Statistical analysis of the generalized additive semiparametric survival model with random covariate

Q¨ UESTII ´ O,vol. 21, 1 i 2, p. 273-291, 1997 STATISTICAL ANALYSIS OF THE GENERALIZED ADDITIVE SEMIPARAMETRIC SURVIVAL MODEL WITH RANDOM COVARIATES V. BAGDONAVIˇ CIUS  University of Vilnius M. NIKULIN  University of Bordeaux-2 Generalizations of the additive hazards model are considered. Estimates of the regression parameters and baseline function are proposed, when covariates are random. The asymptotic properties of estimators are considered. Keywords: Generalized additive model, semiparametric models, survival analysis, regression parameters, baseline function, asymptotical analysis, additive hazards. Clasificaci´ on AMS: 60E15, 62F12, 62J99, 62P10 * V. Bagdonaviˇcius. University of Vilnius, Lithuania. Steklov Mathematical Institute, St.Petersburg, Russia. **M. Nikulin. University of Bordeaux-2, France. – Article rebut el juliol de 1996. – Acceptat el marc¸ de 1997. 273 1. INTRODUCTION Models describing the dependence of lifetimes distributions on explanatory variables have been considered. A number of such models was proposed by Andersen, Borgan, Gill and Keiding (1993), Cox and Oakes (1984), Dabrowska and Doksum (1988), Droesbeke, Fichet and Tassi (1989), Kalbfleisch and Prentice (1980), Lin and Ying (1994),(1996), etc. Bagdonaviˇcius and Nikulin (1994)-(1996) proposed a general approach, which gives the possibility of formulating a number of new models and showing where known models fit into the proposed new classes. Suppose that a time to failure Tx (  ) is a nonnegative random variable with the survival function Sx (  ) ( t ) = P f Tx (  ) > t g which depends on a vector of stresses x: [ 0 ; + ∞ ) ! B  R m : The time to failure Tx (  ) could be called the resource of this item. But the notion of the resource should not depend on x (  ) . So we will call the uniform resource to the random variable RU = 1 ? Sx (  ) ( Tx (  ) ) : It takes values in the interval [ 0 ; 1 ) and does not depend on x (  ) . Note that Tx (  ) = tif and only if RU = 1 ? Sx (  ) ( t ) . So the number 1 ? Sx (  ) ( t ) 2 [ 0 ; 1 ) is called the uniform resource used until the moment t under the stress x (  ) . The concrete item which failed at the moment tunder the stress x (  ) used 1 ? Sx (  ) ( t ) of the resource. Instead of the uniform resource one can define a resource with any probability distribution, so we can consider a whole class of resources. Really, suppose that Gis some fixed survival function, strictly decreasing and continuous on the support [ a ; b ] , ? ∞  a < b  ∞,G ( a ) = 1, G ( b ) = 0. H = G ? 1. The functional fG x (  ) ( t ) = ? H  Sx (  )  ( t ) ; is called the G-transfer functional. The survival function of the random variable RG = fG x (  ) ( Tx (  ) ) is Gand does not depend on x (  ) . The random variable RGis called the G-resource and the number fG x (  ) ( t ) is called the G-resource used till the moment t. Denote by Gthe family of survival functions, continuous and decreasing on their supports. Consider the class of transfer functionals M = f fG ; G 2 G g : Models will be formulated in dependence on properties of the transfer functionals. Note that some assumptions may be satisfied by one transfer functional, but not satisfied by another. This is the cause of considering the whole class of resources. In the case of G ( t ) = e ? t1 [ 0 ; ∞ [ ( t ) the transfer functional has the form fG x (  ) ( t ) = ? lnSx (  ) ( t ) and the rate of resource use is ∂fG x (  ) ( t ) ∂t = αx (  ) ( t ) ; where αx (  ) ( t ) = ? S 0 x (  ) ( t ) = Sx (  ) ( t ) is the hazard rate of Tx (  ) . 274 The additive hazards model (AHM) (see Andersen et al. (1993)) holds on Eif there exist functions λ0and asuch that for all x (  ) 2 E αx (  ) ( t ) = α0 ( t ) + a [ x ( t )] : Now we generalize this model. Definition. The G-generalized additive model holds on E if there exists a function a (  ) on E and a survival function S0such that, for all x (  ) 2 E, the transfer functional fG 2 Msatisfies the differential equation ∂fG x (  ) ( t ) ∂t = ∂fG 0 ( t ) ∂t + a ( x ( t )) (1) with the initial conditions fG 0 ( 0 ) = fG x (  ) ( 0 ) = 0. So the stress influences additively the rate of resource use. Equation (1) implies that Sx (  ) ( t ) = G f fG 0 ( t ) + Z t 0a [ x ( τ )] dτ g : Let us consider some particular models different from AHM. 1. Taking G ( t ) = exp f? exp f t gg , for t 2 R1, we obtain ∂fG x (  ) ( t ) ∂t = αx (  ) ( t ) Ax (  ) ( t ) ; ∂fG 0 ( t ) ∂t = α0 ( t ) A0 ( t ) ; where Ax (  ) ( t ) = Z t 0αx (  ) ( τ ) dτ ; A0 ( t ) = Z t 0α0 ( τ ) dτ are the cumulated hazards rates. So we have the model: αx (  ) ( t ) Ax (  ) ( t ) = α0 ( t ) A0 ( t ) + a ( x ( t )) : 2. Taking G ( t ) = 1 = ( 1 + t ) , for t  0, we obtain αx (  ) ( t ) Sx (  ) ( t ) = α0 ( t ) S0 ( t ) + a ( x ( t )) : 3. Taking G ( t ) = 1 = ( 1 + et ) , for t 2 R1, we obtain αx (  ) ( t ) 1 ? Sx (  ) ( t ) = α0 ( t ) 1 ? S0 ( t ) + a ( x ( t )) : 275 4. Take G ( t ) = 1 ? Φ ( lnt ) ; t  0, where Φ (  ) is the normal N ( 0 ; 1 ) cumulative distribution function. In terms of survival functions we obtain the model : Φ ? 1 ( 1 ? Sx (  ) ( t )) = ln  Z t 0a [ x ( τ )] dτ + exp [ Φ ? 1 ( 1 ? S0 ( t ))]  : 5. Taking G = S0, we obtain Sx (  ) ( t ) = S0 f Z t 0σ [ x ( τ )] dτ g ; where σ [ x ( t )] = 1 + a [ x ( t )] . It is the accelerated life model. Other distributions of the resource can be taken. So a number of alternatives to the AHM is proposed. For example, if at the beginning of life data follow well the AHM but later it is not so, the model of example 2could be choosed. On the contrary, if at the beginning of life data do not follow the AHM but at the end of life this model suits well, the model of example 3 could be choosed. 2. ESTIMATION 2.1. Notations Consider the model (1) with some specified Gand an unknown baseline survival function S0. The function a [ x (  )] is parametrized as follows: a [ x ( t )] = γTx ( t ) ; where γ = ( γ1 ; :::; γm ) Tis the vector of unknown regression parameters. So the following model is considered: for all x (  ) 2 E Sx (  ) ( t ) = G f H ( S0 ( t )) + Z t 0γTx ( τ ) dτ g : (2) Suppose that nindividuals are observed and assume that the vector of covariates for the ith individual is a random process Xi (  ) = ( Xi1 (  ) ; :::; Xim (  )) T. Denote by Ni ( t ) the univariate counting processes. This process counts the numbers of observed failures of each individual in the interval [ 0 ; t ] ; t  0. Denote by 276 Yi ( t ) the numbers of individual “ at risk” (non-censored and non-failed) just prior to tfor each i. Suppose that f Ft ; t  0 g is the filtration generated by f Ni ( s ) ; Yi ( s ) ; Xi ( s ) ; i = 1 ; :::; n; 0  s  t g ; Xiare predictable and the intensities λi, given by λi ( t ) = lim ε # 0 1 εP f Ni ( t + ε ) ? Ni ( t )  1 j Ft ? g ; exist. In this case the compensators of the counting processes Ni ( t ) are Λi ( t ) = Z t 0λi ( s ) ds : Assume that the random intensities λisatisfy the multiplicative intensity model λi ( t ) = αXi (  ) ( t ) Yi ( t ) : (3) Denote N ( t ) = n ∑ i = 1 Ni ( t ) ; Y ( t ) = n ∑ i = 1 Yi ( t ) ; Λ ( t ) = n ∑ i = 1Λi ( t ) ; Mi ( t ) = Ni ( t ) ? Λi ( t ) ; M ( t ) = n ∑ i = 1Mi ( t ) ; J ( s ) = I ( Y ( s ) > 0 ) ; α = ? G 0 G ; ψ = α  H : The hazard rates αXi (  ) ( t ) have the form αi ( t ) = αXi (  ) ( t ) = ψ ( SXi (  ) ( t )) f ( H  S0 ) 0 ( t ) + γTXi ( t ) g : (4) 2.2. Estimating equation and estimators ˜ H0 ˜ H0 ˜ H0,ˆ γ ˆ γ ˆ γand ˆ Sx (  ) ˆ Sx (  ) ˆ Sx (  ) From the Doob-Meyer decomposition N = M + Λ, equalities (3) and (4) imply dN ( t ) = dM ( t ) + n ∑ i = 1ψ ( SXi (  ) ( t )) Yi ( t ) f dH ( S0 ( t )) + γTXi ( t ) dt g and Z t 0 J ( u )( dN ( u ) ? S ( 0 )  ( γ ; u ) du ) S ( 0 ) ( γ ; u ) = Z t 0J ( u ) dH ( S0 ( u )) + Z t 0 J ( u ) dM ( u ) S ( 0 ) ( γ ; u ) ; 277 where S ( 0 ) ( γ ; u ) = n ∑ i = 1Yi ( u ) ψ ( SXi (  ) ( u )) ; S ( 0 )  ( γ ; u ) = n ∑ i = 1 Yi ( u ) ψ ( SXi (  ) ( u )) γTXi ( u ) : Under some mild assumptions Mis a martingale, therefore E Z t 0 J ( u )( dN ( u ) ? S ( 0 )  ( γ ; u ) du ) S ( 0 ) ( γ ; u ) = E Z t 0J ( u ) dH ( S0 ( u )) : (5) If Y ( t ) > 0, then Z t 0J ( u ) dH ( S0 ( u )) = H ( S0 ( t )) : (6) Equalities (5) and (6) imply that a reasonable estimator ˆ H0 ( t ; γ ) for H0 ( t ) = H ( S0 ( t )) (still depending on γ) is determined by the equation ˆ H0 ( t ; γ ) = Z t 0J ( u ) dN ( u ) ? ˜ S ( 0 )  ( γ ; u ) du ˜ S ( 0 ) ( γ ; u ) ; where ˜ S ( 0 ) ( γ ; u ) = n ∑ i = 1 Yi ( u ) α ( ˆ Hi ( u ; γ )) ; ˜ S ( 0 )  ( γ ; u ) = n ∑ i = 1 Yi ( u ) α ( ˆ Hi ( u ; γ )) γTXi ( u ) ; Hi ( u ; γ ) = H0 ( u ) + Z u 0γTXi ( τ ) dτ ; ˆ Hi ( u ; γ ) = ˆ H0 ( u ? ; γ ) + Z u 0γTXi ( τ ) dτ : Denote τ  = sup f t:Y ( t ) > 0 g . Suppose that at the non-random moment τ 2 ] 0 ; ∞ ] all the individuals are censored, i.e., τ   τ. We propose to estimate the parameter γby solving the estimating equations U ( γ ; τ ) = 0 ; where the estimating function is given by the formula below U ( γ ; t ) = n ∑ i = 1 Z t 0J ( u ) Xi ( u ) f dNi ( u ) ? Yi ( u ) α ( ˆ Hi ( u ; γ )) dˆ Hi ( u ; γ ) g : Those equations generalize the estimating equations of Lin and Ying (1994) for the additive hazards model (taking α ( p )  1). 278 If we denote by ˆ γthe estimator of γ, i.e. U ( ˆ γ ; τ ) = 0, then the estimator of the function H0 ( t ) is ˜ H0 ( t ) = ˆ H0 ( t ; ˆ γ ) (7) and the estimator of the survival function Sx (  ) ( t ) under any covariate x (  ) is ˆ Sx (  ) ( t ) = G  ˜ H0 ( t ) + Z t 0ˆ γTx ( u ) du  : (8) Let ˜ S ( 1 ) ( γ ; u ) = n ∑ i = 1 Xi ( u ) Yi ( u ) α ( ˆ Hi ( u ; γ )) ; ˜ S ( 1 )  ( γ ; u ) = n ∑ i = 1Xi ( u ) Yi ( u ) α ( ˆ Hi ( u ; γ )) γTXi ( u ) ; ˜ E ( γ ; u ) = ˜ S ( 1 ) ( γ ; u ) ˜ S ( 0 ) ( γ ; u ) : Then the estimating function U ( γ ; t ) can be written U ( γ ; t ) = n ∑ i = 1 Z t 0J ( u ) f Xi ( u ) ? ˜ E ( γ ; u ) g  dNi ( u ) ? Yi ( u ) α ( ˆ Hi ( u ; γ )) γTXi ( u ) du  : (9) 3. ASYMPTOTIC PROPERTIES OF THE ESTIMATORS 3.1. Asymptotic properties of the estimating function Denote by k A k = supi ; j j aij j the norm of any matrix A, by γ0the true value of the parameter γ, S ( 1 ) ( γ ; u ) = n ∑ i = 1Xi ( u ) Yi ( u ) α ( Hi ( u ; γ )) ; S ( 1 )  ( γ ; u ) = n ∑ i = 1 Xi ( u ) Yi ( u ) α ( Hi ( u ; γ )) γTXi ( u ) ; E ( γ ; u ) = S ( 1 ) ( γ ; u ) S ( 0 ) ( γ ; u ) ; S ( 2 ) ( γ ; u ) = n ∑ i = 1 Xi ( u ) XT i ( u ) Yi ( u ) α ( Hi ( u )) ; 279 ˜ S ( 2 ) ( γ ; u ) = n ∑ i = 1Xi ( u ) XT i ( u ) Yi ( u ) α ( ˆ Hi ( u ; γ )) ; S ( 3 ) ( γ ; u ) = n ∑ i = 1Yi ( u ) α 0 ( Hi ( u ; γ )) ; S ( 3 )  ( γ ; u ) = n ∑ i = 1 Yi ( u ) α 0 ( Hi ( u ; γ )) γTXi ( u ) : Assumptions A 1. Negligibility conditions. There exist a neighborhood Γof γ0and a scalar, vector or matrix functions s ( 0 ) , s ( 0 )  ,s ( 1 ) ,s ( 1 )  ,s ( 2 ) ,s ( 2 )  ,s ( 3 ) ,s ( 3 )  , such that for m = 0 ; 1 ; 2: (a) sup γ 2 Γ ; t 2 [ 0 ; τ ] k 1 nS ( m ) ( γ ; t ) ? s ( m ) ( γ ; t ) k P ! 0 ; sup γ 2 Γ ; t 2 [ 0 ; τ ] k 1 nS ( m )  ( γ ; t ) ? s ( m )  ( γ ; t ) k P ! 0 ; (b) s ( 0 ) ( γ0 ;  ) is bounded away from zero on [ 0 ; τ ] , (c) s ( m ) (  ;  ) ,s ( m )  (  ;  ) are continuous functions of γ 2 Γuniformly in t 2 [ 0 ; τ ] and bounded on Γ  [ 0 ; τ ] . 2. H0 ( τ ) = H ( S0 ( τ )) < ∞ ; 3. αis a positive continuously differentiable on ] 0 ; ∞ [ function, 4. P f sup t 2 [ 0 ; τ ] j Xij ( t ) j < ∞ g = 1 for all i ; j : 5. σ2 ( t ) = Z t 0 s ( 0 ) ( γ ; u ) dH0 ( u ) + s ( 0 )  ( γ ; u ) du s ( 0 ) 2 ( γ ; u ) < + ∞ : Lemma. Suppose that assumptions A hold. Then p n  ˆ H0 ( t ; γ ) ? H0 ( t )  D ! h ( γ ; t ) Z t 0 dV ( u ) h ( γ ; u ) as n ! ∞ ; whereV is the Gaussian martingale with 280 EV ( t ) = 0and Cov ( V ( t ) ; V ( s )) = σ2 ( t ^ s ) ; h ( γ ; t ) = exp  ? Z t 0 1 s ( 0 ) ( γ ; u )  s ( 3 ) ( γ ; u ) dH0 ( u ) + s ( 3 )  ( γ ; u ) du   : Proof: Consider the difference: p n  ˆ H0 ( t ; γ ) ? H0 ( t )  = p n ( Z t 0J ( u ) dN ( u ) ? ˜ S ( 0 )  ( γ ; u ) du ˜ S ( 0 ) ( γ ; u ) ? H0 ( t ) ) = = p n ( Z t 0J ( u ) " dM ( u ) ˜ S ( 0 ) ( γ ; u ) + S ( 0 ) ( γ ; u ) ? ˜ S ( 0 ) ( γ ; u ) ˜ S ( 0 ) ( γ ; u ) dH0 ( u ) + S ( 0 )  ( γ ; u ) ? ˜ S ( 0 )  ( γ ; u ) ˜ S ( 0 ) ( γ ; u ) du #) : Note that 1 p n h ˜ S ( 0 ) ( γ ; u ) ? S ( 0 ) ( γ ; u ) i = 1 n n ∑ i = 1 Yi ( u ) α 0 f Hi ( u ; γ ) g p n  ˆ H0 ( u ? ; γ ) ? H0 ( u )  + op ( 1 ) = s ( 3 ) ( γ ; u ) p n [ ˆ H0 ( u ? ; γ ) ? H0 ( u )] + op ( 1 ) : Similarly 1 p n h ˜ S ( 0 )  ( γ ; u ) ? S ( 0 )  ( γ ; u ) i = s ( 3 )  ( γ ; u ) p n  ˆ H0 ( u ? ; γ ) ? H0 ( u )  + op ( 1 ) and n ˜ S ( 0 ) ( γ ; u ) = 1 s ( 0 ) ( γ ; u ) + op ( 1 ) : So p n  ˆ H0 ( t ; γ ) ? H0 ( t )  = p n Z t 0 J ( u ) dM ( u ) ˜ S ( 0 ) ( γ ; u ) ? Z t 0 p n  ˆ H0 ( u ? ; γ ) ? H0 ( u )  s ( 3 ) ( γ ; u ) dH0 ( u ) + s ( 3 )  ( γ ; u ) du s ( 0 ) ( γ ; u ) + op ( 1 ) : Note that the predictable variation < p n Z t 0 J ( u ) dM ( u ) ˜ S ( 0 ) ( γ ; u ) > = n Z t 0J ( u ) S ( 0 ) ( γ ; u ) dH0 ( u ) + S ( 0 )  ( γ ; u ) du ˜ S ( 0 ) 2 ( γ ; u ) P ! σ2 ( t ) 281 where H0 ( t ) = H ( S0 ( t )) ; ˜ H0 ( t ) = ˆ H0 ( t ; ˆ γ ) ; σ2 2 ( γ0 ; t ) = σ2 1 ( γ0 ; t ) + C ( γ0 ; t )) Σ2 ( γ0 ; τ ) CT ( γ0 ; t ) ? 2C ( γ0 ; t ) Σ ? 1 1 ( γ0 ; τ ) A ( γ0 ; t ) : Proof: By Taylor expansion around γ0the random process p n f ˜ H0 ( t ) ? H0 ( t ) g can be written in the following manner: p n f ˜ H0 ( t ) ? H0 ( t ) g = n1 = 2  Z t 0 J ( u ) ˜ S ( 0 ) ( ˆ γ ; u ) ( dN ( u ) ? ˜ S ( 0 )  ( ˆ γ ; u ) du ) ? H0 ( t )  = n1 = 2  Z t 0 J ( u ) ˜ S ( 0 ) ( γ0 ; u ) ( dN ( u ) ? ˜ S ( 0 )  ( γ0 ; u ) du ) ? Z t 0J ( u ) ∂ ∂γ  1 ˜ S ( 0 ) ( γ  ; u )  dN ( u )( ˆ γ ? γ0 )+ Z t 0J ( u ) ∂ ∂γ ˜ S ( 0 )  ( γ  ; u ) ˜ S ( 0 ) ( γ  ; u ) ! du ( ˆ γ ? γ0 ) ? H0 ( t ) ) = n1 = 2 f C ( γ0 ; t )( ˆ γ ? γ0 ) + Z t 0J ( u ) dM ( u ) ˜ S ( 0 ) ( γ0 ; u ) g + op ( 1 ) ; (17) where γ  is on the line segment between ˆ γand γ0. Taking into account (15), we obtain n1 = 2 ( ˆ γ ? γ0 ) = Σ ? 1 1 ( γ0 ; τ ) n ? 1 = 2U ( γ0 ; τ ) + op ( 1 ) : From theorem 1: n ? 1 = 2U ( γ0 ; τ ) = n ? 1 = 2n ∑ i = 1 Z τ 0Hi ( γ0 ; u ) dMi ( u ) + op ( 1 ) : Therefore the predictable covariation < n ? 1 = 2n ∑ i = 1 Z τ 0Hi ( γ0 ; u ) dMi ( u ) ; n1 = 2 Z t 0J ( u ) dM ( u ) ˜ S ( 0 ) ( γ0 ; u ) > = n ∑ i = 1 Z t 0 Hi ( γ0 ; u ) ˜ S ( 0 ) ( γ0 ; u ) α ( Hi ( γ ; u )) Yi ( u ) ? dH0 ( u ) + γT 0Xi ( u ) du  P ! A ( γ0 ; t ) : The predictable variation < n1 = 2 Z t 0J ( u ) dM ( u ) ˜ S ( 0 ) ( γ0 ; u ) > = n Z t 0J ( s ) S ( 0 ) ( γ0 ; s ) dH ( S0 ( s )) + S ( 0 )  ( γ0 ; s ) ds ˜ S ( 0 ) 2 ( γ0 ; s ) P ! σ2 1 ( γ0 ; t ) : Let H1i ( γ0 ; u ) = C ( γ0 ; t ) Σ ? 1 1 ( γ0 ; τ ) Hi ( γ0 ; u ) + nJ ( u ) = ˜ S ( 0 ) ( γ0 ; u ) : 288 Then p n [ ˜ H0 ( t ) ? H0 ( t )] = n ? 1 = 2n ∑ i = 1 Z t 0H1i ( γ0 ; u ) dMi ( u ) + op ( 1 ) : By assumptions of the theorem for all ε > 0 1 n n ∑ i = 1 Z t 0H2 1i ( γ0 ; u ) I fj H1i ( γ0 ; u ) j > ε g Yi ( u ) αi ( u ) du P ! 0 : The asymptotic normality of p n [ ˜ H0 ( t ) ? H0 ( t )] follows from the theorem of Rebolledo. The proof of theorem 3 is complete.  Consider the asymptotic distribution of the estimator ˆ Sx (  ) of the survival function Sx (  ) ( t ) under any covariate x (  ) . Denote Cx ( γ0 ; t ) = C ( γ0 ; t ) + Z t 0xT ( u ) du ; g = G 0 : Theorem 4. Under assumptions of theorem 1 p n [ ˆ Sx (  ) ( t ) ? Sx (  ) ( t )] D ! N ( 0 ; σ2 x ) as n ! ∞ ; where σ2 x ( γ0 ; t ) = f σ2 1 ( γ0 ; t ) + Cx ( γ0 ; t ) Σ2 ( γ0 ; τ ) CT x ( γ0 ; t ) ? 2Cx ( γ0 ; t ) Σ ? 1 1 ( γ0 ; τ ) A ( γ0 ; t ) g g2 ( H ( Sx ( t ))) : Proof: By Taylor expansion around γ0we obtain (similarly as in the proof of theorem 3): n1 = 2 [ H ( ˆ Sx (  ) ( t )) ? H ( Sx (  ) ( t ))] = n1 = 2  ˜ H0 ( t ; γ0 ) ? H0 ( t ) + Z t 0xT ( u ) du ( ˆ γ ? γ0 )  = n1 = 2  Cx ( γ0 ; t ) Σ ? 1 1 ( γ0 ; t ) U ( γ0 ; τ ) + Z t 0J ( u ) dM ( u ) ˜ S ( 0 ) ( γ0 ; u ) ) + op ( 1 )  : The proof of the asymptotical normality of p n [ H ( ˆ Sx (  ) ( t )) ? H ( Sx (  ) ( t ))] is similar to the proof of the asymptotical normality of p n [ H ( ˆ S0 ( t )) ? H ( S0 ( t ))] in the theorem 3. So we skip it. The asymptotical normality of p n [ ˆ Sx (  ) ( t ) ? Sx (  ) ( t )] is obtained by the functional delta method. The proof is complete.  289 Remark 1. 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