Biomedical Engineering Reference
In-Depth Information
mixed models can be found in [20], and some comparisons of covariance
selection are found in [28].
Example 1. In this example, we generated 200 data sets using Octave
code (a free version of Matlab), each consisting of n = 50 subjects with
each subject having J = 5 observations (i.e, all n
i
equals J = 5), from the
following linear model:
y
ij
= x
ij
+ 3"
ij
;
where = (3; 1; 0; 0; 2; 0; 0; 0)
T
(i.e., there were 5 inactive and 3 active
predictors), and x
ij
N
8
(0; ), where the diagonal elements of all equal
1, and all o-diagonal elements equal 0.6. Furthermore, ("
i1
;; "
iJ
)
T
are
multivariate normal with AR(1) true correlation structure with = 0:7.
In our simulation, we compare the following GEE model selection cri-
teria:
(1) naive AIC ignoring correlation, dened as N log(RSS
S
=N) + 2df
S
;
(2) naive C
p
, dened to be RSS
S
+ 2df
S
2
, where
2
b
b
is the MSE under
the full model;
(3) Cantoni's C
p
dened in (4.3);
(4) Pan's AIC, dened in (4.4);
(5) Fu's penalized GEE with L
1
penalty. The
j
were proportional to the
unpenalized standard errors; their magnitude was chosen using the
modied GCV-like statistic dened in [18].
(6) Penalized GEE with the SCAD penalty. The tuning parameters are
selected by using BIC
1
and BIC
2
tuning parameter selectors described
in Section 4.2. Corresponding to the BIC
1
and BIC
2
, this procedure
is referred to as SCAD
1
and SCAD
2
in Table 1, respectively.
To nd the subset which minimizes AIC and C
p
criteria in (1)|(4),
we exhaustively search all 2
8
possibilities. Thus, the corresponding results
represent best subset variable selection with the underlying criterion.
We compare each variable selection procedure in terms of model com-
plexity and model error, dened by ME(
b
b
b
)
T
E(xx
T
)(
) (see
[15]). Table 1 depicted the mean of model error for each procedure and
summarized model complexity in terms of correct deletions, the average
number per simulation of truly zero coecients correctly estimated as zero,
erroneous deletions, the average number of truly nonzero coecients erro-
neously set to zero, and proportion correct models, the proportion of trials
in which exactly the true subset of nonzero predictors was chosen.
) = (
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