Biomedical Engineering Reference
In-Depth Information
than the GCV selector, in that selecting the tuning parameters by BIC
guarantees that the resulting estimator possesses the asymptotic oracle
property, while using GCV does not. Thus, we will use the BIC selector
for the SCAD in our numerical comparison.
By some straightforward calculation, the eective number of parameters
in the last step of the Newton-Raphson algorithm is
e() = tr[fG() + N
(
)g
1
G()];
where G() = @G()=@, corresponding to the nonzero components of
b
b
b
b
b
) is a diagonal matrix with diagonal elements p
0
j
(j
and
(
j
j)=j
j
jfor
b
nonzero
j
's.
Parallel to two extensions of the BIC proposed in [38], the BIC statistic
can be dened as follows:
X
n
X
n
i
N
log(
1
1
r
ij
) + log(n)e();
BIC
1
() =
(4.6)
N
i=1
j=1
or
X
n
X
n
i
N
log(
1
1
r
ij
) + log(N)e();
BIC
2
() =
(4.7)
N
i=1
j=1
b
where r
ij
is the Pearson residual corresponding to
, given . One may
replace the Pearson residuals with deviance residuals if they are available.
The BIC
2
is more compatible with the original denition of the BIC (and
is equivalent under working independence), but in practice it tends to be
a little too strong, and it tends to give somewhat poorer empirical perfor-
mance than BIC
1
. Both presumably have similar asymptotic behavior if
the n
i
are bounded (see [43]).
We can select by minimizing, say, BIC
1
. To nd an optimal , the
BIC selector needs to be minimized over a d-dimensional space, an unduly
onerous task. However, it is intuitively expected that the magnitude of
j
should be proportional to the standard error of estimate of
j
. Therefore, we
may set = se(
b
b
GEE
) stands for the stan-
dard error of the unpenalized GEE estimate. Thus, we minimize the BIC
score over the one-dimensional space, saving a great deal of computational
cost. This scheme is used in our simulations.
GEE
) in practice, where se(
5. Numerical Comparison
This section presents some comparisons of variable selection procedures for
longitudinal data. Comparisons of variable selection procedures for linear
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