GLM: possible to constrain 1 beta weight?

[kurtio]

Hi,
I am interested in comparing two design matrices in which the parameter values of interest are perfectly negatively correlated. Of course, via traditional GLM, these design matrices yield identical residuals, as the sign of the beta corresponding to the parameter of interest flips between models (but the absolute value remains the same). The direction of these parameter values are informative, and it seems to me that it is, therefore, inappropriate for these to be assigned a negative beta.

I have considered using a non-negative least squares algorithm to solve this problem. However, in this case, beta weights of all predictors become non-negative (specifically, they take on the value of "0"). I wonder if it is possible to constrain only the value of the beta weight corresponding to this particular parameter.

Any advice would be appreciated.
Thanks,
Kurt

[Dason]

I really don't get what you're saying. Can you give a concrete example? It sounds like you're using something like x as a predictor in one and -x as the predictor in the other model (or something similar) - but I don't get what you're trying to do after that.

[kurtio]

Sure, this is fMRI data. The DV is blood flow. I have four predictors, 3 of which are common to the two models: 1) "baseline" 2) main effect of stimuli 3) parametric effect of stimuli 4) the constant.

In a dichotomous response task, I am interested in whether the model that tracks evidence for option "A" provides a better explanation for the data than the model that tracks evidence for option "B."

So, I have a design matrix that might look like this for the model tracking "A"
0.206880000000000 0.587486000000000 -0.336037000000000 1
0.488924000000000 0.363641000000000 -0.227409000000000 1
0.596300000000000 0.351152000000000 -0.285687000000000 1
0.484012000000000 0.536002000000000 -0.419492000000000 1

...and a design matrix that looks like this for model "B"
0.206880000000000 0.587486000000000 0.336037000000000 1
0.488924000000000 0.363641000000000 0.227409000000000 1
0.596300000000000 0.351152000000000 0.285687000000000 1
0.484012000000000 0.536002000000000 0.419492000000000 1

The betas associated with model A are -0.754, 0.409, -0.005, -0.084.
The betas associated with model B are -0.754, 0.409, 0.005, -0.084.

My idea is that it is inappropriate to use a negative beta for the parametric predictor in model A, because I expect there to be less blood flow in response to lesser values. (ie. I would like a negative value in the design matrix to predict a lower value in the DV).

[kurtio]

It has been suggested to me that it may be possible to limit one beta via an iterative nonlinear least squares algorithm. Does anyone have familiarity with this? Does this seem like an appropriate way to address this problem?