Methods

Here, we consider two different FMRI datasets, both of which are simple motor tasks:

• [INDEX] index finger vs. rest tapping task.
• [SEQUENTIAL] sequential finger tapping vs index finger tapping.

Each dataset consists of single sessions for eight different subjects. In both datasets the overall aim is to infer the group means at the top-level.

For each subject, echo planar images (EPI) were acquired using a 3 Tesla system with TR=3 seconds, time to echo (TE) = 30ms, in-plane resolution 4mm and slice thickness 7mm. The first 4 scans were removed and the data was motion corrected using MCFLIRT (16) and high-pass filtered as described in (25).

The overall model for this group experiment consists of two levels. The first-level is a standard FMRI GLM with a design matrix for subject $k$, $X_k$, containing regressors modelling the response to the task within each subject's dataset. The second-level is a GLM which models the group mean of the individual subject's responses to the tasks, via a design matrix $X_g=[1,1,1,1,1,1,1,1]^T$.

To infer on this two-level model we utilise the summary statistic approach we have laid out in this paper. To do this we firstly produce the multivariate non-central t-distribution summary statistics of equation 16 using a first-level analyses of standard generalised least squares (GLS). This GLS analysis was performed using FEAT (FSL). FEAT performs voxel-wise time-series statistical analysis using local autocorrelation estimation to prewhiten the data (25).

To infer the group mean, we now need to infer on the marginal posterior, $p(\beta_g\vert Y)$, using the multivariate non-central t-distribution summary statistics obtained from these first-level analyses (equation 14).

To do this, we use two different approaches. Firstly, the [OLS] approach as described in section 6. Secondly, a hybrid approach which provides a compromise between the fast posterior approximation approach (section 3.5) and the slower but more accurate approach of using Markov Chain Monte Carlo (MCMC) sampling and the fitting of a non-central multivariate t-distribution (BIDET, section 3.7). The [HYBRID] approach is now described in detail. It is this which is implemented as the FLAME (FMRIB's Local Analysis of Mixed Effects) C++ program used for higher-level analyses in FEAT (part of FSL v3.1).