Proportional Hazards Regression Model for Time to Event Breast Cancer Data: A Bayesian Approach

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1 IJIRST Internatonal Journal for Innovatve Research n Scence & Technology Volume 3 Issue 08 January 2017 ISSN (onlne): Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach T. Leo Alexander Assocate Professor Department of Statstcs Loyola College, Chenna , Inda Abstract The paper deals wth a Bayesan based Cox regresson model to consder strateges for performng survvablty of patents wth breast cancer through Bayesan aspects. The proportonal hazards model (PHM) n the context of survval data analyss, s same as Cox model and was ntroduced by Cox (1972) n order to estmate the effects of dfferent covarates nfluencng the tmes-toevent data. It s well known that Bayesan analyss has the advantage n dealng wth censored data and small sample over frequentst methods. Therefore, n ths paper t delberately explores the PHM for rght-censored death tmes from Bayesan perspectve, and compute the Bayesan estmator based on the Markov Chan Monte Carlo (MCMC) method. In partcular t focuses on the approaches based on Gbbs sampler. Such approaches may be mplemented usng the publcally avalable software BUGS. It ams to compare and apply Bayesan models of survvablty for predcton of patents wth breast cancer usng outcome as explanatory varables and to produce better descrptons to survval of patents wth breast cancer and of subgroups of patents wth dfferent survval characterstcs. Keywords: Proportonal Hazards Model, Posteror Dstrbutons, Markov Chan Monte Carlo, Gbbs Sampler, Wnbugs I. INTRODUCTION Survval analyss ams to estmate the three survval (survvorshp, densty, and hazard) functons, denoted by S(t), f(t) and h(t), respectvely (Collet, 1994). There exst parametrc as well as non-parametrc methods for ths purpose (Klenbaum and Klen, 2005). The survval functon S(t) gves the probablty of survvng beyond tme t, and s the complement of the cumulatve dstrbuton functon, F(t). The hazard functon h(t) gves the nstantaneous potental per unt tme for the event to occur, gven that the ndvdual has survved up to tme t (Klenbaum and Klen, 2005).In survval analyss, one must consder censored data. Ths s a key ssue for the analyss of survval data and one of the reasons why survval analyss s a specal topc n statstcs. In essence, censorng occurs when there s some nformaton about ndvdual survval tme, but the survval tme s unknown exactly. Accordng to Mller (1998) and Hougaard (2000) data are sad to be censored f the observaton tme censored survval s only partal, not untl the falure event. There are many reasons for censorng. For examples, the patents can be lost to followup, patents stll alve at the end of the study or patents drop out of the study. There are also several types of censorng, ncludng rght censorng, left censorng, nterval censorng, random censorng, Type I censorng and Type II censorng (Collet, 1994, Hosmer and Lemeshow, 1999, Kalbflesch and Prentce, 2002, Klenbaum and Klen, 2005).Survval tme data have two mportant specal characterstcs (Klenbaum and Klen, 2005) as follows: Survval tmes are non-negatve, and consequently are usually postvely skewed. However, one can adopt a more satsfactory approach as an alternatve dstrbutonal model for the orgnal data. Typcally, some subjects (as mentoned above) have censored survval tmes. There are nonparametrc and parametrc approaches to modellng survval data. A Cox proportonal hazards (PH) model s a popular mathematcal model used for modellng survval. Ths model was proposed by Cox and Oakes (1972) and has also known as the Cox regresson model. The reason why the Cox PH model s so popular because t s a semparametrc and a robust model. The results from ths model wll closely approxmate the results of the correct parametrc model (Klenbaum and Klen, 2005). A survval analyss typcally examnes the relatonshp of the survval dstrbuton to covarates. If the rsk of falure at a gven tme depends on the value of x 1, x2,,x of p predctor varables X1,X2,,Xp, then the value of these varables are assumed to have the tme orgn. If h0(t) s the hazard functon for each object wth the value of all predctor varable X s zero, then the functon of h0(t) s the baselne hazard functon (Collet, 1994). The Cox PH model s usually wrtten n terms of the hazard model as follows: h( t ) h ( t) ex X X... Xp ) (1) p There are two basc assumptons of ths model; the effect of X s lnear and β s constant over tme. The latter beng the assumpton of proportonal hazards. It s also assumed that ndvduals are ndependent and homogeneous (Andersen, 1991). All rghts reserved by 45

2 Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach II. BAYESIAN MODELLING Modern Bayesan analyss began wth a posthumous publcaton n 1763 by Reverend Thomas Bayes, set the theoretcal framework and after a status of around 200 years, publcatons by Geman and Geman (1984) and Bernando and Smth (1994). The dea of Bayesan statstcs wthn the context of lfe data analyss s to ntegrate pror knowledge, along wth a gven set of current observatons, n order to make statstcal nferences. The pror nformaton could come from operatonal or observatonal data, from prevous comparable experments or from engneerng knowledge (Gelman et al., 2004). Ths type of analyss can be manly useful when there s lmted test data for a gven desgn. By ntegratng pror nformaton about the parameters, a posteror dstrbuton for the parameters can be obtaned and nferences on the model parameters and ther functons can be made. Suppose s some quantty that s unknown and let ) denote the pror dstrbuton of. Next, let y be some observed data, whose probablty of occurrence s assumed to depend on. Ths dependence s formalzed by y / ), the condtonal probablty of y for each possble value of, and when consdered as a functon of s known as the lkelhood (Spegelhalter et al., 2004). The probablty for dfferent values of, takng account of y s denoted by / y). Bayes theorem appled to a general quantty says that: y ) ) y) Where and s consdered as normalzng factor to ensure that p ( y) y) d p ( y) d 1. So that, p ( y) y ) ). In classcal nference the sample data y are taken as random whle populaton parameter, of dmenson p, are taken as fxed. Often, a pror amounts to a form of modelng assumpton or hypothess about the nature of parameters, for example, random effect models. In many stuatons, exstng knowledge may be dffcult to summarze n the form of an nformatve pror and to reflect such essentally pror gnorance, resort s made to non-nformatve prors are unformly dstrbuted between - and + and Jeffreys pror, ( ) det{i()} 0.5, Where I() s the expected nformaton matrx. It s possble that a pror s mproper. Such prors may add to dentfablty problems (Gefland and Sahu, 1999), and so many studes prefer to adopt mnmally nformatve prors, whch are just proper. The strategy s consdered n terms of possble pror denstes to adopt for the varance. The gamma process can be descrbed, Let G(,) denote the gamma dstrbuton wth shape parameter >0 and scale parameter >0, where the densty s gven by { x f ( x, ) 0 1 e y) x } / ( ) otherwse, x 0 To reflect pror gnorance whle avodng mproprety, Spegelhalter et al (1996) suggestng a pror standard devaton at least an order of magntude greater than the posteror standard devaton. Pror dstrbutons play a very mportant role n Bayesan statstcs. There are two dfferent types of pror dstrbutons; nformatve and non-nformatve (Gelman et al., 2004, Marn and Robert, 2007, Spegelhalter et al., 2004). Non-nformatve pror dstrbutons (vague, flat, and dffuse) play a mnmal role n posteror nference, and posteror can be senstve to pror (Gelman et al., 2004). The non-nformatve pror dstrbutons can be used to make nferences that are not greatly affected by external nformaton or when external nformaton s not avalable Gelman et al., 2004, Marn and Robert, 2007, Spegelhalter et al., 2004).Accordng to the Bayesan rule, one can express posteror probablty of certan event H gven some data wth the formula P ( data H ) P ( H data ) P ( data ) The probablty of H gven the data s called the posteror probablty of H. The posteror equals to the lkelhood tme the pror dvded by margnal probablty of data. In classcal approaches such as maxmum lkelhood, nference s based on the lkelhood of the data alone. Lndley (1968) used nonnformatve prors and thus, for the lnear case, the posteror means are the MLEs. The posteror mean derved above for the proportonal hazards model s a weghted sum of the maxmum partal lkelhood estmators and the pror means. In Bayesan models, the lkelhood of the observed data y gven parameters, denoted f(y ) s used to modfy the prors belefs ( ), wth the updated knowledge summarzed n a posteror densty, ( y). Thus f ( y, ) f ( y ) ( ) ( y) m( y) and therefore the posteror densty can be wrtten ( y) f ( y ) ( ) / m( y), where m(y) s known as the margnal lkelhood of the data. Ths quantty plays a central role n some approaches to Bayesan model choce, but for the present purpose can be seen as a proportonalty factor, such as ( y) f ( y ) ( ). The Gbbs sampler s a Monte Carlo method for approxmatng jont and margnal dstrbutons by samplng from condtonal dstrbutons. Ths method s well dscussed by Casella and George (1992), Gelfand and Smth (1990) and Geman and Geman (1984), among others. The Gbbs sampler uses only full condtonal denstes n approxmatng jont and margnal denstes. The Gbbs sampler s a way to generate emprcal dstrbutons of two varables from a model. Gbbs samplng s a specal case of the Metropols-Hastngs algorthm, and thus an example of a Markov chan Monte Carlo. The Gbbs sampler ntroduced by Geman and Geman (1984) on MCMC algorthm s known as Gbbs samplng and nvolves successve samplng from a complete condtonal denstes whch condtons on both the data and the other parameters. (Appled Bayesan Modellng P.Congdon Ch-1)... All rghts reserved by 46

3 Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach Let,,..., be a p-dmensonal vector of parameters and let ' D be ts posteror dstrbuton gven the data D. Then, 1 2 p the fundamental format of the Gbbs sampler s gven as Step 1: Select an arbtrary startng pont Step 2: Generate ',...,, 1 1, 1 2, 1 p, 1 ',,..., and set 0 0 1, 0 2, 0 p, 0 Generate,...,,D ; ~ 1, 1 1 2, p, Generate,,...,,D ; ~ 2, 1 2 1, 1 3, p, Generate,,...,,D ; ~ p, 1 p 1, 1 2, 1 p 1, 1 Step 3: Set =+1, and go to step 2 Such successve samples may nvolve smple samplng from standard denstes (gamma, Normal ). An alternatve schemes based on the Metropols-Hastngs algorthm, may be used for non-standard denstes (Mergan, 2000). The program WINBUGS may be appled wth some or all parameters sampled from formally coded condtonal denstes; provded wth pror and lkelhood WINBUGS wll nfer the correct condtonal denstes. III. COMPUTATION OF BAYESIAN APPROACH Bayesan computaton generally explots modern computer power to carry out smulatons (Spegelhalter et al., 2004) based on Markov chans and s known as Markov chan Monte Carlo (MCMC). Monte Carlo methods are technques that have the am of evaluatng ntegrals rather than exact or approxmate algebrac analyss (Spegelhalter et al., 2004). Several MCMC algorthms that are commonly used are Gbbs samplng, Metropols-Hastngs, reversble jump, slce samplng, partcle flters, perfect samplng and adaptve rejecton samplng (Marn and Robert, 2007, Spegelhalter et al., 2004). Gbbs Sampler Survval model can be convenently nspected wth the help of hazard functon. A common approach to handlng the pror probablty for the baselne hazard functon n PHM s a Gamma process pror. However, ths can lead to based and msleadng results (Spegelhalter et al., 1996). The Gbbs samplng s proposed to smulate the Markov chan of parameters posteror dstrbuton dynamcally, whch avods the calculaton of complex ntegrals of the posteror usng WnBUGS package. Gbbs varable selecton was defned by Dellaportas et al. (2000, 2002) usng the BUGS software (Spegelhalter et al., 1996a,b,c). The specfcaton of the lkelhood, pror and pseudo-pror dstrbutons of the parameters as well as the pror term and model probabltes are descrbed. The pror s f ) N (0, c ) (1 ) N (, S ). ( The pror where f ( ) f ( ) potentally makes the method less effcent. The full condtonal posteror dstrbuton s f ( y /, ) N (0, c ), 1 f (,, y ) V (, S ), 0 Bayesan nference has been dscussed by several authors for censored survval data where the ntegrated baselne hazard functon s to be estmated non-parametrcally Kalbflesch (1978),Kalbflesch and Prentce (1980), Clayton (1991), Clayton (1994).Clayton (1994) formulates the Cox model usng the countng process notaton ntroduced by Andersen and Gll (1982) and dscusses estmaton of the baselne hazard and regresson parameters usng MCMC and Gbbs sampler methods. Although these approaches may appear somewhat fxed, t forms the bass for extensons to random effect (fralty) models, tme-dependent covarates, smoothed hazards, multple events and so on. The Cox model n BUGS formulaton s mplemented as below. For subjects = 1,...,n, we observe processes N (t) whch count the number of falures whch have occurred up to tme t. The correspondng ntensty process I (t) s gven by l ( t) dt E ( dn ( t) F ), t Where dn (t) s the ncrement of N over the small tme nterval [t, t+dt), and F t- represents the avalable data just before tme t. If subject s observed to fal durng ths tme nterval, dn (t) wll take the value 1; otherwse dn (t) = 0. Hence E(dN (t) F t- ) corresponds to the probablty of subject falng n the nterval [t, t+dt). As dt tends to zero (assumng tme to be contnuous) then ths probablty becomes the nstantaneous hazard at tme t for subject. Ths s assumed to have the proportonal hazards form l t) Y ( t) ( t) ex Z ), ( 0 Where Y (t) s an observed process takng the value 1 or 0 accordng to whether or not subject s observed at tme t and 0 (t) exβz ) s the famlar Cox regresson model. Thus we have observed data D = N (t), Y (t), z ; = 1,..n and unknown parameters All rghts reserved by 47

4 Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach β and 0 (t) = Integral( 0 (u), u, t, 0), the latter to be estmated non-parametrcally. The jont posteror dstrbuton for the above model s defned by P(β, 0 () D) ~ P(D β, 0 ()) P(β) P( 0 ()). For BUGS, we need to specfy the form of the lkelhood P(D β, 0 ()) and pror dstrbutons for β and 0 (). Under nonnformatve censorng, the lkelhood of the data s proportonal to Ths s essentally as f the countng process ncrements dn (t) n the tme nterval [t, t+dt) are ndependent random varables wth means I (t)d: dn ( t) ~ Posson(l (t)dt). We may wrte l ( t) dt Y ( t) ex Z ) d ( t), Where d 0(t) = 0(t)dt s the ncrement or jump n the ntegrated baselne hazard functon occurrng durng the tme nterval [t, t+dt). Snce the conjugate pror for the Posson mean s the gamma dstrbuton, t would be convenent f 0() were a process n whch the ncrements d 0(t) are dstrbuted accordng to gamma dstrbutons. We assume the conjugate ndependent ncrements pror suggested by Kalbflesch (1978), namely dl 0 (t) ~ Gamma(cd 0 (t), c). 0 Here, dl 0 0 (t) can be thought of as a pror guess at the unknown hazard functon, wth c representng the degree of confdence n ths guess. Small values of c correspond to weak pror belefs. In the example below, we set d 0 0(t) = r dt where r s a guess at the falure rate per unt tme, and dt s the sze of the tme nterval. o IV. APPLICATION TO BREAST CANCER DATA We consder the database consstng of 368 breast cancer women patents dagnosed at Cancer Insttute (WIA), Chenna, Inda and follow-up perod up to 180 months. The event of nterest was tme to death. Overall 187(51%) cases have experenced the event and 63% of 130 are of stage 3B cases. The event experenced cases among age group n more than 50 years s hgher than the less than 50 years (Par Dayal et al., 2013, Leo Alexander et al., 2014). The lnear predctor s set,equal to the ntercept n the reference group (stage = 3)[from the database], ths defnes the baselne hazard. In ths analyss, BUGS program have been used, Spegelhalter et al., (2003). Ths program performs, based on the assumptons of Gbbs sampler by smulatng from the full condtonal dstrbutons. The Bayesan estmators were obtaned through the mplementaton of the Gbbs samplng scheme. It was executed 50,000 teratons of the algorthm and descrbed the frst 1000 teratons as a burn-n. The chans are used to check ts convergence of the Gbbs sampler as recommended by the Spegelhalter et al., (2004). Hence, convergence has been acheved for every 10,000 observatons and s taken from each chan after the burnn perod. The summary (Table1) s showng posteror mean, medan and standard devaton wth a 95% posteror credble nterval along wth MC error, as well as the number of teratons as sample at the fnal after the burn-n perod. The posteror dstrbuton s provded usng the densty opton n the Sample Montor Tool whch draws a kernel densty estmate of the posteror dstrbuton for a chosen parameter, as n Fgure 1. There are varous addtonal optons for dsplayng the posteror dstrbuton. They are quantles, trace and hstory etc., lke the survval curves. Table - 1 WnBUGS output for the Breast Cancer data: Posteror Statstcs Node Mean SD MC error 2.5% Medan 97.5% Start Sample Alpha beta.age beta.stage[2] beta.stage[3] R Sgma Alpha beta.age beta.stage[2] beta.stage[3] R Sgma Alpha beta.age beta.stage[2] All rghts reserved by 48

5 Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach beta.stage[3] R Sgma Table1 presents the posteror statstcs for teratons n three spell of every wth dfferent nodes. The rsk of the stage2 (3A) and stage3 (3B) are compared wth the stage1 (2A) lke ex ) = and ex )= respectvely. WnBUGS as well mplements the Devance Informaton Crteron (DIC) (Spegelhalter et al., 2002 & 2003) for model comparson crteron. Ths s a convenent nformaton crteron measure that trades off goodness-of-ft aganst the complexty of a model. The DIC s computed as DIC D PD Dˆ 2 PD. The Lowest value of the crteron ndcates the better fttng models. D (Dbar) s the posteror mean of -2LL (log lkelhood); Dˆ (Dhat) s the -2LogLkelhood at posteror mean of stochastc nodes. Table - 2 WnBUGS output for the Breast Cancer data Devance Informaton Crteron (DIC) Dbar Dhat pd DIC Sample of 10,000 Iteratons beta.stage t total Sample of 20,000 Iteratons beta.stage t total Sample of 50,000 Iteratons beta.stage t total The DIC values for stage are llustrated n Table2 wth dfferent stages of teratons lke 10000, and respectvely. There are margnal changes n each stage of teratons. The lesser the DIC value wll be consdered as the better model. Snce we have a smple model for ths non-nformatve censored data, t s not requred for model comparson. However, there s no reasonable change n the DIC values after teratons and n fact, t s ncreasng margnally. There are some vsual approxmate estmates as confrmatve measures such as posteror densty or probablty functon, trace plots, posteror percentles, quantles etc. The fgure1 demonstrates all types of vsual approxmate estmates. The frst stage of the graphs s kernel densty. The evoluton for the medan and the 2.5% and 97.5%percentles for each teraton of the algorthm are obtaned by usng ths quantles plot, s n the second stage of the graph. The trace and hstory plots provde an on-lne plot of the generated value as n the thrd and fourth stages of the fgure1. The trace plot shows the full hstory of the samples for any parameter for whch we have prevously set a samples montor and carred out the updates: The trace and hstory are related n several aspects. These plots are called trace of beetles. In fgure1, the chans for whch convergence looks reasonable and the chans whch have clearly reached convergence. All rghts reserved by 49

6 Proportonal Hazards Regresson Model for Tme to Event Breast Cancer Data: A Bayesan Approach Fg. 1: Chans for whch convergence looks reasonable and the chans whch have clearly reached convergence The proposed Bayesan method was used to ft the survval models for non-nformatve rght censored breast cancer data. The results whch are presented n ths paper followed the trend and n fact t showed the realty. The statstcal analyss of hazard functon s assumed for the tme to death. Usng WnBUGS software (Lunn et al. 2000, 2009), the computatonal problem become easly and we proved that wth baselne hazard, the DIC value s smaller than the Gamma process pror for the nonparametrc part n Cox model. WnBUGS s a tool for analyzng survval data n a Bayesan framework usng Markov Chan Monte Carlo (MCMC) wth Gbbs sampler. Usng DIC (devance nformaton crteron); t s suted to compare Bayesan models whose posteror dstrbutons that have been obtaned usng MCMC. DIC has been mplemented as a tool n the BUGS software package. However, much techncal statstcal knowledge s requred for t to be used correctly. These programs provde an alternatve platform that could be used to confrm results of frequentst software. Moreover, for many models, frequentst nference can be obtaned as a specal case of Bayesan nference wth the use of non-nformatve prors (Ibrahm et al., 2001). The Bayesan approach enables us to make exact nference based on the posteror dstrbuton for any sample sze especally when sample sze s too small, whereas the 'frequentst' approach reles heavly on the large sample approxmaton, and there s always the ssue of whether the sample sze s large enough for the approxmaton to be vald (Ibrahm et al., 2001). There s a danger that the addtonal complexty of Bayesan methods could lead to mproper data analyss f t s not used correctly or choosng napproprate pror. Although, ths artcle proposed the pror elctaton for the baselne hazard n the Cox model, t s straght forward to extend to addtve hazard model wth smple modfcaton as suggested by Aalen et al, (2009). REFERENCES [1] Andersen P.K. Survval analyss : The second decade of the proportonal hazards regresson model. Statstcs n Medcne, 10: ,1991. [2] Andersen P.K, and R.D. Gll,.Cox s regresson model for countng processes: A large sample study. Annals of Statstcs, 10 (1982), [3] Aalen.O.O, Andersen P.K., Borgan, R.D.Gll, N. Kedng,. Hstory of applcatons of martngales n survval analyss. Electronc Journal for Hstory of Probablty and Statst. 5, 1 (2009), [4] Besag.J, Green. E, Hgdon D., and Mengersen K. Bayesan computaton and stochastc systems. Statstcal Scence, 10(1):3 41, [5] Chen M. H., Ibrahm J. G. and Snha D. A new Bayesan model for survval data wth a survvng fracton. Journal of Amercan Statstcal Assocaton, 94: , [6] Chen W. C., Hll B. M., Greenhouse J. B. and Fayos J. V. Bayesan analyss of survval curves for cancer patents followng treatment. In: J. M. Bernardo, M. H. Degroot, D. V. Lndley and A. F. M. Smth. Bayesan Statstcs 2, pages , [7] Chb S. and Greenberg E. Understandng the Metropols Hastngs algorthm. Amercan Statstcan, 4: , [8] Collet D. Modellng Survval Data n Medcal Research. Chapman and Hall, 1st edton, [9] Clayton D. A Monte Carlo for Bayesan nference n fralty models. Bometrcs, 47, (1991), [10] Clayton D., Bayesan analyss of fralty models. Techncal Report, Medcal Research Councl Bostatstcs Unt, Cambrdge, [11] Cox D.R., Regresson models and lfe tables (wth dscusson), Journal of the Royal Statstcal Socety, Seres B, 34 (1972), [12] Cox, D.R. (1975). Partal lkelhood. Bometrka, 62, [13] Congdon P. Appled Bayesan Modelng. John Wley & Sons, [14] Congdon P. Bayesan Statstcal Modellng. JohnWley & Sons, 2nd edton, [15] Gelman, A. and Rubn D. B., Inference from teratve smulaton usng multple sequences, Statstcal Scence, 7 (1992), [16] Geman S., and Geman D., Stochastc relaxaton, Gbbs dstrbuton and the Bayesan restoraton of mages, IEEE Transactons on Pattern Analyss and Machne Intellgence, 6 (1984), [17] Hastngs W. K.. Monte carlo samplng methods usng Markov Chans and ther applcatons. Bometrka, 1:97 109, [18] Hosmer D. W. and S. Lemeshow. Appled Survval Analyss: Regresson Modelng of Tme to Event Data. John Wley and Sons, [19] Hougaard P.. Fralty models for survval data. Lfetme Data Analyss, 1(3): , [20] Hougaard P.. Analyss of Multvarate Survval Data. Sprnger-Verlag, 2000 [21] Ibrahm J. G., Chen M., and Snha D., Bayesan Survval Analyss. Sprnger Verlag, New York, [22] Ibrahm J. G., Chen M. H., and Snha D. Bayesan semparametrc models for survval data wth a cure fracton. Bometrcs, 57: , 2001a. All rghts reserved by 50

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