Biomedical Engineering Reference
In-Depth Information
estimating it requires computational eort, this criterion has unfortunately
received little attention.
For linear regression models with independent observations, leave-one-
out cross validation is asymptotically equivalent to two simple and easily
implemented criteria, Mallows' C
p
and AIC. To extend C
p
and AIC for
GEE methods, we need to dene degrees of freedom for model t. A simple
but reasonable (see [52]) denition of degrees of freedom is the number of
regression coecients used in the model, denoted by df
S
; we dene others
later. It is also necessary to choose a goodness of t measure. It is known
that Mallows' classical C
p
for linear regression models estimates a quadratic
predictive risk. Cantoni, Flemming and Ronchetti
9
extended the C
p
crite-
rion to GEE t using a weighted quadratic predictive risk, resulting in a
generalized C
p
, denoted as GC
p
. The weights allow data analysts to incor-
porate their professional experience easily and incorporate robustness (see
[7]), but implementing the GC
p
with a general weighting scheme requires
bootstrapping or Monte Carlo simulation to estimate the eective degrees
of freedom of the model t. This is very computationally expensive and
may become infeasible in practice. Fortunately, the GC
p
with all weights
equal has a simple, closed form. Let
n
n
X
X
M = n
1
D
i
V
i
D
i
;
N = n
1
D
i
A
i
D
i
;
and
i=1
i=1
where V
i
= A
1=2
R
i
A
1=2
. Cantoni, Flemming and Ronchetti
9
set the de-
i
i
grees of freedom for GEE model t to be
df
C
= tr(M
1
N):
The GC
p
is then
n
n
i
n
X
X
X
r
ij
GC
p
=
J
i
+ 2 df
C
:
(4.3)
i=1
j=1
i=1
The denition of degrees of freedom is motivated by the denition of robust
sandwich formula for the GEE estimate, and if working independence is
used then df
C
= df
S
, see [9] for a more detailed derivation of the degrees
of freedom.
The classic AIC is asymptotically equivalent to C
p
for linear models, but
applies more generally. It estimates the relative Kullback-Leibler distance of
the likelihood function specied by a model, from the true likelihood func-
tion which generated the data. AIC cannot be used directly in GEE since
the likelihood is not specied (although a quasi-likelihood may be implicitly
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