Image Processing Reference
In-Depth Information
More generally, within the GLM framework it is possible to create maps
reflecting the locations of significant effects of interest after the other modeled
effects have been taken into account. Let us consider an experiment whose design
matrix can be partitioned into two subsets
X
=
[
X
a
|
X
b
]
(15.16)
with corresponding partition of the regression values
β
=
[
β
a
T
|
β
b
T
].
(15.17)
where
X
a
(
β
a
) indicates the predictors (regression values) corresponding to the
confounds (e.g., mean level, low-frequency fluctuations) and
X
b
(
β
b
) indicates the
effects of interest. Detecting the locations of the brain in which there is a signif-
icant effect of interest corresponds to testing voxel-by-voxel the hypothesis
{
H
b
:
0
} and selecting those voxels in which this hypothesis can be safely
rejected. The extra sum of squares principle provides a means to perform these
tests [27]. Under
H
b
, the model in Equation 15.7 reduces to:
β
b
=
y
=
X
a
β
a
+
e.
(15.18)
The extra sum of squares due to
β
b
after
β
a
is defined as:
X
r
(
β
a
|
β
b
)
=
X
r
(
β
a
) -
X
r
(
β
).
(15.19)
β
a
) denote respectively the residual sum of squares for the
full model and for the reduced model. Under
H
b
,
X
r
(
where
X
r
(
β
) and
X
r
(
β
a
|
β
b
) ~
σ
2
χ
2
independently
of
X
r
(
β
), with L
b
=
rank(
X
)
−
rank(
X
a
) degrees of freedom. Therefore, under
H
b
,
the ratio
(X
()
β
−
X
( )/
β
L
F
=
r
a
r
b
(15.20)
X
()/(
β
TL
−
)
r
b
L
b
degrees of freedom [27].
The desired map can thus be computed using the following steps:
has a central
F
distribution with n
1
=
L
b
and n
2
=
T
−
1.
Calculate the statistic
F
of Equation 15.20 for each voxel.
2.
For a fixed value of false alarm
p
determined by
∞
∫
p
=
fudu
f
() .
(15.21)
F
0
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