Studies on Survival Analysis using Frailty Models under Bayesian Mechanism and its Further Scope

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1 IOSR Journal of Mahemacs (IOSR-JM) e-issn: , p-issn: X. Volume 12, Issue 4 Ver. V (Jul. - Aug.2016), PP Sudes on Survval Analyss usng Fraly Models under Bayesan Mechansm and s Furher Scope Jen Hazara 1 & Ku Kalpa Mahana 2 1 Professor, Deparmen of Sascs, Dbrugarh Unversy, Dbrugarh (Assam), Inda 2 Research Scholar, Deparmen of Sascs, Dbrugarh Unversy, Dbrugarh (Assam), Inda Absrac: In hs paper, an aemp has been made o dscuss fraly models and her furher developmens under he Bayesan approach. The modelng of baselne hazard funcon wh paramerc and semparamerc approach, dsrbuon of fraly varable along wh dervaon of poseror dsrbuon wh he help of he daa lelhood and he pror dsrbuon of he parameers of he fraly dsrbuon wh examples are also dscussed here. Based on hs revew wor s observed ha he Bayesan analyss of fraly models n survval analyss aans vas aenon by he researchers. The sudy of fraly models under Bayesan mechansm has been connung from varous vew pons, bu ye here s a vas scope of exendng he Bayesan approach n survval modelng n presence of unobserved random effec (fraly) n dfferen drecons. Keywords: Survval modelng, unobserved random effec, shared and correlaed fraly models, Marov Chan Mone Carlo mehods, Baselne Hazard funcon I. Inroducon Fraly models, a specfc area n survval analyss, provdes a convenen way o nroduce random effecs, assocaon and unobserved heerogeney no models for survval daa. In s smples form, a fraly s an unobserved random proporonaly facor ha modfes he hazard funcon of an ndvdual, or of relaed ndvduals. In essence, he fraly concep goes bac o he wor of Greenwood and Yule [1920] on accden proneness. The erm fraly self was nroduced by Vaupel e al. [1979] n unvarae survval models and he model was subsanally promoed by s applcaon o mulvarae survval daa n a semnal paper by Clayon[1978] (whou usng he noaon fraly ) on chronc dsease ncdence n famles. Normally n mos clncal applcaons, survval analyss mplcly assumes a homogeneous populaon o be suded. Ths means ha all ndvduals sampled no ha sudy are subjec n prncple under he same rs (e.g., rs of deah, rs of dsease recurrence). In many applcaons, he sudy populaon canno be assumed o be homogeneous bu mus be consdered as a heerogeneous sample.e., a mxure of ndvduals wh dfferen hazards. For example, n many cases s mpossble o measure all relevan covaraes relaed o he dsease of neres, somemes because of economcal reasons, somemes he mporance of some covaraes s sll unnown. The fraly approach s a sascal modelng concep whch ams o accoun for heerogeney, caused by unmeasured covaraes. In sascal erms, a fraly model s a random effec model for me o even daa, where he random effec (he fraly) has a mulplcave effec on he baselne hazard funcon. A fraly model s a mulplcave hazard model conssng of hree componens: a fraly (random effec), a baselne hazard funcon (paramerc or non paramerc), and a erm modelng he nfluence of observed covaraes (fxed effecs). Bayesan approach s a mehod of sascal nference n whch Bayes heorem s used o updae he probably for a hypohess as evdence s acqured. The Bayesan Mechansm refers o pror, poseror and predcve dsrbuons o oban esmaes, compare models and es hypohess and mae predcons condonal on an observed sample. The use of Bayesan mechansm n survval analyss s wde spread. Several researchers have been developng Bayesan mehodologes o analyse survval daa wh dfferen fraly models and pror processes. In hs paper, an aemp has been made o dscuss fraly models n he conex of Bayesan analyss based on pas leraures and her furher developmens. The paper ends wh a bref dscusson on scope for furher sudy. II. The Fraly Model Fraly models are exensons of populaon hazards model whch s bes nown as he Cox model [Cox, 1972], a wdely pursued model n survval analyss. Accordng o Cox he hazard rae of an ndvdual s gven by, μ (,X) = μ 0 ()exp(β T X).(2.1) Where μ 0 () denoes he baselne hazard funcon, assumed o be unque for all ndvduals n he sudy populaon. X s he vecor of observed covaraes and β he respecve vecor of regresson parameers o be DOI: / Page

2 esmaed. The mahemacal convenence of hs model s based on he separaon of he effecs of agng n he baselne hazard μ 0 () from he effecs of he covaraes n he paramerc erm exp(β T X). Fraly models exends he Cox model such ha he hazard of an ndvdual depends n addon on an unobservable random varable Z, whch acs mulplcavely on he baselne hazard funcon μ 0 (), μ (,Z,X) = Z μ 0 ()exp(β T X)..(2.2) Here fraly Z s he random varable varyng over he populaon decreases (Z<1) or ncreases (Z>1) he ndvdual rs. The mos mporan pon here s ha he fraly s unobservable. The respecve survval funcon S, descrbng he fracon of survvng ndvduals n he sudy populaon s gven by 1 S ( Z,X)= exp(-z μ 0 0 (s)ds exp(β T X))..(2.3) The cumulave hazard funcon s gven by M () = μ s ds.. (2.4) 0 III. Classfcaon of Fraly Models The fraly models can be classfed no wo broad classes, 1) Unvarae fraly Models: Models wh an unvarae survval me as endpon and 2) Mulvarae fraly Models: Models whch descrbe mulvarae survval endpons (e.g., compeng rs, recurrence of evens n he same ndvduals, occurrence of a dsease n relaves) 3.1 Unvarae Fraly Models In case of unvarae fraly models a unvarae (ndependen) lfeme s used o descrbe he nfluence of unobserved covaraes n a proporonal hazards model (heerogeney). The varably of survval daa s spl no a par ha depends on rs facor, and s herefore heorecally predcable, and a par ha s nally unpredcable, even when all relevan nformaon s nown. A separaon of hese wo sources of varably has he advanage ha heerogeney can explans some unexpeced resuls or gve an alernave nerpreaon of some resuls, for example, crossng-over effecs or convergence of hazard funcons of wo dfferen reamen arms [Manon and Sallard 1981] or levelng-off effecs - ha means he declne n he ncrease of moraly raes, whch could resul n a hazard funcon a old ages parallel o he x-axs [Aalen and Trel 1999] 3.2 Mulvarae Fraly Models In case of mulvarae fraly models mulvarae survval daa are used. Such nd of daa occurs for example f lfemes (or me of onse of a dsease) of relaves (wns, paren chld) or recurren evens le nfecons n he same ndvdual are consdered. Mulvarae survval mes are consdered for he dependence n clusered even mes, for example n he lfeme of paens n sudy ceners n a mul-cenre clncal ral, caused by cenre-specfc condons [Andersen e al., 1999]. A naural way o model dependence of clusered even mes s hrough he nroducon of a cluser-specfc random effec - he fraly. In oher words, he lfemes are condonal ndependen, gven he fraly. Ths approach can be used for survval mes of relaed ndvduals le famly members or recurren observaons on he same person. Le S ( 1 Z,X 1 ) and S( 2 Z,X 2 ) be he condonal survval funcon of wo relaed ndvduals wh dfferen vecors of observed covaraes X 1 and X 2 respecvely. Averagng over an assumed dsrbuon for he laen varables (e.g., usng a gamma, log-normal, sable dsrbuon) hen nduce a mulvarae model for he observed daa. In he case of pared observaons, he wo dmensonal survval funcon s of he form S ( 1, 2 ) = S ( 0 1 z,x 1 ) S( 2 z,x 2 )g(z)dz.(3.2.1) Where g (.) denoes he densy of he fraly Z. In he case of wns, S ( 1, 2 ) denoes he fracon of wns par wh wn 1 survvng 1 and wn 2 survvng 2. Fraly models for mulvarae survval daa are derved under condonal ndependence assumpon by specfyng laen varables ha ac mulplcavely on he baselne hazard. IV. Shared and Correlaed Fraly Models There are wo ypes mulvarae fraly models avalable n survval analyss,- Shared fraly model Correlaed fraly model The shared fraly model s relevan o even mes of relaed ndvduals, smlar organs and repeaed measuremens. Indvduals n a cluser are assumed o share he same fraly ha s why hs model s called shared fraly model. The survval mes are assumed o be condonal ndependen wh respec o he shared (common) fraly. For example consder he case of groups wh pars of ndvduals (bvarae falure mes, e.g., even mes of wns or paren-chld). Condonal on he fraly Z, he hazard funcon of an ndvdual n a par s of he form Z μ 0 ()exp(β T X), where he value of Z s common o boh ndvduals n he par, and hus s he cause for DOI: / Page

3 dependence beween survval mes whn pars. Independence of he survval mes whn a par corresponds o a degenerae fraly dsrbuon (Z=1, σ 2 =0). In all oher cases wh σ 2 >0 dependence s posve by consrucon of he model. Condonal on Z, he bvarae survval funcon s gven as S ( 1, 2 Z) = S 1 ( 1 ) z S 2 ( 2 ) z (4.1) In mos applcaons s assumed ha he fraly dsrbuon (.e. he dsrbuon of he random varable Z) s a gamma dsrbuon wh mean 1 and varance σ 2. Averagng he condonal survval funcon produces under hs assumpon survval funcons of he form S( 1, 2 )= ( S 1 ( 1 ) σ 2 + S 2 ( 2 ) σ 2-1) 1/σ 2 (4.2) Shared fraly explans correlaons beween subjecs whn clusers. However does have some lmaons (See Wne [2003]) also. The correlaed fraly models were developed for he analyss of bvarae falure me daa, n whch wo assocaed random varables are used o characerze he fraly effec for each par. For example, one random varable s assgned for parner 1 and one for parner 2 so ha hey would no longer be consraned o have a common fraly. These wo varables are assocaed and have a jon dsrbuon. Knowng one of hem does no necessarly mply nowng he oher. There s no more a resrcon on he ype of correlaon. These wo varables can also be negavely assocaed, whch would nduce a negave assocaon beween survval mes. Assumng gamma frales, Yashn and Iachne(1995) used he correlaed gamma fraly model resulng n a bvarae survval dsrbuon of he form S ( ) S ( ) / ( S1( 1) S2( 2) 1) S(, ).. (4.3) One mporan problem n he area of fraly model s he choce of fraly dsrbuon. The fraly dsrbuons mos ofen appled are he gamma dsrbuon (Clayon [1978]; Vaupel e al.[ 1979]), he posve sable dsrbuon (Hougaard [1986a]), a hree-paameer dsrbuon (PVF) (Hougaard [1986b], he compound possson dsrbuon (Aalen [1988,1992] and he log normal dsrbuon (McGlchrs and Asbe, [1991]). V. Bayesan Approach n Fraly Models Bayesan approaches o fraly models wh recen advances are compuaonally feasble n compung echnology. Bayesan analyss of survval daa usng semparamerc models sared mmedaely afer Cox [1972] wh wor of Ferguson [1973] and Kalbflesch [1978]. 5.1 Hazard Funcon Modelng Under Bayesan approach a paramerc as well as a semparamerc approach s possble for modelng he baselne hazard funcon. In he paramerc case, we can model he baselne hazard funcon paramercally. Sahu e al.[1997] used Webull dsrbuon o model he baselne hazard funcon and hey compared wh he flexble baselne hazard model based on correlaed pror process. Le us consder he survval me of h ndvdual (=1,2, n) be T and gven he unobserved fraly parameer denoed by z (for he h ndvdual), he hazard funcon as gven n equaon (2.2) s as follows: μ( x,z ) = μ 0 ( )exp(β T x )z..(5.1.1) Where x s he fxed covarae vecor, β s he regresson parameer and μ 0 (.) s he baselne hazard funcon and z s he fraly parameer for he h ndvdual. In he paramerc approach we can model he baselne hazard funcon paramercally. For example f we use webull dsrbuon o model he webull baselne hazard funcon s gven by μ 0 ( ) = λγ γ-1,, =1,2, n (5.1.2) Where λ,γ>0 ; λ and γ are unnown hypermeers. Now from equaon (5.1.1) he hazard of he h ndvdual s gven by μ ( ) = λγ γ-1 exp(β T x )z...(5.1.3) The cumulave hazard funcon s gven by M () = λ γ exp(β T x )z...(5.1.4) Then he survval funcon s gven by S () =exp (-M ()) = exp (-λ γ exp(β T x )z ).(5.1.5) Le us consder rgh censored survval daa (,δ ), =1,2,,n and assume ha he censorng s non nformave. Le δ be he ndcaor varable ang value 1 f he h ndvdual fals and value 0 oherwse. Hence s a falure f δ =1 and censorng me oherwse. Hence he rple (,δ,x ), =1,2,,n s observed for all n ndvduals. Gven he unobserved fraly z, s are ndependen. Hence he complee daa lelhood s gven by DOI: / Page

4 L (parameer, daa) = n 1 n [ ( )] S ( ).. (5.1.6) 1 T T = [ exp( x ) z ] exp ( exp x ) z 1 Agan recen advances n Bayesan nonparamerc survval analyss gve us wde range of choces for he pror processes o model he baselne hazard funcon. Modelng he baselne hazard usng pror processes s very common; see Snha and Day [1997] for more deals. As for example le us consder a pecewse exponenal model. Pecewse exponenal models and pror processes on he componens provde a very flexble framewor for modelng unvarae survval daa. Le {(,δ,x ), =1,2,,n} be he observed daa. We dvde me no g prespecfed nerval I = ( -1, ) for = 1,2,,g where 0 = 0 < 1 <... < g <, g beng he las survval or censored me and assume ha he baselne hazard o be consan whn each nerval..e, μ 0 ( ) = μ, for ϵ I K.(5.1.7) To correlae he μ s n adjacen nervals, a dscree-me marngale process s used by Arjas and Gasbarra[1994] for unvaae survval models. Gven (μ 1, μ 2,, μ ) hey specfy ha μ μ 1, μ 2,, μ -1 Gamma (, ), = 1,2,,g..(5.1.8) Where μ 0 =1. The parameer α conrols he amoun of smoohness avalable. The correspondng hazard and cumulave hazard funcon for he h ndvdual s gven by μ ( ) = μ exp(β T x )z...(5.1.9) (5.1.10) M () = μ exp(β T x )z Where = 1 0, f 1, f 1 1 1, f If he h ndvdual survved beyond he h nerval.e, hen he lelhood conrbuon s gven by (μ exp(β T x )z ) 0 exp(-μ exp(β T x )z ( 1)) = exp (-μ exp(β T x )z ( 1)) When 1 he lelhood conrbuon s gven by T T x z x z 1 ( exp ) exp( exp( ) ( ) ) Hence we arrve a he followng complee daa lelhood n g 1 1 T T T x z 1 g 1 x z g 1 x z g { exp( exp( ) ( ))}( exp ) exp( exp( ) ( )).. (5.1.11) Where g s such ha ϵ ( g, 1) g Ig 1 Kalbflesch [1978] and Ferguson and Phada [1979] nroduced he gamma pror process for non paramerc Bayesan analyss of rgh censored connuous survval daa. Agan Sar e al[2014] used gamma pror process o he recurrence of nfecon me daa a he pon of caheer nseron for dney paens usng porable equpmen. Neo-Barajas and Yn [2008] oo a mxure pror of a Marov gamma process for full Bayesan analyss of a bone marrow ransplan daa se. Burrdge [1981], Clayon [1991] used Levy process wh posve ndependen ncremens n dsjon nervals o model he pror process of cumulave hazard. Gamerman [1991] presened a lnear Bayesan mehod o analyze unvarae survval daa wh possble rgh censorng usng he auo-correlaed pror process. Snha[1998] also used an auo-correlaed ncomplee pror process for he baselne hazard and a jon pror for unnown regresson parameer and varance (ndependen of he pror process) n he condonal proporonal hazard model. Snha[1993] presened he Bayesan semparamerc condonal posson process model for mulple even-me daa. Ferguson and Phada[1979] and Ioro e al. [2009] wored on Bayesan non paramerc survval analyss usng Drchle pror process. Pennell and Dunson[2006] also used he Drchle pror process for a semparamerc shared fraly and for mulplcave nnovaon on he fraly for mulple even me daa. DOI: / Page

5 5.2 The Fraly Dsrbuon The fraly random varable z s are assumed o be ndependen and dencally dsrbued for each ndvdual havng some paramerc dsrbuon wh un mean (o oban denfably). A few examples of fraly dsrbuons are gven n Clayon and Cuzc[1985], Oaes[1989]. The gamma dsrbuon s mos commonly used o model he fraly [Oaes, 1989; Clayon, 1991]. As for example Aslandou e al.[1998] assumed ha z s o be ndependen wh z Gamma( -1, -1 ), =1,2,,n (5.2.1) so ha s he unnown varance of z s and he densy of Gamma (a,b) s proporonal o x a-1 exp (-bx). Thus, hgher value of sgnfes greaer heerogeney among dfferen ndvduals. 5.3 Dervaon of Poseror Dsrbuons The Bayes heorem leads o he followng jon poseror densy funcon of he parameer vecor f ( daa ) f ( ) f ( daa) =...(5.3.1) f ( daa) The Bayes heorem can also be appled o one of he parameers j whle condonng on he oher parameers. We wll denoe he parameer vecor whou he j h componen j by (-j). For j h componen of he followng condonal poseror densy f ( daa ) f ( ) f ( j daa, (-j) ) = f ( daa ) DOI: / Page j ( j) ( j).(5.3.2) Furhermore, snce he condonng of he pror densy of j on he oher parameers n no nsrumenal (we assume ndependence of pror denses) we drop here hs ype of condonng. In he rghhand sde of he condonal poseror densy (5.3.2) he denomnaor does no depend on j. herefore, he numeraor on he rgh-hand sde of (5.3.2) and he condonal poseror densy of j are proporonal: f ( j daa, (-j) ) f ( daa ) f ( ).(5.3.3) usng he lelhood funcons (5.1.6) and (5.1.11) we wll mae use of he unnormalsed condonal poseror denses of form (5.3.3) for all he parameers snce he normalzng facor (.e, he denomnaor on he rghhand sde of (5.3.2) s n mos cases, dffcul o oban. The daa lelhood of he parameers n he fraly model can be obaned by negrang ou he fraly varable z s from he lelhoods (5.1.6) and (5.1.11). The fnal forms of he daa lelhoods afer negraon are oo complcaed o wor wh. Thus, s no easy o evaluae he margnal poseror dsrbuons of he parameers analycally. To crcumven hs problem, we use a Marov chan Mone Carlo mehod Gbbs sampler, see e.g., Gelfand and Smh [1990] and Gls e al. [1996] wh he daa augmenaon mehod (Tanner and Wong, 1987) o generae samples from he approprae margnal poseror dsrbuons. 5.4 MCMC Mehods n Fraly Models Aslandou e al.[1998] analysed a mulvarae survval daa from a Bayesan perspecve usng Marov Chan Mone Carlo(MCMC) mehods. Mallc e al. [1999] used a Bayesan approach o model unvarae and mulvarae survval daa wh censorng. They descrbed a MCMC algorhm o oban he esmae of he hazard funcon as well as he survval curve. Waler and Mallc [1999] used a Bayesan semparamerc approach for acceleraed falure me model and hey descrbed a MCMC algorhm o oban a predcve dsrbuon for a fuure observaon gven boh uncensored and censored daa. They explaned how he resulan model s mplemened va MCMC mehods. Snha and Ma [2004] developed a Bayesan analyss based on a MCMC algorhm for he analyss of Panel- Coun Daa wh Dependen Termnaon of a daa se from a clncal ral. They used Bayesan approach usng MCMC mehods wh he help of a comprehensve smulaon sudy. Yn and Ibrahm [2006] used he MCMC algorhm for a real daa se from a melanoma clncal ral. Agan Basu and Twar [2010] developed a model ha unfes he mxure cure and compeng rs approaches and ha can handle he mased causes of deah n a naural way. In hs developmen he Marov Chan samplng s used. VI. Furher Scope of he Sudy From he above dscusson, s observed ha he Bayesan analyss of fraly models n survval analyss aans vas aenon by he researchers. The sudy of fraly models under Bayesan mechansm has been connung from varous vew pons, bu ye here s a vas scope of exendng he Bayesan approach n survval modelng n presence of unobserved random effec (fraly) n dfferen drecons. Moreover, s o be noed ha he behavor of he unobserved covaraes are dfferen accordng o dfferen soco-demographc varables, mehod of dagnoss of he dsease and effec of correlaed varables. j

6 So far our nowledge goes sll here are some fraly models n survval analyss whch are remaned undagnosed under Bayesan mechansm. For example some daa arses n real lfe whch s nown o be symmerc ofen exhbs deparure from symmery.e, presence of sewness. In mos of he cases s observed ha sewness s also presen n survval daa. An ordnary mehod for reducng he sewness s he use of square roo or logarhm ransformaons of responses. In mos of he survval daa ses, because of hgh rae of mssngness, sewness s remaned even afer usng he ransformaons. An alernave s o develop sew models (by nroducng sewness) sarng from symmerc models, whch can effecvely deal wh nroduced sewness n oherwse symmerc daa. Sew normal dsrbuon of Azzaln (1985) and s furher exenson by Charabory e al. [2015] are such nd of dsrbuons (for more deals abou sewed dsrbuons see Charabory and Hazara [2011]). A few number of wors are done usng sewed dsrbuon o model survval daa. For example Callegaro [2012] used Log-sew-normal dsrbuon o f acceleraed falure me models. Agan Baghfala and Ganjal [2015] used a sew-normal mxed effec model for longudnal measuremens and a Cox proporonal hazard model for me o even varable usng a real HIV daa se. Bayesan approach usng Marov chan Mone Carlo s adoped by hem for parameer esmaon. Also Sahu and Day [2004] developed a class of log-sew- dsrbuon for he fraly. Ths class ncludes he log-normal dsrbuon along many oher heavy aled dsrbuons e.g. Cauchy or log- as specal cases. Bu, mos of he above sewed dsrbuons are no consdered under Bayesan mechansm. So here s a vas scope o exend he wor of fraly models o some sewed dsrbuons under Bayesan mechansm. References [1] Aalen, O.O. Heerogeney n survval analyss, Sascs n Medcn [2] Aalen, O.O. Modellng Heerogeney n Survval Analyss by he Compound Posson Dsrbuon, Annals of Appled Probably 4(2), [3] Aalen, O.O., Trel, S. Analysng ncdence ess cancer by means of a fraly model,cancer Causes and Conroll, [4] Andersen, P.K., Klen, J.P., Zhang, M. J. Tesng for cenre effecs n mul-cenre survval sudes: A Mone Carlo Comparson of Fxed and Random Effec, Tess Sascs n Medcne 18, [5] Arjas, E. and Gasbarra, D., Nonparamerc Bayesan nference from rgh censored survval daa, usng he Gbbs sampler. Sasca Snca, [6] Aslandou, H., Dey D. K. and Snha D. Bayesan analyss of mulvarae survval daa usng Mone Carlo mehods, The Canadan Journal of Sascs, 26(11),1998,33-48 [7] Azzaln, A. A class of dsrbuons whch ncludes he normal ones, Scandnavan journal of sascs, [8] Basu, S. and Twar, R.C. Breas Cancer survval, compeng rss and mxure cure model: a Bayesan analyss J.R. Sas. Soc. A 173, 2010, [9] Baghfala, T., & Ganjal, M. A bayesan approach for jon modelng of sew-normal longudnal measure-mens and me o even daarevsa Sascal Journal, 13(2), 2015, [10] Burrdge, J. Emprcal Bayes analyss of survval me daa. 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