Biomedical Engineering Reference
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servation given all other observed data under the assumed model. The larger
the value of CPOi,
i
, the more the i-th observation supports the fitted model.
Sinha et al. (1999) showed that CPOi
i
can be computed as
n
h
io
1
Pr
T
i
2 [L
i
;R
i
) j;x
i
1
j D
obs
CPO
i
=
E
;
(7.17)
where Pr
T
i
2 [L
i
;R
i
) j ;x
i
is defined in Equation (7.2). The expectation
in Equation (7.17) is taken with respect to the joint posterior distribution of
given the observed data D
obs
. As discussed in Chen et al. (2000), CPOi
i
can be calculated as the harmonic mean of copies of Pr
T
i
2 [L
i
;R
i
) j;x
i
evaluated at MCMC samples from the posterior distribution of . The CPO
i
value can be summed over all subjects to form a single summary statistic
LPML,
X
LPML =
log CPOi.
i
:
(7.18)
i=1
Models with larger LPML values are preferred over models with lower LPML
values. According to Gelfand and Dey (1994), LPML implicitly includes a
similar dimensional penalty as AIC (Akaike, 1973) asymptotically.
7.4.2
Deviance Information Criterion
The DIC proposed by Spiegelhalter et al. (2002) is another useful tool in
Bayesian model comparison. For interval-censored data, we define a deviance
as
D() = 2 log
L( j D
obs
)
;
where L( j D
obs
) is dened in Equation (7.3). Let and D = E
D() j
D
obs
denote the posterior mean of and D(), respectively. According to
Spiegelhalter et al. (2002), the DIC measure is defined as
DIC = D( ) + 2p
D
;
(7.19)
where p
D
= DD( ) is the eective number of model parameters. The rst
term in Equation (7.19) measures the goodness-of-t. The smaller the D( ),
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