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Subject:

Re: individual differences analyses and contrasts for multiple regression

From:

Roberto Viviani <[log in to unmask]>

Reply-To:

[log in to unmask]

Date:

Fri, 8 Jul 2011 13:15:58 +0200

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text/plain (78 lines)

<...>
> Our concern with this approach is related with using the contrast images at
> the second level. We have realized that if we use a certain measure A, and
> subtract a control task B from it, we end up with a measure that usually
> correlates positively with A, but *correlates negatively* with B. Therefore,
> when we use the contrast resulting from a subtraction of parameter estimates
> at the first level, the values at each voxel are negatively correlated with
> our control measure. We would like, however, to be able to use a measure
> that is *independent* of B.

Under the null, the [A-B] constrast is uncorrelated with your  
individual variable; A and B in isolation are also uncorrelated, if  
they are centered. This can be shown agebraically. What you mean is  
that if the null is rejected at the regression of the individual  
variable on the contrast image, then you do not know if you have a  
positive association with A or a negative association with B.

>
> Ideally, we would like to be able to look for the relationship of brain
> response A to certain behavior while *regressing out* the brain responses to
> B, and not subtracting those responses. It seems that only by doing this, we
> would be able to effectively remove the influence of measure B. Would it be
> possible to do this in SPM?

Treating A and B quantitatively, as you are proposing to do here, is  
difficult due to the non-quantitative nature of the BOLD-EPI signal.  
In other words, you cannot view A and B as separate measures of  
activity; only the difference is valid. Furthermore, it breaks the SPM  
approach (because the regressor now including B differs at each voxel  
-- so no, you can't do it with SPM). You could adjust for B, however,  
in a ROI-averaged signal where the ROI is selected by a contrast  
orthogonal to the factor with levels A B.

Apart from the limitations of BOLD-EPI, the issue you are raising  
bears some relation to interventional studies, where for example you  
test the effect of therapy (the 'control variable') on a pre- and  
post-therapy measurement (A and B). In this case, it is indeed common  
to adjust for pre- instead of regressing on [post - pre]. The reason  
is that the contrast [post - pre] is like having pre in the regression  
model with a fixed coefficient of one; if you adjust, instead, you  
estimate this coefficient from the data, i.e. the model is more  
general. The issue, however, is far from clearcut; much depends on the  
existence of a causal association between pre levels and your 'control  
variable'. If this is the case, then contrary to your reasoning you do  
introduce bias in the estimate of the control variable. See basic  
background in S. Greenland, J. Pearl, J. M. Robins 1999, ‘Causal  
diagrams for epidemiologic research’. Epidemiology, 1:37-48, and more  
specifically
M. M. Glymour, J. Weuve, L. F. Berkman, I. Kawachi, J. M. Robins 2005,  
When is baseline adjustment useful in analyses of change? An example  
with education and cognitive change’. American Journal of  
Epidemiology, 162:267-278.

A discussion of the generality of adjusting vs. regressing on the  
contrast is in the book by Hill & Gelman, Data Analysis Using  
Regression and Multilevel/Hierarchical Models. Cambridge (UK):  
Cambridge University Press

The algebra of individual testing at the 2nd level is in  
doi:10.1016/j.neuroimage.2009.10.085

Best wishes,
Roberto Viviani
Dept. of Psychiatry
University of Ulm, Germany


>
> I'm writing to ask if this problem has been considered before, and whether
> you have any suggestion about how we can deal with it?
>
> I'd be very grateful for any thoughts that you have about these issues.
>
> With many thanks and kind regards,
>
> Lucia Garrido
>

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