Hi Greg,
The varcope is the expected variance of the estimated contrast of
parameter
estimates (cope). This is not the same as the residual variance
which is
the variance of the data after having all the regressors removed.
In maths this can be derived from the GLM starting with the basic
equation:
Y = X*beta + e
where Y is the voxel timeseries data, X is the design matrix, beta is
the
true parameter vector and e is the residual noise. Thus the expected
value of the data is: E(Y) = X*beta and the variance of the residual,
Var(e) = sigma^2, which is the same as the variance of the data, Y,
about
its expected value E(Y). So, by getting the best unbiased estimate for
beta as: betahat = (X'*X)^{-1} * X'*Y
then X*betahat can be removed from the data, leaving the residuals, from
which sigma is estimated.
Then, if a contrast vector, C, is specified, the cope is given by:
cope = C * (X'*X)^{-1} * (X'*Y)
and the expected variance of the cope is:
Var(cope) = Var(C*(X'*X)^{-1}*(X'*Y))
= C*(X'*X)^{-1}*(X'*Var(Y)*X)*(X'*X)^{-1}*C'
= C*(X'*X)^{-1}*(sigma^2)*(X'*I*X)*(X'*X)^{-1}*C'
= (sigma^2)*C*(X'*X)^{-1}*C'
In words, the residual variance tells you about the variance of e, while
the varcope tells you about the variance of the cope, which is
related to
Y (and hence e) by the matrix expressions for the estimate of the cope.
Typically, simple means of parameters are more robust to noise (and have
lower varcopes) than differences between parameters. You can use the
estimability feature of FEAT to get a feeling for this.
As for your unpaired t-test, I think it is reasonable for the
varcopes to
be the same if the number of individuals in each group is similar,
although
it would depend on whether you've modelled a different variance for each
group or not. There should be no problems with running any of the
sort of
tests you are talking about. All the relevant corrections are taken
care of
within the code, and you can ask any sort of contrast you like -
especially
single group means, as that is quite standard.
Hope this helps.
All the best,
Mark
On 27 Apr 2007, at 23:35, Greg Burgess wrote:
> Hi FSL list,
>
> I'm a bit confused about the difference between the sigmasquareds and
> varcope maps. Specifically, why is it necessary to have a different
> error
> variance (varcope) for each contrast, as opposed to using the overall
> residual (sigmasquareds)? How are the individual varcopes estimated
> (i.e.,
> what makes error variance specific to one contrast and not another)?
>
> Lastly, I'm comparing two groups in an unpaired t-test
> (http://www.fmrib.ox.ac.uk/fsl/feat5/
> detail.html#UnpairedTwoGroupDifference)
> with different values for the group membership variable, and two
> additional
> contrasts to test each individual group mean (C3: 1 0 ; C4: 0 1).
> Shouldn't
> the varcope for the contrast of each group mean (i.e., C3 and C4) be
> different (they're not)? What is the best way to determine whether
> the error
> variance does indeed differ for the two groups? Are these tests of the
> individual group means valid when they're conducted within the
> context of
> the unpaired t-test model?
>
> Thanks,
> Greg
>
> ______________________________________________________________________
> _____
> Greg Burgess, Ph.D.
> Research Associate, Institute of Cognitive Science
> University of Colorado - Boulder
> Phone: 303-735-5802
> Email: [log in to unmask]
>
> Department of Psychology
> Muenzinger Hall
> UCB 345
> Boulder, CO 80309
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