Hi all, have mixed effects models (as described in NI 24:244-252) been
implemented in SPM2 or perhaps in SPM5?
Thanks,
joe
On Wed, 22 Dec 2004 11:37:29 -0500, Thomas E. Nichols <[log in to unmask]> wrote:
>Stephen,
>
>> > If the extra covariates are orthogonal with respect to the
>> > covariate of interest, then it makes no difference whether you
>> > include the extra covariates or not. [The extra covariates can
>> > reduce the intrasubject variance estimate, but, as you point out,
>> > this has no impact in the SPM summary statistic approach to group
>> > modeling.]
>>
>> I did think of putting in a caveat that there's also the issue of
>> orthogonality, but left it out. Below, assume all the added
>> regressors are orthogonal to the covariate that's being brought to
>> the second level.
>>
>> While it's clear there's no *explicit* effect on the summary
>> statistic of changing the subject-level model by adding more
>> regressors, I'm still wondering if it might affect things
>> implicitly. Looking at equation (10) in Penny and Holmes,
>> "Random-Effects Analysis,"
>> http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch12.pdf
>> the total variance of the estimator of the population coefficient is
>> the sum of inter- and intra-subject variance. So what I'm thinking
>> is that the latter would decrease if more error variance were
>> explained by regressors added at the single-subject (i.e., first)
>> level, and I don't see where the penalty in terms of DOF lies.
>
>The confusing thing is the distinction between the true variance and
>the estimation of the variance. Using the Penny & Holmes notation,
>the *true* variance is given by equation (10). But how is this
>estimated?
>
>In the summary statistic "Holmes & Friston" approach, we estimate the
>Var[\hat{d}_pop] quantity at the second level, with the sample
>standard deviation at the {c_i}, where {c_i} is the set of contrasts,
>one for each subject. Since the c_i's are unaffected by the inclusion
>of orthogonal covariates at the 1st level, this estimate of (10) is
>likewise unaffected. This is a very handy aspect of the summary
>statistic approach.
>
>Now, if you were taking a careful mixed effects approach, as FSL does
>(and as I belive Karl et al just published, NI 24:244-252), then you
>must estimate sigma_b and sigma_w separately. In this case it *does*
>matter wheter you include the nuisance covariates at the first level;
>if you leave them out then sigma_w will become positively biased. In
>this respect the full mixed model approach is less robust.
>
>
>Does this clarify things? I agree, it subtle, but it's the essence of
>statistics. :)
>
>-Tom
>
>
> -- Thomas Nichols -------------------- Department of Biostatistics
> http://www.sph.umich.edu/~nichols University of Michigan
> [log in to unmask] 1420 Washington Heights
> -------------------------------------- Ann Arbor, MI 48109-2029
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