Biomedical Engineering Reference
In-Depth Information
basis toward zero. This property is promising, as it allows one to use many
knots for modeling flexibility but prevents over-fitting problems.
A normal prior N(m
0
;
0
) is used for the additional parameter
0
under
the PO and Probit models. These priors are quite general and are selected
based on the expanded likelihoods from a computational perspective.
6.3.1
Under the Probit Model
Let t
i
= R
i
1
(
i1
=1)
+ L
i
1
(
i1
=0)
. Consider the following data augmentation:
z
i
= (t
i
) + x
0
i
+
i
with constraint zi
i
2 C
i
;
i
N(0; 1); i = 1;:::;n;
where C
i
is an interval taking (0;1) if
i1
= 1, ((L
i
) (R
i
); 0) if
i2
= 1,
and (1; 0) if
i3
= 1.
The augmented likelihood function is
Y
(z
i
(t
i
)x
0
i
)f1
(z
i
>0)
g
i1
f1
((L
i
)(R
i
)<z
i
<0)
g
i2
f1
(z
i
<0)
g
i3
;
L
aug
=
i=1
which is a product of normal density functions. Integrating out all zi,
i
, one ob-
tains the observed likelihood as in Equation (6.1). The following Gibbs sampler
was developed based on the augmented data likelihood and the specified priors
(Lin and Wang, 2010):
1. Sample latent variables zi
i
N((t
i
) + x
0
i
; 1)1
C
i
for i = 1; ;n.
2. Sample
0
from N(E
0
;W
1
), where W
0
= v
0
+ n and
0
l
b
l
(t
i
) x
0
i
o
n
z
i
X
k
X
E
0
= W
1
v
0
m
0
+
:
0
i=1
l=1
3. Sample
l
for l = 1; ;k. For each l 1, let W
l
=
P
i=1
b
l
(t
i
).
(a) If Wl
l
= 0; sample
l
from the prior E().
(b) If Wl
l
> 0, sample
l
from N(E
l
;W
l
)1
(
l
>d
l
)
, where
X
n
o
z
i
0
X
l
0
6=l
E
l
= W
1
l
l
0
b
l
0
(t
i
) x
0
i
b
l
(t
i
)
;
i=1
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