AMS588.
Oneway Repeated Measures ANOVA:
The Univariate and the Multivariate Analysis Approaches
Suppose there are
k
regions of interest (ROI’s) and
n
subjects. Each subject was scanned
on baseline (soda) as well as after drinking alcohol. Our main hypothesis is whether the
change between baseline and alcohol is homogeneous among the ROI’s. That is
0
1
2
:
k
H
μ
μ
μ
=
=
=
L
, where
j
μ
is the effect of alcohol on the
j
th ROI,
1,
, .
j
k
=
L
The Univariate Analysis Approach
For subject
i
, let
Y
ij
denote the paired difference between baseline and alcohol for the
jth
ROI, then the (univariate) repeated measures ANOVA model is:
ij
j
i
ij
Y
S
μ
ε
=
+
+
, where
j
μ
is the (fixed) effect of ROI
j
,
i
S
is the (random) effect of subject
i
,
ij
ε
is the random
error independent of
i
S
. With normality assumptions, we have
(
29
2
~
0,
iid
i
s
S
N
σ
and
(
29
2
~
0,
iid
ij
N
ε
ε
σ
. Let
(
29
'
1
2
,
,
,
i
i
i
ik
Y
Y
Y
Y
=
L
, we have
(
29
~
,
iid
i
k
Y
N
μ
Σ
,
1,
,
i
n
=
L
, where
(
29
'
1
2
,
,
,
k
μ
μ
μ
μ
=
L
and
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
s
s
s
s
s
s
s
s
s
ε
ε
ε
ρ
ρ
σ
σ
σ
σ
ρ
ρ
σ
σ
σ
σ
σ
ρ
ρ
σ
σ
σ
σ
+
+
Σ =
=
+
L
L
L
L
M
M
O
M
M
M
O
M
L
L
with
2
2
2
s
s
ε
σ
ρ
σ
σ
=
+
and
2
2
2
s
ε
σ
σ
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 Fall '10
 WeiZhu
 Yi, oneway repeated measures

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