Biomedical Engineering Reference
In-Depth Information
bivariate current status data. Let T
1
and T
2
be the two related failure times
of interest and suppose that both variables are only observed at a monitoring
time C. That is, the only information available for them is C,
1
= I(T
1
C)
and
2
= I(T
2
C), indicating whether the survival events represented by T
1
and T
2
have occurred before C. Note that here, for simplicity, we assume that
T
1
and T
2
have the same observation time and the methodology given below
can be easily generalized to situations where they have different observation
times. Also it will be assumed that all T
1
, T
2
, and C are continuous variables.
Let Z be a vector of covariates and S
k
(t) denote the marginal survival
function of T
k
, k = 1; 2. To describe the covariate eects on T
k
, it will be
assumed that given Z, S
k
(t) has the form
S
k
(t)
1 S
k
(t)
S
0k
(t)
1 S
0k
(t)
;
= exp(Z
0
)
(4.4)
where S
0k
is an unknown baseline survival function and denotes the vector
of regression parameters. That is, T
k
follows the proportional odds model
marginally. Note that in the model in Equation (4.4), without loss of generality,
it is supposed that the covariate effects are the same for T
1
and T
2
. If they
are dierent, one can easily dene a common through the introduction of
extra type-specific covariates. Define O
k
(t) = S
k
(t)=f1S
k
(t)g and O
0k
(t) =
S
0k
(t)=f1 S
0k
(t)g. Then we have
S
k
(t) =
exp(x
0
)O
0k
(t)
1 + exp(x
0
)O
0k
(t)
:
It will be assumed that T
1
and T
2
are independent of C given covariates.
To model the joint survival function of T
1
and T
2
, several approaches can
be applied. A common one, which will be used here, is the copula model
approach that assumes
S(s;t) = P(T
1
> s;T
2
> t) = C
(S
1
(s);S
2
(t)) :
(4.5)
where C
: [0; 1]
2
! [0; 1] is a genuine survival function on the unit square
and is a global association parameter. The copula model has attracted con-
siderable attention in failure time data analysis (Genest and MacKay (1986);
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