Let me say at the outset that I applaud any conscientious effort to examine the
plausibility of nonlinear and/or nonadditive models. Notwithstanding that, my
answer to R. Allen Reese's question, "Do cont*cont interactions make sense?" is
still yes. If it's always a leap of faith to have a linear additive model, then
perhaps one might want to think twice about creating interaction terms that
assume linear relationships. In the absence of a solid theory that suggests an
effect is anything but linear, however, a linear specification is often a very
good approximation to the true form of the relationship. It is also quite
possible for a linear relationship to appear nonlinear in a particular sample
and estimating it as such amounts to misspecification error and over fitting the
data. Only a strong theory can get around the problem. It would be nice if we
had theories that provided the rationale and clearly described the shape of
relationships, but more often than not we usually work with theories that are no
more clear than "Y increases when X increases." A linear relationship in this
case is as "correct" as any other and is conceptually and mathematically simpler
than just about any other specification.
Now, the fact that one can create simple interactions with multiplicative terms
doesn't mean you have to limit yourself to that form. You can, for instance,
construct interactions with squared and cubed effects to test for some uniform
nonlinearities. In essence, a simple multiplicative term says that the *effect*
of one constituent variable on the outcome of interest depends on the value of
the other constituent variable in a linear manner. I could easily be mistaken,
but I don't think it means "that a k increase in X_i together with an l increase
in X_j should give the same additional increase as an l increase in X_i with a k
increase in X_j." In fact, I wouldn't mind seeing the algebra that led to this
assertion.
Dave Moore
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]]On Behalf Of David Airey
> Sent: Thursday, June 26, 2003 8:59 AM
> To: [email protected]
> Subject: re: st: interactions
>
>
[clip]
> > Do cont*cont interactions make sense? It is often an act of faith
> > that a
> > unit increase in X_i produces a Beta_i increase in Y and that the Betas
> > are additive. It seems wildly optimistic that a k increase in X_i
> > together with an l increase in X_j should give the same additional
> > increase as an l increase in X_i with a k increase in X_j.
> >
> > This seems in general an approach based on having columns of figures
> > so a
> > computer will always do the arithmetic, whether or not the sum is
> > justified.
>
>
> I'm guilty of that last criticism.
>
> Is the reason xi was designed to give the interactions it does based on
> any truth to your criticism?
>
> -Dave
>
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