Linear mixed-effects model equation for correlated and uncorrelated random slopes

Question

I've been asked to provide a linear equation for a lme4:lmer() model that I report in one paper.

I tried to adapt examples from http://rpsychologist.com/r-guide-longitudinal-lme-lmer but this is not my area of expertise and I would like to avoid any guesswork.

Given a repeated measurement experiment dataset where:

• A and B are 2-level factors fully balanced within-subjects
• C is a dichotomous (yes or no response) independent variable
• Y is the outcome
• The 3-way interaction within fixed effects is of main interest

How can I write down a regression style statistical equation for two lmer() mixed-effects models where:

1. random slopes are correlated
   lmer(Y ~ A*B*C + (A*B*C | Subj), data)
2. random slopes are uncorrelated due to convergence errors
   lmer(Y ~ A*B*C + (A*B*C || Subj), data)

PS. Reproducible example of a dataset that simulates my real experiment is posted here: Compute partial η2 for all fixed effects anovas from a lme4 model

Answer 1

The variant with correlated random slopes is your model 1. The variant without correlation is not easily possible using lmer, due to A and B being categorical.

My package afex offers two solutions:

1. lmer_alt which is a wrapper for lmer that allows this. Note that in this case care should be taken that the factor codings are appropriate for interactions with categorical covriates (e.g., "contr_sum").

2. mixed(.., expand_re = TRUE). In your case:
   mixed(Y ~ A*B*C + (A*B*C || Subj), data, expand_Re = TRUE).
   The benefit of using mixed is that it automatically uses appropriate orthogonal contrasts and calculates p-values for the effects.

Information on how afex achieves this is given in section Expand Random Effects in ?afex::mixed.