# three way interaction in lmer

I have two continuous variables (cfreq, and LanPro), and one catergorical variable (cond_aud, as shown in the picture). The summary of a lmer model shows a three way interaction of aud (EA and NoA) x LanPro x cfreq. I used emmip function to make an interaction plot, which shows that the difference between different levels of "LanPro" is smaller when "cfreq" is higher in EA than in NoA, but I am not sure whether my understanding is correct and don't know how to interpret this interaction and test the significance for the difference. Any advice would be much appreciated.

A few things are evident from the plot:

• In all cases association of cfreq with the outcome is negative and linear
• Increasing values of LanPro are associated with lower values of the outcome.
• The differences between the levels of LanPro are lower at higher values os cfreq, however this is more pronounceed in cond_aud = EU than the other two groups.

The latter point seems to be the focus of the question. Looking at the 1st and 3rd plot, the differences between the levels of LanPro are lower at higher values os cfreq, however this appears to be very small, whereas in the middle plot it is far more pronounced. That is the two-way interaction between cfreq and LanProshould be greater at the EA level of cond_aud than the other two levels. That means we would be looking for a meaningful 3-way interaction term involving the EA level of cond_aud. Specifically, if CA is the reference level for cond_aud then the model would estimate two 3-way interaction terms: cfreq:LanPro:cond_audEA and cfreq:LanPro:cond_audNoA, and we would expect the former to be meaninfully larger than the latter

Assuming that you have repeated measures within subjects, or some other kind of clustering (otherwise why are you using lmer?), then you can explore this with the following model:

Y ~ cfreq * LanPro * cond_aud + (1|subject)


This will estimate, assuming CA is the reference level for cond_aud:

• the overall intercept which will be the estimates mean response for CA when cfreq and LanPro are zero.

• 4 main effects (one each for cfreq and LanPro and 2 for each non-reference level of cond_aud) for each of the variables which will tell, for cfreq and LanPro the association of a 1 unit change with the outcome when LanPro is at it's reference level (ie the respective slopes), and for LanPro the asssociation of the difference between its reference level and the other two levels, with the outcome, when cfreq and LanPro are zero.

• 5 2-way interactions: cfreq:LanPro, cfreq:cond_audEA, cfreq:cond_audNoA, LanPro:cond_audEAand LanPro:cond_audNoA, which can be interpreted as the interaction involving those variables when the 3rd variable is zero (or for cfreq:LanPro when cond_aud is CA.

• 2 three way interactions which can be interpreted as above to answer the research question

• Your explanation is very helpful. Based on your model above (except that I use NoA as baseline), does a main effect of LanPro indicate that the outcome decreases when the values of LanPro increases regardless of the cfreq and the cold_aud conditions? Or does it only apply to the reference level (i.e., when conditions_aud is NoA)? Aug 5, 2020 at 1:47
• You are welcome. It is conditional on cond_au being at its reference level (and cfreq being at 0). This interpretation of main effects holds when it is involved in an interaction. However, it is obvious from the plots that it also applies unconditionally. You don't need a test. Aug 5, 2020 at 4:10
• Hi would you be able to help with a similar issue here: stats.stackexchange.com/questions/529850/…? Really appreciate your expertise. Jun 8, 2021 at 13:59