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SPM  March 2009

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Subject:

Re: How do regressors compete for variance if not orthogonalized?

From:

cyril pernet <[log in to unmask]>

Reply-To:

cyril pernet <[log in to unmask]>

Date:

Wed, 4 Mar 2009 13:38:01 +0000

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text/plain

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Dorian P. wrote:
> Dear Cyril,
>
> What I mean in point 2 is that regressor A overlap with B. Together
> they explain 50%, separately A = 30%, B = 40%. This means they have
> 20% in common (we're talking about a voxel's variance to be precise).
>
> >From what you say I immagine that when put in the same GLM this 20%
> will be lost; --- A will have its remaining 10% and B its remaining
> 20%. Comparing them in the 2nd level means that B = 2A (quite good to
> detect this voxel is explained better by B).
>   
yep that's right :-)

> Alternatively, putting them in separate GLMs will simply reproduce
> their original variance, A=30% and B=40%. Comparing them in the 2nd
> level will produce a weaker result,  B = 4/3A. Am I correct with this
> assumption? Does the common variance really get lost (would be nice if
> so)? So, I should I keep them in the same GLM un-orthogonalized to
> have stronger results.
>   
yes the common variance get 'lost' ..
note it is only one way to deal with this - over methods exist ; see 
stuffs about types of sums of squares
an easy read about this is http://www.statsoft.co.uk/textbook/stathome.html
> About point 3, I dont need the main condition at all. If it absorbs
> its variance without *stealing* it from modulations for me is ok, but
> if it gets variance from modulations I would like to keep it
> not-orthogonalized. I have collapsed all trials in that condition to
> play around with parametric modulations and thus don't need the
> condition itself. Any suggestion here can be useful.
>   
Well I'm not sure about your design but if you have only 1 modulation 
you could use the modulation parameters when modeling each trial ..
There was a recent paper, in neuroimage I think, where RT of each trials 
was used to model the hrf (instead of 0 modeling an inpulse) and this 
model was compared with a standard approach (1 column for the regressor 
+ modulation by RT) -- clearly it was better to directly 'modulate' the 
1st regressor, but it may not always be possible to do so ..  cannot 
think of anything else here ...

Good luck
Cyril



> Thank you for answering.
> Dorian.
>
>
>
> 2009/3/4 cyril pernet <[log in to unmask]>:
>   
>> Hi Dorian
>>     
>>> Dear all,
>>> I was discussing this in private with another member of the list but
>>> we cannot fully understand it.
>>>
>>>       
>> ok I'll try then ..
>>     
>>> 1.
>>> When we bypass orthogonalization the variance of the model is
>>> explained by all regressors in a kind of *competition*. I don't
>>> understand how this competition works statistically but actually I
>>> need the regressors to compete as much as possible with each other.
>>> This way I can compare them in a paired t-test at the 2nd level in
>>> order to find areas where one explains more variance than the other
>>> (independently of the order I put them in SPM). Does this make sense
>>> to you?
>>>
>>>       
>> Without orthogonalization (the usual stuff) each regressors fits the data
>> but you are only looking at the 'unique part of variance' for each of them
>> -- with orthogonalization, the order matters because you attribute the
>> maximum of variance to the 1st then 2nd etc ...(it's like performing a
>> simple linear regression with the 1st regressor then another simple linear
>> regression with the 2nd regressor on the residuals of the 1st fit etc .. )
>>
>> The way I think of this is using a diagram: think you have 3 conditions
>> represented by 3 circles. Case 1, the 3 circles do not overlap - easy each
>> condition got its own part of variance explained. Case 2, the circles
>> overlap - well that's where you have several options (that also relates to
>> the different sum of squares options in the statisitcal packages) ; the
>> unique part of variance like in SPM is that you estimate the effect of each
>> circle removing the overlapping part ; now if you orthogonalize you give to
>> the 1st regressor its full variance (full circle) then to the 2nd the full
>> variance minus the overlap with the 1st circle etc ...  (hope that makes
>> sens to you :-\ )
>>
>> Note that in all cases (orthogonalization or not) you can perform a 2nd
>> level analysis.
>>     
>>> 2.
>>> Also discussing with my friend, I thought having one model with 5
>>> non-orthogonalized (i.e. independent) parametric modulations is like
>>> having 5 GLMs. Apparently this is not true because in the first case
>>> we have X variance exlpained by 5 modulations, while in the second
>>> case we have X variance explained by 1 modulation each time. But
>>> wouldn't the comparison in a paired t-test produce the same *winner* ?
>>> It make sense logically: if two collinear regressors A and B explain
>>> 50% variance, with regressor A 30%, and regressor B 40%, their
>>> variance overlaps but regressor B will result with higher T values, no
>>> matter of measured in the same GLM or in two separate GLMs. So is it
>>> better to keep them in the same GLM or split them up? Would the result
>>> be the same?
>>>
>>>       
>> didn't quite understand this -- the sum square of the effect is computed via
>> a single design matrix no matter you orthogonalize or not ..
>> regarding the T value it depends on the error; if orthogonalized A=30% and
>> B=40% they make up 70% of the variance but if not orthogonalized they may
>> make up more (the overlapping bit) so T values will vary
>>     
>>> 3.
>>> Somebody advised to orthogonalize modulations (not with each-other
>>> but) with the main condition . Does this produce any benefit?
>>>
>>> Help is highly appreciated from experts or non-experts. :)
>>>
>>>       
>> well orthogonalization is a matter of theory more than stats; it all depends
>> how you are thinking of your data ..
>> say you use a standard regressor and a some orthogonalized modulation then
>> you mean to model the BOLD response the standard way and the remaining
>> variance by the modulation .. now if you do not, well, as you said above,
>> they 'compete' and one voxel will have some variance explained by the
>> standard model and some other part of variance explained by the modulator ;
>> the problem is that they are extremely correlated (large overlap) and thus
>> the unique part of variance will be small .. i.e. it is likely (I think)
>> that you end up having nothing significant ..
>>
>> Hope this helps
>> Cyril
>>
>>
>>
>> --
>> The University of Edinburgh is a charitable body, registered in
>> Scotland, with registration number SC005336.
>>
>>
>>     
>
>
>   


-- 
Dr Cyril Pernet,
fMRI Lead Researcher SINAPSE
SFC Brain Imaging Research Center
Division of Clinical Neurosciences
University of Edinburgh
Western General Hospital
Crewe Road
Edinburgh
EH4 2XU
Scotland, UK

[log in to unmask]
tel: +44(0)1315373661
http://www.sbirc.ed.ac.uk/cyril
http://www.sinapse.ac.uk/


The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.

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