Biomedical Engineering Reference
In-Depth Information
z
i
P
l
0
6=l
l
0
fb
l
0
(R
i
) b
l
0
(L
i
)g
b
l
(R
i
) b
l
(L
i
)
d
l
= max(c
l
; 0) and c
l
=
max
i:
i2
=1
:
+
P
i=1
x
0
i
x
i
)
1
4. Sample from N(
~
;
~
), where
~
= (
1
0
and
X
~
=
~
0
0
+
(z
i
(t
i
))x
0
i
:
i=1
5. Sample from G(a
+ k; b
+
P
l=1
l
):
6.3.2
Under the PO Model
As seen in Section 6.1, there is a strong connection between the PO model
and the logistic distribution. Also, a logistic distribution can be expressed as
a mixture of scaled normal distributions (Devroye, 1986; Holmes and Held,
2006). Motivated by such a relationship, Wang and Lin (2011) proposed the
following data augmentation:
z
i
= (t
i
) + x
0
i
+
i
with constraint zi
i
2 C
i
;
i
N(0;
i
);
i
= (2
i
)
2
;
i
KS;
where C
i
is the same as that under the Probit model and KS denotes the
Kolmogorov{Smirnov distribution. Similar to that under the Probit model,
the augmented likelihood function is a product of normal densities but with
different variance components
i
. The following Gibbs sampler (Algorithm
6.1) was proposed by Wang and Lin (2011) based on the augmented likelihood
function:
1. Sample latent variables zi
i
N((t
i
) + x
0
i
;
i
)1
C
i
for i = 1; ;n.
), where W
0
= v
0
+
P
i=1
1
2. Sample
0
from N(E
0
;W
1
0
and
i
o
n
X
k
X
E
0
= W
1
1
i
l
b
l
(t
i
) x
0
i
v
0
m
0
+
z
i
:
0
i=1
l=1
3. Sample
l
' for l = 1; ;k. For each l 1, let W
l
=
P
i=1
1
b
l
(t
i
).
i
Search WWH ::
Custom Search