Hi Arkan,

Some thoughts on your questions below
Best
Alfonso
Originally posted by Arkan Al-Zubaidi:
If you were to use a single regressor then only the average BOLD signal effect across all of the outlier scans would be modeled (one effect). By entering one regressor per outlier scan then the BOLD signal effect of each outlier scan will be modeled (one effect per outlier). We want to model the effect of each outlier scan separately because it is not reasonable to assume that different outliers will produce the same effect on the BOLD signal.

Typically the ART files will be entered as a first-level covariate (e.g. 'scrubbing') and that covariate will be added to the 'confounds' list in the Denoising step. That will estimate and remove the modeled effects from the BOLD signal timeseries (for every voxel and for every ROI). More precisely, a linear model of the form

 Y = X * b

will be estimated, where Y is a BOLD signal timeseries (e.g. from a given voxel), X are the effects entered in the confounds list (e.g. ARTcovariates), and b is estimated using linear regression. Then the 'corrected' BOLD timeseries will be:

  Y(corrected) = Y-X*b;

If you are using SPM, for example, to analyze task-related effects, then SPM is creating a design matix characterizing your task-effects (e.g. hrf-convolved blocks, in a block design). Then the additional covariate regressors (e.g. ART regressors) are added as additional columns to this design matrix. SPM then estimates jointly the beta values of this model (one beta value for each regressor, i.e. task effects as well as covariates of no interest effects) using linear regression. The beta volumes associated with your covariates of no interest (e.g. ART regressors) you can typically disregard, but the important thing is that now the beta volumes associated with your task-related effects (effects of interest) will be unaffected by the outlier scans BOLD signal values (since those are being modeled by the 'covariates of no interest' columns of your extended design matrix). You can then typically enter those beta values associated with the task- effects into the standard second-level / between-subjects model without having to worry about the effect of outlier scans. 

In more precise terms, if you have a design matrix A containing your task-related effects (e.g. hrf-convolved timeseries), and a B matrix containing your ART parameters (one regressor per outlier scan), then SPM will estimate the following GLM model:

 Y = [A B] * [betaA; betaB]

where Y is again the BOLD signal within a given voxel, betaA are your estimated task-effects of interest (the ones you want typically to pass to your second-level analyses), and betaB are the (typically disregarded) estimated outlier-scan efects. The nice thing about this model is that changing the value of Y associated with a modeled outlier-scan is not going to change the estimated value of betaA, so betaA becomes invariant to the BOLD signal values within the outlier scans (the desired behavior to remove the influence of these outlier scans on your analyses)

Hope this helps
Alfonso