> Hi, I have a question regarding the interpretation of contrasts among
> main effect coefficients when the same parametric modulator (PM) is
> also included in the model for both main effects. Given the following
> estimated coefficients (two conditions and the same PM for both):
> A AxPM B BxPM
> My understanding is that A reflects the the effect of the condition
> that is NOT modulated by the PM (i.e., the effect of A cleaned of any
> variability due to the PM). Same for B, of course.
Actually, it's more like:
A - models the average effect of A
AxPM - models deviations around this average that are explained by PM
The parameter estimate for A reflects the effect of A that can't be
explained by the parametric modulator, and the parameter estimate for
AxPM explains the effect of the parametric modulator that can't be
explained by A (just as it would be with any two regressors in the
GLM). However, since the parametric modulator is mean-centered on the
main effect, these effects shouldn't be confounded anyway (i.e. the
mean of a condition doesn't depend on how the response varies as a
function of some other factor).
> 1. How should I interpret the contrast among A and B ([1 0 -1 0]). Can
> this be interpreted as differences among A and B that cannot be
> attributed to differences among the PM?
> 2. If not, then what is the appropriate interpretation?
This contrast tells you the difference for the average effects of A >
B. Unless there are issues with collinearity, I would expect this to
be comparable to that from the model that includes the parametric
modulators and a similar model that does not (since the inclusion of
the PMs, especially orthogonalized, should not affect the estimation
of the parameter estimate for the mean effect).
> 3. If not, would the appropriate model be a single predictor
> estimating the general effect of A and B (so, A+B) with two PMs
> modeling (1) the differences among A and B and (2) the PM in the
> original model. I understand that, in this case, the first effect
> would actually need to be entered last given serial orthogonalization
> in spm_orth.
I think this would be the way to go. In this case, yes, I think that
a contrast looking at the parameter estimate for the "difference
between A and B" modulator would give you want.
Though, note that in the case where the parametric modulator isn't
systematically related to your data, I wouldn't expect results any
different than the model that doesn't include the parametric
modulator. If the parametric modulator IS systematically related to
your conditions, then—although this parametric modulator approach is
one way to somewhat mitigate that effect—I think you'd still be
susceptible to seeing the effects of that difference. I.e., you'd
have correlated regressors (condition and modulator), and so you
wouldn't be able to have a completely clean interpretation.
If anyone else has other opinions, though, I'd be very interested to hear them.
Hope this helps!
Dr. Jonathan Peelle
Department of Neurology
University of Pennsylvania
3 West Gates
3400 Spruce Street
Philadelphia, PA 19104