# [R-sig-ME] Interpreting three-way interaction with categorical predictors and binary outcome (GLMER)

Emmanuel Curis emmanuel.curis at parisdescartes.fr
Fri Feb 26 13:47:55 CET 2016

```If you're patient, you may use the tedious but sure way of
1) check how the model is defined exactly
2) write each cell mean using this model
3) write down the exact comparison you want in term of cell means
4) using 2), rewrite this comparison in term of the model
coefficients
==> this will give you the linear combination to test (using
multcomp:glht for instance).

This apply for any model that has a linear part: lm, glm [replace µ by
ln odds in that case], lmer...

Exemple, for a 2-way 2×2 design with interaction, with cell means

A    B
1  µ1A  µ1B
and comparison to do let's say (µ2A - µ1B) =?  (µ1A-µ1B)
2  µ2A  µ2B

Using the default contrats with a reference cell (A,1), the
coefficients in the model are µ0, dB, d2, d2:B with
µ1A = µ0
µ1B = µ0 + dB
µ2A = µ0 + d2
µ2B = µ0 + d2 + dB + d2:B

Then, you're comparison is
(µ2A - µ1B) - (µ1A-µ2B) =
((µ0 + d2) - (µ0 + dB)) -( (µ0) - (µ0 + d2 + dB + d2:B) ) =
d2 - dB - d2 + dB - d2:B =
d2:B

and your « contrast » to use to write your test is
µ0  dA  d2  d2:B
0    0   0     1

This is quite tedious with a three-way interaction and several levels,
and you need to known how were defined each coefficient in the model,
but at least you're certain of what you're testing in the end...

On Fri, Feb 26, 2016 at 12:28:37PM +0100, Francesco Romano wrote:
« One more question regarding the interpretation of interactions.
« I am having difficulty finding information on how to obtain a specific
« contrast in a 3-way interaction of three categorical variables, syntax
« (levels s and of), animacy (levels +AN -AN and -AN +AN), and group (int,
« adv, ns), predicting a binary outcome (correct or inversion).
«
« The anova yielded a significant effect for animacy and an interaction
« between syntax:group. Everything else was non-significant. As I am
« interested in comparing two cells at a time from the three-way table, I
« kept all 3 predictors and their interactions within the model. The best fit
« was given by inclusion of random intercepts for participants and items only.

--
Emmanuel CURIS
emmanuel.curis at parisdescartes.fr

Page WWW: http://emmanuel.curis.online.fr/index.html

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