Two-level GLM

Consider an experiment where there are $N$ subjects and that for each subject, $k$, the preprocessed FMRI data is $Y_k$ (a vector of $T$ time points), the design matrix is $X_k$ and the parameter estimates are $\beta_k$ (for $k=1,\ldots,N$). The two-level model for this experiment is

$$Y_k = X_k \beta_k + \epsilon_k$$ (1)

$$\beta = X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+ \eta$$ (2)

where $\epsilon_k$ specifies the single-subject residuals, $\eta$ specifies the residuals of the group activation (parameter) scores, and where

$$\textrm{E}(\epsilon_k) = 0 \; , \qquad \textrm{Cov}(\epsilon_k) =... ... \begin{array}{c} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_N \end{array} \right]$$

denotes the combined first-level parameter estimates of the whole group, assembled into a single vector, $X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}$ is the group-level design matrix (e.g. separating controls from normals) and $\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}$ is the final vector of group-level parameters.