Thanks for that algebraic legerdemain ? it certainly clarifies why the u^2 in c on u,u^2 models the change in slope with change in u.
You say ?This is because a1 is only the slope of the function at x = 0?
This makes partial sense to me inasmuch as a1 = a01+a10 and a01 is the slope in the ?intercept? function A0 = a00+a01*x. But why are you free to disregard a10 here? Is it necessarily 0 when x is 0? If so, the reason why is eluding me.
Putting aside for a moment the matter of whether we need a ?meaningful? definition for a1, may I operate with a2 the same way I?ve operated with usual slope coefficients to develop the 2-ways?
That is, is it permissible to develop the 2-way a2 interactions involving coreXcomp and nonrandXrand (for each of the dicodon sets 1,2,3) using the same mechanics I?ve previously used for linear regression slopes of the usual type (first roll-up across length intervals within fold and then roll-up across folds)?
If so, I?d like to go ahead and do that, in order to see whether a2 behaves as one might expect it to, given our current understanding of the probable relationship between c and u( namely that c varies directly with u.)
Regarding a ?meaningful? defintion for a1, it would be useful to have one for two reasons:
a) to see what happens with its coreXcomp and nonrandXrand 2-ways
b) to see if this behavior of a1 helps to validate the use of residuals from c on (u,u^2) in the construction of predictors for logistic regressions involving structural alignability.
Does anything you've said change if we use your suggested u/(1+u) instead of u itself?