Semi-Competing Risks on A Trivariate Weibull Survival Model

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1 Semi-Compeing Risks on A Trivariae Weibull Survival Model Jenq-Daw Lee Graduae Insiue of Poliical Economy Naional Cheng Kung Universiy Tainan Taiwan ROC Cheng K. Lee Loss Forecasing Home Loans & Insurance Bank of America Charloe NC USA lee.abcde@gmail.com Absrac A seing of a rivairae survival funcion using semi-compeing risks concep is proposed in which a erminal even can only occur afer oher evens. The Sanford Hear Transplan daa is reanalyzed using a rivariae Weibull disribuion model wih he proposed survival funcion. Keywords: semi-compeing risks; rivariae Weibull. 1. Inroducion Fine Jiang and Chappell (2001) inroduced he erm semi -compeing risk in which one even censors he oher bu no vice versa. In heir aricle he bivariae Clayon survival funcion was used o demonsrae he concep. Even before hem Li (1997) has worked on he same concep using he bivariae Weibull survival model by Lu e al (1990). In his disseraion Li described he censoring even as erminaion even and as did Fine e al (2001) he bivariae survival funcion was divided ino wo componens wih lower wedge and upper densiies. Epsein e al (1996) Shen e al (1998) and Dignam e al (2007) worked on he likelihood funcion of a bivarirae survival model wih four ypes of censoring evens. Differen from hose auhors in his aricle a rivariae survival funcion for semi-compeing risks wih wo-faal and one non-faal evens is firs consruced and hen followed by he likelihood funcion. In Secion 2 he Sanford Hear Transplan Daa is reconsruced for he analysis of semi-compeing risks. In Secion he rivariae Weibull survival funcion is proposed followed by he likelihood funcion. Secion 4 shows he resuls of he analysis of he daa. Finally he aricle is concluded wih some discussion in Secion Daa Srucure The Sanford Hear Transplan Daa has been analyzed by Aikin e al (198) using he Weibull lognormal and piecewise eponenial models wih he

2 Jenq-Daw Lee Cheng K. Lee consideraion of pre-ransplan and pos ransplan survival. They also surveyed he lieraure of analyzing he same daa. In addiion he same daa was sudied by Miller e al (1982) using four regression echniques. The analyses by Noura (1990) and Loader (1991) are o find he change poin of he hazard rae. The Sanford Hear Transplan Daa analyzed in his sudy is from he aricle by Crowley e al (1977) in which hey denoe T 1 he dae of accepance o he sudy T 2 he dae las seen and T he dae of ransplanaion. T 2 is less han or equal o he las day of he daa collecion or he las day of he sudy which was April An individual was o eperience hree evens when he individual was acceped o he sudy. The evens are deah before ransplan ( E 1 ) ransplan (E 2 ) and deah afer ransplan (E ). Therefore E 1 and E are erminal evens or faal evens and E 2 is an inermediae even. Le 1 be he ime o E 1 2 be he ime o E 2 and be he ime o E. 1 2 and begin a T 1 and are in days. The hree evens E 1 E 2 and E are compeing wih each oher for he occurrence o each individual. However hese hree evens mus occur in some cerain orders. When E 1 occurs firs neiher E 2 nor E will occur because E 1 is a erminal even. When E 2 occurs E may occur laer bu E 1 will no occur because E 1 and E 2 are defined muually eclusive. When E occurs E 2 mus occur firs because E is he even defined o occur afer E 2. Wih hese orders an individual mus fall ino one and only one of he following four cases. Firs case an individual eperiences E 1 and herefore no possibiliy for he occurrence of E 2 or E. The individual is said o be uncensored due o E 1. In his case 1 = T 2 T 1 and 2 and do no eis. Second case an individual eperiences E 2 firs and hen E wih no occurrence of E 1. The individual is said o be uncensored due o E 2 and E. In his case 2 = T T 1 = T 2 T 1 and does no eis. Third case an individual eperiences only E 2 before he end of he sudy. The individual is said o be uncensored due o E 2 and censored due o E. In his case 2 = T T 1 = T 2 T 1 and 1 does no eis. Fourh case an individual does no eperience E 1 E 2 or E before he end of he sudy. The individual is said o be censored due o E 1 E 2 and E. In his case 1 = T 2 T 1 2 = T 2 T 1 and = T 2 T 1. The original daa of Crowley and Hu conains 10 observaions. Afer deleing observaions wih 0 in 1 2 or and 4 observaions of ransplan wih no mismach score here are oal 96 observaions included in his sudy.. The Model and he Likelihood Funcion The Weibull disribuion model is chosen for he marginal disribuion of 1 2 and as he model was adoped by Aikin e al (198) and Noura (1990). In order o sudy he relaion among he hree random variables he following rivariae Weibull survival funcion is derived using Clayon copula (Bandeen-Roche & Liang 1996) S S S S (1) 78

3 Semi-Compeing Risks on A Trivariae Weibull Survival Model where S Ep 1 S Ep 2 S Ep 1 0 < < 0 < γ 1 γ 2 γ < and 1 θ <. 1 2 and are independen when θ = 1. One of he feaures of he Clayon copula is ha i allows posiive and negaive associaion beween he random variables. To accoun for he effecs of covariaes le 1 be he eponenial funcion of age a accepance and previous surgery and le boh 2 and be he eponenial funcion of age a accepance previous surgery and mismach score. Noe ha only individuals receiving ransplan had mismach score. The parameers in he proposed rivariae survival model are o be esimaed by maimizing he likelihood funcion. When each of he hree evens can censor and be censored by oher evens he proposed rivariae Weibull survival model is one of he componens in he likelihood funcion. Tha is he componen accouns for individuals of los-ofollow-up or being censored a he end of he sudy. However when semicompeing risks eis wih he orders discussed in Secion 2 he survival funcion for individuals censored a he end of he sudy or los o follow-up becomes S + S + 1 S d (2) 2 2 The deailed derivaion is in he Appendi. As did Lawless (1982) he componen in he likelihood for case 1 is he negaive derivaive of equaion (2) wih respec o 1. The componen for case 2 is he derivaive of equaion (2) wih respec o 2 and. The componen for case is he negaive derivaive of equaion (2) wih respec o 2. And he componen for case 4 is equaion (2) iself. Therefore he likelihood funcion for he proposed rivariae survival funcion wih he four cases is N L( )= f 1 1 i1 i pi f 2 i i qi S i i ri 1 pi qi ri S 1 i S i i S 2 d () i 2 where are p q and r are even indices and denoes he survival ime. 79

4 Jenq-Daw Lee Cheng K. Lee 4. Resuls The esimaes and heir corresponding 95% confidence inervals of he parameers are in he following able. The asympoic covariance mari is approimaed by he inverse of he negaive Hessian. Parameer Esimae 95% Confidence Inerval p-value θ ( ) < age a accepance ( 1 ) ( ) < previous surgery ( 1 ) 1.16 ( ) γ ( ) < age a accepance ( 2 ) ( ) < previous surgery ( 2 ) ( ) mismach score ( 2 ) 0.06 ( ) γ ( ) < age a accepance ( ) 0.11 ( ) < previous surgery ( ) 1.99 ( ) mismach score ( ) 0.40 ( ) γ ( ) < The resuls indicae only he age a accepance is significanly differen from zero a significance level of 0.05 for 1 2 and while Crowley e al (1977) showed an insignifican resul. The covariae previous surgery of 1 2 and and he covariae mismach score of 2 and are boh insignifican ha agrees o he findings of Crowley e al (1977) and Noura (1990). The overall associaion parameer θ is ha indicaes 1 2 and are no much correlaed. 5. Conclusions In his aricle he Clayon rivariae Weibull survival model wih Weibull marginals is applied o he Sanford Hear Transplan daa. Due o he order of he occurrences of he hree evens a new formaion of he likelihood funcion is proposed for he hree random variables ha are allowed for differen ses of covariaes. The work of his aricle can be epanded o accommodae more variables. Acknowledge The auhor hanks Mr. Daniel Warren Whiman for his proofreading his aricle. Appendi The fourh facor in he likelihood funcion is he probabiliy ha an individual is censored a T 2 he dae las seen. Afer he censoring alhough is unobservable 80

5 Semi-Compeing Risks on A Trivariae Weibull Survival Model he individual may eperience E 1 only or E 2 followed by E. Suppose he individual is censored a ime hen he probabiliy for he occurrences of he hree evens is Pr( < 1 ) + Pr( < 2 < ). Pr( < 1 ) is simply equal o S. and Pr( < 2 < )= f d d 2 2 f d f d d = = S f d d 2 2 f 2 d2 Considering f d 2 2 = f 2 2 d = () 0 f d 2 2 f 0 = F F 2 2 d2 2 F 2 2 ( 2 = denoes ha 2 is replaced by ) = F F 2 = S Noe ha S = 1 F F F (Li (1997)) w here 2 F 2 F and F 2 are respecively he cumulaive densiy funcion of 2 and 2 and. Then Pr( < 2 < ) = S + S d. 2 2 Therefore he fourh facor in he likelihood funcion is Pr( < 1 ) + Pr( < 2 < ) = S 1 + S 2 + S 2 d. 2 81

6 Jenq-Daw Lee Cheng K. Lee References 1. Aikin M. Laird N. Francis B. (198). A reanalysis of he Sanford hear ransplan daa. Journal of he American Saisical Associaion 78: Bandeen-Roche K. Liang K. (1996). Modeling Failure-Time Associaions in Daa wih Muliple Levels of Clusering. Biomerika 8: Crowley J. Hu M. (1977). Covariance Analysis of Hear Transplan Survival Daa. Journal of he American Saisical Associaion 72: Dignam J. Wieand K. Rahouz P. (2007). A missing daa approach o semi-compeing risks problems. Saisics in Medicine 26: Epsein D. Muñoz A. (1996). A Bivariae Parameric Model for Survival and Inermediae Even Times. Saisics in Medicine 15: Fine J.P. Jiang H. Chappell R. (2001) On Semi-Compeing Risks Daa. Bimokerika 88: Lawless J. F. (1982). Saisical Models and Mehods for Lifeime Daa John Wiley and Sons. New York USA. 8. Li C. L. (1997). A Model for Informaive Censoring. Ph. D. Disseraion The Universiy of Alabama a Birmingham. 9. Loader C. R. (1982). Inference for A Hazard Rae Change Poin. Biomerika 78: Lu J-C. Bhaacharyya G. (1990). Some New Consrucions of Bivariae Weibull Models. Annals of The Insiue of Saisical Mahemaics 42: Miller R. Halpern J. (1982). Regression wih Censored Daa. Biomerika 69: Noura A. A. (1990). Proporional Hazards Chanepoin Models in Survival Analysis. Applied Saisics 9: Shen Y. Thall P. (1998). Parameric likelihoods for muliple non faal compeing risks and deah. Saisics in Medicine 17: