In order to calculate the average group activation, we model the individual subject activation as being normally distributed according to
where
represents the average group activation and is usually estimated as
We will model the first-level within-subject covariances to be subject-specific and model the between-subject variances (from the group mean) as equal across the group. That is
Then the adjusted second-level covariance matrix is
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Define
, which is the sum of the within- and between-subject covariances.
Then the estimate of the group parameter writes as
Hence we see that in the general framework, the mean group
activation parameter is a weighted average of the combined subject-
specific activations, where the weights are inversely proportional
to the subject-specific variances.
This adjustment is advantageous in the case where
the individual time-series model does not fit well for a particular
subject, , generating an unusual value for
(an
outlier) but also a large
. If no correction
for the first-level variance is done, then this outlier can
significantly affect the estimation of the group (between-subject)
variance. If, however, this first-level correction is performed, the
increased variance in this parameter will effectively de-weight the
contribution of this outlier to the group variance estimate, since we
use General Least Squares estimation.
In the much simpler case, where the within-subject covariances are
, and the
are normalised, such that
for all
, then
The test for significance is then carried out in a -test where