Hi Russ
No, weighting by the within-subject variance *only* shouldn't bias the
statistics - if it increases the efficiency or not I don't know, it will
probably depend on the ratio of fixed- and random effects variance and
I'm not sure you'll necessarily gain anything.
As Tim already said: the maths of things suggest that in order to get
the best linear unbiased estimator (maximally efficient) you will need
to include the lower-level variances in the estimation steps at the
higher-level.
cheers
Christian
Russell Poldrack wrote:
> christian - thanks for the helpful post. My understanding from Tim's
> previous posting is that performing the second-level mixed-effect
> analysis using unweighted contrast images (i.e., each COPE weighted
> identically) is unbiased but not optimally efficient. My question is:
> would weighting the contrast images by the within-subject variance bias
> the resulting statistic, and if not, would it increase the efficiency
> over the unweighted version (even though it's still not minimum
> variance)?
>
> thanks
> russ
>
> On Thursday, October 17, 2002, at 03:58 AM, Christian Beckmann wrote:
>
> > Dear Raj,
> >
> > >Am I right in thinking that, without needing to switch to FSL
> > >or wait till SPM2 comes out, one could therefore make some
> > >kind of variance-weighted contrast-vector to solve the above
> > >problem within the existing SPM99 framework?
> > >If so, what exactly would the formula be for how to weight
> > >the contrast vector? (I've looked at the Tech.Report that
> > >Stephen Smith links to below, and I expect that the formula
> > >is in principle derivable from that paper, but I confess that
> > >it's not entirely clear to me how to do it).
> >
> > There are quite a few issues here. The technical report deals with the
> > case where (for whatever reason) you want to carry out analysis in a
> > multi-level GLM (estimate group paramteres from lower-level parameter
> > estimates and (co-)variance estimates...)
> >
> > It certainly is the case that you can stack all data into one big GLM
> > and start estimating group parameters of interest from the complete set
> > of data. There are, however, plenty of issues you need to consider
> > (like
> > how to constrain AR estimates to only use the relevant parts of the
> > data, how to filter and de-mean the data etc) but I'm not familiar
> > enough with the implementational details of SPM99 to tell you how
> > exactly go about making it work.
> >
> > In the case where you decide to use a multi-level approach we give an
> > example on how to calculate the mixed-effects average group activation
> > in section 8.1 of the technical report: in the multi-level case the
> > individual \hat{\beta}_i need to be weighted by u_i^{-1}, which is a
> > function of the within-subject variance *and* the between subject
> > variance \sigma_s^2. Naturally the second quantity is unknown and needs
> > to be estimated - so the answer to your question is no, you cannot
> > simply weight the first level contrasts that feed into RFX analysis. It
> > all needs to be part of the estimation at a higher level (simply
> > weighting by what is known at the time - the fixed-effects variance -
> > isn't sufficient...).
> >
> > hope this makes it clear
> >
> > Christian
> >
> >
> > --
> > Christian F. Beckmann
> > Address: Oxford University Centre for Functional
> > Magnetic Resonance Imaging of the Brain,
> > John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK
> > Email: [log in to unmask] http://www.fmrib.ox.ac.uk/~beckmann/
> > Phone: +44(0)1865 222782 Fax: +44(0)1865 222717 Mob: +44(0)7811
> > 189123
> >
> ---
> Russell A. Poldrack, Ph.d.
> Assistant Professor
> UCLA Department of Psychology
> Franz Hall, Box 951563
> Los Angeles, CA 90095-1563
>
> phone: 310-794-1224
> fax: 310-206-5895
> email: [log in to unmask]
> web: www.poldracklab.org
>
>
--
Christian F. Beckmann
Address: Oxford University Centre for Functional
Magnetic Resonance Imaging of the Brain,
John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK
Email: [log in to unmask] http://www.fmrib.ox.ac.uk/~beckmann/
Phone: +44(0)1865 222782 Fax: +44(0)1865 222717 Mob: +44(0)7811 189123
--
Christian F. Beckmann
Address: Oxford University Centre for Functional
Magnetic Resonance Imaging of the Brain,
John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK
Email: [log in to unmask] http://www.fmrib.ox.ac.uk/~beckmann/
Phone: +44(0)1865 222782 Fax: +44(0)1865 222717 Mob: +44(0)7811 189123
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