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).
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:
Both groups contain subjects having the SAME random subject effect (e.g., random subject effect = 0 for subjects in both groups);
Group A has x = something; Group B has x = something + 1 (so they differ by 1-unit in the value of x);
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.)
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?
Here is what I managed to find in Jon Wakefield's 2013 book on Bayesian and Frequentist Regression Methods (published by Springer):