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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

$\displaystyle Y_k$ $\displaystyle =$ $\displaystyle X_k \beta_k + \epsilon_k$ (1)
$\displaystyle \beta$ $\displaystyle =$ $\displaystyle X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+ \eta$ (2)

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

$\displaystyle \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.

Christian Beckmann 2003-07-16