# Collinearity of polynomial terms in the case of multiple run

Hi,
Please forgive me if this is a silly question.

If there are two runs of fMRI and we want to model the constant and first-order drift for each run as part of baseline, it is obvious that the correlation between the constant term of run#1 and run#2 is -1.

e.x., cor(c(0,0,0,1,1,1), c(1,1,1,0,0,0)) = -1

How to avoid the Collinearity issue?

Does AFNI removed the linear trend first for each run and then model the task regressors?
Or AFNI just put the design matrix as a whole into the regression model?

I am confused about the underlying mechanism of 3dDeconvolve to deal with this issue.

Thanks!
lz

lz,

Multicollinearity can have several negative effects on a regression model, but not all cases lead to numerical problems. In the case of separately estimating the baseline/intercept for each of the two individual runs, the corresponding regressors are indeed anti-correlated. However, this does not pose any numerical issues. These two regressors are essentially dummy-coding for a factor with two levels and are orthogonal to each other.

Does AFNI removed the linear trend first for each run and then model the task regressors? Or AFNI just put the design matrix as a whole into the regression model?

Indeed, a single design matrix is constructed and assembled for all runs within a single model.

Gang Chen

Gang,

If set the intercept of regression model as 0, the two regressors of constant term was indeed dummy-coding.

If the fMRI only have two conditions (A vs. B) with no gap between trials (e.x., single-trial design with duration of 16 seconds), there was also a negative serious correlation between A and B, which can be -1 if using boxcar function without convolving the canonical HRF.

Dose this case lead to numerical problems, especially using the boxcar function with height of 1, which also create a dummy-coding?

Please forgive me for this stupid question. We were taught not to do that in designing fMRI studies. However, I want some essential understanding.

Thanks!

LZ

Hi LZ,

It seems like these 2 conditions of interest span all time, and so (ignoring the initial ramp-up) the sum of those regressors is basically equal to the constant baseline terms. While we can call this multicollinearity, it might be more descriptive to say that we would not be able to tell the difference between a brain voxel that was very responsive to the 2 tasks (equally responsive), and a voxel with absolutely no response to the tasks, even a non-brain voxel. We could not distinguish between a beta pair of 0, 1, and another beta pair of -5, -4. All we could evaluate is the contrast between them.

Effectively, one of those 2 conditions would need to be called the baseline condition (and so not even explicitly modeled), and the other would be the only condition of interest. And then (after the multicollinearity was removed) the result would still be just the contrast, for example, between task A and baseline condition B.

If you want a beta for both A and B, then there needs to be additional non-task time to compare against.

Does that seem reasonable?

• rick

Hi rick,