Biomedical Engineering Reference
In-Depth Information
heterogeneity in the
t
-test is Welch's [Welch, 1938] correction of the number of de-
grees of freedom. The test can be formulated as follows for each of the resels
(
i
,
j
)
.
The null hypotheses
H
i
,
j
0
and alternative hypotheses
H
i
,
j
1
:
H
i
,
j
0
:
X
(
j
)
=
X
(
i
,
j
)
(4.27)
b,tr
tr
H
i
,
j
1
:
X
(
j
)
=
X
(
i
,
j
)
(4.28)
b,tr
tr
where:
b,tr
is the normalized energy in the baseline time averaged across base-
line time and trials, and
X
(
j
)
X
(
i
,
j
)
tr
is the normalized energy in resel
(
i
,
j
)
averaged
across trials. The statistics is:
)=
X
(
j
)
b,tr
−
X
(
i
,
j
)
tr
t
(
i
,
j
(4.29)
s
Δ
where
s
Δ
is the pooled variance of the reference epoch and the investigated resel. The
corrected number of degrees of freedom ν is:
s
n
1
+
s
n
2
2
ν
=
(4.30)
s
n
1
2
s
n
2
2
+
n
1
−
1
n
2
−
1
where
s
1
is the standard deviation in the group of resels from the reference period,
n
1
=
·
N
b
is the size of that group,
s
2
is the standard deviation in the group of resels
from the event-related period, and
n
2
N
=
N
is the size of that group.
to be distributed normally, we estimate the distribu-
tion of the statistic
t
from the data using (4.29), separately for each frequency
j
:
If we cannot assume
X
n
(
i
,
j
)
1. From the
X
(
i
,
j
)
,
i
∈
t
b
draw with replacement two samples:
A
of size
N
and
B
of size
N
·
N
b
)=
X
A
−
X
B
2. Compute
t
as in (4.29):
t
r
(
j
,where
s
Δ
is pooled variance of
s
Δ
samples
A
and
B
.
...and repeat steps 1 and 2
N
rep
times. The set of values
t
r
(
j
)
approximates the
distribution
T
r
at frequency
j
. Then for each resel the actual value of (4.29) is
compared to this distribution:
(
j
)
p
(
i
,
j
)=
2min
{
P
(
T
r
(
j
)
≥
t
(
i
,
j
))
,
1
−
P
(
T
r
(
j
)
≥
t
(
i
,
j
))
}
(4.31)
for the null hypothesis
H
ij
0
yielding two-sided
p
(
i
,
j
)
. The relative error of
p
is (c.f.
[Efron and Tibshirani, 1993])
σ
p
p
=
(
−
)
1
p
err
=
(4.32)
pN
rep
Search WWH ::
Custom Search