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Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms

From   Maarten buis <>
Subject   Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms
Date   Fri, 20 Aug 2010 07:29:39 +0000 (GMT)

--- On Thu, 19/8/10, wrote:
> -mfx does not recognize interaction terms and thus does not
> calculate the marginal effects for interaction terms in the
> correct way (the problems resulting are explained in Ai/
> Norton and Greene (2010).

The problem is that marginal effects try to force a from of
effect onto a model that is not native to that model. The 
consequence is that it will always give some problems, like
the difficulty you get with getting interaction effects. It
is like fitting a square peg into a round hole, you might 
force it in with a hammer (possible add a bit of duct tape),
but the result will never be nice. The obvious solution is
to use the form of effect that is native to the model, unless
you for substantive reasons really need that other type of
effect. In case of (multilevel) logistic regression that means 
that you need to look at odds ratios, and the interaction 
effects will give you ratios of odds ratios. This is discussed 
in Buis (2010).

So, based on my 2010 article I recommend that you just report
odds ratios, and interpret the interaction terms of ratios of
odds ratios. This article gives you a concrete example of what
such a discussion of results could look like, it is actually
much easier than many people think.

This article has led to quite a bit of private communication
with people who are so used to the idea that interactions are
very hard in non-linear models that they are surprised that
the solution can be so easy. So below I will give a bit longer
explanation that liberally borrows from these private 

Remember that an effect is nothing other than a comparison of
the expected outcome across real or counter-factual groups. We 
observe incomes for a set of males, we find a comparable set
of females who work (that can be hard, which is why we have 
models like -heckman-) and a comparison of the average wages
of the males and females is the effect of gender. We can 
compare average wages by looking at the difference or at the
ratio, i.e. women earn x euros/pounds/yen/... less then men, 
or women earn y% less then men. Both are completely valid
quantifications of the effect of gender on income. 

We usually start our statistical (econometric/psychometric/
...) modelling education with a linear model. Linear models 
are naturally designed for a comparison of groups in terms of 
differences.  If we add a continuous variable x than its 
parameter says if we get one more unit of x we can expect b 
more units of y, regardless of how many units of y you had to 
start with. So we took all possible comparisons that where one 
unit apart and constrained the differences in expected y to be 
equal. We can relax such assumptions, but it shows that effects
in terms of differences is the native way of thinking about 
effects in linear models. 

The native way of thinking about effects is in terms of ratios in 
non-linear models that include a log in their link function 
(e.g. (multi-level) logistic regression that models the log(odds),
Poisson that models the log(count), survival models that model the
log(hazard) or log(time), etc.). If we add a continuous variable
in one of those models we say that expected value of y increases 
by a factor of exp(b) when you get an additional unit of x, 
regardless of how many units of y you had to start with. Again 
this assumption can be relaxed, but it shows that effects in terms 
of ratios are the native way of thinking in this type of non-linear 

Now you can use effects in terms of ratios in linear models (e.g.
elasticities) and effects in terms of differences in non-linear 
models (marginal effects), but since they are not the native way 
of thinking in those models and thus there will always be some 
friction. You cannot represent an effect that is constant in 
terms of ratios with one effect in terms of differences, and
vice versa (except for the trivial case where there is no effect). 
This type of problems multiply when looking at interaction 
effects. This is why Ai & Norton (2004) had to go through all the 
effort and than present their interaction effects in terms of 
graphs, while I could just exponentiate my coefficient and had my 
interaction effect as one number.

The mathematical proof is straightforward, and can be directly 
derived from the properties of the logarithm. A version of that 
proof can actually be found in the Stata Journal article of Edward 
Norton, Hua Wang, and Chunrong Ai (2004), where they discuss the 
implementation of their technique in Stata. In section 2.6 of 
(Norton et al. 2004) they showed that the exponentiated 
interaction effect is the ratio of odds ratios. They claim that 
nobody can understand that, in my article (Buis 2010) I show that 
that is actually not that hard.

In short Norton et al. and I don't disagree on the math. The choice
between which method to choose is really one that needs to be based
on substantive and pragmatic reasoning. If your theory gives you a
clue whether or not you want to control for differences in the
baseline odds, than the choice is easy: with level effects you don't
control for such differences, with ratio effects you do. When your
theory is not that precise, then you need to use pragmatic reasoning.
With odds ratios you need to put a bit of effort into explaining
your results as a lot of people find them hard. With marginal effects
you have the awkwardness of trying to force a linear model on top
of a non-linear one. Sometimes it is nice to present both, in the
way that I have done in my article.

Hope this helps,

M.L. Buis (2010) "Stata tip 87: Interpretation of interactions in 
non-linear models", The Stata Journal, 10(2), pp. 305-308.

Edward Norton, Hua Wang, and Chunrong Ai (2004) "Computing 
interaction effects and standard errors in logit and probit 
models" The Stata Journal 4(2):154--167.

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


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