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Re: st: Poisson model with interaction term
On Feb 26, 2008, at 4:49 AM, Maarten buis wrote:
The problem I have is this: In the case of non-linear models you
would expect the effect of x1 to change when x2 changes even if we
do not enter the interaction term.
<snip>
I think (but I am not sure) that the method by Norton et al. gives
the combined change in the effect of x1, i.e. the change in effect
of x1 that would have occured anyhow and the change in effect due
to the interaction term together. I think that in many case this
would be reasonable, but I can also imagine situations where you
just want to know the effect of the interaction term net of the
change in effect that would occur anyhow.
Just to make sure I understand the question, if you define the
interaction effect as Norton et al. do (i.e., as the mixed partial
derivative of the mean of y), then, as they note, this may be non-
zero even if the coefficient on the interaction term itself (i.e.,
the interaction term in the linear predictor) is zero. I believe
that this is what you are referring to in the first quote above.
Now, if you fit a model without the interaction term (i.e.,
constraining its coefficient to be zero), you can compute the
interaction effect (i.e., the mixed partial derivative); call this
A. You can then refit the model with the interaction term included
(let's assume the estimated coefficient is non-zero), and recompute
the interaction effect (call this B). What you are asking, I think,
is if there is a way to decompose B into A plus another part that is
due to the addition of the interaction term to the model. Is that
correct?
If so, I don't think it can be done. This is because, when you add
the interaction term to the model, the coefficients on the first
order terms will change. IOW, A and B are computed with different
values for the same quantities, and thus you can't decompose B in
terms of A. You could, I suppose, simply compare A and B
numerically, however, if the model without the interaction term
doesn't fit the data well, interpretation of A is problematic.
Don't know if that helps at all...
-- Phil
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