Fixed subject effect

Firstly, we will rewrite the single-group study. One has for each $i=1\cdots n$ the GLM models:

$$y_i=X\beta_i+\epsilon_i$$ (30)

with $E(\epsilon_i)=0$ and $var(\epsilon_i)=\sigma_i^2I\!d_T$. The case of auto-correlation $var(\epsilon_i)=\sigma_i^2V_i$ will be examined later on. It is rather straightforward to recognise the fixed effect in the following model:

$$\left ( \begin{array}{c } y_1 \\ y_2 \\ \vdots\\ ... ...\epsilon_2 \\ \vdots\\ \epsilon_n \end{array} \right )$$ (31)

or $y=(I\!d_n\otimes X)\beta+\epsilon$ with obvious definitions for $y$, $\beta$ and $\epsilon$ with $var(\epsilon)=diag({\sigma_{\epsilon_i}^2}_{[1..n]})\otimes I\!d_T$. One then derives easily the OLS for $\beta$ which gives the vector of the OLS's from each model (30):

$$\hat{\beta}= \left ( \begin{array}{c } \hat{\beta_1} \\ ... ...[1..n]}}^2)\otimes(^tXX)^-=diag(var(\hat{\beta}_i)_{[1..n]})$$ (32)

Note that if one assumes that $var(\epsilon)=\sigma^2I\!d_{nT}$ then:

$$\hat{\sigma}_\epsilon^2=\frac{^t\epsilon\epsilon}{trace(P_{... ...nT-n rank(X)}=\overline{\hat{\sigma}_{{\epsilon_i}_{[1..n]}}^2}$$ (33)

which is the pooled estimate of variances, then:

$$var(\hat{\beta})=\sigma_\epsilon^2I\!d_n\otimes(^tXX)^-$$ (34)

In either cases testing $L\beta=0$, with $L=(1/n,\cdots,1/n)\otimes c$, where $c$ is the contrast used on a single model to assess the activation looked for, with the statistic (19) will give the statistic used in (4), and if $c$ is a matrix ($C$) containing more than one contrasts the $F$ statistic (22) will be used.