Semi-Competing Risks on A Trivariate Weibull Survival Model Cheng K. Lee Department of Targeting Modeling Insight & Innovation Marketing Division Wachovia Corporation Charlotte NC 28244 Jenq-Daw Lee Graduate Institute of Political Economy National Cheng Kung University Tainan Taiwan 700 ROC SUMMARY A setting of a trivairate survival function using semi-competing risks concept is proposed. The Stanford Heart Transplant data is reanalyzed using a trivariate Weibull distribution model with the proposed survival function. KEY WORDS: semi-competing risks; trivariate Weibull; terminal event. INTRODUCTION Fine Jiang and Chappell [] introduced the term semi-competing risk in which one event censors the other but not vice versa. In their article the bivariate Clayton survival function was used to demonstrate the concept. Even before them Li [2] has worked on the same concept using the bivariate Weibull survival model by Lu and Bhattacharyya []. In his dissertation Li described the censoring event as termination event and as did Fine Jiang and Chappell [] the bivariate survival function was divided into two components with lower wedge and upper densities. Epstein and Muñoz [4] Shen and Thall [5] and Dignam Weiand and Rathousz [6] worked on the likelihood function of a bivarirate survival model with four types of censoring events. Different from those authors in this article a trivariate survival function for semi-competing risks with twofatal and one non-fatal events is first constructed and then followed by the likelihood function. In Section 2 the Stanford Heart Transplant Data is reconstructed for the analysis of semicompeting risks. In Section the trivariate Weibull survival function is proposed followed by the likelihood function. Section 4 shows the results of the analysis of the data. Finally the article is concluded with some discussion in Section 5. 2. DATA STRUCTURE The Stanford Heart Transplant Data has been analyzed by Aitkin Laird and Francis [7] using the Weibull lognormal and piecewise eponential models with the consideration
of pre-transplant and post transplant survival. They also surveyed the literature of analyzing the same data. In addition the same data was studied by Miller and Halpern [8] using four regression techniques. The analyses by Noura [9] and Loader [0] are to find the change point of the hazard rate. As Boardman [] pointed out this classical data set has been intensively studied in the past and the data set really does not have too much going for it. Therefore the results of this article are not to be compared with previous studies. Instead the main purpose of this article is analyzing the same data from a new approach using a trivariate Weibull model with the concept of semi-competing risks. The Stanford Heart Transplant Data analyzed in this study is from the article by Crowley and Hu [2] in which they denote T the date of acceptance to the study T 2 the date last seen and T the date of transplantation. T 2 is less than or equal to the last day of the data collection or the last day of the study which was April 974. An individual was to eperience three events when the individual was accepted to the study. The events are death before transplant (E ) transplant (E 2 ) and death after transplant (E ). Therefore E and E are terminal events or fatal events and E 2 is an intermediate event. Let X be the time to E X 2 be the time to E 2 and X be the time to E. X X 2 and X begin at T and are in days. The three events E E 2 and E are competing with each other for the occurrence to each individual. However these three events must occur in some certain orders. When E occurs first neither E 2 nor E will occur because E is a terminal event. When E 2 occurs E may occur later but E will not occur because E and E 2 are defined mutually eclusive. When E occurs E 2 must occur first because E is the event defined to occur after E 2. With these orders an individual must fall into one and only one of the following four cases. First case an individual eperiences E and therefore no possibility for the occurrence of E 2 or E. The individual is said to be uncensored due to E. In this case X = T 2 - T and X 2 and X do not eist. Second case an individual eperiences E 2 first and then E with no occurrence of E. The individual is said to be uncensored due to E 2 and E. In this case X 2 = T - T X = T 2 - T and X does not eist. Third case an individual eperiences only E 2 before the end of the study. The individual is said to be uncensored due to E 2 and censored due to E. In this case X 2 = T - T X = T 2 - T and X does not eist. Fourth case an individual does not eperience E E 2 or E before the end of the study. The individual is said to be censored due to E E 2 and E. In this case X = T 2 - T X 2 = T 2 - T and X = T 2 - T. The original data of Crowley and Hu contains 0 observations. After deleting observations with 0 in X X 2 or X and 4 observations of transplant with no mismatch score there are total 96 observations included in this study.. THE MODEL AND THE LIKELIHOOD FUNCTION The Weibull distribution model is chosen for the marginal distribution of X X 2 and X as the model was adopted by Aitkin Laird and Francis [7] and Noura [9]. In order to study the relation among the three random variables the following trivariate Weibull survival function is derived using Clayton copula [].
X X X X 2 X 2 θ θ θ θ S = S + S + S () γ γ 2 γ 2 where SX = Ep SX = Ep 2 λ SX = Ep λ 0 < λ λ 2 λ < 2 λ 0 < γ γ 2 γ < and θ <. X X 2 and X are independent when θ =. One of the features of the Clayton copula is that it allows positive and negative association between the random variables. To account for the effects of covariates let λ be the eponential function of age at acceptance and previous surgery and let both λ 2 and λ be the eponential function of age at acceptance previous surgery and mismatch score. Note that only individuals receiving transplant had mismatch score. The parameters in the proposed trivariate survival model are to be estimated by maimizing the likelihood function. When each of the three events can censor and be censored by other events the proposed trivariate Weibull survival model is one of the components in the likelihood function. That is the component accounts for individuals of lost-to-follow-up or being censored at the end of the study. However when semi-competing risks eist with the orders discussed in Section 2 the survival function for individuals censored at the end of the study or lost to follow-up becomes SX ( t ) + SX t t + S X 2 d. (2) t 2 = The detailed derivation is in the Appendi. As did Lawless [4] the component in the likelihood for case is the negative derivative of equation (2) with respect to. The component for case 2 is the derivative of equation (2) with respect to 2 and. The component for case is the negative derivative of equation (2) with respect to 2. And the component for case 4 is equation (2) itself. Therefore the likelihood function for the proposed trivariate survival function with the four cases is N L(θ)= f X ( t ) i= i pi f X t t 2 i i qi S ( ) X 2 2 2 = t 2 = t i i ri pi qi ri SX ( t ) i + SX t i ti + SX 2 d ti 2 = () where are p q and r are event indices and t denotes the survival time. 4. RESULTS
The estimates and their corresponding 95% confidence intervals of the parameters are in the following table. The asymptotic covariance matri is approimated by the inverse of the negative Hessian. Parameter Estimate 95% Confidence Interval θ.677 (.47 2.208) age at acceptance (X ) 0.087 ( 0.060 0.4) previous surgery (X ) -.6 (-5.527 2.894) γ 0.42 ( 0.258 0.425) age at acceptance (X 2 ) 0.076 ( 0.06 0.09) previous surgery (X 2 ) 0.96 (-0.65.045) mismatch score (X 2 ) -0.06 (-0.52 0.460) γ 2 0.7 ( 0.64 0.852) age at acceptance (X ) 0. ( 0.098 0.65) previous surgery (X ).99 (-0.242 4.228) mismatch score (X ) 0.40 (-0.787.467) γ 0.422 ( 0.22 0.52) The results indicate only the age at acceptance is significantly different from zero at significance level of 0.05 for X X 2 and X. The overall association parameter θ is.677 that indicates X X 2 and X are not much correlated. 5. DISCUSSION In this article the Clayton trivariate Weibull survival model with Weibull marginals is applied to the Stanford Heart Transplant data. Due to the order of the occurrences of the three events a new formation of the likelihood function is proposed. The correlation coefficients between pairs of random variables can be obtained eplicitly or numerically. The work of this article can also be epanded to higher dimensions. ACKNOWLEDGE The author thanks Mr. Daniel Warren Whitman for his proofreading this article. APPENDIX The fourth factor in the likelihood function is the probability that an individual is censored at T 2 the date last seen. After the censoring although is unobservable the individual may eperience E only or E 2 followed by E. Suppose the individual is censored at time t then the probability for the occurrences of the three events is S t. Pr(t < X ) + Pr(t < X 2 < X ). Pr(t < X ) is simply equal to X And Pr(t < X 2 < X )= t t ( ) f d d X 2 2
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