> From my reading, the F statistic associated with -fcoef is the significance of
> the overall regression, not the significance of the individual parameter. By
> contrast, the t-statistic associated with -tcoef is the significance of the
> individual parameter.

You're right. I totally forgot about that. Yes, -tcoef is what you want.

> Now imagine a situation where a voxel has some type of waveform in between
> a linear and a quadratic. The ttest for that voxel might be significant for both
> regressors. Then the question is: Which regressor does it look most like? Does
> it look significantly more linear than quadratic or more quadratic than linear?
>
> How do I test each voxel to find out which ones have significantly more
> variance accounted for by the linear regressor than the quadratic regressor
> and which ones have significantly more variance accounted for by the
> quadratic regressor than the linear regressor?

OK, that's clearer. In your model

Y = b_0 + b_1 * X + b_2 X^2 + e

you have a t-statistic for b_1 and one for b_2. The t-statistic squared would be the corresponding F-statistic, which can be interpreted as the amount of variability accounted for each regressor (trend in this case) among the total variability in the data in marginal sense (a full model versus one with that regressor removed). So you could compare the relative magnitude between the two F-statistics, but I can't think of a way to test the significance about their difference.

> Would a ttest on the parameter estimates themselves be correct? In that
> case, I am worried that the parameter estimates for a linear versus
> quadratic trend might be affected by the different scales used to code for the
> linear vs the quadratic waveforms.

Yeah, the problem about testing b_1 - b_2 is that they have different dimensions and it's not mathematically interpretable.

Gang