Dear Darren,
>I agree for SPM99, and this was my precise concern when I asked the
>question, though not with so many symbols. However, is it true that the use
>of the non-sphericity correction in SPM2 will deal with this?
Yes. In fact this was one of the principle motivations for introducing
the non-sphericity option. Tom refers to a fairly subtle form of
non-sphericity induced by correlations among contrasts from each
subject. In other situations, the variance of the contrasts may differ
(e.g. contrasts testing for the hrf and its derivative) which produces
another, simpler, form of non-sphericity (heteroscedasticity).
In short, allowing for non-sphericity enables mulitple contrasts from
a single subject to be taken up to the second level without assuming
sphericity. This, in turn, enables conjuctions of contrasts at the
second-level.
I hope this helps - Karl
At 11:47 26/06/2002 -0500, Darren R. Gitelman wrote:
>Tom:
>
>
>Darren
>
>At 10:09 AM 6/26/2002 -0400, Thomas E. Nichols wrote:
>>Hi all,
>>
>>
>>I've noticed some SPM help email exchanges on conjunctions with random
>>effects contrasts. In discussions locally, we've decided this is a bad
>>thing to do. I'm interested in what people think of the following...
>>
>>
>>The uncorrected and corrected conjunction p-values rely on an assumption
>>of independence between contrasts (test statistics). It would seem,
>>however, that independence is not a tenable assumption for RFX
>>conjunctions, making their validity questionable.
>>
>>
>>To set notation, consider the conjunction of hypotheses represented by
>>contrast estimate c1 and c2. In the random effects setting, each of these
>>contrast estimates is an average of each subject's contrast estimates,
>>say c1_j and c2_j for subject j. The random effects t-statistic for
>>each contrast estimate will be (for, say, a single voxel)
>>
>> t1 = c1/s1 c1 = Mean{c1_j} s1 = Stdev{c1_j},
>> t2 = c2/s2 c2 = Mean{c2_j} s2 = Stdev{c2_j}.
>>
>>In SPM99, the conjunction is assessed with min{t1,t2}, using an assumption
>>of independence between t1 and t2. However, in the random effects context
>>("unconditionally"), c1_j and c2_j are not independent since they are from
>>the same subject, and hence c1 and c2 are not independent. In stat-ese
>>
>> cor(c1_j,c2_j) <> 0 => cor(c1,c2) <> 0 => cor(t1,t2) <> 0
>>
>>
>>If this seems like statistical pedantry, consider that it is often the
>>case that there are ``good activators'' and ``bad activators''. (You
>>know who the good grad student activators are, because they're the
>>ones asked to come back again and again for studies.) If subject j is
>>a good activator, then both c1_j and c2_j are likely to be large,
>>where if subject j is ``bad'', both c2_j and c2_j will be relatively
>>small. At each voxel{c1_j} and {c2_j} will be correlated, and hence
>>so will c1 and c2, and so will t1 and t2 (s1 and s2 are independent of
>>c1 and c2, and hence don't play into the argument).
>>
>>Also, the particular cognitive details of the contrasts involved may
>>suggest a different correlation, where big responders on one are the
>>low responders on another contrast. So there might be negative
>>correlation instead.
>>
>>The impact of positive correlation will be generally increase the
>>value of min{t1,t2}, while negative correlation will decrease
>>min{t1,t2}. So positive autocorrelation will make inference on the
>>minimum conservative, while negative correlation can make the test
>>invalid, anticonservative
>>
>>Have people thought about this? We've avoided RFX conjunctions for
>>this reason.
>>
>>-Tom
>>
>>
>> -- Thomas Nichols -------------------- Department of Biostatistics
>> http://www.sph.umich.edu/~nichols University of Michigan
>> [log in to unmask] 1420 Washington Heights
>> -------------------------------------- Ann Arbor, MI 48109-2029
>
>
>-------------------------------------------------------------------------
>Darren R. Gitelman, M.D.
>Cognitive Neurology and Alzheimer¹s Disease Center
>E-mail: [log in to unmask]
>WWW: http://www.brain.northwestern.edus
>Voice: (312) 908-9023
>Fax: (312) 908-8789
>Northwestern Univ., 320 E. Superior St., Searle 11-470, Chicago, IL 60611
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