Semi-Competing Risks on A Trivariate Weibull Survival Model

Semi-Competing Risks on A Trivariate Weibull S urvival Model Cheng K. Lee Department of Targeting Mo deling Insight & Innovation Marketing Division Wachovia Corporation Charlotte, NC 28244 Jenq-Daw Lee Graduate Institute of Poli tical Economy National Cheng Kung Uni versity Tainan, Taiwan 70101 ROC SUMMARY A setting of a trivairate survival function u sing semi-competing risks con cept is proposed. The Stanford Heart Transplant data is reanal y zed usin g a trivariate Weibull distribution mo del with the proposed survival functi on. KEY WORDS: semi -competing risks; trivariate Weibull ; terminal event 1. INTRODUCTION Fine, Jiang and Chappell [ 1] introduced the term “semi-competing risk” in which one event censors the other but not vice versa . In their article, the bivari ate Clayton survival function was used to demons trate the concept. Even before them, Li [2] has worked on the same concept using the bivariate W eibull survival m odel by Lu and Bhattacharyya [3]. In his dissertation, Li described the censoring event as “term ination event” and, as did Fine, Jiang and Chappell [ 1], the bivariate survival function was di vided into two components with lower wed ge and upper densities. Epst ein and Muñoz [4], Shen and Thall [5], and Di gnam, Weiand and Rathousz [6] worked on the likelihoo d function of a bivarirate survival model with four types of censoring events. Diff erent from thos e authors, in this article, a t rivariate survival function for semi -competing risks with two- fatal and one non-fatal event s is first constructed and then followed b y the likelihood function. In Section 2, the Stanford Heart Transplant Data is reconstructed for the analysis of semi- competing risks. In Section 3, t he trivariate Weibull survi val function is proposed followed b y the likelihood functio n. Section 4 shows the results of the an aly sis of the data. Finally, the article is concluded with some discuss ion in Section 5. 2. DATA STRUCTURE The Stanford Heart Tran splant Data has been analyzed b y Aitkin, Laird, and Francis [7] using the Weibull, lognorm al, and piecewise exponential models with the consideration of pre-transplant and post transplant survival. They also surve yed the literature of analyzing the same data. In addit ion, the same data was studied b y Miller and Halpern [8] using four regression techn iques. The analyses by Noura [9] and Loader [10] are to find the change point of the hazard rate. As Boardman [11] pointed out, this classical data set has been intensivel y studied in the past, and “the data set real ly does not have too much going for it”. The refore, the results of this article are not to be compared wit h previous studies. Instead, the main purpose of this article is anal y zing t he same data from a new approach usin g a trivariate Weibull model with the concept of semi -competing risks. The Stanford Heart Tran splant Data analyzed in thi s study is from the article by Crowle y and Hu [12] in which they denot e T 1 the date of acceptance to the stud y , T 2 the date last seen, and T 3 the date of transplantation. T 2 is less than or equal to the last da y of the data collection or the last da y of the study which was April 1, 1974. An i ndividual was to experience three events when the indivi dual was accepted to the stud y. The 3 events are death before transplant ( E 1 ), transplant ( E 2 ), and death after transplant ( E 3 ). Therefore, E 1 and E 3 are terminal events or fatal events , and E 2 is an intermediate event. Let X 1 be the time to E 1 , X 2 be the time to E 2 , and X 3 be the time to E 3 . X 1, X 2, and X 3 begin at T 1 and are in days. The three events E 1 , E 2 , and E 3 are competing with each other for the occurrence to each indi vidual. However, these three events must occur in some certain orders. When E 1 occurs first, neither E 2 nor E 3 will occur because E 1 is a terminal event. When E 2 occurs, E 3 may occur later but E 1 will not occur because E 1 and E 2 are defined mutually exclusi ve. When E 3 occurs, E 2 must occur first because E 3 is the event defined to occur after E 2 . With these orders, an in dividual must fall int o one and only one of the following four cas es. First case, an individual ex periences E 1 , and, therefore, no possibilit y for the occurrence of E 2 or E 3 . The individual is said t o be uncensored due to E 1 . In this case, X 1 = T 2 - T 1 , and X 2 and X 3 do not exist. Second case, an individual experiences E 2 first, and then E 3 with no occurrence of E 1 . The individual is said t o be uncensored due to E 2 and E 3 . In this case, X 2 = T 3 - T 1 , X 3 = T 2 - T 1 , and X 3 does not exist. Third case, an individual ex periences only E 2 before the end of the stud y. The individual is said to be uncensored due to E 2 , and censored due to E 3 . In this case, X 2 = T 3 - T 1 , X 3 = T 2 - T 1 , and X 1 does not exist. Fourth case, an indivi dual does not experience E 1 , E 2 or E 3 before the end of the study. The individual i s said to be censored due to E 1 , E 2 and E 3 . I n this case, X 1 = T 2 - T 1 , X 2 = T 2 - T 1 , and X 3 = T 2 - T 1 . The original data of Crowle y and Hu contains 103 observations . After deleting 3 observation s with 0 in X 1 , X 2 or X 3 , and 4 observations of transplant wit h no mismatch score, there are total 96 observati ons included in this stud y . 3. THE MODEL AND T HE LIKELIHOOD FUNCT ION The Weibull dist ribution model is chosen for the marginal dist ribution of X 1, X 2, and X 3 as the model was adopted by Aitkin, Laird, and Francis [7 ] and Noura [9]. In order to stud y the relation among the three random variables, the follo wing trivariate Weibull survival function is derived usin g Clayton copula [13]. ( ) 1 2 3 1 2 3 1 1 1 1 1 , , 2 X X X X X X S S S S θ θ θ θ − − − − = + + − (1) where 1 1 1 1 X x S Exp γ λ       = −         , 2 2 2 2 X x S Exp γ λ       = −         , 3 3 3 3 X x S Exp γ λ       = −         , 0 < λ 1 , λ 2 , λ 3 < ∞ , 0 < γ 1 , γ 2 , γ 3 < ∞ , and 1 ≤ θ < ∞ . X 1, X 2, and X 3 are independent when θ = 1. One of the features of the Cla yton copula is that it allows positive and negative asso ciation between the random variables. To account for the effects of covariates, let λ 1 be the exponential function of age at acceptance and p revious surgery, and let bot h λ 2 and λ 3 be the exponential function of a ge at acceptance, previous surgery and mismatch score. Note that only individuals rece iving transplant had mismatch score. Th e parameters in the proposed trivariate survi val model are to be estimated b y maximizing the likelihood function. When each of the three events c an censor and be censored b y other events, the proposed trivariate Weibull survival model is one of the components i n the likelihood function. That is the com ponent accounts for individuals of l ost-to-follow-up or being censored at the end of the stud y. However, when semi-competing risks exist with the orders discussed in Section 2, the survival function for indi viduals censored at the end of the study or lost to follow-up becomes ( ) 1 X S t + ( ) 2 3 , , X X S t t + ( ) 2 3 2 3 , 2 3 3 3 , X X t x x S x x dx x ∞ =   ∂   ∂   ∫ . (2) The detailed derivation is in the Appendix . As did Lawless [14] , the component in t he likelihood for case 1 is th e negative derivati ve of equation (2) with respect to x 1 . The component for case 2 is t he derivative o f equation (2) with respe ct to x 2 and x 3 . The component for case 3 is t he negative d erivative of equation (2) with respect to x 2 . And, the component for case 4 is equation (2) itself. Therefore, the likelihood function for the propos ed trivariate surviv al function with the four cases is L ( θ θ θ θ )= ( ) 1 1 1 i i N p X x i f t = ∏ × ( ) 2 3 2 3 , , i i i q X X x x f t t × ( ) 2 3 2 3 2 3 , 2 3 2 , , i x x i i r X X x t x t S x x x = =     ∂   −     ∂     × ( ) ( ) ( ) 1 1 2 3 2 3 2 3 , , 2 3 3 3 , , p q r i i i i X i X X i i X X t x x S t S t t S x x dx x − − − ∞ =     ∂   + +     ∂     ∫ (3) where are p , q , and r ar e event indices and t denot es the survival tim e. 4. RESU LTS The estimates and their corresponding 95% con fidence intervals of the parameters are in the following table. The asymptotic covarian ce matrix i s approximated by the inverse of the negative Hessian. Parameter Estimate 95% Co nfidence Interval θ 1.677 ( 1.1 47, 2.208) age at acceptance ( X 1 ) 0.087 ( 0.060, 0.114) previous surger y ( X 1 ) -1.316 (-5.527 , 2.894) γ 1 0.342 ( 0.2 58, 0.425) age at acceptance ( X 2 ) 0.076 ( 0.061, 0.091) previous surger y ( X 2 ) 0.196 (-0.653, 1.045) mismatch score ( X 2 ) -0.036 (-0.532, 0.4 60) γ 2 0.733 ( 0.6 14, 0.852) age at acceptance ( X 3 ) 0.131 ( 0.098, 0.165) previous surger y ( X 3 ) 1.993 (-0.242, 4.228) mismatch score ( X 3 ) 0.340 (-0.787, 1.467) γ 3 0.422 ( 0. 322, 0.523) The results indicate onl y the age at acceptance i s significantl y different from zero at significance level of 0.05 for X 1, X 2, and X 3 . The overall association parame ter θ is 1.677 that indicates X 1, X 2, and X 3 are not much correlated. 5. DISCUSS ION In this article, the Cla yton trivariate Weibull survi val model with W eibull marginals is applied to the Stanford H eart Transplant data. Du e to the order of the o ccurrences of the three events, a new form ation of the likeli hood function is proposed. T he correlation coefficients between pair s of random variables ca n be obtained explicit ly or numerically. The work of this arti cle can also be expanded to hi gher dimensions . ACKNOW L EDGE The author thanks Mr. Daniel Warren Whi tman for his proofre ading this article. APPEND IX The fourth factor in the li kelihood function is the probabilit y that an individual is censored at T 2 , the date last seen. After the c ensoring, althou gh is unobservable, the individual ma y experience E 1 only, or E 2 followed b y E 3 . Suppose the indi vidual is censored at time t , t hen the probability for the oc currences of the three ev ents is Pr( t < X 1 ) + Pr( t < X 2 < X 3 ). Pr( t < X 1 ) is simpl y equal to ( ) 1 X S t . And, Pr( t < X 2 < X 3 )= ( ) 3 2 3 , 2 3 2 3 , x X X t t f x x dx dx ∞ ∫ ∫ = ( ) ( ) 2 3 2 3 3 , 2 3 2 , 2 3 2 3 , , X X X X t t x f x x dx f x x dx d x ∞ ∞ ∞   −       ∫ ∫ ∫ = ( ) 2 3 , , X X S t t - ( ) 2 3 3 , 2 3 2 3 , X X t x f x x dx dx ∞ ∞ ∫ ∫ Considering ( ) 2 3 3 , 2 3 2 , X X x f x x dx ∞ ∫ , ( ) 2 3 3 , 2 3 2 , X X x f x x dx ∞ ∫ = ( ) 2 3 , 2 3 2 0 , X X f x x dx ∞ ∫ - ( ) 3 2 3 , 2 3 2 0 , x X X f x x dx ∫ = 3 3 ( ) X f x - ( ) 3 2 3 , 2 3 2 2 3 0 , x X X F x x dx x x     ∂ ∂       ∂ ∂     ∫ = ( ) 3 3 3 X F x x ∂ ∂ - ( ) 2 3 2 3 , 2 3 3 , X X x x F x x x =   ∂   ∂   ( x 2 = x 3 denotes that x 2 is replaced b y x 3 ) = ( ) ( ) ( ) 3 2 3 2 3 3 , 2 3 3 , X X X x x F x F x x x =   ∂ −   ∂   = ( ) ( ) 2 3 2 3 , 2 3 3 , X X x x S x x x =   ∂ −   ∂   Note that ( ) 2 3 , 2 3 , X X S x x =1- ( ) 2 2 X F x - ( ) 3 3 X F x + ( ) 2 3 , 2 3 , X X F x x ([2]) where 2 X F , 3 X F and 2 3 , X X F are, respectivel y, the cu mulative densit y function of X 2 , X 3 , and X 2 and X 3 . 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