Biomedical Engineering Reference
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equations:
X
n
Q()
de
=
G() + N
_
P() =
D
i
A
1=2
R
i
A
1=2
_
P() = 0;
(y
i
i
) + N
i
i
i=1
(4.5)
where
_
P() = [p
0
1
(j
1
j)sgn(
1
);; p
0
d
(j
d
j)sgn(
1
)]
T
.
Fu
18
studied the asymptotic properties of these penalized generalized
estimating equations with L
q
penalties, including L
1
as a special case. He
further addressed practical implementation issues and recommmended an
adaptation of the GCV (see [11]) to select the regularization parameter
j
.
As demonstrated in Fan and Li
15
, the SCAD penalty dened in Section
2.1 retains the main virtues of L
1
while reducing estimation bias. Here
we suggest using the SCAD penalty instead of the L
1
penalty in (4.5).
The oracle property for penalized GEE with the SCAD penalty can be
established using the same strategy as that in [15], see [13] for more details.
Since the SCAD penalty is singular at the origin, and is nonconvex over
(0;1), it is not straightforward to solve the penalized GEE with the SCAD
penalty.
For practical implementation, we use a modied Newton-Raphson algo-
rithm to solve the penalized GEEs, with iterative local quadratic approx-
imation (LQA, [15]) to approximate the SCAD penalty. Given an initial
value
(0)
that is close to the solution of (4.5), for coecient estimates not
too close to zero (j
b
j
j where in practice could be .001, or smaller if
the standard error of
b
is very small), the penalty p
j
(j
j
j) can be locally
approximated by the quadratic function as
[p
j
(j
j
j)sgn(
j
)fp
0
j
(j
(0)
j)=j
(0)
j
j)]
0
= p
0
j
(j
j
jg
j
:
With the local quadratic approximation, the Newton-Raphson algorithm
can be implemented directly to solve the penalized GEE (4.5). When the
algorithm converges, the solution satises the penalized GEE equations
Q() = 0. Of course, forj
j
b
b
j
jto zero. Following conventional
techniques in GEE approaches, we may use a sandwich formula to estimate
the standard error of the coecient estimates in the nal model. A similar
LQA algorithm can be used to nd L
q
-penalized estimates also; this is very
similar to the adjusted iterative algorithm mentioned by Fu
18
.
To implement penalized GEE in practice, it is desirable to have an
automatic data-driven method for selecting the tuning parameters =
(
1
;;
d
). Fan and Li
15
chose the tuning parameters by minimizing a
GCV criterion. Wang, Li and Tsai
47
later proposed a BIC-like tuning pa-
rameter selector. They demonstrate that the BIC selector performs better
j
j< we setj
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