Biomedical Engineering Reference
In-Depth Information
is given by
(
Y
)
i
p
ik
1k
d
ik
2k
L
i
(jR
i
)
=
k=1
1i
+T
T
2i
)
e
Z
0
i
f
i
1
+d
i
(1
i
)
2
+d
i
i
3
g
e
R
i
(
i
+d
i
)
e
(T
NT
0i
+T
NT
0i
= e
z
i
1
+R
i
P
k=1
1k
T
NT
1i
= e
z
i
2
+R
i
P
k=1
2k
T
NT
where T
NT
ik
; T
NT
ik
and
2i
= e
z
i
3
+R
i
P
k=1
1k
T
ik
. Then the conditional distribution of Ri
i
is de-
ned as
T
WT
f(R
i
; )L
i
R
1
1
f(R
i
; )L
i
dR
i
f
R
i
jO
i
(R
i
) =
;
(5.1)
where the denominator in Equation (5.1) does not have a closed form, unlike
a gamma frailty. In this chapter, a numerical integration such as a Gauss-
hermite algorithm is applied.
At the second stage, the unknown quantities by unknown tumor onset
time are estimated. Denote O = (O
1
; ;O
n
) with O
i
= (c
i
;R
i
;z
i
) as the
observed data. Then the conditional expectations E(N
k
jO;), E(T
N
k
jO;)
and E(T
k
jO;) are calculated as follows (Lindsey and Ryan, 1993):
X
E(N
k
jO;) =
i
p
ik
;
i=1
where p
ik
= I(t
i
2 I
k
) is the conditional probability that a tumor occurs
to the i-th subject in the k-th interval given that a subject with tumor was
sacrificed or dead at ci,
i
,
8
<
R
s
k
s
k1
q(u;c
i
)ds=
R
c
0
q(u;c
i
)ds if ci
i
> s
k
;
R
c
i
s
k1
q(u;c
i
)ds=
R
c
0
q(u;c
i
)ds if ci
i
2 I
k
;
0
p
ik
=
:
otherwise;
where
q(t;c
i
) =
01i
(t; z
i
) expf
R
s
0
01i
(u; z
i
)+
02i
(u; z
i
)gdu(c
i
; z
i
) expf
R
c
i
=
t
12
(u; z
i
)gdu:
For the duration time for each state, the following conditional expectation
is applied:
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