Biomedical Engineering Reference
In-Depth Information
Also define
S
00
(;t) = P(T
1
> t;T
2
> t) = C
()
(S
1
(;t);S
2
(;t)) ;
S
10
(;t) = P(T
1
< t;T
2
> t) = S
2
(;t) C
()
(S
1
(;t);S
2
(;t)) ;
S
01
(;t) = P(T
1
t;T
2
t) = S
1
(;t) C
()
(S
1
(;t);S
2
(;t)) ;
and
S
11
(;t) = P(T
1
t;T
2
t)
= 1 S
1
(;t) S
2
(;t) + C
()
(S
1
(;t);S
2
(;t)) :
Then it can be easily shown that for j = 0; 1 and m = 0; 1, the intensity
function for N
jm
is given by Y (t)
c
(t) S
jm
(;t), where Y (t) = I(C t), and
c
(t) denotes the hazard function for C. The log-likelihood function based on
fC;
1
;
2
;Xg can be written as
Z
1
X
1
X
l(;O
01
;O
02
) =
logfS
jm
(;t)gdN
jm
(t)
j=0
m=0
because S
jm
(t) depends on O
0k
(t). Zhang et al. (2009) showed that the MLE of
the regression parameter
^
n
from the ecient score function is ecient in the
sense that
p
n(
^
n
0
) ! N(0;
1
), where is the information matrix of
and the variance-variance matrix of
^
n
asymptotically achieves the information
lower bound.
4.6
Illustrative Examples
In this section, two examples will be given to illustrate the approaches intro-
duced in the previous sections. First, we look at the tumorigenicity data given
in Table 1.3 of Sun (2006). Tumorigenicity experiments are usually designed
to determine whether a suspected agent or environment accelerates the time
until tumor onset in experimental animals. In these situations, the time to
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