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Re: Interpretation of coefficients in multiple regressions which
model linear dependence on an IV
Posted:
Dec 2, 2012 5:48 PM
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On Nov 21, 8:13 pm, djh <halitsk...@att.net> wrote:
> In a different thread, Ray Koopman explained that if one
> suspects these regressions to be dependent on the IV ?u?:
>
> c on u
> c on e
> c on (e,u)
>
> then under the usual initial assumption that the dependence
> is linear, these three regressions should be modified to:
>
> 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)
In this post I want to talk about only your third case:
y = a0 + a1*x1 + a2*x2 + a3*x1*x2 + a4*x2^2.
(As before, I use the usual generic variable names so as not to
get caught up in any peculiarities of your particular variables.)
One interpretation of the model is that y is a linear function of x1
and x2: y = A0 + A1*x1 + A2*x2, with A0, A1, and A2 all being linear
functions of x2. (I'll let you do the algebra; it parallels that in
the first two models.) Note that this model does not treat x1 and x2
symmetrically: it matters which x is 1 and which is 2.
There are two average slopes. Call them Av1 and Av2:
dy/dx1 = a1 + a3*x2
dy/dx2 = a2 + a3*x1 + 2*a4*x2
Let m1 = mean_x1 and m2 = mean_x2.
Av1 = a1 + a3*m2
Av2 = a2 + a3*m1 + 2*a4*m2
var[Av1] = var[a1] + var[a3]*m2^2 + 2*cov[a1,a3]*m2
var[Av2] = var[a2] + var[a3]*m1^2 + 4*var[a4]*m2^2 +
2*cov[a2,a3]*m1 + 4*cov[a2,a4]*m2 + 4*cov[a3,a4]*m1*m2
cov[Av1,Av2] = cov[a1,a2] + var[a3]*m1*m2 +
cov[a1,a3]*m1 + cov[a2,a3]*m2 +
2*cov[a1,a4]*m2 + 2*cov[a3,a4]*m2^2
df = n-5.
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