Tom,
> Suppose I have an experiment with 10 subject and 4 conditions (A, B,
> C, D) and I want to test the conjunction of A-B and C-D. If I enter
> this in SPM, I get a t-statistic with 18 df, suggesting that SPM
> treats each of the 20 first level contrasts (ie, the A-B and C-D
> contrasts) as coming from a different subject.
*If* you have correctedly specified the nonsphericity, then 18 df is
correct and SPM is appropriately accounting for the repeated-measures
nature of the data.
> My questions: Is this interpretation correct? If so, is this
> assumption necessary because a minimum-t distribution (as used when
> testing against the global null) requires its component
> t-distributed variables to be independent?
While the estimatation and inference process accounts for the
correlation in the data, it cannot change the fact that the two
effects that you are interested in (A-B and C-D) may in fact be
correlated.
To get technical, SPM whitens the model and data as
K Y = K X beta + K epsilon
such that Var( K epsilon ) = I sigma^2, however, this does not
guarantee that Var( betahat ) or Var( C betahat ) is diagonal.
A little of algebra scratchwork shows that if the two effects are
dependent to start with (as we would assume, since pairs of
measurements come from the same subject) then so are the two contrasts
that you want to conjoin. This would seem to doom the global null
conjunction with second level data.
Well, not always. If the correlation between A-B and C-D is due to
the design (i.e. from an non-orthogonal design) then, yes, you are
stuck, there will always be correlation which invalidates the global
null conjunction test. However if you used an orthogonal design then
*under* the global null hypothesis there is no activation to correlate
the tests and the global null conjunction will be valid.
There are a lot of details there... I hope this helps more than confuses.
-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
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