Interpreting interactions in linear mixed effect model using lmer

Question

I am building a linear mixed effect model using the lmer function from the lme4 package in R but I am struggling to interpret the interactions terms in the model. I have used the following syntax:

mod2 <- lmer(post.diff ~ #my predicted DV
            course * group
             #my fixed effects
             + (1|bib)
             #my random effects
             , dat, REML = FALSE)

The two factors - course and group - are dummy coded variables. This gives me the following output:

Fixed effects:
               Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)    -0.26080    0.18036 56.22506  -1.446    0.154    
courseB         0.87647    0.09257 94.00000   9.468 2.49e-15 ***
courseC         2.38860    0.09257 94.00000  25.802  < 2e-16 ***
group1         -0.20996    0.26361 56.22506  -0.796    0.429    
courseB:group1  0.09664    0.13531 94.00000   0.714    0.477    
courseC:group1  0.10678    0.13531 94.00000   0.789    0.432   

Using the anova() function, I can see that there is no main effect of group nor any interactions:

Type III Analysis of Variance Table with Satterthwaite's method
              Sum Sq Mean Sq NumDF DenDF  F value Pr(>F)    
course       142.305  71.152     2    94 664.1974 <2e-16 ***
group          0.034   0.034     1    47   0.3188 0.5750    
course:group   0.081   0.041     2    94   0.3795 0.6853   

Still, I want to better understand how I can interpret the output from my model. From my understanding:

-0.26080 (intercept) is the estimated mean for the group codes with 0 in course A

0.87647 is the estimated simple slope for the group codes with 0 in course B

2.38860 is the estimated simple slope for the group codes with 0 in course C

-0.20996 is the estimated simple slope for the group codes with 1 in course A

My question is how I should interpret the interactions terms in the model. I hope someone can help me so that I can see it an equation form.

Answer 1

I'm going to change the terminology a bit as I find that talking consistently and explicitly about coefficients representing differences in associations with outcome under treatment/dummy coding help a lot when you get to the level of interactions. I'll keep your interpretation of the intercept, but from there on I recommend saying:

0.87647 is the estimated difference between course B and course A for group 0;

2.38860 is the estimated difference between course C and course A for group 0;

-0.20996 is the estimated difference between group 1 and group 0 in course A.

Then the interpretation of interaction coefficients is just in terms of the next level of differences in estimates.

The courseB:group1 interaction of 0.09664 is the extra estimated difference from course A and group 0 beyond the sum of individual differences associated with group B and group 1 above, and:

the courseC:group1 interaction of 0.10678 is the extra estimated difference from course A and group 0 beyond the sum of individual differences associated with group C and group 1 above.

Some find that writing things out explicitly in terms of the 0/1 False/True entries used in the dummy-coded design matrix helps with understanding. For fixed effects you have the following model, with 0/1 coding except for intercept, and * representing actual multiplication (not the R expansion into individual and interaction terms):

outcome ~ intercept + isCourseB + isCourseC + isGroup1 + (isCourseB * isGroup1) + (isCourseC * isGroup1)

with the corresponding coefficients in the order shown in your summary display. Each interaction term is non-zero only when both of the individual dummy codings is non-zero. Reading left to right shows how each coefficient represents a difference from what had already been predicted. This can extend to higher-order interactions if present in a model.