Biomedical Engineering Reference
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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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