(Revised version; one sentence was garbled & I forgot three bits)
Steve,
> Can someone give me definitions for the corrected and uncorrected p-values
> at the cluster level, appearing in spm99 output? I read the Friston et al
> 1994 HBM paper and looked at spm_list.m and spm_P.m, but it's still not
> clear to me what these 2 p values constitute.
I'll take a stab at this, but I wonder what others think.
Setup
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V Number of voxels
W Smoothness (W = [w1 w2 w3], FWHM of spatial ACF)
S Search volume (a list of x,y,z points, or a mask image)
u Threshold
k cluster size
p_uc uncorrected p-value
p_c corrected p-value
Take I. Statistically Anal Description
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Let's say I randomly generate statistic images, over and over again;
each image has V voxels and smoothness W, and has same collection of
brain voxels S. They are "null" images, in that they have no
signal--the null hypothesis is true everywhere. I threshold each at u.
In the long-run, on average, the proportion of random images that have
a cluster of size k *or* *larger* is p_c. The clusters under
consideration are anywhere in the brain; this search over the whole
brain is what makes these p-values corrected p-values.
The uncorrected p-values remove the search over the whole brain. I
assume it goes something like this: For each random image, uniquely
identify a single cluster (e.g. the cluster nearest to point (x,y,z));
if no cluster matches the criterion *throw* *out* that random image
(e.g. if there are no clusters at all).
Then, in the long-run, on average, the proportion of random images
whose uniquely identified cluster is of size k or larger is p_uc.
There is no search over the brain, and hence these are not corrected
p-values.
Note that *nowhere* is magnitude mentioned (except in defining the
cluster above threshold u). These p-values only assess the size of a
cluster, not the magnitude of statistic image in that cluster.
Also note that the unique identification of a cluster must be made a
priori, before you see the data. The cluster size itself cannot be
part of a criterion. For example, a criterion that selects the
largest cluster in a given region won't be appropriate, since the
p-value doesn't account for that selection-by-size.
Finally, remember that these results are only valid for stationary
random fields (the smoothness is W everywhere in the brain). This
makes the cluster size p-values invalid for VBM.
Take II. Informal Description
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The corrected p-value p_c measures how likely a cluster of size k is
(under the null hypothesis of no signal), accounting for a search
over the whole brain.
The uncorrected p-value p_uc does the same but does not account for the
search over the brain; it assumes that you have uniquely identified
this cluster somehow.
What do people think?
-Tom
-- Thomas Nichols -------------------- Department of Biostatistics
http://www.sph.umich.edu/~nichols University of Michigan
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-------------------------------------- Ann Arbor, MI 48109-2029
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