One or two Groups of Subjects

A one-group analysis is a multi-subjects analysis with the obvious restriction that every subject of the random sample studied is a member of this group. Two-group analysis is carried out with a two-sample $t$-test under the same framework:

$$\widehat{b1}_i\sim N(b1_i \, , \, \hat{\sigma}^2_{b1_{\eps... ... \, , \, \hat{\sigma}^2_{b2_{\epsilon_i}}) \quad i=1\cdots n2$$

All the first level analysis has been done for every subject in each group. Note that this looks very similar to the original simple single-subject analysis, but here the sample size of groups may be quite different (a situation to avoid if possible). So one will test $\theta=m_{b1}-m_{b2}=0$ with the statistics:⁷

$$t_o(fixed)=\frac{ \overline{\widehat{b1}} - \overline{\wideha... ..._{b_\epsilon}^2/n_2)}} \approx t_{dist}((n_1+n_2)(T_A+T_B -2))$$ (12)

or

$$t_o(random)=\frac{ \overline{\widehat{b1}} - \overline{\wideh... ...{(b_\eta +b_\epsilon)}^2/n_2)}} \approx t_{dist}((n_1+n_2) -2)$$ (13)

with the same definitions as in section 3 for each group. Note that here no conjunction analysis approach can be made unless the same subjects are in both groups (for example, before and after medical treatment) as pairing of subjects across the groups would be required - in fact this then ends up reverting to a one-group analysis (with a paired $t$-test). The ``variance-ratio method'' can, however, be performed. When $g$ groups are studied one can compare them two by two, then introducing a multiple comparison (not as problematic as the statistical-map one). If equal variances are assumed, a pooled variance of all $g$ samples must be used for either method, instead of just the two considered. To do more advanced comparisons ⁸ one has to return to the GLM or ANOVA to be able to test, for example, if all the groups have the same activation, or if there is a trend in the groups, as one would expect for groups defined by increasing doses of a treatment. These would involve either F statistics and/or using a linear function of the parameters estimated in the model, i.e. contrasts. Remarks: The distributions for the statistics given here are approximations as the variance estimates are given under unequal variances assumption and in that case the Satterthwaite formula (3) should be used for the degrees of freedom (for the fixed effect one has to re-integrate first levels in the formula beforehand, see appendix). To use the degrees of freedom given one has to replace (either in the fixed or random) the variance in each group by their pooled

$$\widehat{\sigma_{pooled}}^2= \frac{\sum_{u=1}^g(n_u-1)\widehat{\sigma_u}^2}{((\sum_u n_u) -g)}$$

Notice that for equal sample sizes ($n_1=n_2$) the pooled estimate of the variance would give the same result for $t_o$as the unequal variance and so the given distributions become exact.