Biomedical Engineering Reference
In-Depth Information
In practice, we may replace
2
by its MLE under the full model, denoted by
b
F
. To select the best subset of random eect covariates, we then minimize
n
X
b
x
is
z
is
k
2
+ 2
F
ky
i
s
b
is
b
b
i=1
over all possible subsets, where
is an estimate of ; this approach is similar
to the C
p
criterion for the linear regression model
31
.
Alternatively, we may replace
2
by its conditional MLE, the maximizer
of the conditional likelihood (2.8), i.e.,
b
X
n
1
N
b
cs
=
x
is
z
is
k
2
;
b
ky
i
s
b
is
i=1
P
n
i=1
n
i
. Then we nd a subset of x and z which minimizes
N log
where N =
cs
+ 2
b
b
and this can be viewed as an extension of AIC for linear regression models.
Vaida and Blanchard
46
also proposed a nite sample correction for cAIC;
here we omit the details.
Chen and Dunson
10
propose a hierarchical Bayesian model to identify
random eects having zero variance. A key step in their approach is to
apply a modified Cholesky decomposition for the covariance matrix A
of the random eects:
A = D
T
D;
(2.9)
where D = diagfd
1
;; d
q
gis a diagonal matrix, and is a lower tri-
angular matrix with one on its diagonal. Represent
i
= Dv
i
, where
v
i
= (v
i1
;; v
iq
)
T
is a vector of independent standard normal latent vari-
ables. Thus, model (2.1) can be rewritten as
y
i
= X
i
+ Z
i
Dv
i
+ "
i
:
Thus, we can select signicant random eects variables by identifying
nonzero diagonal elements of D. This can be done by choosing mixture
priors with positive point mass at zero for d
j
under the Bayesian variable
selection framework. Following standard convention, Chen and Dunson
10
choose conjugate priors for and
2
. The modied Cholesky decomposi-
tion allows us to choose the prior for nonzero o-diagonal elements of ,
given d
1
;; d
q
, to be a normal distribution. With these priors, we are
ready to run MCMC to get the posterior distribution of the parameters,
including posterior probabilities for models. Since the priors for the d
j
's
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