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Dear Ged, Thank you very much for your help :) cheers, Hans. On Mon, Jul 20, 2009 at 4:39 AM, DRC SPM <[log in to unmask]> wrote: > Dear Hans, > > Sorry it's taken me two weeks to reply... > > > once you permute these transformed residuals and > > premultiply by U we get U*P*U'*y. Now these residuals have covariance > > sigma^2 U*P*U' where sigma^2 is the true noise variance. If you permute > the > > raw residuals you get the covariance sigma^2 P*U*U'*P'. Is this correct? > > Your second expression is correct, writing sigma^2 as v, the original > residuals have covariance v*U*U', and if you permuted these, you would > get v*P*U*U'*P' as you say. However, your first expression is wrong; > the covariance of U*P*U'*y is: > U*P*U' * v * (U*P*U')' > U*P*U' * v * U*P'*U' > v * U*P*U'*U*P'*U' > v * U*P*P'*U' > v * U*U' > since both U'*U and P*P' equal the identity, and hence the > back-transformed Huh-Jhun permuted residuals have the same (non-white) > covariance as the original residuals. I think this is the best that > one can hope for, since I believe Theil proved that you cannot find > linear (in the data) unbiased residuals with scaled identity > covariance without reducing the dimensionality. (And in case it's not > obvious, the only design matrix I can think of where dimensionality > reduction wouldn't screw it up, is a plain one-sample t-test, which > has no nuisance variables, and an exact permutation test anyway.) > > By the way, if you're keen to read up more on this kind of thing, then > another key paper (though a very difficult one, at least for me) is > Welch's 1990 JASA paper, http://www.jstor.org/stable/2290004 > > Best, > Ged >