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
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-------------------------------------- Ann Arbor, MI 48109-2029
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