Biomedical Engineering Reference
In-Depth Information
L
1
= e
01
(0;c)
02
(0;c)
;
(2) DNT : (;d) = (0; 1)
L
2
= e
01
(0;c)
02
(0;c)
02
(c);
(3) SWT : (;d) = (1; 0)
Z
c
e
01
(0;x)
02
(0;x)
01
(x) e
12
(x;c)
dx;
L
3
=
0
(4) DWT : (;d) = (1; 1)
Z
c
e
01
(0;x)
02
(0;x)
01
(x) e
12
(x;c)
dx
L
4
=
12
(c);
0
then the likelihood is
Y
L
(1
i
)(1d
i
)
1
L
(1
i
)d
i
2
L
i
(1d
i
)
3
L
i
d
4
:
L() =
i=1
In this study, we consider two distributions of frailty effect: (i) gamma, Ri
i
Gamma(
1
;) and (ii) normal R
i
N(0;
2
). For each distribution, intensities
are defined and estimation procedures are derived.
A. Gamma frailty effect.
Given Ri,
i
, the conditional intensities 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;
where
01
() and
02
() are baseline intensities of tumor onset and death,
respectively. By integrating a gamma distribution, the marginal distributions
are derived as follows:
01i
(tjZ
i
) = [1 + w(t;t)]
1
01
(t)exp(
0
1
Z
i
);
02i
(cjZ
i
) = [1 + w(c;c)]
1
02
(c)exp(
0
2
Z
i
);
12i
(cjt;Z
i
) = (1 + )[1 + w(t;c)]
1
02
(t)exp(
0
3
Z
i
);
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