Biomedical Engineering Reference
In-Depth Information
6.3.3
Under the PH Model
The PH model is currently the most popular semiparametric model in the
survival literature. Existing Bayesian methods on interval-censored data in-
clude Sinha et al. (1999) and Yavuz and Lambert (2011), among others. In
an unpublished manuscript, Lin et al. (2012) developed an ecient Bayesian
method that requires much less computational effort than any other existing
Bayesian work under the PH model.
A novel two-stage data augmentation was developed by Lin et al. (2012)
utilizing the interval-censored data structure and the PH model properties. Let
t
i1
= R
i
1
(
i1
=1)
+L
i
1
(
i1
=0)
and t
i2
= R
i
1
(
i2
=1)
+L
i
1
(
i3
=1)
. In the rst stage,
two sets of Poisson latent variables are introduced: zi
i
Pf
0
(t
i1
) exp(x
0
i
)g
and w
i
P[f
0
(t
i2
)
0
(t
i1
)gexp(x
0
i
)], where P(a) denotes the Poisson dis-
tribution with mean parameter a. We also use P() for the Poisson probability
mass function in the remainder of this chapter.
The augmented data likelihood function is thus
Y
P(z
i
)P(w
i
)
i2
+
i3
if
i1
1
(z
i
>0)
+
i2
1
(z
i
=0;w
i
>0)
+
i3
1
(z
i
=0;w
i
=0)
g;
L
aug1
=
i=1
which is a product of Poisson probability mass functions. Integrating all zi
i
and w
i
in the above likelihood function leads to the observed likelihood in
Equation (6.1).
In the second stage, each z
i
and w
i
is decomposed as the sum of k inde-
pendent Poisson latent variables as follows: zi
i
=
P
l=1
z
il
and w
i
=
P
l=1
w
il
,
where z
il
Pf
l
b
l
(t
i1
) exp(x
0
i
)g and w
il
P[
l
fb
l
(t
i2
) b
l
(t
i1
)gexp(x
0
i
)]
for l = 1;:::;k:
Then, the new augmented likelihood function is
Y
k
Y
P(z
il
)P(w
il
)
i2
+
i3
L
aug2
=
(6.6)
i=1
l=1
subject to the following constraints
P
l
z
il
> 0 if
i1
= 1,
P
l
z
il
= 0 and
P
l
w
il
> 0 if
i2
= 1, and
P
l
z
il
= 0 and
P
l
w
il
= 0 if
i3
= 1 for each i.
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