# How to interpret intercept in linear mixed effects with two categorical predictors with three levels?

I'm having a bit of a hard time interpreting the results of this linear mixed-effects model:

happy_prob ~ height_shuffle * height_original + (1 | template)


Where height_shuffle and height_original are two categorical variables with three levels (high, mid, low) referring to vowel height (e.g., /i/ is high while /a/ is low).

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)                           4.993e-02  5.475e-03  8.349e+02   9.119   <2e-16
height_shufflelow                    -2.762e-03  3.203e-03  4.692e+03  -0.863   0.3885
height_shufflemid                    -9.682e-04  5.184e-03  4.727e+03  -0.187   0.8519
height_originallow                   -3.693e-03  7.857e-03  5.312e+02  -0.470   0.6386
height_originalmid                    8.528e-04  5.643e-03  3.857e+02   0.151   0.8800
height_shufflemid:height_originallow  1.273e-02  7.346e-03  4.737e+03   1.733   0.0831


I'm not really getting how to interpret the intercept. What I am trying to understand is whether shuffling a vowel in a name predicts happiness as a function of the original vowel. Now given that the shuffling follows a logic, if this is the case, then it means that vowel height plays a role. In addition to that, I am considering the template of the word as a random effect.

What I am not getting is how I should interpret the reference group since there are three levels in both categorical variables. For example, is it my intercept height_shuffle_high:height_original_high?

The intercept is the expected value when both categorical variables are set at their 'reference' level. The choice of reference level is done by your stats program if you don't specify it. R chooses it based on alphabetical order, so 'high' is the reference level for both variables because h(igh) comes before l(ow) and m(id). It's possible to change this reference to anything you desire but it doesn't really matter (especially if you plot your model output to understand it, which I recommend).

In this case, the intercept therefore represents the expected value when height_shuffle = high and height_original = high.

Note that there's nothing specific to mixed models here, this applies to standard regression as well.

• Thank you for your answer! I am struggling a bit with visualizing the results since I have 258 nested groups (the template). Do you recommend anything? Commented Jun 22, 2023 at 9:51
• @MarcoB That does sound tricky, and I'm not sure there is a great solution. I would start by visualising the fixed effects and random effects separately.
– mkt
Commented Jun 22, 2023 at 9:55
• @MarcoB But that sounds like a worthwhile question in itself - why don't you post it as a new one? Ideally with a reproducible example.
– mkt
Commented Jun 22, 2023 at 9:57
• Yes, you're right. I will add a separate question. I was thinking that maybe I could start by showing only some templates to get an idea. Let's see if someone has a better idea. Thanks again! Commented Jun 22, 2023 at 13:25