Biomedical Engineering Reference
In-Depth Information
7.3.2
Sample Baseline Hazards and Regression Coecients
As discussed in Wang et al. (2011a) and following Green (1995), the reversible
jump algorithm requires three types of moves: birth move, death move, and
update move. In the birth move, a new jump time is randomly selected from
the set of grid points that are not currently a jump point. In the death move,
a jump time is randomly selected from the current set of jump times and re-
moved. In the update move, both J and the jump points are fixed, while the
constant pieces are updated based on their full conditional posterior distribu-
tions. For each time-varying function '(t), only one type of move is performed
in a given MCMC sampling iteration. The probabilities of birth, death, and
update moves are set to be 0.35, 0.35, and 0.4, respectively.
7.3.2.1
Update Move
When an update move is selected, the dimension of the model is unchanged.
That is, both J and jump times are fixed. Let Y
ik
be the at-risk indicator. For
R
i
< 1, if e
ik
= 1 for some value k, then Yil
il
= 1 for l < k, Yil
il
= 0 for l > k
and Y
ik
= (T
i
a
k1
)=
k
. For R
i
= 1, Y
ik
= 1fa
k
L
i
g and in this case,
we also define e
ik
= 0 for k = 1;:::;K. Let 1fAg be the indicator function
such that 1fAg = 1 if A is true and 0 otherwise. The full conditional posterior
distribution of '(
j
) given jump times
1
< ::: <
J
= a
K
is
/ exp
n
('(
j
)
j
)
2
2
j
o
'(
j
) jnf'(
j
)g;w
2
;D
aug
exp
n
X
i
o
;
X
1fa
k
2 [
j1
;
j
)
k
k
exp
x
0
i
(a
k
)
Y
ik
k
(7.14)
where D
aug
= (D
obs
;f(T
i
;e
i
) : R
i
< 1g), and
j
and
j
are calculated as
follows. Let Zi
i
= 1 when '(
j
) is a piece of log baseline and Zi
i
= x
ik
when
'(
j
) is a piece of the k-th component of . When J = 1, it corresponds to the
time-independent-coecient model. There is only one piece of '(t) to sample
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