Hi.
On Mon, 4 Feb 2002, John Herrington wrote:
> I have a question about orthogonality in the design matrix used to compute
> statistical contrasts between explanatory variables, in FEAT. I just
> wanted to confirm something that is implied, but as far as I could tell
> not directly addressed, in the FSL website documentation. FEAT runs
> statistical analyses on the entire design matrix of contrasts
> simultaneously, as opposed to running each contrast individually, correct?
Yes - but I should start by clearing up terminology: I think you are
asking about different columns in the design matrix (also known as
different "covariates" or different "effects of interest" or different
"regressors" or different "explanatory variables" (EVs) ) and not
contrasts; contrasts are calculated AFTER the model fitting is completed
and are linear combinations of the parameter estimates (each EV results
in its own parameter estimate (PE)) - in the general linear model
data = EV1*PE1 + EV2*PE2 .... + error
a contrast is a linear combination of PE's - eg PE2-PE1 or just PE1
and - yes, this is multiple regression, ie all columns in the design
matrix are fit together.
> If this is the case, a lack of orthogonality among the contrasts *can*
> lead to interpretive problems in the analysis, as is typically the case in
> this sort of regression, correct? I'm assuming both of the above are
> true, based on the fact that FEAT has an option to "orthogonalize" the
> contrasts, but I wanted a confirmation of this from the source...
As long as the different EVs are not very close to co-linear then the
model fit will still be "unique" and there will not be an interpreation
problem. They do not have to be totally orthogonal. Two EVs can be
somewhat similar, and the model fit still be well-conditioned.
The option to orthogonalise is in case you are wanting to force this to be
the case; but this needs to make sense in terms of what you are trying to
fit to the data - it's not _necessarily_ what you want to do.
Hope this all makes sense!
Stephen M. Smith
Head of Image Analysis, FMRIB
Oxford University Centre for Functional MRI of the Brain
John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK
+44 (0) 1865 222726 (fax 222717)
[log in to unmask] http://www.fmrib.ox.ac.uk/~steve
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