Within the general linear model, the
data from an individual subject at time and voxel location
is expressed as:
In order to compare the variance terms, assume that and . Then the mixed-effects error corresponds to the error term in equation 8. In contrast to the PARAFAC/tensor ICA model, however, equation 14 uses a-priori specified design matrices and . When fixing the GLM group-level design to a column vector of ones in order to calculate mean group activation size, the resulting group-level error, , has associated voxel-wise random-effects variance. That is, the multi-level GLM permits different voxel locations to have different random-effects variance. Even in the case where two voxels show similarly significant amplitude modulation to the same regressor, , their random-effects variance contribution is allowed to be different. As such, the multi-level GLM random-effects variance is the variance of the individual subjects' responses around the expected population mean response at a given voxel location, i.e. the random-effects variance is averaged over time but not over space. By comparison, in equation 8 represents the amplitude of signal modulation of the entire spatio-temporal process defined by and for different processes, , and subjects , independent of voxel location . As such, the variance of a single vector signifies the variance of the individual subjects' responses around the expected population mean response for the entire spatio-temporal process described by and . Within the standard multi-level GLM this quantity is not readily available but can be approximated by the spatially averaged random-effects variance weighted by the normalised voxel-wise parameter estimates within post-thresholded clusters [Smith et al., 2004].