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I have read many times that the interpretation of fixed effects estimates differs in a mixed model, compared to a model without random effects. For example this answer from Dimitris Rizopoulos:

Interpretation of fixed effect coefficients from GLMs and GLMMs

However, in GLMMs and because there you do have random effects, the inverse-link transformed regression coefficients have an interpretation for the for the mean of the outcome conditional on the random effects. Most often you are interested in the marginal mean of the outcome averaged over the random effects distribution, but the coefficients you obtained from the GLMMs do not have this interpretation.

I have couple questions about this.

  1. I assume that this apply to generalised mixed models only, and not to linear models. Is that correct ?

  2. What exacting is meant by "conditional on the random effects", and why is this important ? An example of this would be teriffic.

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The interpretation gets complicated when you use a nonlinear link function to connect the mean of the outcome variable with the linear predictor of the model that includes fixed and random effects. Hence, in the standard linear mixed-effects models fitted by, e.g., lmer() (package lme4) or lme() (package nlme), you do not have this problem because they use the identity link function. But if you define a model with a nonlinear link function and normal error terms, you will still have the same issue. You will also have the same problem in some formulations of nonlinear mixed effects models with normal error terms fitted by nlmer() or nlme().

An example and more information on the conditional on the random effects interpretation is given in this post.

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    $\begingroup$ Thank you. I have checked the linked answer, where the example was a patients in hospitals - in a GLMM setting with random interrepts for hospitals - the interpretation of the regression coefficients (fixed effects) is for those subjects in the same hospital - not averaged accross all hospitals. I get that, but what about when there are repeated measurements in patietnts ? Then the estimates are only applicable for the same patient - not averaged over all patients ? Is that right ? $\endgroup$
    – underflow
    Dec 14, 2023 at 10:34
  • $\begingroup$ That's correct. For example, the coefficient for sex in a mixed-effects logistic regression would tell what is the log odds ratio if a particular patient changed sex. It is not the log odds ratio between the group of males vs. the group of females. $\endgroup$ Dec 14, 2023 at 11:30

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