Biomedical Engineering Reference
In-Depth Information
Step 3. Sample each w
2
corresponding to each component in given . As
the inverse gamma prior for w
2
is conjugate, w
2
is sampled as follows:
'(
k
) '(
k1
)
2
2
w
2
j;D
obs
IG
0
+
J
:
+
X
k2
2
;
0
+
'(
1
)
2
2c
0
The algorithm is initiated by drawing Ti
i
uniformly from interval [Li,
i
;R
i
) for
all Ri
i
< 1. The last step that samples w
2
's is straightforward. The details of
the first two steps are discussed below.
7.3.1
Sample Event Times
Because the baseline hazard function and regression coecients are piecewise
constant, we draw event time Ti
i
in two steps:
(i) Determine which grid interval contains the event time. Using Equa-
tion (7.3), we have
S(L
i
jx
i
;) S(R
i
jx
i
;) = S(a
`
i
jx
i
;) S(a
r
i
jx
i
;)
r
X
=
S(a
k
jx
i
;) S(a
k1
jx
i
;):
k=`
i
+1
Let p
ik
= 0 if k `
i
or k > r
i
and
p
ik
=
S(a
k
jx
i
;) S(a
k1
jx
i
;)
S(a
`
i
jx
i
;) S(a
r
i
jx
i
;)
;
`
i
< k r
i
:
Then we sample ei
i
= (e
(ii)
;e
Ti
;:::;e
iK
)
0
from a multinomial distribution
with size 1 and probability vector (pi1,
(ii)
;p
Ti
;:::;p
iK
). The event time Ti
i
is in the k-th grid interval such that e
ik
= 1.
(ii) Sample the exact time within the selected grid interval. If e
ik
= 1, the
event time Ti
i
follows a doubly truncated exponential distribution on
[a
k1
;a
k
) with distribution function
1 exp
k
(ua
k1
) exp
x
0
i
(a
k
)
1 exp
k
k
exp
x
0
i
(a
k
)
:
Thus, it is straightforward to sample Ti
i
via the inverse distribution func-
tion method.
F(u) =
Search WWH ::
Custom Search