Biomedical Engineering Reference
In-Depth Information
where w(t;c) = A
01i
(t) + A
02i
(c) + [A
12i
(c) A
12i
(t)]; 0 < t c: In a tu-
morigenicity study, therefore, the relative risk with respect to death is defined
as
12i
(c)=
02i
(c) = (1 + )exp(
1
2
)Z
i
=
i
, which measures the lethal-
ity of tumor on death. Thus, the ratio of corresponding survival functions,
S
12
(c
i
)=S
02
(c
i
) = exp(
i
).
B. Normal frailty effect. With a normal frailty, the conditional intensities
with a shared frailty effect are given by
01i
( t jZ
i
;R
i
) =
01
(t)exp(
0
1
Z
i
+ R
i
);
02i
( c jZ
i
;R
i
) =
02
(c)exp(
0
2
Z
i
+ R
i
);
12i
( c jZ
i
;R
i
) =
02
(c)exp(
0
3
Z
i
+ R
i
); t < c:
Unlike a gamma distribution, an integration with respect to frailty effect has
no closed form, and suitable integration techniques such as the MCMC tech-
nique or a numerical integration technique are adopted. In the next section,
the estimation procedure is discussed in detail.
5.3
Estimation
Following Lindsey and Ryan (1994), a piecewise constant model
is assumed
on the baseline hazard functions,
01
and
02
. Divide the time axis into K
time points and set I
k
= (s
k1
;s
k
] for k = 1;:::;K:
01
(s) =
1k
for s 2 I
k
and
02
(t) =
2k
for t 2 I
k
Let = (
11
;:::;
1K
;
21
;:::;
2K
;
1
;
2
;
3
;(
2
)) be the vector of param-
eters. Now dene N
k
as the number of subjects with tumor at interval I
k
,
T
NT
ik
as the time the i-th subject spends without tumor at interval I
k
, and
T
ik
as the time the i-th subject spends at interval I
k
with tumor. While the
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