Biomedical Engineering Reference
In-Depth Information
−
1
(
)
(
)
−
1
⎡
⎣
−
1
⎤
⎦
T
T
T
T
ββ
=+
XX
C CXX
C
C
β
(12.18)
In general, for number of parameters
=
p
and number of contrasts
=
q
, the error
2
variance
σ
is estimated as
(
)(
)
(
)
2
σ
=−
YX YX
β
−
β
np
−
(12.19)
2
which is
χ
distributed with (
n
p
) degrees of freedom. It is always true that
(
) (
)
T
QYX YX
=−
β
−
β
1
is less than
T
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
QYXYX
=−
β
−
β
(12.20)
0
β
because the
's are unconstrained.
When the null hypothesis is true (regions where there is no spike-correlated acti-
vation), an estimate of the error variance using
Q
1
and
Q
0
can be obtained as
(
)
QQ
q
−
σ
1
0
2
=
(12.21)
If
H
0
is true, then the ratio
(
)
≡=
−
QQq
Qnp
σ
σ
2
1
0
(12.22)
F
(
)
2
−
0
will be distributed according to an
F
-distribution with
q
and (
n
−
p
) degrees of free-
dom. For (
q
1), as in the EEG-fMRI case illustrated,
F
reduces to the square of the
t
-random variable with
n
=
p
degrees of freedom.
If
H
0
is true, then the numerator and denominator are both estimating
−
2
σ
, so the
value of
F
tends to be
1. So a standard practice in EEG-fMRI analysis is to assume
that
H
0
is true, calculate the
F
-value, and compare the computed
F
-value against the
critical value in an
F
-table with
q
and
n
-
p
degrees of freedom. If the computed
value is larger than the critical threshold
F
-value, then one can reject the null
hypothesis and accept the decision that the voxel shows a positive area of activation
correlated to the EEG spikes. Figure 12.15 [49] shows an example of spike-corre-
lated fMRI of an epileptic seizure.
≤
12.2.3.3 Evoked Potentials and Event-Related Potentials
Simultaneously recorded EEG and fMRI offers the possibility of mapping the neural
substrates of evoked activity with a higher spatiotemporal resolution than is possi-
ble by either modality alone. Previously, various research groups have relied on sep-
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