Date: Nov 27, 2012 12:44 AM
Author: Ray Koopman
Subject: Re: Them there is some neat algebraic mechanics !

On Nov 26, 5:06 am, djh <halitsk...@att.net> wrote:
> 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.
>
> Some questions.
>
> 1.
>
> 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.


slope = dy/dx = a1 + 2*a2*x

>
> 2.
>
> 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)?


Yes. Think about a2 as if it were c1-b1, which "uses up" the u-level
factor. After summing over length intervals and folds, you have a 3-
way design: set X subset X method. (Note that "X" in this context
usually denotes a Cartesian product. Your use of X in "coreXcomp" and
"nonrandXrand" may be misunderstood by others. It would be better to
change the "X" to something else, say "/".)

For each set, you would have only 4 values, say A,B,C,D:

non ran
core A B
comp C D

The 2-way (core/comp X non/ran) interaction is A-B-C+D.
It corresponds to the 3-way (core/comp X non/ran X u-lev)
interaction when your were dichotomizing u.

>
> 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.)
>
> 3.
>
> 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.


a1 is well defined. You need to define a measure of "average slope".
Think about what you want it to mean, what properties you want it to
have and not have. For instance, you might get the slope at each data
point and then use their literal average, a1 + 2*a2*mean_x. But what
if cells with different x-means give the same a1 and a2? Should their
"average slope" measures be the same or different? That's the kind of
question you have to ask yourself.

>
> 4.
>
> Does anything you've said change if we use your suggested u/(1+u)
> instead of u itself?


No. That's why I called the variables 'x' and 'y'.
Nothing I said was specific to your variables.