Biomedical Engineering Reference
In-Depth Information
Oakes (1989); Wang (2003)) and includes many useful bivariate failure time
models as special cases. For example, one special case is the Clayton model
given by
C (u;v) = (u 1 + v 1 1) 1=(1)
(Clayton (1978)) and a more general example is the Archimedean copula fam-
ily defined as
C (u;v) = 1 ( (u) + (v)) ; 0 u; v 1;
where () is a decreasing convex function dened on [0; 1] with (1) = 0.
The global association parameter is related to the Kendall's through
= 4 R 1
0
R 1
0 C (u;v)dudv 1.
One special and desirable feature of the copula model is that one can
model the association and the marginal survival functions separately. This is
convenient as sometimes may depend on the covariates. In the following, we
will suppose that
= exp(X 0 ) + 1;
(4.6)
where is a vector of regression parameters representing the eects of covari-
ates on the association between T 1 and T 2 .
Dene 0 = ( 0 ; 0 ) and the following counting processes (see Andersen
and Gill (1982)):
N 00 (t) = 1 2 I(C t) ;
N 10 (t) = (1 1 ) 2 I(C t) ;
N 01 (t) = 1 (1 2 )I(C t) ; N 11 (t) = (1 1 )(1 2 )I(C t) :
 
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