Biomedical Engineering Reference
In-Depth Information
for k = 1; 2;:::;K. Then, Sinha et al. (1999) specied a gamma prior for each
k
,
k
Gamma(
k
;
k
);
(7.6)
where Gamma(
k
;
k
) denotes a Gamma distribution with probability density
function (
k
) /
k
k
exp(
k
k
). In practice, a set of guide values for the
hyperparameters
k
and
k
, namely,
k
= 0:2 and
k
= 0:4 for k = 1;:::;K
are given in Sinha et al. (1999) when no prior information is available. Under
the specification of Equation (7.5), the cumulative hazard function can be
written as
H(tjx
i
;) =
k
(ta
k1
) expfx
0
i
(a
k
)g
k
X
g
(a
g
a
g1
) expfx
0
i
(a
g
)g
+
(7.7)
g=1
for a
k1
t < a
k
. Further properties of the piecewise exponential model for
h
0
are discussed in detail in Ibrahim et al. (2001).
7.2.4
Gamma Process Prior Model for H
0
The use of the gamma process prior for the cumulative hazard function dates
back to Kalbfleisch (1978) for right-censored data. Further properties of the
gamma process prior for modeling the survival data can be found in Sinha et al.
(2003). The cumulative baseline function H
0
(t) follows a gamma process (GP),
denoted by H
0
GP(H
0
; c), if the following properties hold: (i) H
0
(0) = 0;
(ii) H has independent increments in disjoint intervals; and (iii) for t > s,
H
0
(t) H
0
(s) Gamma(c[H
0
(t) H
0
(s)]; c), where c > 0 is a constant and
H
0
(t) is an increasing function such that H
0
(0) = 0 and H
0
(1) = 1. It is
easy to see that E[H
0
(t)] = H
0
(t) and Var(H
0
(t)) = H
0
(t)=c. Thus, H
0
(t) is
the mean of H
0
(t) and c is a precision parameter that controls the variability
of H
0
about its mean. As c increases, the variability of H
0
decreases.
For a
k1
t < a
k
, we observe that the cumulative hazard function
Search WWH ::
Custom Search