
In this section we illustrate the potential benefit of a heteroscedastic variance model (i.e. by allowing for separate firstlevel variances) compared to the homoscedastic variance model (otherwise known as OLS for randomeffects analysis of FMRI data [9]).
Both the heteroscedastic model (equation 16) and the homoscedastic model (equation 17) provide an unbiased estimate of the group level parameter of interest . They differ, however, in their associated variance, , and therefore will give different or statistic. The associated variance for the heteroscedastic model is always less than or equal to that of the homoscedastic model, as can be shown using Jensen's inequality. This will result in an increase in the expected statistics values for the model in equation 16:
As a quantitative example of this increase we have generated a simulated group FMRI study, with a known set of firstlevel variances , together with a secondlevel variance . From these known values we calculate the expected percentage increase in  statistic (see above) and show this versus changes in the ratio of the secondlevel variance to the mean firstlevel variance (the assumed variance in the homo scedastic model). Firstlevel variances were taken from a real FMRI group study (simple motor paradigm) estimated using the GLS prewhitening approach as implemented in FILM (part of FSL [7,17]). These estimates of firstlevel variances are realistic representatives of typical firstlevel variance structures and are defined to be groundtruth in the following simulation.
Figure 1 shows the expected percentage increase in scores for three different sets of firstlevel variances, each of them containing one or more 'outliers' (see boxplots). As can be seen from the first three images, the expected percentage increase, for a given firstlevel variance structure, only depends on the ratio of secondlevel variance to mean firstlevel variance ( ) and is independent of the grouplevel effect size . As the second level variance becomes considerably larger than the mean first level variance, the expected increase in score tends to zero. When these variances are approximately equal, the heteroscedastic model allows for a increase in score, increasing to much larger values as the secondlevel variance decreases relative to the mean firstlevel variance. The grouplevel effect size determines the actual level, as shown by the overlaid contour plots (solid lines show  score levels for the heteroscedastic model while dashdotted lines show the score levels for the homoscedastic model). Over a large range of variance ratios the improvements gained by the heteroscedastic model have substantial implications on post thresholded results, e.g. typical subthreshold values of increase to superthreshold values of .
Figure 2 shows similar plots for cases where the firstlevel variances do not contain large outliers, that is, they are close to the homoscedastic model. In this case, the expected percentage score increase is smaller, but still potentially important.
It is clear from these figures, that the exact configuration of firstlevel variances has a major impact on the improvements gained by using the heteroscedastic model. This dependency is further investigated in figure 3, where we show the expected (log) increase in score for a set of variance configurations estimated from voxels in a real group study, assuming equal secondlevel and mean firstlevel variances. This histogram shows that approximately voxels have a increase in expected score.

