Thanks a lot for considering the ?u-relativization? proposal.
Please confirm that what you mean by this:
?The usual first step is to take it to be linear, which would add u^2 to the regression of c on u, u & u*e to the regression of c on e, and u*e & u^2 to the multiple regression of c on e & u.?
are the three regressions:
c on (u,u^2) instead of c on u
c on (e, u, u*e) instead of c on e
c on (e, u, u*e, u^2) instead of c on (e,u)
Or did you mean ?add? arithmetically, e.g. c on u+u^2 instead of c on u, etc.?
(Fortunately, I?m sufficiently automated now to do these new regressions with only the obvious changes to one program in which I call the GSL regression functions; so the re-reruns will not be onerous.)
II. Question about the centroids
I was a little taken aback by your reaction to the ?centroids? suggestion:
> ?So all you're suggesting is to look at some linear transformations > of the centroids of the points at each L. That can never tell you > anything about the within-L relations or if/how they change with L.?
until I realized that:
i) in my own mind, I was proposing the centroid analysis as part of an entirely different approach to the problem;
ii) I hadn?t yet communicated this new approach to you yet (Whoops!)
So, to keep things simple, suppose we agree for the sake of discussion that:
a) there IS a discoverable function F such that L = F(c,e,u);
b) it?s worth trying to discover F because knowing how to predict L from (c,e,u) would be an important first step toward predicting structural alignability (this is the part I won?t bother to explain now ? again, to keep things simple.)
Then, given the proposed transformation of the (L,c,e,u) 4-space into a 3-space containing a 2-space projection of (c,e,u) and an L-axis orthogonal to this space:
c) isn?t it at least possible that as L increases, the path traversed by the centroid might provide some clues as to the nature of F? (E.g. might even be something like some kind of smooth space curve?)
Thanks as always for considering this question ? hope it?s not too stupid.