Hi Andrew,
> Suppose that I am reviewing a paper that has necessary design
> multicollinearity (e.g., Cue always precedes response), but wants to
> make explicit claims about one portion of the trial (Cue). Most
> methods sections do not report measures such as VIF. But,
> multicollinearity at the time-series level cannot be assessed at the
> group-level SPM (inflates first level standard errors, but does not
> bias point estimates).
If you have both cues and responses in the first level design matrix
X, and they are to some extent collinear, then this will affect second
level too: the data at the second level are the coefficient estimates
from the first level, and their variance is affected by inv(X'X) or
its GLS equivalent. (the variance at the second level has two terms,
one imported from the first level, the second arising from
subject-to-subject variance).
In theory, collinearity at the first level leads to lower t's at the
second level: when you estimate second-level residual variance, this
variance will be larger for estimates coming from quasi-collinear
models.
Note, however, that this kind of setup may rely on distributional
assumptions that become increasingly unrealistic, i.e. the behaviour
of such highly variable coefficient estimates may produce tails which
are not correctly assessed by a parametric model, and which are
dredged up by the whole-brain search for high t values required by
multiple comparison procedures.
Something that I have been thinking about lately is a related problem,
i.e. the fact that in such a setting we carry out at the second level
the one-sample t test for cue and response separately... These
estimates are bound to be anticorrelated, so it seems wrong not to
consider this, and adopt a setup instead in which both coefficient
estimates are brought to the second level.
>
> Obviously, multicollinearity measures like VIF can be
> requested--when is it appropriate to do this? (when a paper has
> plausible multicollinearity? Why not all the time?)
I am not aware that something like VIF has been systematically
examined in the context of neuroimaging (I could be wrong here of
course). Since everything goes through multiple comparison procedures,
native indices from univariate tests are difficult to evaluate.
Best wishes,
Roberto Viviani
Dept. of Psychiatry III
Ulm
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