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
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