Hi Bas/others
The way I understand it (but one of the pros might like to chip in as my ML
terminology may be a bit dodgy) is as follows:
1. If OLS is used and there is no autocorrelation (iid) then OLS estimators
are maximally efficient (and == to WLS estimates as in 4)
2. If OLS is used and there is autocorrelation then OLS estimators are
unbiassed but not optimally efficient estimators of the 'real' parameters
3. If WLS is used but the autocorrelation model is incorrect, the same thing
happens as in 2
4. With the correct autocorrelation model, WLS gives the ML estimate and
this is the best linear unbiased estimate
So no, I don't think the betas are biassed - just not optimally efficient -
although 1st level stats will be. The impact of unmodelled HRF variability
is qualitatively the same as that of other sources of autocorrelation that
are not modelled by AR(1)+w (in the case of current SPM methods)
Friston et al, 2000 'to smooth or not to smooth' is a good reference for
this, although it only deals with 1st level inference
I will be interested to hear if you get data on the magnitude of the impact
of these things on 2nd level inference...
Alexa
>
> 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
>
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