Hi Rik, list,
Sure, got that point, phrased my question wrong. Of course you would
need correlated regressors for different Betas in both cases when using
OLS, I should have added that right away. But arent they always
correlated to some extent?
But the most important is that you might not estimate stuff well under
OLS, as you indicate. It would be therefore interesting to see, and
validate experimentally, whether under WLS the inclusion of derivatives
improves 2nd level RFX analyses on the HRF betas alone because improved
1st level estimation.
Cheers,
Bas
Rik Henson schreef:
>
> Bas -
>
> I am pretty sure the addition of extra regressors, providing they are
> orthogonal to existing regressors, does not affect the expectation of
> the parameters for the existing regressors under OLS (though the
> variance associated with those parameter estimates is likely to change).
>
> This is simply because B = pinv(X)*y under OLS, so B is independent of
> the residuals.
>
> In your example, there will be changes in the residuals (including
> structure), but this does not bias the parameters, it only affects
> their associated variance.
>
> The addition of correlated regressors, or use of WLS, will affect
> the expectation of the parameters however. Under WLS (ML estimation)
> for example, B = inv(X'*inv(C)*X)*X'*inv(C)*y, where C is the error
> covariance estimated from the resiudals.
>
> But perhaps I misunderstood your question?
>
> Rik
>
>
> ----------------------------------------
> Dr Richard Henson
> MRC Cognition & Brain Sciences Unit
> 15 Chaucer Road
> Cambridge
> CB2 7EF, UK
>
> Tel: +44 (0)1223 355 294 x522
> Fax: +44 (0)1223 359 062
>
> http://www.mrc-cbu.cam.ac.uk/~rik.henson
> <http://www.mrc-cbu.cam.ac.uk/%7Erik.henson>
> ----------------------------------------
>
>
>
> ----- Original Message -----
> *From:* Bas Neggers <mailto:[log in to unmask]>
> *To:* [log in to unmask] <mailto:[log in to unmask]>
> *Sent:* Tuesday, October 24, 2006 1:34 PM
> *Subject:* Re: [SPM] Time & Dispersion Derivatives
>
> Dear Alexa/Rik/List,
>
> we are considering to test the effect of inclusion of more basis
> function on 2nd level RFX analyses, see my most recent email in this
> thread on the list. Is it really the case that betas are not
> affected by
> the inclusion of extra basis functions, even under OLS? When not
> including them AND when HRF does not capture the BOLD response well,
> the residuals have more structure, are not that independent
> anymore, and
> hence the Beta estimates have a larger bias (e.g., are more different
> from the 'real' beta), in the worst case? That would also affect 2nd
> level stats on the betas of the HRF alone...
> I am not a statistician, but one should be able to prove or disprove
> that analytically to some extent.
>
> But perhaps I am talking nonsense here.
>
> Cheers,
>
> Bas
>
>
>
>
>
> Alexa Morcom schreef:
> > Rik/Bas/List
> >
> > Has anybody actually tested in a quantitative way the impact of
> these
> > somewhat (we know - but how much?) inefficient first level betas
> on a 2nd
> > level inference?
> >
> > It also seems a recurring question how white one should wash one's
> > residuals...
> >
> > Alexa
> >
> >>> In the absence of correlation, the inclusion of extra basis
> functions
> >>> can reduce the residual error in 1st-level models, and hence
> improve
> >>> T-contrasts on one basis function alone (eg the canonical HRF).
> >>> However, the inclusion of such extra basis functions will not
> affect
> >>> 2nd-level analyses on only one basis function in SPM99, or only
> >>> minimally so in SPM2/5, because this orthogonality means that the
> >>> parameter estimates are not affected under OLS, and only minimally
> >>> under WLS. Thus it is a common misapprehension that including,
> eg the
> >>> temporal derivative of the canonical HRF in 1st-level models
> somehow
> >>> allows for latency differences in 2nd-level analyses on the
> canonical
> >>> HRF alone. It does not. The answer is to take (contrasts of)
> all basis
> >>> functions to 2nd-level analyses, and again perform F-contrasts.
> >>>
> >
> >
>
>
> --
> -------------------------------------------------
> Dr. S.F.W. Neggers
> Division of Brain Research
> Rudolf Magnus Institute for Neuroscience
> Utrecht University Medical Center
>
> Visiting : Heidelberglaan 100, 3584 CX Utrecht
> Room A.00.1.24
> Mail : Huispost A.01.126, P.O. Box
> 3508 GA Utrecht, the Netherlands
> Tel : +31 (0)30 2503386 Fax : +31 (0)30 2505443
> E-mail : [log in to unmask] <mailto:[log in to unmask]>
> Web : http://www.fmri.nl/people/bas.html
> --------------------------------------------------
>
--
-------------------------------------------------
Dr. S.F.W. Neggers
Division of Brain Research
Rudolf Magnus Institute for Neuroscience
Utrecht University Medical Center
Visiting : Heidelberglaan 100, 3584 CX Utrecht
Room A.00.1.24
Mail : Huispost A.01.126, P.O. Box
3508 GA Utrecht, the Netherlands
Tel : +31 (0)30 2503386 Fax : +31 (0)30 2505443
E-mail : [log in to unmask]
Web : http://www.fmri.nl/people/bas.html
--------------------------------------------------
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