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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