Biomedical Engineering Reference
In-Depth Information
There are two kinds of variable selection problems for linear mixed ef-
fects models: identifying signicant xed-eects variables when the random
eects are not subject to selection, and identifying both signicant xed
eects and random eects. We may select signicant xed eects covariates
by using a relatively straightforward penalized likelihood approach, but se-
lection of signicant random eects is more challenging. Let us begin with
selection of xed eects covariates.
2.1. Fixed eects variable selection
When the random eects are not subject to selection, we rewrite the mixed
eects model as
y
i
= X
i
+
i
;
(2.2)
N(0;
i
()) with
i
() = Z
i
AZ
i
+
2
I. Here consists of
all unknown parameters in A and
2
. Thus, (2.2) is a linear model with
correlated errors. This enables us to modify variable selection procedures for
ordinary linear regression models to select signicant variables for (2.2). Liu,
et al.
30
proposed a leave-one-out cross validation method to estimate the
predicted residual sum of squares (PRESS) and select signicant variables
for model (2.2) via minimizing the PRESS. Let us assume for the moment
that is known. Let e
i
= y
i
where
i
b
b
being the maximum likelihood estimate (MLE, equivalently the weighted
least squares estimate) of . Dene e
(i)
= y
i
X
i
be the ordinary residuals, with
b
X
i
(i)
to be the deleted
b
residual with
(i)
dened as the parameter estimate when the i-th subject
is deleted from the analysis. Dene
X
n
k
2
;
PRESS =
ke
(i)
(2.3)
i=1
wherekkis the Euclidean norm. As for the ordinary linear regression
model, a fast algorithm can be developed here to calculate the PRESS
statistic. Let X = (X
1
;; X
n
)
T
, = diagf
i
();;
n
()g, H =
X(X
T
1
X)
1
X
T
1
g, with i; j = 1;; n, be one version of
the hat matrix for model (2.3), where H
ij
is n
i
=fH
ij
n
j
. Dene Q = IH and
let Q
ii
be the i-th diagonal block of Q. As shown in [30],
e
(i)
= Q
1
ii
e
i
:
Thus,
X
n
kQ
1
ii
e
i
k
2
:
PRESS =
(2.4)
i=1
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