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Dear Ged,

I am trying to think of a Fundamental Topologist's reply :

Perhaps the simplest is that if you want to model an image as a bag 
of correlated
voxels, then you should report and discuss every voxel in every 
cluster. Unless you
do this, you are a closet topologist.

I hope this helps :)

Karl

PS To assign a p-value to a voxel is a category error. The p-value is 
an attribute of
the Euler characteristic, which is an attribute of connected voxels 
(not a single voxel).






At 18:08 06/05/2010, Justin Chumbley wrote:
>Hi Ged,
>
>On 6 May 2010 14:21, DRC SPM 
><<mailto:[log in to unmask]>[log in to unmask]> wrote:
>Hi Guillaume,
>
>This is an interesting philosophical point... Personally, I am
>slightly more "Voxelist" than I think Justin, Karl, and perhaps you
>are... You might be surprised to hear that Keith was also Voxelist in
>one pratical regard too, as SurfStat can indeed produce maps of RFT
>corrected p-values for "peaks" and clusters, where these maps are
>defined over all vertices or voxels. In the cluster case,
>vertices/voxels have the uniform p-value of the cluster they are
>contained in (or p=1, if they are outside a significant cluster), but
>in the peak case, despite the arguments from the "Topologist" school,
>Keith did assign FWE-corrected p-values to every vertex/voxel, and not
>just the local maxima.
>
>In fact, based on the lack of information within the clusters, Keith
>came up with a nice visualisation which combines cluster and
>vertex-wise significance, see e.g.
> 
><http://www.stat.uchicago.edu/%7Eworsley/surfstat/figs/Pm-f.jpg>http://www.stat.uchicago.edu/~worsley/surfstat/figs/Pm-f.jpg
>I don't think he got around to implementing a similar visualisation
>for voxel-wise data (SurfStatP returns the peak and cluster results
>necessary, but I think you are on your own as to how to visualise
>these), but I've seen no evidence that he had a philosophical
>objection to this (especially not one that was somehow specific to the
>voxel-wise but not vertex-wise case).
>
>Similarly, in permutation testing, comparison to the null distribution
>of the maximum over the image yields FWE-corrected p-values for every
>voxel; you can choose to look at these only at local maxima voxels if
>you wish, but no topological assumptions are required to control FWE.
>In fact, being able to interpret individual voxels as significant is a
>key distinction between weak and strong control of FWE made by e.g.
>Nichols and Hayasaka (2003), p.422
> 
><http://dx.doi.org/10.1191/0962280203sm341ra>http://dx.doi.org/10.1191/0962280203sm341ra
>Of course, this is all assuming that you can declare voxels as true or
>false positives, which Justin and Karl have argued against... However,
>I don't think their arguments have entirely convinced me that you can
>declare local maxima or clusters as true or false either, if you can't
>do so for voxels, since the same arguments about continuous and
>infinitely extended signal would seem to screw up *all* notions of
>type I and type II error, not just the voxel-wise ones.
>
>
>
>Smoothing images with broad support (e.g. Gaussian) Kernels rides 
>roughshod over the aspiration for strong control. Beyond this 
>nit-picking, some of this depends on definition. Traditional RFT 
>defines and controls false-positives under the null SPM. Under the 
>null SPM all positives - no matter where they occur spatially - are 
>false-positives. We considered a definition of false-positives that 
>is more general, applying also under the alternative SPM (i.e. in 
>the presence of experimentally-induced activations, even when these 
>extend across the whole image). We followed the intuition that a 
>false positive must generally be spatially removed from any 
>underlying activation. To formalise this, take the example of peak 
>inference. First interpret significant SPM peaks as indicating the 
>existence of true signal peaks.  Let x indicate the distance between 
>a discovered peak and the nearest true peak. Then any discovered 
>peak beyond (predefined) distance x>c from a true peak is defined as 
>false-positive (otherwise it is true-positive). Under the null SPM, 
>we define x=inf for all discovered peaks (there are no underlying 
>peaks). All discovered peaks are therefore spatial false-positives, 
>in accordance with non-spatial definition of error. Importantly, 
>false-positives are now also defined under the alternative 
>SPM:  i.e. observed peaks farther than c from a true peak. 
>Familywise false-positive error-rate and false-discovery rates can 
>now be defined under the alternative SPM.
>
>Note that this definition of a procedure's spatial error-rate is 
>derived from true/false classification of peaks. This classification 
>is based on the spatial accuracy with which the procedure identifies 
>target peaks in the underlying signal. Spatial accuracy can be 
>therefore be examined and discussed per se. We took this perspective 
>in the work Guillaume cited.
>
>Finally, while the analysis of spatial error (or accuracy), applies 
>most naturally to peak-level inferences using RFT, one may appraise 
>the spatial accuracy of other procedures. Keith seemed happy with 
>this (he was a co-author!), but I think it is rather questionable to 
>preempt how he would have contributed to this debate now....
>
>
>With my very best wishes
>
>JC
>
>Perhaps a fundamentalist Topologist will reply to put me in my place?! ;-)
>
>Best wishes,
>Ged