2 and g groups multi-subject analysis

For 2 groups or $g$ groups this framework is used and the models are the same. As we have already seen, to compare two groups, a two-sample $t$-test is used, and that will be similar here, as the use of a contrast to compare two groups will form the difference of the means (of the ``activation contrasts'' applied to each subject). If one has $g$ groups, multiple comparisons of pairs of groups might be used (with some multiple comparisons corrections if needed) but an $F$-test might be used for an overall (or more than two) group effect. With $g$ groups with the same number of subjects $n$ in each group, the model will have the form:

$$y=(I\!d_g \otimes I\!d_n\otimes X)\beta+\epsilon$$ (41)

with ${}^ty=({}^ty_1,{}^ty_2, \cdots, {}^ty_g)$ and similar for $\beta$ and $\epsilon$. The general form of the ``design matrix'' would be a diagonal matrix of $g$ blocks of the form $I\!d_{n_k}\otimes X$ where $n_k$ is the sample size of the $k$th group (notice that for the random model when estimating the variance components the blocks are $1_{n_k}\otimes X$). As the groups are independent $var(\epsilon)=I\!d_g \otimes V$ where $V$ will have the form of either the fixed or random approach in a 1 group analysis, or more generally $var(\epsilon)=diag(V_{k_{[k=1\cdots g]}})$ if the $V_k$ are not assumed equal or the groups are unbalanced. For a two-group comparison in a $g$-group analysis, the applied hypothesis is:

$$L_{12}\beta=(0 \cdots 0, L_1,0 \cdots 0,- L_2,0 \cdots 0)\beta=0$$ (42)

where $L_k$ is a $1 \times n_kT$ contrast, providing the mean for this group of the activation contrast used in a single-subject model as for a 1-group analysis. Then the $t$-map is derived using the same framework as in the 1-group analysis, using the estimates for $\hat{V}$ according to the approach and the model chosen. If one wants to have a global assessment of group differences (among more than 2 groups) the $F$ statistic given in (22) has to be used with a contrast matrix, $e.g.$ to compare all the groups $2\cdots g$ with a control group, $c=1$, the matrix L of $(g-1)$ contrasts ${\;}^tL=({\;}^tL_{2c},{\;}^tL_{3c},\cdots,{\;}^tL_{gc})$ is involved. Remarks: The degrees of freedom (GLS approach) to apply for $t$-test (2 groups comparison), for the fixed approach will be $df(fixed)=\sum_{k=1}^g n_k(T-rank(X))$, will be $T\sum_{k=1}^g n_k-rank(X)$ for the pooled Random and fixed model, and for the random model with the compound symmetry covariance structure $df(random)=\sum_{k=1}^g n_k - g$.