Dear Chih-Chen,
Quoting Chih-Chen Wang <[log in to unmask]>:
> As I am new to SPM the following idea may sounds odd, so please
> correct me if it is wrong.
>
> I tried to model the hemodynamic response to the onset of each
> event type
> with two basis
> functions in SPM2: a canonical HRF and a delayed HRF, shifted to
> onset 2
> sec(1.33 TR)
> later than the canonical HRF. According to Henson'
> paper(Confidence in
> Recognition Memory
> for Words: Dissociating Right Prefrontal Roles in Episodic
> Retrieval), the
> covariates for
> the late HRF need to be orthogonalized with respect to those for
> the early
> HRF.
There is no reason per se for having to orthogonalize the "early"
and "late" basis functions. That they are correlated poses no
problem per se for estimation. The reason stated in Henson et al.
for orthogonalizing is:
"Given that the early and late HRFs were correlated,
covariates for the late HRF were orthogonalized
with respect to those for the early HRF using a Gram–
Schmidt procedure (loadings on the early covariate
thus represent variance that is not shared with the
orthogonalized late covariate, Andrade, Parades, Roulette,
& Poline, 1999)."
No offense of any kind at all intended, but this statement seems to
misrepresent or somewhat obfuscate the properties of least-squares
estimation. In particular the sentence suffers from a non-sequitir
(i.e., it does not follow that correlated covariates need to be
orthogonalized). Expectations of "loadings" (I take "loadings" to
mean linear model parameters) on any covariate always depend on
what that covariate can explain uniquely in the context of all the
other covariates, even if the covariate in question is correlated
with others (i.e., the partial correlation interpretation of
regression coefficients). Now, orthogonalizing the late component
with respect to the early can change the respective loadings,
because what each covariate can uniquely explain has changed. For
example, a true late response will load both on the early basis
function and the orthogonalized late basis function (in a model
using these two basis functions), but will load only on the
non-orthogonalized late basis function (in a model using early and
late basis functions). The net fit (i.e., the contribution to the
fit from both basis functions) will not be changed at all by
orthogonalization, nor will an F-test assessing the two loadings.
The variance of the early loading will be smaller when one does
orthogonalize; the variance of the late loading will be unaffected
by orthogonalizing.
As a final note, I think there might be a small pocket of
misunderstanding in the neuroimaging community regarding this issue
as a reviewer asked me to explain why I did not orthogonalize two
correlated covariates, as if it conveyed a universal benefit or
were somehow the status quo or the proper way to do things. Rather,
the correct point made by Andrade et al. was that orthogonalizing
changes the interpretation of regression loadings and that
therefore one should think about the consequences of
orthogonalizing versus not (not that one per se needs to or should
orthogonalize as a rule). In fact, when fMRI basis functions are
theoretically modeling different neural components (e.g., early and
late) it is proper to not orthogonalize in order to get correct
estimation of the amplitude of those components.
Eric
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