Two-level GLM

Consider an experiment where there are $N_K$ first-level sessions and that for each first-level session, $k$, the preprocessed FMRI data is a $T\times 1$ vector $Y_k$, the $T\times P_K$ design matrix is $X_k$, and $\beta_k$ is a $P_K\times 1$ vector of parameter estimates ( $k=1,\ldots,N_K$). The preprocessed FMRI data, $Y_k$, is assumed to have been prewhitened (25,4). An individual GLM relates first-level parameters to the $N_k$ individual data sets:

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

where $\epsilon_k\sim N(0,\sigma_k^2 I)$. In this paper we consider the variance components as unknown with the exception of the first-level FMRI time-series autocorrelation. The residuals $\epsilon_k$ are assumed to be prewhitened data and as a result are uncorrelated. This inherently means that we assume that the autocorrelation is known with no uncertainty, an assumption which is commonly made in FMRI time-series analysis (8,25,4). Note that the first level design matrices, $X_k$, do not need to be the same for all $k$.

Using the block diagonal forms, i.e. with

$$Y\! =\! \left[\! \begin{array}{c} Y_1 \\ Y_2 \\ \vdots \\ Y_{N_K}... ...begin{array}{c} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_{N_K} \end{array} \right]$$   and$$\quad \epsilon\! =\! \left[\! \begin{array}{c} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_{N_K} \end{array}\! \right]$$

the two-level model is

$$Y = X \beta_K + \epsilon_K$$ (2)

$$\beta_K = X_g \beta_g + \epsilon_g$$ (3)

where $X_g$ is the $N_K\times P_G$ second-level design matrix (e.g. separating controls from normals or modelling different sessions for subjects), $\beta _g$ is the $P_G\times 1$ vector of second-level parameters, and $\epsilon_g \sim N(0,\sigma_g^2 I)$ and where $\epsilon_K \sim N(0,V_K)$ with $V$ denoting the diagonal form of first-level covariance matrices $\sigma_k^2 I$. We call $\sigma _g^2$ the random effects variance.