Peter Flom wrote earlier today (in part):
"[...]
Recently, I came across an article claiming that this [dealing with
interactions in a logistic model much as one dies in linear regression] is
incorrect. The article is in the STATA Journal, and was sent to me by a
client, I don't use STATA. The full cite is
Norton, EC; Wang, H; Ai, C (2004). Computing interaction effects and
standard errors in logit and probit models. The Stata Journal, v 4 no 2, p
103-116.
They claim that one needs an entire new method to deal with interaction
terms in logit and probit models;
[...]"
I'm replying to the list, not because I want to start a discussion, but
because the article raises issues about how interactions are defined, and I
want to check whether there's something drastic missing in my understanding
of all this.
There is a copy of the article available on the Internet (which can be found
by doing a Google search for its title). Very briefly, the authors proceed
as follows. They note that, in a linear model where the interaction of two
explanatories x_1 and x_2 is included, the mean response as a function of
the explanatory variables x_1, x_2, ... Is
f(x_1, x_2, ...)=beta_1 x_1 + beta_2 x_2 + beta_12 x_1 x_2 + more terms,
and that the interaction parameter beta_12 is the cross partial derivative
of f with respect to x_1 and x_2 (i.e. partial d^2 f by dx_1 dx_2). (Or
something similar in terms of differences for categorical explanatories.)
Then they point out that, in nonlinear models such as probit models or
logistic models, this isn't the case --- the mean function is
f(x_1, x_2, ...)=g(beta_1 x_1 + beta_2 x_2 + beta_12 x_1 x_2 + more terms),
where g is the inverse of the link function (if it's a GLM) and so the cross
partial derivative of f is no longer the interaction coefficient beta_12,
it's something more complicated.
This is of course true and unexceptionable. The bit I don't get is that they
seem to be saying that the 'full interaction effect' is *defined* as this
cross partial derivative, and that people have made mistakes by saying that
the interaction 'is' beta_12 in such a model, and they describe a STATA
command that allows one to calculate the 'full interaction effect' properly
in certain models. Well, I would not of course want to deny that people make
misinterpretations involving interactions, but until I read the paper I was
not aware that anyone did define interactions in terms of partial
derivatives in this way. Have I been missing something, and if so, can
anyone give me relevant references?
Regards,
Kevin McConway
Senior Lecturer in Statistics
Department of Statistics
The Open University
Walton Hall
Milton Keynes MK7 6AA, UK
Phone: +44-1908-653676
Fax: +44-1908-652140
email: [log in to unmask]
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