Bayesian Inference in a Joint Model for Longitudinal and Time to Event Data with Gompertz Baseline Hazards

Transcription

1 Modern Appled Scence; Vol. 12, No. 9; 218 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton Bayesan Inference n a Jont Model for Longtudnal and Tme to Event Data wth Gompertz Baselne Hazards Josua Mwanyekange 1, Samuel Mwall 1,2 & Oscar Ngesa 1,2,3 1 Pan Afrcan Unversty Insttute for Basc Scence, Technology and InnovatonPAUSTI), Department of MathematcsStatstcs), Narob, Kenya 2 Jomo Kenyatta Unversty of Agrculture and TechnologyJKUAT), Department of Statstcs and Actuaral Scence, Narob, Kenya 3 Tata Taveta Unversty, Narob, Kenya Correspondence: Josua Mwanyeaknge, Pan Afrcan Unversty Insttute for Basc Scence, Technology and InnovatonPAUSTI), Department of MathematcsStatstcs), Narob, Kenya. E-mal: josua@ams-cameroon.org Receved: June 19, 218 Accepted: June 25, 218 Onlne Publshed: August 24, 218 do:1.5539/mas.v12n9p159 URL: Abstract Longtudnal and tme to event data are frequently encountered n many medcal studes. Clncans are more nterested n how longtudnal outcomes nfluences the tme to an event of nterest. To study the assocaton between longtudnal and tme to event data, jont modelng approaches were found to be the most approprate technques for such data. The approaches nvolves the choce of the dstrbuton of the survval tmes whch n most cases authors prefer ether exponental or Webull dstrbuton. However, these dstrbutons have some shortcomngs. In ths paper, we propose an alternatve jont model approach under Bayesan prospectve. We assumed that survval tmes follow a Gompertz dstrbuton. One of the advantages of Gompertz dstrbuton s that ts cumulatve dstrbuton functon has a closed form soluton and t accommodates tme varyng covarates. A Bayesan approach through Gbbs samplng procedure was developed for parameter estmaton and nferences. We evaluate the fnte samples performance of the jont model through an extensve smulaton study and apply the model to a real dataset to determne the assocaton between markerstumor szes) and tme to death among cancer patents wthout recurrence. Our analyss suggested that the proposed jont modelng approach perform well n terms of parameter estmatons when correlaton between random ntercepts and slopes s consdered. Keywords: jont modelng, Longtudnal data, tme to event data, Bayesan nference, gompertz dstrbuton, gbbs samplng, MCMC 1. Introducton In many clncal studes, longtudnal outcomes are collected alongsde wth tme to event data. For example, we collect repeated patents nformaton such as heghts, weghts, blood pressures and many other varables over tme and at the same tme we are also nterested n the tme to an event of nterest e.g., death). For such data, the focus s always on how longtudnal outcomes nfluence the tme to an event. Approaches for analyzng these two processes separately have been extensvely dscussed n the lterature. The most common methods used to analyze longtudnal outcomes are the random effect model proposed by Lard and Ware 1982), lnear mxed effects model by Verbeke 1997) and Generalzed lnear models by Lang & Zeger 1986), Zeger et al. 1988). On the other hand, event tme process s analyzed by makng use of cox proportonal hazard models Hougaard 212). However, analyss of longtudnal and tme to event process separately have receved many crtcsms as they bear neffcent and bas results. Therefore, t s very mportant to take nto account both processes n analyss. Over the past two decades, extensve efforts have been made on statstcal methods that smultaneously analyze longtudnal outcomes and tme to event data. These methods offers great advantages over separate analyzng of each process. A comprehensve overvew of jont modelng for longtudnal and tme to event data was gven n Tsats & Davdan 24). More basc concepts and methods for jont models can be found n Ibrahm et al. 21). In all these revews, the two processes are jontly modeled by lnkng them together through a common latent structure. That s the longtudnal and survval process both share the same random effects. Extensons to earler developments of jont modelng framework has been proposed. More recent developments wth applcatons ncludes, Sweetng & Thompson 211), Rzopoulos et al. 212), McCrnk et al. 213) and Yang et al. 216). Some authors have already proposed jont models for multple longtudnal outcomes and repeated events of dfferent outcome Ch & Ibrahm.,26; Musero et al.,215; Huang & 159

2 Elashoff.,21). Up to date, there are three frequently used types of estmaton approaches for jont models of longtudnal and tme to event data, namely: maxmum lkelhood approach, two stage approach and Bayesan approach. Accordng to Lang 211), parameters of the jont model can be smultaneously estmated by ncorporate both processes n a log jont lkelhood functon and use EM algorthm. The EM algorthm s commonly used for lkelhood nferences of jont models due to ts stablty and generalty Tsats & Davdan 24), Tseng et al. 25). However, the method can be computatonally challengng when unobserved random effects have hgh dmensons. The two stage approach comprses two stages. The frst stage fts lnear mxed effects model to the longtudnal data and parameter estmate are obtaned based on the ftted model wthout consderng the survval model. In the second stage, survval model s ftted separately wth longtudnal values predcted n stage one used as true observed values. Even though, two stage approach seem to more computatonally frendly, the approach may lead to based nferences. On the other hand, Bayesan approach has been studed by a number of authors such as Faucett & Thomas 1996), Ibrahm et al. 24), Martns et al. 21), Haung et al. 211) and Rzopoulos & Ghosh 211). Jont models may have too many unknown parameters whch n some cases can be dffcult to solve analytcally. Bayesan approach uses pror nformaton by ncorporate them nto current analyss called posteror dstrbuton. Then Markov chan samples can be drawn from posteror dstrbuton for nference. Conversely, the approach may not provde good nferences as the results may heavly affected by the pror nformaton. Nevertheless, Bayesan s easy to mplement whch make t more useful for nference of jont models. Prevous work on jont model for longtudnal outcomes and tme to event data wth tmes varyng covarates consdered lnear mxed effects models for longtudnal process and a proporton hazards model wth specfed baselne functon for tme to event process, Yu et al. 26); Song et al. 22). The most frequently used beelne functons for proportonal hazards models assumes that survval data follow exponental or Webull dstrbuton. Some authors suggested that proportonal hazard model wth Webull baselne functon may be very flexble when tme dependent covarates are ncluded n the model Casellas, 27). However, generatng survval data from such model wth tme varyng covarates can be too complex. Accordng to Austn 213), one of the dsadvantages of Webull proportonal hazards model wth tme dependent covarates s that ts cumulatve dstrbuton functon can not be derved n a crossed form. Therefore, to compute the cumulatve ncdence and the hazard functon, we need to make use of numercal ntegraton method. On the other hand, proportonal hazard model wth exponental baselne functon also has dsadvantages as t assumes constant hazards. For these reasons, researchers wllng to generate survval tmes that nvolve tme dependent covarates are advsed to consder usng Gompertz dstrbuton event tmes data Austn, 212). Proportonal hazards models wth baselne functon from Gompertz dstrbuton seems to be more flexble as one can compute survval functon wthout hgh programmng needs. The Gompertz dstrbuton was frst ntroduced by Gompertz 1825). It s descrbed as one of the fundamental mathematcal models that accurately represents survval functon based on the laws of mortalty. That s the force of mortalty or survval tmes tends to ncrease exponentally over tme. It has been extensvely used as growth model n many cancer assessments. It s for these reasons that Gompertz dstrbuton plays an mportant role n modelng human mortalty. Its earler applcatons can be found n Ahuj & Nash 1967). To the best of our knowledge, there has been no jont model for longtudnal outcome and tme to event data wth proportonal hazard model assumed to have Gompertz dstrbuton propertes. Therefore, n ths artcle, we propose such model wth tme varyng covarates under the framework of Bayesan nferences. Our jont model s made of two submodels lnked together by common random effects. The frst submodel s a lnear mxed effects model for modelng true and unobserved markers. The nterrelatonshp between measurements and subject specfc effects s accounted by random ntercept and slope. The second submodel s the proportonal hazards model wth Gompertz baselne functon. To estmate the parameters of the jont model, we use both R and WnBUGS, Ntzoufras 211) softwares. The rest of ths artcle s organzed as follows; In secton 2 we present the notatons and formulatons of the jont model. Secton 3 presents estmatons and nferences, whch ncludes jont lkelhood and Bayesan parameter estmaton procedures such as pror and jont posterors specfcatons and model selecton. In secton 4, we present some smulaton studes n order to access the performance of the model. Secton 5, presents the applcaton of the model to real datasets. The last secton presents a dscusson. 2. Method 2.1 Longtudnal Sub-Model Suppose we have observatons for n subjects under study. Let y t j ) be n 1 column vector of random varables representng the observed longtudnal outcomes for subject measured at tme ponts t j t1, t 2,..., t n ), where j = 1, 2,..., n ). Here n represents the numbers of repeated measurements for subject whch vares among subjects. In practce, we may have mssng observatons for some subjects. Ths happens due to the fact that some subjects may decde to drop out of the study for reasons not related to the occurrence of event of nterest. Therefore, n ths study, we assume that these 16

3 mssng values n longtudnal measurements trajectory are mssng ndependently of the unobserved measurements. To analyze the longtudnal process, we defne the dstrbuton of y j by a Lnear Mxed EffectLME) model, where, y j = µ t j) + ɛ j, 1) µ t ) ) j) = x t j βl + η t j s the true value of y j and the outcome varable x s the n p) desgn matrx of fxed effects whch ncludes possble tme ) ) dependent covarates; β l s a p 1) correspondng column vector of the fxed effect coeffcents β L,s ); η t j = z t j w, where z denotes q 1) desgn matrx for the random effects; w MVN, A ) and ɛ s a n 1) column vector of the resduals whch represents the part of y j whch s not accounted by the model: x ) ) t j βl +η t j such that ɛ j N ), σ 2 ɛ I n. It should be noted here that A represents the varance co-varance matrx n whch correlatons among the repeated measurements and wthn subject correlaton measurement values are represented. On other hand resduals among subjects and random effects w are ndependent of each others. The varance covarance matrx s defned as ) σ A = 2 w ρσ w σ w1 ρσ w σ w1 σ 2 w 1 or wth no correlaton. 2.1 Tme to Event Sub-Model A = ) σ 2 w σ 2 w 1 We assumed a proportonal hazards model wth Gompertz baselne functon. Let λ t) = λ t) exp ξ v + ψµ t)) 2) denote the hazards functon for subject at tme t, where λ t) = α exp γt) s a baselne functon wth α > and γ takes any value. When γ > or γ <, the hazards functon n monotonencreasng or decreasng). On the other hand, when γ =, the hazards functon s equvalent to hazards functon from an exponental dstrbuton. v represents the vector of prognostc factor assocated wth vector of coeffcent ξ and ψ quantfes the strength between the two processes. Furthermore, the cumulatve hazards/ncdences functon s defned as follows: Equaton 3) can be smplfed as follows: Λ t) = t Λ t) = α α expγs) exp ξ v + ψµ s)) s 3) t exp γs + ξ v + ψµ s) ) s 4) From equaton 4), we deduce [ exp γ + ψµ )s) Λ t) = α expξ v ) γ + ψµ ] t [ exp γ + ψµ ] )t) 1 = α expξ v ) γ + ψµ 5) So, the nverse cumulatve hazards functon s gven as Λ 1 t) = 1 γ + ψµ γ + ψµ ) log )t α expξ v ) + 1 6) and the ndvdual survval functon s expressed as S t) = exp α expξ v ) [ exp γ + ψµ )t) 1 γ + ψµ ])). 7) 161

4 Consequently, the ndvdual event tmes can be generated as 1 γ + ψµ ) [ T = γ + ψµ log ) logu)) exp γ + ψµ ] )t) 1 logu)) + 1, f logu) < α expξ v α expξ ) v ) γ + ψµ, 8) where T stands for the survval for subject and U s a random varable unformly dstrbuted wth U Un f orm, 1). The dstrbuton functon of the tme to event based on the proportonal hazards functon s [ exp γ + ψµ ])) )t) 1 F t) = 1 exp α expξ v ) γ + ψµ and the probablty densty functon of tme to event s 3. Estmaton and Inferences 3.1 The Jont Lkelhood Functon f t) = α exp γt + ξ v + ψµ t)) exp α expξ v ) [ exp γ + ψµ )t) 1 γ + ψµ 9) ])). 1) Let C be the censorng tme for subject. Then, we have T = mn T, C ) denotes the th subject s observaton tme, where T s the true event tme for that subject. Furthermore, we denote δ = I ) T = T, were I s the ndcator functon. It follow that, δ s equal to 1 f T s an event tme and f T rght censored for subject. Lettng φ = { β L, α, ψ, γ, A, σ 2 ɛ, ξ } denote the vector of all parameters n equaton to be estmated. Note that y j s the longtudnal outcomes for subject measured at tme t j j = 1, 2,..., n ). Then, gven subject- specfc random effects, the two process are sad to be ndependent of each other. Hence, the jont lkelhood functon of the longtudnal and tme to event sub-models based on all observed data D obs = { y j, T, δ }, x, v s defned by n n Lφ w, D obs ) = f y y j w, φ) { f T T, δ w, φ) } f w w φ) w, 11) where =1 f y y j w, φ) = 2πσ 2 ɛ j=1 ) n /2 exp { 1 / 2σ 2 ɛ ) [ y j x t j)β L + η t j ) ) ] 2 } s the probablty densty functon of the longtudnal outcomes condtonally on the random effects w and f w w φ) = 2π) q/2 A 1/2 )) exp 1/2) w A 1 w s defned as the probablty densty functon of the random effects wth q as dmenson of the covarance matrx A. The lkelhood of the survval sub-part f T T, δ w, φ ) = λ T w, φ ) δ S T w, φ ) whch s a verson of equaton 7) n secton Bayesan Inferences The soluton to the jont lkelhood functon 11) can not be acheved wth the normal standard maxmum lkelhood procedure due to the fact that t nvolves ntegraton of the longtudnal and survval components over the subject-specfc random effects w. Ths requres hgh programmng needs. The best way to overcome such dffcult s to make use of Bayesan approach. Faucett and Thomas 1996), proposed a Bayesan Markov Chan Monte Carlo MCMC) approach that smulates samples from posteror dstrbuton and estmate all the unknown parameters by usng non-nformatve pror pφ). Takng equaton 11) and pφ), we can defne jont posteror of φ as product of the lkelhood of the observed data and some prors as Lφ w, D obs )pφ) pφ D obs ) = n=1 f y,t y j, T φ, δ, x, v )pφ) φ. 14) 12) 13) If the observed data s fxed, then we have pφ D obs ) Lφ w, D obs )pφ.) 15) 162

5 Therefore, ths translate our jont posteror dstrbuton of the parameter φ nto n n pφ D obs ) f y y j w, φ) { f T T, δ w, φ) } f w w ; φ) pφ). 16) =1 j=1 Due to computatonal complexty, log jont posteror dstrbuton s more preferable. Hence, n n log pφ D obs ) log f y y j w, φ) + { log f T T, δ w, φ) } + log f w w ; φ) + pφ). 17) =1 j= Full Condtonal Dstrbutons In order to mplement the Bayesan procedure, we need the full condtonal dstrbuton of each of unknown parameters of the model. Then Gbbs sampler can be used to generate MCMC samples from the jont posteror densty pφ D Obs ). In Bayesan framework, ths procedure nvolves teratvely samplng from ts full condtonal dstrbutons wth the remanng componentsothers) fxed to ther current value. Hence, the full condtonal dstrbuton of the coeffcent β L of the lnear mxed effect model s pβ L others) n n [ exp 1/2σ2 ɛ ) y j µ t j) ] 2 α exp γt + ξv + ψµ t))) δ exp Λ t)) =1 where µ βl and τ βl are the parameters of the pror of β L. The full condtonal dstrbuton of ξ s pξ others) n =1 j=1 exp 1/2)β L µ βl ) τ 1 β L β L µ βl ), 18) α exp γt + ξv + ψµ t))) δ exp Λ t)) exp 1/2)ξ µ ξ ) τ 1 ξ ξ µ ξ), 19) where µ ξ and τ ξ are the parameters of the ndependent normal pror of ξ. The full condtonal dstrbuton of γ takes the form pγ others) n =1 α exp γt + ξv + ψµ t))) δ exp Λ t)) exp 1/2)γ µ γ ) τ 1 γ γ µ γ ) ), 2) where µ γ and τ γ are the parameters of the ndependent normal pror of γ. The full condtonal dstrbuton of shape parameter, α n the survval sub-model s gven by pα others) n α exp γt + ξv + ψµ t))) δ exp Λ t)) α a 1 exp αb o ), 21) =1 where a and b are the parameters of the ndependent gamma pror of α. The full condtonal dstrbuton of the assocaton between the two processes ψ s pψ others) n =1 α exp γt + ξv + ψµ t))) δ exp Λ t)) exp 1/2)ψ µ ψ ) τ 1 ψ ψ µ ψ ) where µ ψ and τ ψ are the specfed parameters of the ndependent normal pror of ψ. The full condtonal dstrbuton of nverse varance, 1 σ 2 ɛ ) 1 p others σ 2 ɛ takes the form n ) 2πσ 2 n /2 n ɛ exp 1/2σ2 ɛ ) =1 where a and b are the parameters of the nverse gamma of σ 2 ɛ. j=1 22), [ y j µ t j) ] 2 ) σ 2 a1 1 ) ɛ exp b1 /σ 2 ɛ, 23) 163

6 The full condtonal dstrbuton of the nverse covarance varance, A 1 takes the form pa 1 others) n A n /2 )) exp 1/2) w A 1 w A ν +q+1)/2) exp 1/2) tr A A 1)), 24) =1 where ν and A are the parameters of the nverse wshart pror of A. More smplfed full condtonal dstrbuton can be derved the way as n Faucett & Thomas 1996) and Zhang et al. 217). 3.3 Model Selecton In ths paper, we propose two forms of the random effects terms n the jont model: a) Correlated random ntercepts and slope - Case I b) Uncorrelated random ntercepts and slopes - Case II For the model selecton, we consder the Devance nformaton crteradic)spegelhalter et al., 22) whch defned as a verson of well known Akake s nformaton crteraaic) and Bayesan crterabic). The dea of usng of DIC s that, DIC uses the posteror dstrbuton whch allows t to take nto account the pror nformaton of both longtudnal and tme to event sub-models. Consder φ, the collecton of parameters n the jont model and D obs = { y j, T, δ, x, v }, the observed data. Now, let specfy the devance functon as Dφ) = 2 n log f D obs φ) = 2 logφ w, D obs ), 25) =1 where logφ w, D obs ) s the lkelhood n equaton11). Defne Dφ) = E [ 2 n =1 log f y j, T, δ, x, v ; φ) ] as the expectaton of the devance under posteror and φ = E [ φ ] posteror means of the parameters. Accordng to Spegelhalter et al., 22, the dfference between two measures denoted by p D = Dφ) Dφ) can be nterpreted as the posteror estmate of the effectve number of parameters and t measures the complexty of the model. Hence addng t to the posteror mean Devance, gves a measure of ft that penalzed for complexty. Therefore, DIC = Dφ) + 2p D = Dφ) + p D. 26) Based on the DIC, the model wth the smallest DIC value s consdered to be the model that would best predct a replcated dataset whch has the same structure as the current observed dataset. However, as stated by Geedpally at el., 214, the model s penalzed by the Dφ) whch wll decrease as the number of the parameters n the model ncreases and p D, whch compensates for ths effect by favorng model wth a smaller number of parameters. Therefore, t s very mportant to note that the way the model s parameterzed wll nfluence the outcomes of the Devance Informaton Crteron DIC) values. 4. Smulaton Studes In ths secton, we performed two sets of smulatons studes under Markov Chan Monte Carlo MCMC) n order to assess the performance of the proposed methodology. 4.1 Smulaton Study 1 The data are generated from a jont model wth a sngle longtudnal varable and a tme to event varable. Each subject s expected to have a record of n = 5 bomarker values recorded at baselne and thereafter 4 vsts are scheduled at equally spaced tme nterval t j = {.,.15,.3,.45,.6}. After ths perod, subjects are sad to be censored nonnformatvely. Specfcally, we frst generate data usng a Lnear Mxed Effects model for the longtudnal sub-model; y t j ) = µ t j) + z t j)w + ɛ j, 27) where µ t j) s the true unobserved bomarker value and ɛ t j ) N, σ 2 ɛ ), where σ 2 ɛ =.5 s the measurement error varance. We then smulate longtudnal data from a lnear curve, µ t j) = β + β 1 x + β 2 t j + w + w 1 t j, where the vector of random effects w = w, w 1 ) s smulated ) from a multvarate) normal dstrbuton MVN, A), wth σ varance covarance matrx A = 2 w ρσ w σ w ρσ w σ w1 σ 2 = w Note that ρ =.2. We set coeffcents β = β, β 1, β 2 ) = 3.,.5, 5.). 164

7 For the survval sub-model, we have λ t v, µ t)) = λ t) exp ξv + ψµ t)) 28 where λ t) = α expγt), wth α =.2 and γ = 1.2, v s a vector of baselne covarates wth assocated coeffcents ξ = ξ, ξ 1 ), wth ξ =.74 and ξ 1 =.15. The values of baselne covarates were smulated as follows: x N12, 4) and v Bn1,.5) The parameter ψ was set at.2. Snce our nterest s n the tme to event, we generate the vector of true event tmes T by frst smulate the survval probablty, U, from Un f orm, 1) for each subject and solve for T from the followng equaton: { T } U = exp α expγs) expξv + ψµ s)) s 29) To obtan the cumulatve functon, we let Λ t v, µ t)) = T α expγs) exp ξ + ξ 1 v + ψ β + β 1 x + β 2 s + w + w 1 s) ) s = T α exp γs + ξ + ξ 1 v + ψ β + β 1 x + β 2 s + w + w 1 s) ) s = α exp ξ + ξ 1 v + ψβ + β 1 x + w )) T exp γ + ψβ 2 + ψw 1 )s) s [ ] T exp γ + ψβ2 + ψ h w 1 )s) = α exp ξ + ξ 1 v + ψβ + β 1 x + w )) γ + ψβ 2 + ψw 1 exp ) γ + ψβ 2 + ψw 1 )T 1 = α exp ξ + ξ 1 v + ψβ + β 1 x + w )) γ + ψβ 2 + ψw 1 γ + ψβ 2 + ψw 1 = α exp ξ + ξ 1 v + ψβ + β 1 x + w )) [ exp γ + ψβ 2 + ψw 1 )T ) ] 1 3) γ + ψβ 2 + ψw 1 Hence, we have U = exp α exp ξ + ξ 1 v + ψβ + β 1 x + w )) [ exp γ + ψβ 2 + ψw 1 )T ) ] ) 1 31) γ + ψβ 2 + ψw 1 Now, solvng for T, from equaton 29), we have logu ) = α exp ξ + ξ 1 v + ψβ + β 1 x + w )) [ exp γ + ψβ 2 + ψw 1 )T ) ] 1 γ + ψβ 2 + ψ h w 1 ) T 1 γ + ψβ 2 + ψw 1 = log γ + ψβ 2 + ψw 1 α exp ξ + ξ 1 v + ψβ + β 1 x + w )) logu ) ) We draw censorng tme C from an unform dstrbuton U f orm.5, 8). Then, we compute T = mnt, C ) and event ndcator δ = 1 f T C and otherwse. The censorng was recored between 4% 45%. 4.2 Smulaton Study 2 In ths smulaton study, subjects are expected to have 1 repeated measurements taken at each vst between t We generated longtudnal data from the followng model: y j = β + β 1 x 1 + β 2 t j + β 3 x 2 + β 4 x 3 + w + w 1 t j + ɛ j = 1, 2,..., n, j = 1, 2,..., n 33) where < t j 1, β = {5., 2., 1., 2.5}, x k, k = 1, 2, 3 were generated from a Bernoull dstrbuton wth dfferent success probabltes and ɛ t j ) N, σ 2 ɛ ), wth σ 2 ɛ =.2. For the vector of random effects w = w, w 1 ) wth no correlaton between random ntercept and slope was generated from a multvarate normal dstrbuton MVN, A 2 ), where A = dag ) σ w, σ 2 w1, wth σ w =.75 and σ w1 =

8 Also, a proportonal hazards model wth Gompertz baselne was consdered: λ t) = α expγt) exp ξ + ξ 1 v 1 + ξ 2 v 2 + ψ β + β 1 x 1 + β 2 t + β 3 x 2 + β 4 x 3 + w + w 1 t)) 34) where γ = 1.5 and ξ = { 4., 1., 2.1}, α =.15, ψ =.2, the covarate v 1 and v 2 were generated from normal dstrbuton.e., N25, 6)) and unform dstrbuton.e., Un f orm, 3)), respectvely. The true event tmes was then generate from T ) U = exp α expγs) exp ξ + ξ 1 v 1 + ξ 2 v 2 + ψ β + β 1 x 1 + β 2 s + β 3 x 2 + β 4 x 3 + w + w 1 s)) s ) T 1 = γ + ψβ 2 + w 1 ) log γ + ψβ 2 + w 1 )) α exp ξ + ξ 1 v 1 + ξ 2 v 2 + ψ β + β 1 x 1 + β 3 x 2 + β 4 x 3 + w )) logu ) ) 36) We generate censorng tmes, C from an exponental dstrbuton wth mean Smulaton Study Results The am of the smulaton study was to nvestgate the performance of the jont model. The results are summarzed n Table 1 and 2. We have consdered several quanttes to determned the behavor of the estmators ˆφ by comparng to the true φ as follows; 1) The estmated bas: Bφ) = 1 N ˆφ φ = φ φ, where a negatve bas ndcates an underestmaton whle a postve bas ndcates an overestmaton. 1 2) The Root Mean Square Error: RMES ˆφ) = ˆφ φ )2, ths measures the accuracy of the estmates. The N lower the RMSE, the more accurate effects estmates. 3 The standard error: S.E ˆφ) = σ n, defned as equal to standard devaton dvded by the square root of the sample sze. Ths mples that the larger the sample sze, the smaller the standard error. 4) The 95% coverage probabltycp): ˆφ ± 1.96 se ˆφ), the proporton of 1 smulated data sets for whch 95% confdence ntervals ncluded the true estmates. The closer the outcomes to 95%CP.95), the more accurate the estmates. Table 1. Smulaton study 1 results wth correlaton random effects from 1 replcatons of 2 and 5 subjects. n = 2 Parameter Trueφ) Mean ˆφ) Bas S.E RMSE 95%CP β β β σ w σ w ρ σ ɛ ξ ξ ψ α γ n = 5 Mean ˆφ) Bas S.E RMSE 95%CP Table 2. Smulaton study 2 results wth uncorrelated random effects 166

9 n = 2 Parameter Trueφ) Mean ˆφ) Bas S.E RMSE 95%CP β β β β β σ σ σ ɛ ξ ξ ξ ψ α γ n = 5 Mean ˆφ) Bas S.E RMSE 95%CP The summary statstcs of the estmated regresson coeffcents are summarzed n table 1 and 2, respectvely for n = 2 and n = 5 subjects. In each smulaton, 1 replcatons were performed. We can clearly see that the proposed methodology performed well n terms of parameter estmatons. The bases are relatvely small and the probablty of 95% credble ntervals dwells around the.95 values. On the other hand, smulaton wth large sample sze have smaller standard error, hence the root mean square error. Overall, better results were obtaned wth jont model wth correlated random effects. That s, the correlaton between random ntercept and slope postvely nfluences the estmates. 5. Real Data Applcaton 5.1 Data In ths secton, we apply the jont model to the FFCD 2-25 mult-center phase III clncal tral of patents dagnosed wth metastatc colorectal cancer. The study was conducted between February 22 and January 27 n France by Federaton Francophone de Cancerologe DgestveFFCD). The man am of the study was to examne the effcacy of two treatment effects: Sequental arms) and combnaton armc). We consder datasets presented by Kro l et al. 216 & 217), n whch 15 patents were randomly selected from the same clncal tral. The data contans ndvdual progresson of dsease such as tumor sze, tme of new lesonsrecurrent events), baselne covaratesage, WHO performance status and prevous resecton: combnaton arm vs sequental arm), tme to death or the last observed tme for rght censored. A total of 96 tumor sze measurements were recored at subject specfc follow-up tme. Durng the study, 289 recurrences and 121 deaths were also recorded. We chose to model the longtudnal outcomes together wth tme to death wthout recurrence. In our model, we ncluded a total of 716 tumor sze measurements of 41 deaths and 19 rght censored. The am of ths applcaton s to examne the effects of longtudnal dynamcs and baselne covarates on subjects who ded wthout experencng any recurrence. Fgure 1. Indvdual profles for Longtudnal measurementsrght) and the Kaplan-Mer estmates of the survval functon among patents wth no recurrenceleft) Longtudnal Sub-Model 167

10 For the longtudnal process, we ncludes the followng covarates: Treatment1: Sequental, 2: Combnaton), Age1: < 6 years, 2: 6 69 years, 3: > 69 years), who.ps1: status. 2: status 1 and 3: status 2). Y j = β + β 1 Age + β 2 who.ps + β 3 Year Treatment + w + w 1 Year + ɛ j, 37) where y j represents tumor sze, whch s the transformed sum of the longest dameters S LD = S LD.3 1)/.3) measured durng the vst, w and w 1 are the subject specfc random effects for ntercept and slope respectvely Survval Submodel The followng proportonal hazards model wth Gompertz baselne functon and Prev-resecton: No, 1: Yes) covarate was used λ t) = α exp γtme1 +ξ +ξ 1 Prev.resecton+ψ β +β 1 Age +β 2 who.ps +β 3 tme1 Treatment +w +w 1 tme1 )) 38) In the Gbbs sampler algorthm, we ran three MCMC chans wth random generated ntal values for 25 teratons. In order to avod nfluences of pre-convergence on our fnal posteror nferences, 55 teratons were dscarded as burnn. Therefore, our posteror nferences were based on the last 15 teratons. As presented n subsecton 3.2.2, we consder nformatve prors for models coeffcents, that s β L, ξ and γ are assumed to have normal dstrbuton wth mean zero and a large varance β L N, 1), ξ N, 1), γ N, 1) ). We have also assume an ndependent gamma pror for α, α G.1,.1). For the random ntercept σ w and slope σ w1, we use nverse-wshart pror dstrbuton A Inv Wshart A, q ), where A s the pror dstrbuton for covarance matrx wth q = 2 degrees of freedom. Table 3. Posteror mean, standard error and 95% credble nterval for the jont model Case I Case II Parameters Estmate SE 95%CI Estmate SE 95%CI Longtudnal submodel Fxed Effect Interceptβ ) ,3.98) ,3.925) Ageβ 1 ) ,.221) ,.27) Who.PS β 2 ) ,.636) ,.597) Year Treatmentβ 3 ) ,-.269) ,-.338) Random Effect ρ ,.258) σ 2 w ,1.4) ,11.4) σ w σ w ,11.11) σ 2 w ,1.976) ,2.49) σ 2 ɛ ,.876) ,.874) Tme to death wthout recurrence Interceptξ ) ,6.74) ,3.854) Prev.resectonξ 1 ) ,.87) ,.148) T meγ) ,.675) ,.665) α ,1.85) ,2.18) ψ ,.56) ,.54) Model selecton D ,1978.) , 1975.) p D DIC In table 3, we present the posteror estmates, standard error and 95% credble nterval of the estmates based on the proposed jont models. Although, there s lttle dfference between parameter values n both cases, we can clearly see that all the parameters are statstcally sgnfcant by lookng at the value of standard error. For the longtudnal outcome, we see that there s a decrease n tumor sze measurements contrbuted by age and tme varyng treatment. We also observed a postve effect of prevous resecton on the rsk of death. On the other hand, the value of assocaton parameter, ψ n both cases ndcate that there s a postve assocaton between the two processes. Based on Devance nformaton crtera, a jont modelcase I) wth correlated random ntercepts and slopes seems to be the model that would best ft the data as t has relatvely small DIC value compared to case II wth uncorrelated random ntercepts and slopes. 168

11 Fgure 2. Posteror margnal dstrbuton trace and densty plots for β, β 1, β 2 and β 3. Fgure 2 show the the trace and densty plots for posteror margnal dstrbutons of selected parameters. We clearly see that, the MCMC of all parameters have converged to ther target posteror dstrbutons. 169

12 6. Dscusson Jont modelng for longtudnal outcome and tme to event data has ganed ncreasng popularty n lterature. However, when t comes to the choce of baselne functon on survval sub-part, many authors assumed that the survval tmes follow exponental or webull dstrbutons. In ths paper, we developed a jont model under a Bayesan prospectve assumng that the proportonal hazards model for the survval tmes has a Gompertz baselne hazard functon. We thnk that generatng survval tmes from a Gompertz dstrbuton have more advantages over Webull dstrbuton as ts cumulatve dstrbuton has a closed form soluton whch make t more easer to smulate survval data n presence of tme varyng covarates. We started buldng separate models for each process and then lnk them together through a common latent varable. Our model ncorporate both tme nvarantfxed) and tme varyng covarates that forces the hazards of the outcomes to change over tme. On the other hand, the nter-relatonshp between markers was accounted by subject-specfc random effects. Due to hgh programmng needs for fttng the jont lkelhood functon, we proposed a Bayesan approach that estmates the parameters by smulatng samples from posteror dstrbuton. Specfcally, Gbbs sampler was used for posteror nferences as t provdes convenently way to the ft of complex models. We have conducted an extensve examnaton of the model parameter estmaton through smulaton studes smulaton study 1- correlated random effects, and smulaton study 2- uncorrelated random effects). The smulaton results for both smulaton studes are presented by lookng at several quanttes such as Bas-the dfference between the average estmate over all smulatons and the true parameter value, S.E- standard error of the estmates that measures the accuracy of predctons, RMSE-the square root of the mean error and CP- the coverage probablty. The results from the two smulaton studes, ndcated adequate performance of the jont model. They hghlghted however some weakness of the model when few sample szes were used. There s ample of addtonal work needed n jont modelng framework. Ths paper covered only a small area of ths fast growng feld of research. Therefore, ths work can be extended further to accommodate multple longtudnal outcomes and competng rsks as done by Musoro 214). In future, we plan to develop a jont model for longtudnal and tme to event data assumng that survval tmes follow a Generalzed Gompertz dstrbuton wth three parameters Hale et al.. 216). In short, we have ntroduced a more flexble jont model for longtudnal and tme to event data assumng that the survval tme follow a Gompertz dstrbuton. We further demonstrated that ths model can easly be used n practce through a study, the FFCD 2-25 mult-center phase III clncal tral of patents dagnosed wth metastatc colorectal cancer. In both smulaton and applcaton studes, two cases case I wth correlated random ntercepts and slope and case II wth uncorrelated random ntercepts and slopes) were consdered. It s clear that our work contrbuted to ths fascnatng research area by makng use of more flexble methodology to develop a jont model. Acknowledgements Ths work was supported by the Pan Afrcan Unversty PAUSTI) n collaboraton wth Jomo Kenyata Unversty of Agrculture and Technology JKUAT). The authors are grateful to the anonymous revewers for ther valuable comments and suggestons. References Ahuja, J., & Nash, S. W. 1967). The generalzed gompertz-verhulst famly of dstrbutons. Sankhya: The Indan Journal of Statstcs, Seres A, Retreved from Austn, P. C. 212). Generatng survval tmes to smulate cox proportonal hazards models wth tme-varyng covarates. Austn, P. C. 213). Correcton: generatng survval tmes to smulate cox proportonal hazards models wth tme-varyng covarates. Statstcs n Medcne, 326), Casellas, J. 27). Bayesan nference n a pecewse webull proportonal hazards model wth unknown change ponts. Ch, Y. Y., & Ibrahm, J. G. 26). Jont models for multvarate longtudnal and multvarate survval data. Bometrcs, 622), Faucett, C. L., & Thomas, D. C. 1996). Smultaneously modellng censored survval data and repeatedly measured co- varates: a gbbs samplng approach. Statstcs n medcne, 1515), Geedpally, S. R., Lord, D., & Dhavala, S. S. 214). A cauton about usng devance nformaton crteron whle 17

13 modelng traffc crashes. Safety scence, 62, Gompertz, B. 1825). On the Nature of the Functon Expressve of the Law of Human Mortalty, and on a New Mode of Determnng the Value of Lfe Contngences. Phlosophcal Transactons of the Royal Socety of London, 115, Retreved from Hale, S., Jeong, J. H., Chen, X., & Cheng, Y. 216). A 3-parameter gompertz dstrbuton for survval data wth competng rsks, wth an applcaton to breast cancer data. Journal of Appled Statstcs, 4312), Hougaard, P. 212). Analyss of multvarate survval data. Sprnger Scence & Busness Meda. Retreved from Huang, X., L, G., & Elashoff, R. M. 21). A jont model of longtudnal and competng rsks survval data wth heterogeneous random effects and outlyng longtudnal measurements. Statstcs and Its Interface, 32), Huang, Y., Dagne, G., and Wu, L. 211). Bayesan nference on jont models of hv dynamcs for tme-to-event and longtudnal data wth skewness and covarate measurement errors. Statstcs n Medcne, 324), Ibrahm, J. G., Chu, H., & Chen, L. M. 21). Basc Concepts and Methods for Jont Models of Longtudnal and Survval Data. Journal of Clncal Oncology, 2816), Ibrahm, J., Chen, M., & Snha, D. 24). BAYESIAN METHODS FOR JOINT MODELING OF LONGITUDINAL AND SURVIVAL DATA WITH APPLICATIONS TO CANCER VACCINE TRIALS. Statstca Snca, 143), Retreved from Journal of Anmal Breedng and Genetcs, 1244), Kro ' l, A., Ferrer, L., Pgnon, J. P., Proust-Lma, C., Ducreux, M., Bouch e, O., Mchels, S., & Rondeau, V. 216). Jont model for left-censored longtudnal data, recurrent events and termnal event: Predctve abltes of tumor burden for cancer evoluton wth applcaton to the ffcd 2 5 tral. Bometrcs, 723), Kro ' l, A., Mauguen, A., Mazrou, Y., Laurent, A., Mchels, S., & Rondeau, V. 217). Tutoral n jont modelng and predcton: a statstcal software for correlated longtudnal outcomes, recurrent events and a termnal event. arxv preprnt arxv: Lard, N. M., & Ware, J. H. 1982). Random-effects models for longtudnal data. Bometrcs, pages Lang, K. Y., & Zeger, S. L. 1986). Longtudnal data analyss usng generalzed lnear models. Bometrka, 731), Martns, R., Slva, G. L., and Andreozz, V. 217). Jont analyss of longtudnal and survval ads data wth a spatal fracton of long-term survvors: A bayesan approach. Bometrcal Journal, 596), do.org/1.12/bmj McCrnk, L. M., Marshall, A. H., & Carns, K. J. 213). Advances n jont modellng: a revew of recent developments wth applcaton to the survval of end stage renal dsease patents. Internatonal Statstcal Revew, 812), Musoro, J. Z., Geskus, R. B., & Zwnderman, A. H. 215). A jont model for repeated events of dfferent types and multple longtudnal outcomes wth applcaton to a follow-up study of patents after kdney transplant. Bometrcal Journal, 572), Ntzoufras, I. 211). Bayesan Modelng Usng WnBUGS. Wley Seres n Computatonal Statstcs. Wley. Rzopoulos, D. 212). Jont models for longtudnal and tme-to-event data: Wth applcatons n R. CRC Press. Rzopoulos, D., & Ghosh, P. 211). A bayesan semparametrc multvarate jont model for multple longtudnal outcomes and a tme-to-event. Statstcs n medcne, 312), Song X, Davdan M, Tsats A. 22). An estmator for the proportonal hazards model wth multple longtudnal covarates measured wth error. Bostatstcs, 43),

14 Spegelhalter, D. J., Best, N. G., Carln, B. P., & Van Der Lnde, A. 22). Bayesan measures of model complexty and ft. Journal of the Royal Statstcal Socety: Seres B Statstcal Methodology), 644), Sweetng, M. J., & Thompson, S. G. 211). Jont modellng of longtudnal and tme-toevent data wth applcaton to predctng abdomnal aortc aneurysm growth and rupture. Bometrcal Journal, 535), Tseng, Y. K., Hseh, F., & Wang, J. L. 25). Jont modellng of accelerated falure tme and longtudnal data. Bometrka, 923), Tsats, A., & Davdan, M. 24). Jont Modelng of Longtudnal and Tme-To-Event Data: An Overvew. Statstca Snca, 143), Retreved from Verbeke, G. 1997). Lnear mxed models for longtudnal data. Sprnger, page Yang, L., Yu, M., and Gao, S. 216). Jont models for multple longtudnal processes and tmeto- event outcome. Journal of statstcal computaton and smulaton, 8618), Yu, M., Taylor, J. M. G., & Sandler, H. M. 28). Indvdual predcton n prostate cancer studes usng a jont longtu- dnal survval cure model. Journal of the Amercan Statstcal Assocaton, 13481), Zeger, S. L., Lang, K. Y., & Albert, P. S. 1988). Models for longtudnal data: a generalzed estmatng equaton approach. Bometrcs, Zhang, D., Chen, M. H., Ibrahm, J. G., Boye, M. E., & Shen, W. 217). Bayesan model assessment n jont modelng of longtudnal and survval data wth applcatons to cancer clncal trals. Journal of Computatonal and Graphcal Statstcs, 261), Copyrghts Copyrght for ths artcle s retaned by the authors), wth frst publcaton rghts granted to the journal. Ths s an open-access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense 172