Hi Gina, Satoru, everyone,
[Gina:]
>> I would like to correlate two activation maps to test the hypothesis
>> that the spatial distribution of activation over a predefined region-
>> of-interest is very similar under two conditions.
I'm also interested in this, for the purposes of comparing different
analysis methods. A paper related to this issue
http://dx.doi.org/10.1006/nimg.1999.0472
reports a "concordance correlation coefficient" (which is closely
related to the standard Pearson correlation) between each pair of
images, but avoids attempting to derive significance values for them.
Since the paper is pretty highly cited and includes some notable
statisticians among the authors, it's tempting to think that this is:
a) fair enough
b) as far as you can go (e.g. perhaps valid significance values are
very difficult to derive).
>> Now, the problem is that using N-2 (where N is the
>> number of voxels) as degrees of freedom in testing the significance
>> of the correlation coefficient is clearly uncorrect, because nearby
>> voxels are not independent (spatial autocorrelation). My intuition is
>> that degrees of freedom should be adjusted, e.g. using the gaussian
>> field theory... but how?
I don't think GRF would be immediately applicable, since the statistic
you are interested in (correlation between two maps) is not itself a
field.
There are other alternatives to the rho -> t (-> p) transformation
though, which wouldn't involve the concept of degrees of freedom.
Permutation/randomisation based testing of the correlation might be
one option, however, I'm not confident that exchangeability of the
voxels could reasonably be assumed -- Satoru, I'd be interested in
your thoughts on this, or anyone else's.
I wonder if a non-parametric correlation such as Spearman's R or
Kendall's Tau would be appropriate? (p-values can be derived for
these; exactly for small samples.) The only assumption that seems to
be mentioned below is that the data can be ranked; nothing is said
about homogeneity of variance, autocorrelation or anything like that:
http://www.statsoft.com/textbook/stnonpar.html
But I'm not completely confident that these methods would be okay.
Anyone care to comment?
>> Note that a related problem has been posted some times ago, but in
>> that case the aim was to correlate values derived from two contrasts
>> over a set of subjects/scans, voxel by voxel, ending up with a
>> correlation map. The question was: is the activation of each voxel in
>> contrast A correlated to the activation of the same voxel in contrast
>> B (over a set of subjects/scans)? My question is instead: are the two
>> contrast images similar as to the spatial distribution of the
>> activation?
I think this is an important distinction, and your case is much more
general than the former. E.g. you might be interested in correlation
between contrast images from studies with different numbers of
scans/subjects, or where there isn't a corresponding matching of the
scans.
It's also very different to be able to report a single correlation
value rather than an image. The former is much more useful if several
methods are compared -- a whole bunch of subtly different correlation
maps would be very difficult to interpret, but a table of values would
be quite nice.
[Satoru:]
> [...] I think creating a voxel-by-voxel correlation
> map would be a lot easier in your case, and it could answer your
> question of whether voxel values in contrast A can predict voxel values
> in contrast B.
[...]
> You can create a voxel-by-voxel correlation map using the BPM toolbox.
> You can just calculate voxel-wise correlation between two sets of
> contrast images (A and B).
This presumably requires that the sets A and B have the same number of
images, and that they are paired in some way. E.g. if one group of
subjects were scanned under two different conditions. But I can't see
how to generalise the approach...
Thanks in advance to anyone who can contribute to the discussion; I
hope my ramblings are of some use/interest to you, Gina!
Best,
Ged.
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