The previous section assumed that all the first-level parameter estimates would be used in the second-level model. This is often not true, as parameters of no interest (confounds) are often present in the first-level to remove unwanted signals (e.g. motion estimates as regressors to remove motion-related artefacts). This section shows that, when confounds are present in the first level, the model equivalence theorem still holds. That is, the second-level (group) analysis does not need to know about the confounds defined in the first level and only requires the estimates and covariance estimates of the parameters of interest. The proof of this result follows.
Consider a first-level GLM including confounds. This model can be written as
The parameter estimates in this case are
It is always possible to re-write such a model so that the confounds are orthogonal to the signals of interest in the pre-whitened space, without affecting the estimates of the signals of interest. This result is formally stated and proven in the appendix as Theorem A. Therefore, we can impose
In the second-level, only the parameters of interest, , appear. That is
which is identical to equation 6, so that the parameter estimates are exactly the same as those given in equation 7.
On the other hand, the single-level model with confounds is written as
Using equation 11 from the previous section, together with the orthogonality condition in equation 12, it is easy to show that:
This gives the parameter estimates as:
Therefore the model equivalence result still holds when confounds are present.