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I'm conducting LMM by adding three independent variables (A, B, C) and a covariate (E) as the fixed effect, and the random intercept for each subject as the random effect in the model, as shown here below:

model <- lmer(RT_log ~ A * B * C * E + (1|Subject)

The results showed a significant three-way interaction between A, B, and E, of which the covariate E is a continuous variable (0-100). As we know that if E has only two or three levels, we can further conduct two LMM analyses, by adding two dependent variables (A, B) as the fixed effect and the same random effect as here above, for the two levels of E, separately. I was wondering how should I do if the E is a continuous variable in my case? Thanks in advance for your any suggestion!

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    $\begingroup$ Why do you speak of dependent variables when they are on the right-hand side of the equation, and what is the difference between those and the "covariate" E? $\endgroup$
    – stefgehrig
    Dec 31, 2020 at 11:18
  • $\begingroup$ Sorry, it should be "independent variables", rather than "dependent variables" as I stated before. The difference between them and the "covariate" E is that independent variables are what we manipulated in the experiment, each have two levels; whereas "covariable" E is the subjective score reported by participants, which is a continuous number. As I'm interested in whether this score interact with other independent variables, it is therefore included in the fixed effects as well. $\endgroup$
    – Qian
    Jan 3, 2021 at 6:11
  • $\begingroup$ @Qian: Please do these corrections/clarifications as an edit to the post, not only in comments. Not everybody reads comments ... $\endgroup$
    – kjetil b halvorsen ♦
    Jan 8, 2021 at 18:33

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The distinction between independent variable and covariate is unimportant for the way you specify the model. It also doesn't matter whether E is continuous or categorical. If E is continuous then it makes the interpretation less cumbersome, however it is important to note that when a variable is involved in an interaction, then the estimates are conditional on the other variable(s) it is interacted with being zero. Since E has a scale of 0 to 100 it may make more sense to centre this variable around zero (which will correspond to 50 on the original scale)

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  • $\begingroup$ Thanks a lot for your useful suggestions Robert, all variables are mean-centered. I have a further question, as I found the three-way interaction between A, B, and E, in order to explore the-way interaction pattern, can I divide subjects into two groups according to the median value of E, and then explore the interaction pattern between A and B in the two groups, separately? $\endgroup$
    – Qian
    Jan 9, 2021 at 8:43
  • $\begingroup$ You're welcome. If you include E in an interaction, then there is no point in splitting the dataset by E. Also, spliting the dataset will result in a loss of statistical power. $\endgroup$
    – Robert Long
    Jan 9, 2021 at 18:54

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