On Thu, 26 Feb 2009 14:48:39 +0000, Stephen J. Fromm
<[log in to unmask]> wrote:
>On Wed, 25 Feb 2009 17:29:12 +0000, cyril pernet <[log in to unmask]>
>wrote:
>
>>Hi there,
>>
>>I'm stuck on some GLM computation and hoping to get an answer here ..
>>In SPM (the book) it is stated that a contrast is valid if C =
>>C*pinv(X'*X)*X'*X.
>>If I run a model by hand and use say C = [1 -1] that's fine and the
>>result in meaningful.
>>BUT, how does SPM handles the case C = 1? the contrast isn't valid
>>according to the above definition and the result is wrong when done by
>>hand .. however the contrast manager takes it and returns a map ...
>>
>>thanks in advance for any help (especially if you've got the equations)
>>cheers
>
>I've always been fond of this criterion: c is valid if c is orthogonal to the null
>space of the design matrix.
>
>The value of the estimated contrast is
> c'*beta_hat
>where beta_hat is the estimated reg. coeff. If beta_0 is in the null space of
>the design matrix, then (beta_hat + beta_0) gives the same y_hat. If the
>value of the contrast shouldn't depend on the choice of beta_0, then we
need
> c'*(beta_hat + beta_0) = c'*beta_hat
>or
> c'*beta_0 = 0
>
>Haven't had the time to check that it's equivalent to the criterion you list,
>though I assume it must be.
OK, the Wikipedia page on the Moore-Penrose pseudoinverse, at
http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
claims that
I - pinv(A)*A
is the orthogonal projector onto the null space of A. That matches your
criterion, where A = X'*X (design matrix is X).
That says that c must be orthogonal to Null(X'*X).
I'm pretty sure Null(X'*X) = Null(X), though. Clearly Null(X) is contained in Null
(X'*X). And if beta is in Null(X'*X), then
X'*X*beta = 0
beta'*X'*X*beta = 0
(X*beta)'*(X*beta) = 0
X*beta = 0
Not clear to me what's going on (i.e, why use criterion about pinv(X'*X)*X'*X
rather than pinv(X)*X); either there's a mistake here somewhere, or perhaps
the authors of the book you cited used the trick that dealing with X'*X can be
easier than dealing with X, if X has many more rows than columns.
>Best,
>
>S
>
>>Cyril
>>
>>--
>>Dr Cyril Pernet,
>>fMRI Lead Researcher SINAPSE
>>SFC Brain Imaging Research Center
>>Division of Clinical Neurosciences
>>University of Edinburgh
>>Western General Hospital
>>Crewe Road
>>Edinburgh
>>EH4 2XU
>>Scotland, UK
>>
>>[log in to unmask]
>>tel: +44(0)1315373661
>>http://www.sbirc.ed.ac.uk/cyril
>>http://www.sinapse.ac.uk/
>>
>>
>>The University of Edinburgh is a charitable body, registered in
>>Scotland, with registration number SC005336.
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