# Question on Interpretation of Between-Subject Effect in a Mixed Effects Model

Setup:

Assume I have a linear mixed effects model of the form: outcome ~ x + f + (1|ID), where x is a between-subjects predictor variable, f is a within-subject factor with 2 levels and ID is a subject ID. Each subject has 2 outcome values (one for each level of f).

Question 1:

What is the correct way to interpret the effect of x in this model?

My Stab at Answering Question 1:

My current interpretation is as follows:

If we compare two groups of subjects, A and B, in the target population represented by the subjects in our study such that:

1. Both groups contain subjects having the SAME random subject effect (e.g., random subject effect = 0 for subjects in both groups);

2. Group A has x = something; Group B has x = something + 1 (so they differ by 1-unit in the value of x);

3. Groups A and B have the same value of f.

then the slope of x in the model y ~ x + f + (1|ID) captures the difference in the mean value of y among groups A and B.

Does this interpretation make sense? If not, what is the proper interpretation of the slope of x? (I should add that, in my case, x is something I have to control for and the effect of f is what I am really interested in.)

Question 2:

Given the above setup, would it make sense to modify the model to allow for a smooth, non-linear effect of x via a smooth of x, so that the modified model looks like outcome ~ s(x) + f + (1|ID)? If yes, what would be the correct way to interpret s(x) in the modified model?  • In the case of a model with an identity link, the coefficient of $x$ is interpreted as you would in a linear regression model, based on this answer. And to question 2: I don't see any reason why not. Personally, my default assumption is that of nonlinear effects, especially for a control variable such as age. I always include those flexibly using splines. Frank Harrell has written a lot about that. Aug 3, 2022 at 19:20
• I might miss something but the number of values per subject seems irrelevant to me. The smooth will fit over the range of $x$ of all subjects (e.g. for age). Does the value of $x$ differ within a subject? Aug 3, 2022 at 19:48
• Thanks for this information. As Dimitris Rizopoulos explains in the linked answer from my first comment: The effect of $x$ is only unconditional on the random effects in the case of an identity link function. In a model with a nonlinear link function, the coefficient is dependent on the random effect and thus has a subject-specific interpretation. I don't see any changes for question 2, however. Aug 3, 2022 at 20:12