Appendix C: Group statistics - background and theory

For higher-level analysis (e.g. analysis across sessions or across subjects) FEAT uses FLAME (FMRIB's Local Analysis of Mixed Effects). FLAME uses sophisticated methods for modelling (see related technical report TR01CB1) and estimating the inter-session or inter-subject random-effects component of the mixed-effects variance, by using MCMC to get an accurate estimation of the true random-effects variance and degrees of freedom at each voxel (see related technical report TR03MW1).

FEAT offers both fixed effects (FE) and mixed effects (ME) higher-level modelling. FE modelling is more "sensitive" to activation than ME, but is restricted in the inferences that can be made from its results; because FE ignores cross-session/subject variance, reported activation is with respect to the group of sessions or subjects present, and not representative of the wider population. ME does model the session/subject variability, and it therefore allows inference to be made about the wider population from which the sessions/subjects were drawn.

The remainder of this section discusses FLAME's mixed effects modelling.

"Mixed-effects" (ME) variance is the sum of "fixed-effects" (FE) variance (the within-session across-time variances estimated in the first-level analyses) and "random-effects" (RE) variance (the "true" cross-session variances of first-level parameter estimates). Note that the labels "mixed effects" and "random effects" are often (incorrectly) used interchangeably, partly because they are in practice often (but, importantly, not always) quite similar.

One factor that makes FEAT's approach to higher-level modelling particularly powerful is that it is easy to model and estimate different variances for different groups in the model. For example, an unpaired two-group comparison (e.g. between controls and patients) can be analysed with separate estimates of variance for each group. It is simply a case of specifying in the GUI what group each subject belongs to. (Note - FLAME does not model different group variances differently in the case of higher-level F-tests, due to the complexity of the resulting distributions; this may be addressed in the future.)

A second sophistication not normally available in multi-level analyses is the carrying-up of the first-level (FE) variances to the higher-level analyses. This means that the FE component of the higher-level ME variance can be taken into account when attempting to estimate the ME variance. One reason why it is suboptimal to simply use the directly-estimated ME variance is that this is often in practice lower than the estimated FE variance, a logical impossibility which implies negative RE variance. FEAT forces the RE variance in effect to be non-negative, giving a better estimate of ME variance.

Another reason for wanting to carry up first-level variances to the higher-level analyses is that it is not then necessary for first-level design matrices to be identical (ie "balanced designs" - for example having the same number of time points or event timings). (Note though: the "height" of design matrix waveforms at first-level must still be compatible across analyses.)

A third advantage in higher-level analysis with FEAT is that it is not necessary for different groups to have the same number of subjects (another aspect to design balancing) for the statistics to be valid, because of the ability to model different variances in different groups.

The higher-level estimation method in FEAT (FLAME) uses the above modelling theory and estimates the higher-level parameter estimates and ME variance using sophisticated estimation techniques. First, the higher-level model is fit using a fast approximation to the final estimation ("FLAME stage 1"). Then, all voxels which are close to threshold (according to the selected contrasts and thresholds) are processed with a much more sophisticated estimation process involving implicit estimation of the ME variance, using MH MCMC (Metropolis-Hastings Markov Chain Monte Carlo sampling) to give the distribution for higher-level contrasts of parameter estimates, to which a general t-distribution is then fit. Hypothesis testing can then be carried out on the fitted t-distribution to give inference based upon the best implicit estimation of the ME variance.