# Re: st: Poisson model with interaction term

 From Maarten buis <[email protected]> To [email protected] Subject Re: st: Poisson model with interaction term Date Tue, 26 Feb 2008 10:49:03 +0000 (GMT)

```Just to make things more complicated, I have a problem with the
approach by Norton and collegues.

Say we have two explanatory variables, called x1 and x2, than an
interaction effect is, how much does the effect of x1 change when x2
changes. Norton et al. deal with the case when we have non-linear model
and we are interested in the effect on the untransformed dependent
variable than the computation.

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.
This is most easily seen in a graph. In case of a logistic regression
the marginal effect of x1 is the slope of the curve of the probability
against x1 (In case of poisson it the the slope of the curve of the
rate against x1) In the graph that is created by the code below you can
see the marginal effects of x1 when x1 == 0 when x2==0 and x2 == 1 when
the logistic regression equation is:

invlogit(pr) = x1 - 2*x2

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.

-- Maarten

*-------------- begin graph ----------------------------
// Marginal effects at x = 0
local marg1 = invlogit(-2)*invlogit(2)*2
local marg2 = invlogit(0)*invlogit(0)*2

// graph
twoway function y = invlogit(2*x-2), range(-2 2)    ///
lpattern(shortdash)                       || ///
function y = invlogit(2*x), range(-2 2)   || ///
function y = invlogit(-2) + `marg1'*x,       ///
range(-.5 .5) lpattern(solid)             || ///
function y = invlogit(0) + `marg2'*x,        ///
range(-.5 .5) lpattern(solid) xline(0)       ///
xtitle(x1) ytitle(probability)               ///
legend(order( 1 "effect when" "x2==1"        ///
2 "effect when" "x2==0"        ///
3 "marginal" "effects" ))
*--------------- end graph -----------------------------
(To see the graph, run this in Stata as described in
http://home.fsw.vu.nl/m.buis/stata/exampleFAQ.html#work )

--- Maarten buis <[email protected]> wrote:

> The formulas can be found in section 2.3 here:
> http://www.unc.edu/%7Eenorton/NortonWangAi.pdf
>
> Where in case of a poisson with a standard link function:
> F(u)=exp(u); f(u)=F'(u)=exp(u); f'(u)=F''(u)=exp(u)
>
> Hope this helps,
> Maarten
>
> --- Lloyd Dumont <[email protected]> wrote:
>
> > Actually, I am running count models on panel data
> > using xtpoisson and xtnbreg, but have the exact same
> > question.  (But mine really does include a count as a
> > dep var.)  How do I make sense of the interaction
> > term?  I am fairly sure I cannot just add the
> > coefficients of main effects and two-way interactions
> > in this case.  I don't even think I can take the
> > significance of the coefficient on the interaction
> > term seriously.
> >
> > Thanks as always.  Lloyd Dumont
> > --- [email protected] wrote:
> >
> > > thank you Maarten, hence I can simply apply the
> > > linear case formulas...
> > >
> > > thanks again
> > > Maria
> > > Citazione Maarten buis <[email protected]>:
> > >
> > > > --- [email protected] wrote:
> > > > > I�m estimating a Poisson model, which includes
> > > an interaction term
> > > > > and I need to compute the impact (marginal
> > > effect) of x1 on lnY.
> > > > >
> > > > > I have found on SJ an article �Computing
> > > interaction effects and
> > > > > standard errors in Logit and Probit models�, by
> > > Norton, Wang and Ai
> > > > > (2004), who warn that for nonlinear models
> > > > <snip>
> > > > >  Please notice that my interest is  computing
> > > the effect of x1 on lnY
> > > > > ,  I�m not interested in the marginal effect of
> > > the interaction term,
> > > > > nor in the effect of  x1 on E(Y), because my
> > > dependent variable is
> > > > > not a count.
> > > > <snip>
> > > >
> > > > If you are only interested in the effect on ln(y)
> > > than it is no longer
> > > > a non-linear model, so the article by Norton et
> > > al. is no longer
> > > > relevant.
> > > >
> > > > -- Maarten

-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands

Buitenveldertselaan 3 (Metropolitan), room Z434

+31 20 5986715

http://home.fsw.vu.nl/m.buis/
-----------------------------------------

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