Dear Eric and Rik,
I am so sorry to make a wrong conclusion on Rik's paper. He surely
didn't say
orthogonalization of regressors is always needed.
Thanks for your emails clarifying the use of the
orthogonalization.
Chih-Chen Wang
Rik Henson wrote:
> Eric -
>
> I fully endorse your comments that orthogonalisation of regressors is not
> normally necessary. As you are aware, orthogonalising A wrt B will not
> change the parameter estimate (loading) for A - this already reflects the
> orthogonal (unique) part of A. Moreover, it will not affect any T-tests
> performed on A, or F-tests on A and B.
>
> However, it will affect T-tests on B, whose parameter estimate will change
> (even though it was regressor A that was changed, somewhat
> counterintuitively). Effectively, you have chosen to assign the "common"
> variance to B.
>
> For a graphical illustration of the effects of orthogonalisation, see
> Lecture 2 here:
>
> http://www.mrc-cbu.cam.ac.uk/Imaging/Common/spm-minicourse.shtml
>
>
> You are also correct that regressors that represent basis functions do not
> normally need orthogonalisation, given that one normally performs F-tests
> over them.
>
> HOWEVER, there WAS a good reason for our orthogonalisation in Henson et al
> (2000). This is because we performed separate T-tests (in separate second
> level models) on the early and late HRFs. We did not combine them in a
> single model for an F-test, because corrections for nonsphericity were not
> available in SPM at the time. We elected to assign the common variance to
> the early HRF, since that was our expected response for most voxels. If we
> had not done this, our T-test on the early HRF would not have been as
> powerful (because it would have only revealed what could be NOT be explained
> by the late).
>
> Hope that clarifies things for the record.
>
> Rik
>
> --------------------------------------------------------
> DR RICHARD HENSON
> MRC Cognition & Brain Sciences Unit
> 15 Chaucer Road, Cambridge,
> CB2 2EF England
>
> EMAIL: [log in to unmask]
> URL: http://www.mrc-cbu.cam.ac.uk/~rik.henson
>
> TEL +44 (0)1223 355 294 x522
> FAX +44 (0)1223 359 062
> MOB +44 (0)794 1377 345
> --------------------------------------------------------
>
>
>
>>-----Original Message-----
>>From: SPM (Statistical Parametric Mapping)
>>[mailto:[log in to unmask]] On Behalf Of Eric Zarahn
>>Sent: 16 December 2005 11:46
>>To: [log in to unmask]
>>Subject: Re: [SPM] Model event types with two basis functions
>>
>>
>>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
>>
--
Chih-Chen Wang
Associate in Research
Center for Cognitive Neuroscience, Duke University
Box 90999, LSRC Bldg., Room B243Q
Durham, NC 27705
[log in to unmask]
(919) 668-2299 www.cabezalab.org
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