Ah. Yeah, I have poor intuition about matrix math. Thanks, Steve.
Could you help me understand, in more descriptive terms, what's
happening with the varcopes?
It seems like:
(c1'*pinvX*[pre-whiteningstuff]*pinvX'*c1)
is essentially determining the portion of the residuals that should be
used as the denominator for the statistical images.
Because the residual is a scalar, it's not trying to sample the
residuals that were generated while the regressor/contrast was active
(in other words, it's not using the 4dres), but it's estimating how
much of the *total* modeled activity was accounted for by this
regressor's activity, and using that fraction of the variance in the
sigmasquareds. Is that right? And, if so, why not use the residuals at
each time point? It seems like it's valuable information if one of
your regressors has a terrible fit, and another has a perfect fit.
Also, for my own sake of completeness so that I can reconstruct this
process with my data, is there somewhere I can look for the
pre-whitening values at each voxel?
Finally, others have mentioned that the magnitude of the varcopes is
essentially the square of the magnitude of the copes, but I don't have
an intuition for why that is. Is there a short version, or could you
point me to a source that would explain it?
Thanks so much,
Todd
On Wed, Feb 24, 2010 at 1:33 PM, Stephen Smith <[log in to unmask]> wrote:
> Hi - no, that is changing because the contrasts (c1, c2) themselves interact
> with this calculation - this is matrix maths not scalar.
> Cheers.
>
> On 24 Feb 2010, at 18:19, Todd Thompson wrote:
>
> Ok, that makes sense. I'd thought it might be something like that.
> So, the other thing I noticed was that the *ratio* of my varcopes
> wasn't constant, either.
>
> In other words, if, at any given voxel,
> vc1 = (c1'*pinvX*[pre-whiteningstuff]*pinvX'*c1) * sigsq
> and
> vc2 = (c2'*pinvX*[pre-whiteningstuff]*pinvX'*c2) * sigsq
>
> then, if the design matrix, pre-whitening, and sigmasquareds are
> constant for each voxel, I expect
> vc2/vc1 to have the same value at every voxel in the brain. But that's
> not what I see...
>
> (The contrasts I tried this with were [1 0] and [0 1], in case that
> matters.)
>
> Any idea what's going on? And is there some place to see the values of
> the pre-whitening kernel?
>
> Thanks,
> Todd
>
>
>
> On Wed, Feb 24, 2010 at 1:03 PM, Stephen Smith <[log in to unmask]> wrote:
>
> Hi - in the case of FILM the thing you're missing is the pre-whitening
>
> kernel, which appears in the middle of this varcope expression, and is
>
> estimated separately for each voxel.
>
> Cheers, Steve.
>
> On 23 Feb 2010, at 23:56, Todd Thompson wrote:
>
> Hi, all. Thanks for the previous help with varcope explanations.
>
> I'm a little stumped on something related, and hoping you can help
>
> point out where I've gone astray.
>
> As I understand it, the varcope for a contrast is calculated by:
>
> varcope(i)=c'*pinvX*pinvX'*c*sigsq
>
> where c is the contrast vector, pinvX is the pseudoinverse of the
>
> design matrix, and sigsq is the variance of the data.
>
> So, within a varcope, every voxel of the brain should have the same
>
> value for (c'*pinvX*pinvX'*c), presumably, since neither the contrast
>
> nor the model know anything about the data.
>
> Here's my confusion:
>
> If I take the varcope and divide it by the sigmasquareds, why don't I
>
> get a uniform volume back out representing that (c'...c) value?
>
> Instead, for a sample varcope I try this on, I get a range of values
>
> (from 0.8 to 3.5, in this case).
>
> (The command I ran was fslmaths varcope1.nii.gz -div
>
> sigmasquareds.nii.gz output.nii.gz).
>
> Thanks!
>
> Todd
>
> (previous varcope information taken from:
>
> https://www.jiscmail.ac.uk/cgi-bin/webadmin?A3=ind0907&L=FSL&E=7bit&P=1423692&B=--Apple-Mail-14-987887678&T=text%2Fplain;%20charset=US-ASCII
>
> )
>
>
>
> ---------------------------------------------------------------------------
>
> Stephen M. Smith, Professor of Biomedical Engineering
>
> Associate Director, Oxford University FMRIB Centre
>
> FMRIB, JR Hospital, Headington, Oxford OX3 9DU, UK
>
> +44 (0) 1865 222726 (fax 222717)
>
> [log in to unmask] http://www.fmrib.ox.ac.uk/~steve
>
> ---------------------------------------------------------------------------
>
>
>
>
>
>
>
> ---------------------------------------------------------------------------
> Stephen M. Smith, Professor of Biomedical Engineering
> Associate Director, Oxford University FMRIB Centre
>
> FMRIB, JR 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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