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Multi-Level Linear Modelling for FMRI Group Analysis Using Bayesian Inference

FMRIB Technical Report TR03MW1
(A related paper has been accepted for publication in Neuroimage)

Mark W. Woolrich *, Timothy E.J. Behrens *, Christian F. Beckmann, Mark Jenkinson and Stephen M. Smith

Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB),
Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital,
Headley Way, Headington, Oxford, UK
* First two authors contributed equally to this work
Corresponding author is Mark Woolrich:


Functional magnetic resonance imaging studies often involve the acquisition of data from multiple sessions and/or multiple subjects. A hierarchical approach can be taken to modelling such data with a General Linear Model at each level of the hierarchy introducing different random effects variance components. Inferring on these models is non-trivial with frequentist solutions being unavailable. A solution is to use a Bayesian framework. One important ingredient in this is the choice of prior on the variance components and top-level regression parameters. Due to the typically small numbers of sessions or subjects in neuro-imaging the choice of prior is critical. To alleviate this problem we introduce to neuro-image modelling the approach of reference priors, which drives the choice of prior such that it is non-informative in an information-theoretic sense. We propose two inference techniques at the top-level for multi-level hierarchies (a fast approach and a slower more accurate approach). We also demonstrate that we can infer on the top-level of multi-level hierarchies by inferring on the levels of the hierarchy separately and passing summary statistics of a non-central multivariate t-distribution between them.

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Next: Introduction