Thanks for the response. I may not be fluent in your language, but I think I understood it :).

So if I understand you correctly, if I run 3dDeconvolve with the -svd option, with the X matrix we have been discussing with a column of ones, the individual betas will be indeterminate, but I could run a GLT that contrasts certain conditions against other conditions, provided that the contrast meets the criteria you laid out.

I was talking about taking the dot product of the X matrix and c', and passing this product to 3dDeconvolve. This way my betas reflect the differences between the conditions in the original X matrix, and would presumably be equivalent to output of a GLT using c' as a GLT contrast, provided that the contrast meets the criteria you laid out above.

My interest is in now extracting the similarity structure from these data. I was pre-multiplying the X matrix bya contrast matrix because I could then divide the resulting beta vector by the contrast matrix to extact a beta estimate for each of my original, pre-contrasted regressors. So, for instance:

X = [
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 1 0
1 0 0 0
0 0 0 1
];

c = [
3 -1 -1 -1
0 2 -1 -1
0 0 1 -1
];

X*c' = [
3 0 0
-1 2 0
-1 -1 1
-1 -1 -1
-1 2 0
-1 -1 1
3 0 0
-1 -1 -1
];

If I pass this new matrix through 3dDeconvolve, perhaps I get the following beta matrix:
beta = [ .3 .2 .1];

If we divide the beta matrix by c', we get beta estimates for the 4 original conditions:

beta_cond' = [.18, .15, .1, 0]

We have lost the last column, but the pattern across beta_cond remains intact; we've just subtracted beta 4 from all of the betas.

The goal was to get a relative measure of each individual regressor, despite their high level of collinearity. I mean relative here in the sense that it the estimate for each beta doesn't really matter; what matters is that condition 1 showed a greater effect than condition 2 and condition 3, and so forth, and that our estimates of the magnitude of these differences are accurate.

I suppose my original question was "If I pass an X matrix with zero singular values to 3dDeconvolve, will the -svd option cause the betas for these conditions to be interpretable" and if I understand correctly, the answer is no. I would have to run a contrast matrix, either by multiplying X by c' prior to passing it to 3dDeconvolve, or inside 3dDeconvolve itself. Does that sound correct?

Thanks again!
-cdm