Multi-subject analysis

Because of the separability aspect of the analysis, one easy implementation is to perform a two-stage procedure. At the first stage the analysis above is performed for each of the $n$ subjects: following the previous section (or, for the more general case, see the GLM section), at each voxel an estimation of activation parameter can be made: $\hat{b}_i=\bar{y}_{iB}-\bar{y}_{iA}$ with standard error $\hat{\sigma}_{b_{\epsilon_i}}=\sqrt{\hat{\sigma}_{\epsilon_i}^2(\frac{1}{T_B} + \frac{1}{T_A})} \quad i=1\cdots n$.
So for each of the $n$ subjects can be built the statistic map

$$t_o(i)=\frac{\hat{b_i}}{ \hat{\sigma}_{b_{\epsilon_i}}}$$

to test $b_i=0$. How do we combine these maps to conclude that there is activation in the population from which these subjects come? This is the purpose of multi-subject analysis, i.e. the second stage. From the experiment a different conclusion is drawn for the population according to the method chosen: fixed subject effect analysis allows a conclusion limited to the sample studied; a random subject effect allows a population conclusion. Because the latter can be ``too conservative'' due to large estimated subject-subject variability (partially due to small sample sizes generally studied ), some alternative approaches have also been developed: ``conjunction analysis'', and ``variance ratio smoothing''.