Biomedical Engineering Reference
In-Depth Information
H(tjx
i
;) can be expressed as
Z
t
Z
a
g
k
X
H(tjx
i
;) = expfx
0
i
(a
k
)g
expfx
0
i
(a
g
)g
h
0
(u)du +
h
0
(u)du
a
k1
a
g1
g=1
= expfx
0
i
(a
k
)g[H
0
(t) H
0
(a
k1
)]
k
X
expfx
0
i
(a
g
)g[H
0
(a
g
) H
0
(a
g1
)]:
+
(7.8)
g=1
If H
0
GP(H
0
; c), then we have
H
0
(t) H
0
(a
k1
) Gamma(c[H
0
(t) H
0
(a
k1
)]; c)
for a
k1
t < a
k
and
H
0
(a
g
) H
0
(a
g1
) Gamma(c[H
0
(a
g
) H
0
(a
g1
)]; c)
for g = 1;:::;k 1. From Equation (7.7), we see that there is a connection
between the GP prior and the piecewise exponential prior, namely,
k
= [H
0
(t) H
0
(a
k1
)]=(ta
k1
); a
k1
< t < a
k
:
That is,
k
is the average cumulative baseline hazard function over the time
interval [a
k1
;t) for a
k1
< t < a
k
. When
k
= 0:2 and
k
= 0:4, the piecewise
exponential prior in Equation (7.6) is equivalent to the GP prior with H
0
(t) =
H
0
(a
k1
) +
0
for a
k1
t < a
k
for k = 1; 2;:::;K, H
0
(0) = 0,
0
= 0:5,
and c = 0:4. In this sense, the piecewise exponential prior may be viewed as a
discretized version of the GP prior (Sinha et al., 1999). Nevertheless, the GP
prior provides a more flexible class of priors.
7.2.5
Autoregressive Prior Model for
On grid G, Sinha et al. (1999) proposed an AR model for (t). Let
j
(t) be
the j-th component of (t) for j = 1;:::;p. In Equation (7.4), we write
j;k
=
j
(a
k
); k = 1;:::;K:
(7.9)
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