Dear Tom,
> > Given two volumes V_1 and V_2 sampled at discrete points, in both V_1
> > and V_2 homologous locations (A_1 and A_2) show the same mean and the
> > same activation effect, where A_1 and A_2 are the only locations showing
> > an activation effect. Let the voxel intensities in V_1 and V_2 around
> > A_1 and A_2 be different. Now apply a spatial convolution L to both V_1
> > and V_2.
> >
> > Note that the estimated mean intensities in the convolved V_1 and V_2 at
> > voxel A_1 and A_2 will be different, whereas the measured activation
> > effect will not be a function of the intensities around A_1 and A_2.
>
> But the assuption that "A_1 and A_2 are the only locations showing an
> activation effect" seems a bit dodgy... we never have exactly zero
> activation anywhere.
>
> What we get after convolution is a mix of everything from the
> local neighborhood... the estimated baseline at A_1 after convolution is
> a mixture of baseline around A_1... the estimated activation a mix
> activation around A_1.
>
> > This example can be generalized to a range of other spatial
> > configurations. In other words, relating the signal change to the voxel
> > mean intensity can be misleading, if some low-pass filter L has been
> > applied to the image prior to the statistical analysis.
>
> Smoothing the data means you're smoothing both the signal and the
> baseline. Maybe there is a more compelling example where one would
> get misleading answers?
I think Stefan's point is simpler and more fundamental than it may
appear. Any discrepancy between the effects of spatial convolution
(e.g. partial volume effects of voxels in fMRI, PSF in PET or applied
smoothing) on the background signal and the change induced by neuronal
responses will render the ratio between change and background signal
uninterpretable (strictly speaking). This discrepancy is unavoidable
in fMRI because the signal has at least three components (vascular
deoxyhemogloblin, intra-voxel magnetic feild inhomogenieties and
conventional relaxation) which themselves change from voxel to voxel.
This means any smoothing operator will have a different effect on the
signal and the change in signal (because that change comes from only
one signal component). A simple example of this would be a false
reduction in proportional signal change caused by having two much CSF
under the smoothing kernel relative to another equally activated
region. In PET the same happens when the smoothing kernel is large
relative to the activation. A focal activation of 2% in the center of
the head of the caudate will be, apparently, reduced to 1% at the
edge, if one uses a local ratio approach. This problem is eschewed if
one uses a reference that is much greater than the size of the kernel
(e.g. global reference).
Thats my take anyway!
With very best wishes - Karl
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