol. 84, No. 407, pp , Erkanl, Laplace approxmatons for posteror expectaton when the model occurs at the boundary of the parameter space, Journal of the Amercan Statstcal Assocaton, Vol. 89, , Ntzoufras, Bayesan Modellng usng WnBugs, John Wley & Sons, D. Spegelhalter, N. Best, B. Carln, and A. van der Lnde, Bayesan measures of model complexty and ft, Journal of the Royal Statstcal Socety, B Vol. 64, , Copyrght to IJIRSET DOI: /IJIRSET