[VoxBo] Beta values, covariates, % change

[Geoffrey Aguirre]

	I wanted to take a few lines to clarify some issues in GLM design and 
the interpretation of beta values.

	As many of you know, the GLM provides an estimate of the relationship 
between a covariate and data in the form of a beta value. The beta 
value is a scalar that is multiplied by the covariate to make it best 
fit the data (in the context of the other covariates in the model). The 
beta value provides the numerator of the t-statistic.

	Clearly, the excursion of the values in the covariate will have an 
impact upon the size of the beta value. If you create a covariate which 
models the "off" condition as a zero, and the "on" condition as one, 
you might obtain a beta value of X, but if you model the "on" condition 
as two, then the beta values that result will be 1/2 X.

	In one sense, the particular scaling that you select for a covariate 
is arbitrary. Whether the excursion of your covariate is zero to one or 
zero to two, the t-statistic that results will be the same -- although 
the betas are different, the error term is scaled appropriately.

	There are situations, however, where the particular scaling of your 
covariates is of some importance. For example, suppose you have 
conditions A, B and R(est). You create covariates A-R and B-R. After 
completing your GLM, you now wish to compare A vs. B, perhaps testing 
the idea that the magnitude of the fMRI response to condition A was 
greater than that of condition B. It is important in this case that the 
scale of the A-R and B-R covariates are the same, in that one unit of 
neural activity is represented by one unit of covariate excursion, 
prior to convolution with a hemodynamic response function. If not, then 
the same magnitude of fMRI response in A and B would lead to different 
sized beta values. For example, if A-R was modeled as one for A and 
zero for R, and B-R was modeled as two for A and zero for R, then a 
contrast between A and B would yield a difference, even if one was not 
present.

	This can be a complicated matter. Suppose condition A consists of 
trials of stimulus presentations that are 1 second in duration, while 
condition B presents the stimulus for 2 seconds. You model condition A 
initially as a square-wave of neural activity that is one unit in 
amplitude and 1 second in duration, and you model condition B as a 
square-wave that is also one unit in amplitude but 2 seconds in 
duration. When these covariates are convolved with a hemodynamic 
response function to yield appropriate predictors for fMRI data, the 
magnitude of the covariate for B-R will be twice that of A-R. The 
null-hypothesis that will be tested using this model when you compare A 
versus B is that there is no difference in the intensity of neural 
activity evoked per unit time. Conversely, if you scaled the covariates 
AFTER convolution with a hemodynamic response function so that both had 
the same degree of excursion, then the null-hypothesis being tested 
when you compare A versus B is that there is no difference in the 
total, integrated amount of neural activity evoked by each stimulus.

	There are some further nuances:

	- To provide for beta values that are meaningful in terms of 
percentage change, then the final covariate that is placed in the G 
matrix (after convolution with a hemodynamic response function) should 
range from zero to one.

	- One might also scale covariates so that, regardless of their 
excursion, their total variance is unity, so that the beta value 
represents a measure of the amount of variance that the covariate 
explains

	I have added a menu item to the G Design widget to make it easier to 
scale covariates to have unit excursion. This will be available in the 
next VoxBo release (I will commit it to the CVS tonight).  As the 
preceeding discussion suggests, however, you should consider the impact 
of scaling covariates when those covariates might differ in amplitude 
in a meaningful way that reflects durations of modeled neural activity.

Geoff 	

--

Geoffrey Karl Aguirre, M.D., Ph.D.
University of Pennsylvania      Center for Cognitive Neuroscience
3815 Walnut Street              Fax: (215) 898-1982
Philadelphia, PA 19104-6196     mailto:aguirre at neuro.med.upenn.edu
http://ccn.upenn.edu/~aguirre


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