Date: Dec 1, 2012 4:46 PM
Author: Ray Koopman
Subject: Re: Interpretation of coefficients in multiple regressions which<br> model linear dependence on an IV

On Dec 1, 11:00 am, Ray Koopman <koop...@sfu.ca> wrote:
> [...]
>
> There are two average slopes. Call them Av1 and Av2:
>
> dy/dx1 = a1 + a3*x2 ==> Av1 = a1 + a3*mean_x2
>
> dy/dx2 = a2 + a3*x1 ==> Av2 = a2 + a3*mean_x1
>
> To estimate the coefficients, give your regression program
> three predictors: x1, x2, and x3 = x1*x2. Then
>
> var[Av1] = var(a1) + 4*var(a3)*(mean_x2)^2 + 4*cov(a1,a3)*mean_x2,
>
> var[Av2] = var(a2) + 4*var(a3)*(mean_x1)^2 + 4*cov(a2,a3)*mean_x1,


Those are a good example of the sort of mistake it is easy to
make when cutting and pasting instead of typing from scratch.
The correct variances are

var[Av1] = var(a1) + var(a3)*(mean_x2)^2 + 2*cov(a1,a3)*mean_x2,

var[Av2] = var(a2) + var(a3)*(mean_x1)^2 + 2*cov(a2,a3)*mean_x1.

Also, there is a covariance that might be needed sometime:

cov[Av1,Av2] = cov[a1,a2] + var[a3]*mean_x1*mean_x2 +
cov[a1,a3]*mean_x1 + cov[a2,a3]*mean_x2.