Biomedical Engineering Reference
In-Depth Information
10). We generated correlated binary response with marginal distribution:
y
ij
jx
ij
Bernoullifp(x
ij
)g;
where p(x) = exp(0:5 + x
i
)=f1 + exp(0:5 + x
i
)g, is the same as
that in Example 1, x
ij
N
8
(0; ), where the diagonal elements of all
equal 1, and all o-diagonal elements equal 0.5. The binary responses were
exchangeably correlated (i.e., with compound symmetry).
Since the classic C
p
criterion is not well dened for binary data, we
do not include it in our comparison. Simulation results are summarized in
Table 2, in which MSE is the mean squared error in coecient estimation.
Table 2.
Comparison of GEE Model Selection (Binary Response)
Criterion
10
Mean MSE
Prop. True Models
Working Corr. Matrix:
Ind
AR
CS
Ind
AR
CS
Full
1.11
0.87
0.82
0.00
0.00
0.00
Na
ive AIC
0.88
0.62
0.66
0.38
0.53
0.46
AIC (Pan)
0.88
0.64
0.69
0.38
0.41
0.31
LASSO (Fu)
0.77
0.65
0.63
0.01
0.00
0.00
SCAD
1
0.78
0.62
0.66
0.69
0.72
0.70
SCAD
2
0.86
0.76
0.79
0.61
0.57
0.56
Correct Deletions
Erroneous Deletions
Working Corr. Matrix:
Ind
AR
CS
Ind
AR
CS
Full
0.00
0.00
0.00
0.00
0.00
0.00
Na
ive AIC
4.09
4.41
4.34
0.07
0.10
0.10
AIC (Pan)
4.07
4.22
4.05
0.07
0.08
0.09
LASSO (Fu)
1.92
1.70
1.63
0.00
0.01
0.01
SCAD
1
4.83
4.93
4.88
0.23
0.22
0.23
SCAD
2
4.94
4.97
4.98
0.38
0.41
0.43
The SCAD with BIC
2
had a false deletion rate higher than we would
wish; the SCAD with BIC
1
had almost as good a correct deletion rate but
a noticeably lower false deletion rate. In general, then, the lighter version
seems to be better. Pan's AIC does fairly well. The LASSO provides good
estimation performance but not very sparse models.
6. Variable Selection for Other Models
Although mixed eects models and GEE are very popular formulations for
analyzing longitudinal data, many other models have also been used. In this
section, we will briey introduce variable selection procedures for a partial
linear model, a semiparametric approach highly relevant to longitudinal
data (see [51], [34], [29], and [16]).
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