BrainVoyager version: 22.4
Datasets used: ses-01_task-Localizer_run-01_bold and ses-01_task-Localizer_run-02_bold of sub-01 and sub-02 of the newbi4fmri datasets (Jody Culham and Kevin Stubbs and Ethan Jackson and Rebekka Lagace Cusiac (2020). newbi4fmri2020 Localizer. OpenNeuro. [Dataset] doi: 10.18112/openneuro.ds003433.v1.0.1, accessed 31 January 2023)

If your research aim is to draw conclusions about the broader population beyond the participants in your study, then a random effects analysis is the most suitable approach. In this method, the individuals involved are considered as random samples from the larger pool of potential participants, which differs from the fixed effects analysis where the subject factor remains constant. Consequently, the design matrix is different.

It is important to note that including an adequate number of subjects is crucial for generalizing the findings to the population level. For instance, in non-fMRI studies, it is typically recommended to have at least 50 subjects per experimental group in order to estimate population effects reliably. With a small number of subjects, it becomes exceedingly challenging to determine effects that hold true for the broader population. The minimum sample size for a random effects analysis in fMRI data may vary depending on factors such as effect size, desired statistical power, and analysis complexity. However, a commonly suggested guideline is to have group sizes of around 20 participants.

In BrainVoyager there are two possibilities to run a random effects analysis:

RFX approach

This analysis immediately calculates beta weights per subject and predictor. It allows the calculation of very large studies in a reasonable time (which might present a problem when following the second approach described below).

In order to successfully separate the subject predictors, BrainVoyager has to know which VTC files belong to the same participant. To ensure this, the names of the included VTC files must follow a simple rule:

All VTC file names belonging to the same participants have to begin with the same text string to code the subject ID, e.g. sub-01.

This subject coding section of a VTC file name is defined as all characters from the beginning of the file name up to the first encountered underscore ("_"). An example of a correct naming scheme is shown here:

sub-01_ses-01_task-Localizer_run-01.vtc
sub-01_ses-01_task-Localizer_run-02.vtc
sub-02_ses-01_task-Localizer_run-01.vtc
sub-02_ses-01_task-Localizer_run-02.vtc
...
In this case, the first two file names would be assigned to subject "sub-01" and the third and fourth file name would be assigned to subject "sub-02". The file name scheme used in the example above is

"<SUBJECT-ID >_<SESSION-NUMBER>_<TASK-NAME>_<RUN-Number>.vtc

Only the first section up to the underscore is crucial for the subject sorting.

To compute the RFX-analysis you have to use the RFX GLM checkbox in the Multi Study GLM dialog. This automatically checks the %-transform (timecourse normalization) and the Separate subjects predictors checkmarks:

We can now inspect the multi study design matrix reflecting the separate subject predictor settings. Click the Design Matrix button and in the appearing dialog the Design Matrix Plot button to see the entire design matrix of the specified model:

As you can see, there are now two sets of the four main predictors. Each predictor set defines a time course (non-zero values) for two studies each belonging to one subject but contains zero values (dark grey color) for all other studies. Each subject, thus, has its own set of separated predictors. Study 1 and study 2 belong to subject sub-01, the latter two studies belong to subject sub-02.

The separation of predictors for each subject means that the signal changes at a voxel time course are estimated by 8 (2*4) values (beta weights of the main predictors) across the concatenated data points plus the respective constant terms. Since the predictors for a particular study contain, however, only zero values for the other studies, only five values (beta weights of the four main predictors plus constant term) actually estimate the time course of the studies belonging to a single subject. Although the runs belonging to the same subject are explained by one set of predictors, the signal level confounds are still defined separately for each study (see design matrix).

To run the RFX-GLM click the GO button. In the example below the same RFX analysis was computed for 17 participants. In this case the Overlay General Linear Model dialog will look as follows:

The signal level confound are represented in yellow. The 68 rows represent the four main predictors for each subject (N=17) of the multi study design matrix. The four main predictors are appropriately labeled to reflect the subject to which they belong. If you now specify contrasts, the same predictors will automatically be included for all subjects. The resulting contrast map now shows a comparison of the individual betas of all subjects. This way, the variability between the different subjects can be calculated and thus conclusions can be drawn for the general population.  You will notice the degrees of freedom in the right lower corner to be number of subjects-1, in our case 16:

Thus one can easily understand, why an RFX-analysis with less than 10 subjects does not make sense for a real study.

Separate Subject Predictors approach

This way is very instructive, so we explain it here, though for most applications we recommend the RFX approach. The Separate Subject Predictors approach is very similar to the RFX approach, but it uses a two step procedure. It pools the predictors of all studies (runs) belonging to the same subject like before. Check the Separate subject predictors option in the General Linear Model: Multi Study, Multi Subject dialog, then calculate the GLM.

In the second step of this approach (the so called summary statistic), a contrast is specified for all participants, using CTRL + left click, and the Random Effects Analysis is checked in the Options of the Overlay GLM approach.  Below you will see a Separate Subjects GLM that has been computed for the same 17 participants as above:

This second step defines a t-test for a specific contrast only, so no random effects GLM is computed that includes all predictors, like in the RFX-analysis. For this it uses the beta weights that have been calculated in the first step of the separate subjects GLM .