Biomedical Engineering Reference
In-Depth Information
Then, the AR prior in Sinha et al. (1999) assumes
j;k
=
j;k1
+
jk
;
jk
N(0;w
jk
); k = 1;:::;K;
(7.10)
where
j;0
and w
jk
are prespecied hyperparameters for k = 1;:::;K and
j = 1;:::;p. In Equation (7.10), it is also assumed that
jk
are independent
for all j and k. Wang et al. (2011a) extended Equation (7.10) to a hierarchical
AR prior given by
j;1
=
j;0
+
j1
;
j1
N(0;c
j0
w
j
);
j;k
=
j;k1
+
j;k
;
jk
N(0;w
j
); k = 2;:::;K;
(7.11)
w
j
IG(
j0
;
j0
);
where c
j0
> 0 is a hyperparameter, IG(
j0
;
j0
) denotes an inverse Gamma
distribution with shape parameter
j0
and scale parameter
0
such that the
mean is
j0
=(
j0
1) for j = 1;:::;p. In Equation (7.11), we further assume
that w
j
are independent across all j. Note that when
j0
1, the mean of
the inverse Gamma IG(
j0
;
j0
) does not exist. The prior for
j;1
is specified
to be more noninformative by multiplying a known constant c
j0
> 1. The
hierarchical AR prior in Equation (7.11) avoids elicitation of w
jk
in Equa-
tion (7.10), whose specification would become an enormous task when K is
large. In addition, w
j
in Equation (7.11) controls the strength of borrowing
from adjacent intervals for
jk
and as w
j
in Equation (7.11) is unspecified,
the magnitude of w
j
is automatically determined by the data.
7.2.6
Dynamic Model for h
0
and
For the cure rate model, Kim et al. (2007) developed a dynamic model for the
baseline hazard function h
0
. A dynamic model for a time function is charac-
terized by its data-driven, time-varying nature and, hence, the number and
locations of the knots of the model are dynamically adapted to the extent that
is needed by the data. We consider a dynamic model for the baseline hazard
function and each component of (t). Because log h
0
(t) can be considered the
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