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I understand, that in (generalized) mixed models, the calculated $\beta$s are not population ones, but rather "conditional to the random effect". Let's say, I have a model, where I assess a set of animals, several times, so the model includes the animal ID as the random effect: response ~ Treatment + Time + (1|AnimalID).

I was told, that the interpretation will be: the change (say ratio of the log odds) between two treatments for a certain animal. OK, but WHICH CERTAIN animal? I have just single $\beta_{treatment}$ returned by the GLMM.

Yes, there is a long list of model coefficients (intercept and slopes) per EACH ANIMAL, returned by functions like the lmer in R - so the variance of them can be calculated, but I am asking about the part displayed by default:

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -1.3605     0.2276  -5.978 2.26e-09 ***
trtB         -0.9762     0.3033  -3.219 0.001288 **

If there is just one $\beta$, it must be, somehow, averaged over the animals. How does it work exactly?

Response_for_animal_1 averaged with Response_for_animal_2...? But how this differs from GEE, where it's averaged over all animals?

What I obtain is not:

  • $\beta_1$ for change in animal #1
  • $\beta_2$ for change in animal #2
  • $\beta_...$ for change in animal #...
Gerard
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  • Check this post: https://stats.stackexchange.com/questions/451373/what-does-it-mean-exactly-to-have-marginal-population-averaged-or-subject-spec and this one: https://stats.stackexchange.com/questions/365907/interpretation-of-fixed-effects-from-mixed-effect-logistic-regression/365918#365918 – Dimitris Rizopoulos Feb 27 '20 at 11:38
  • Does this answer your question? [Interpretation of Fixed Effects from Mixed Effect Logistic Regression](https://stats.stackexchange.com/questions/365907/interpretation-of-fixed-effects-from-mixed-effect-logistic-regression) – Dimitris Rizopoulos Feb 27 '20 at 11:39
  • I'm also interested in the same question. The links, that you mention, dear Dimitris, say "particular patient" in a centre. OK, now our cluster is the patient, having multiple assesments. I can relate to the asker in this thread, as I also have hard time to get this idea. Given just a single beta coefficient, to which "particular" patient does it refer? It has to refer to all of them, so this seems to be somehow averaged over them. The change in response in patient_1 is rather different than patient_2. So how can a single number describe them both, of not by averaging? But that's the GEE... – Damasco Feb 27 '20 at 22:08
  • I can see, reading the questions, that more people struggle with this concept, even if it is trivial to people who already know it and find it obvious :) There is a difference, sure, I saw the formulas (when the link is non-identity), but I'm stuck with understanding how just a single beta in mixed model is "conditional to a patient" and is the same (single beta is reported) for all of them. I run a linear mixed model in R and there was an item, in the model object, with separate betas for each patients. It's clear to me-own regression lines. But now it's about the single set of betas for all. – Damasco Feb 27 '20 at 22:14
  • @Damasco even though the beta coefficient is one, it is **not** averaged over the subjects. This is the same situation as in a simple additive model in which you have age and sex as predictors. You get one coefficient for age, and it has the interpretation of the effect of age for subjects of the *same* sex. It is not averaged over males and females. Nor you get separate coefficients for males and females. – Dimitris Rizopoulos Feb 29 '20 at 07:57
  • @DimitrisRizopoulos Thank you, dear Dimitris, for the answer. So does it mean, that if my "cluster" is the same patient, the interpretation is: "a unit change in X1 causes _certain change_ (depending on the model) in the response for a certain patient" - but since the beta is same for all patients, it means that the relationship is valid for all of them (but this is not averaged, even if single). And in GEE it means "a unit change in X1 causes _certain change_ in the response in average" - there's still single beta, but now it's calculated differently, yes? Am I right, or went far away? :) – Damasco Feb 29 '20 at 18:22

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