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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?

Addendum

Here is what I managed to find in Jon Wakefield's 2013 book on Bayesian and Frequentist Regression Methods (published by Springer):

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    $\begingroup$ 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. $\endgroup$
    – COOLSerdash
    Commented Aug 3, 2022 at 19:20
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    $\begingroup$ Thank you very much for your comment, @COOLSerdash. I asked Question 2 on Twitter and the feedback I got there was that it doesn't make sense to fit a smooth with just 2 values of x per subject. But my take is similar to yours - that it would make sense, provided we adopt an interpretation similar to what I outlined in my own stab at answering Question 1. I just wanted to check that I am not off in my interpretation - this has been driving me crazy for a while now. $\endgroup$
    – Isabella Ghement
    Commented Aug 3, 2022 at 19:44
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    $\begingroup$ 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? $\endgroup$
    – COOLSerdash
    Commented Aug 3, 2022 at 19:48
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    $\begingroup$ The value of x is constant within a subject since x is a between-subject predictor. Each subject has 2 values of y (outcome), 2 values of f (factor; within-subject predictor) and 1 value of x (continuous; between-subject predictor). I kept things simple but the additional complication is that y does not have a (conditional) Gaussian distribution - it actually has a zero-and-or-one-inflated beta distribution. But Questions 1 and 2 would be similar. $\endgroup$
    – Isabella Ghement
    Commented Aug 3, 2022 at 20:05
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    $\begingroup$ 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. $\endgroup$
    – COOLSerdash
    Commented Aug 3, 2022 at 20:12

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