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3.1 Relation to mixed-effects GLMs

Within the general linear model, the data from an individual subject $k$ at time $i$ and voxel location $j$ is expressed as:

\begin{displaymath}
x_{ijk} = \sum_r^R\bar{\mbox{\protect\boldmath$A$}}^{(k)}_{ir}\bar{\beta}^{(k)}_{jr}+\bar{\epsilon}^{(k)}_{ij}.
\end{displaymath} (12)

Here, $\bar{\mbox{\protect\boldmath$A$}}^{(k)}$ denotes the (potentially subject-specific) lower-level GLM design matrix containing $R$ regressors, $\bar{\beta}^{(k)}$ denotes the subject specific lower-level linear model parameters at voxel location $j$ and $\bar{\epsilon}^{(k)}_{ij}$ is the subject-specific (fixed-effects) error. Typically, the subject specific linear model parameters are then related to group parameters in a second linear model6:
\begin{displaymath}
\bar{\mbox{\protect\boldmath$\beta$}}^{(k)}_{jr}=\bar{\mbox{...
...}}_{kr}\bar{\mbox{\protect\boldmath$B$}}_{jr}+\bar{\eta}_{jr},
\end{displaymath} (13)

where now $\bar{\mbox{\protect\boldmath$C$}}_{kr}$ denotes the $k^{\mbox{\tiny th}}$ entry of the group-level design matrix for lower-level regressor $r$ (which, in the case of mean group activation studies, is a vector of ones for each $r$), $\bar{\mbox{\protect\boldmath$B$}}_{jr}$ is the group effect size at voxel location $j$ and where $\bar{\eta}$ is the random-effects variance contribution of the lower-level regressor $r$ at voxel location $j$. Within this model, the estimated random-effects variance contribution is given by the variance of the estimated parameter estimates around their mean. The mixed-effects two-level GLM relates the group level parameters of interest to the original data as
\begin{displaymath}
x_{ijk}=\sum_r^R \bar{\mbox{\protect\boldmath$A$}}^{(k)}_{ir...
...}}_{kr}\bar{\mbox{\protect\boldmath$B$}}_{jr}+\bar{\nu}_{ijk},
\end{displaymath} (14)

where $\bar{\nu}$ is the combined error term with associated mixed-effects variance. Similar to the generative model of equation 1[*], the two-level mixed-effects GLM expresses the data via a tri-linear decomposition.

In order to compare the variance terms, assume that $\bar{\mbox{\protect\boldmath$A$}}^{(k)}=\mbox{\protect\boldmath$A$}, \bar{\mbox{\protect\boldmath$B$}}=\mbox{\protect\boldmath$B$}$ and $\bar{\mbox{\protect\boldmath$C$}}=\mbox{\protect\boldmath$C$}$. Then the mixed-effects error $\bar{\nu}$ corresponds to the error term $\widetilde{E}$ in equation 8[*]. In contrast to the PARAFAC/tensor ICA model, however, equation 14[*] uses a-priori specified design matrices $\bar{\mbox{\protect\boldmath$A$}}$ and $\bar{\mbox{\protect\boldmath$C$}}$. When fixing the GLM group-level design $\bar{\mbox{\protect\boldmath$C$}}$ to a column vector of ones in order to calculate mean group activation size, the resulting group-level error, $\bar{\eta}$, 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, $r$, 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, $\mbox{\protect\boldmath$C$}_r$ in equation 8[*] represents the amplitude of signal modulation of the entire spatio-temporal process defined by $\mbox{\protect\boldmath$A$}_r$ and $\mbox{\protect\boldmath$B$}_r$ for different processes, $r$, and subjects $k$, independent of voxel location $j$. As such, the variance of a single vector $\mbox{\protect\boldmath$C$}_r$ signifies the variance of the individual subjects' responses around the expected population mean response for the entire spatio-temporal process described by $\mbox{\protect\boldmath$A$}_r$ and $\mbox{\protect\boldmath$B$}_r$. 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].


next up previous
Next: Data pre-processing for tensor-PICA Up: Tensor PICA Previous: Tensor PICA
Christian Beckmann 2004-12-14