Biomedical Engineering Reference
In-Depth Information
results in a marginal t
8
distribution for
i
, and
i
=a is used to approximate a
standard logistic random variable.
This data augmentation allows us to ignore the constant a in the MCMC
first and then adjust the outputs for estimation. Specifically, define z∗i
i
= az
i
,
(t) = a(t), and
= a, and run Algorithm 6.1 above with the only
change being sampling
i
from G(4:5; 4 + 0:5(z
i
(t) x
0
i
)
2
) instead
for each i in Step 6.
The regression parameters and the baseline CDF F
0
(t) under the PO
model can be estimated by
X
X
1
M
F
0
(t)
1
M
expf
(k)
(t)=ag
1 + expf
(k)
(t)=ag
;
(k)
=a and
k=1
k=1
respectively, where
(k)
(t) and
(k)
denote the k-th Gibbs sampler outputs
using the modified Gibbs sampler after the burn-in stage.
Both of these two algorithms rely on expressing the logistic distribution
as a scaled-normal mixture, either exactly or approximately. A much simpler
algorithm is to use the relationship between a standard normal distribution
and logistic distribution directly. It is well-known that (t) G(a
0
t), with the
maximum dierence less than 0:01 for any t, where G here denotes the CDF
of a standard logistic distribution and a
0
= 1:70 is a scale constant. Thus, one
can run the Gibbs sampler under the Probit model directly and then adjust
the MCMC output by the scale constant a
0
to get the estimates of interest
under the PO model. Specifically, let
(k)
(t) and
(k)
denote the k-th output
from the Gibbs sampler under the Probit model after the burn-in stage. Then
we can estimate and F
0
(t) under the PO model by
P
k=1
(k)
a
0
=M
and
X
X
expf
(k)
(t) a
0
g
1 + expf
(k)
(t) a
0
g
:
F
0
(t)
1
M
1
M
f
(k)
(t)g or
k=1
k=1
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