Multi-subject analysis with GLM

The single-subject analysis seen at the beginning of this paper is a GLM with particular choices for $X$ and $V$ ( $V=\sigma^2I\!d_T$ and $X$ with 2 columns; one for the mean and one for the box-car, which is equivalent to the model where $y$ and $X$ are demeaned, reducing $X$ to just one demeaned box-car). The GLM approach enables one to take into account in the model correlated errors, and also to provide different statistical maps according to the hypothesis being tested. These models can be used as a first step or first stage of a multi-subject analysis, $i.e$ for every subject, as is done in section 3. In fact a GLM can also be used at a second level of the multi-subject analysis, but usually the second level model will be simply to estimate a mean or difference of means of the contrasts estimated at the first level. One must remark that it is at the second level where the distinction between the fixed subject effect approach and the random subject effect approach is made. The fixed approach will suppose a known error given by the pool of the errors from all the first stage models and thereby will not take into account the subject sampling variation, and the random approach will re-estimate the whole error variance. The purpose of this section is to rewrite fixed and random analysis described in the previous section within the GLM model, to enable one to carry out the analysis in one step (although there may be practical reasons for not doing so).