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st: Re: Stata tip 87: Interpretation of interactions in non-linear models

From   Maarten Buis <>
Subject   st: Re: Stata tip 87: Interpretation of interactions in non-linear models
Date   Wed, 18 May 2011 17:55:02 +0200

On Wed, May 18, 2011 at 4:48 PM,  andre ebner wrote:
> Regarding your concern to interpret differences between APE please allow
> me one more consideration:
> When comparing differences you write:
>> In your case (with a panel model) this would be a mixture of average
> partial effects while fixing the unobserved group constants at the
> average. This is not very pretty, especially be-cause the variability of
> the individual partial effects for interaction effects is so high that
> it really matters.
> If I understood you correctly, your worry is that this approach does not
> take into account that mean and variance of RE are likely to be different
> across groups (by interacting two discrete variables I get four groups)
> but instead fixes the unobserved group constants at the average.

No I worry about the fact that there is variance, regardless of
whether that variance is the same or different across groups. Think of
the random effects as group specific constants. The value of a
marginal or partial effect depends on that constant. To get _average_
partial effects we need to compute the partial effect for every
individual and than average those individual partial effects. What you
are doing is you first fix the group specific constants at their
average, than compute partial effects and than you average those (over
the distribution of the observed variables). The two are not the same,
as there is a non-linear transformation involved. As long as
everything is nice and (approximately) linear all this does not matter
(much). Unfortunately that is not the case when dealing with
interaction effects in non-linear models. It is not uncommon for
estimated marginal effects for interaction terms to vary from
significantly positive to non-significant to significantly negative
depending on the values of the other covariates. In essence you are
computing an uncomfortable mix between average partial effects and
marginal effects at average values of the covariates.

Anyhow, my main point is that you should _not_ average the individual
partial effects and certainly not compute the marginal effects at the
average. Instead you should take Norton et al.'s point seriously and
accept that when you want to present your results as marginal effects,
the marginal effects of interaction terms will vary widely from
individual to individual, even though the underlying ratio of odds
ratios is constant. This variation tends to be so extreme that trying
to summarize it with one number just does not make sense.

The problem is that you'll end up with a conclusion like "the
interaction term is significantly positive, significantly negative,
and not significant, depending on the values of the covariates", which
is not much of a conclusion, especially if you consider that the
underlying ratio of odds ratios leads to a single unambiguous

Hope this helps,

Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
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