[R-sig-ME] interpreting interactions

Ben Bolker bbolker at gmail.com
Tue Apr 8 02:50:55 CEST 2014

Joshua Hartshorne <jkhartshorne at ...> writes:

> A colleague recently made the argument that interaction terms in logistic
> regression are uninterpretable, citing Ai & Norton (2003)
>  Interaction terms
> in logit and probit models. On reading the paper, it seems to make the
> weaker claim that interaction terms of continuous predictors may be
> calculated incorrectly in 2003-era STATA, and that one should take care to
> calculate them correctly.
> But this did make me wonder whether there are any issues in interpreting
> interpreting interaction terms for 'binomial' models in lmer. Can anyone
> comment?
> Josh

  This topic was new to me.  As far as I can tell from my reading of
the paper, it's extremely important to make the distinction between
interaction _terms_ and interaction _effects_.  Again as far as I can
tell, the interaction _terms_ correspond exactly to the estimated
coefficients, and are relevant on the scale of the linear predictor
(where everything is indeed linear).  The interaction _effects_,
in contrast, seem to be defined on the response scale. Because there
is a nonlinear transformation between these scales, there is
not necessarily an intuitive correspondence between expected 
differences-in-difference (cross derivatives) on the linear predictor
scale (terms) and the response scale (effects).

  Not being an applied econometrician, I don't really understand why
one would want to do a statistical test of an interaction _effect_
rather than an interaction _term_.  To me it makes most sense to
do statistical tests on the scale of the linear predictor where
everything is linear and (relatively) simple ...

  As far as how this applies to GLMMs; I don't know, but
there is an additional level of variation and/or averaging that may raise
issues depending on whether you're trying to understand 
population-level, conditional, or marginal effects ...

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