Hi Mark,
Thanks for you're detailed explanation. I'll need to sit with my GLM and
matrix algebra books for a while to absorb it fully, but I've understood
most of it. In the meantime, I wanted to move things down to a practical
level for a bit.
You said that the varcopes might be the same for the two groups if the
number of subjects was similar in the two groups. We have the same number of
Ss in each group, so similar varcopes might be expected. However, our
varcopes for the two groups are identical. I don't think that is expected
unless the groups are precisely identical, which they aren't.
Also, you stated that the varcopes depend on whether we modelled variances
for each group separately. It wasn't clear from your original reply, but I
assume that if we modelled each group variance separately, that the varcopes
should differ for each group. The website states that modelling separate
variances for each group "is simply a case of specifying in the GUI what
group each subject belongs to." We've done that, but I still don't see
differences in the varcopes for the two groups, and the varcopes for the
individual group contrasts in the unpaired t-test model are different from
the varcopes in a single-group average (one-sample t-test) model, even
though the COPEs are identical for those two models. How do I know whether
the variances were modelled for each group separately other than specifying
different group membership in the GUI?
Thanks in advance,
Greg
On Sat, 28 Apr 2007 00:02:36 +0100, Mark Jenkinson <[log in to unmask]> wrote:
>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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