# What does it mean exactly to have marginal (population-averaged) or subject-specific effects, say in mixed models vs. GEE?

I know that mixed-effect models produce subject-specific answers and the GEE produces marginal effect. But I don't get it very well still, I'm sorry for a silly question. Let me explain what exactly bothers me.

Let's assume we have a

mixed model: Response ~ Covariate_A + Time_point + (Time_point | SubjectID). (random slopes and residuals, more or less the unstructured covariance)

GEE Response ~ Covariate_A + Time_point + (1 | SubjectID)

So, a subject is assessed multiple times (t1, t2... tN). The responses can be correlated across the subject.

Now:

• from the mixed model I get 2 outputs:

1. the standard fixed effects part with SE "corrected' for the covariance + random effects (variances). Are they the subject-level or population-averaged effects? If not, what are they?

2. the individual slopes + intercepts (per each subject). Are they the subject-level effects?

• from the GEE I get only 1 output:

1. the standard fixed effects (adjusted for the intra-covariance in GLS and raw estimates but with sandwich SEs in GEE). Are they the marginal (population-averaged) effects?

Do I interpret those "fixed parts" in a mixed model like subject-specific? So what are the subject-specific slopes and intercepts, returned as a long list (row per subject) by lmer() in R?

Am I right, that in linear models (and not in GLM) the GEE is equivalent to a mixed model in that the population-averaged effects are equivalent to the subject-level ones, only the estimation method changes (and standard errors, as GEE is robust and LMM are model-based)?

What is the sense in having a single number (fixed effect) in LMM and say it's subject-specific, then have also a single number in GEE and say it's population-averaged? Isn't this single number in LMM already averaged over all subjects?

• In a mixed model the fixed effects are conditional on the random effects. Feb 26, 2020 at 13:50

## 2 Answers

A few points:

• In linear models with an identity link function, both linear mixed models and GEEs give coefficients that have the same interpretation. Namely, an interpretation for groups of subjects. Say for example that you want to interpret the coefficient for the sex covariate. Then, the interpretation of this coefficient is the difference in the average outcome between the group of males and the group of females.
• In models with a nonlinear link function there is a difference between the interpretation of the coefficients from the two approaches. In this case, mixed models give coefficients with a subject-specific interpretation. Continuing on the same example as above, and say for a dichotomous outcome for which you have fitted a mixed effects logistic regression, the interpretation of the coefficients for sex is the difference in the log odds if a particular subject changed sex. On the contrary, the GEEs will give you the difference in the log odds between the groups of males and females (as in the linear case).
• Because of this more strange interpretation with nonlinear link function, mixed models have been criticized that are less useful for non-normal data. Nonetheless, there are approaches to solve this issue in mixed models. One recent one is implemented in the marginal_coefs() function of the GLMMadaptive package in R.
• Check also this answer.
• I understand how the fixed-part is subject specific in the odds/probabilities scale because the difference in the expectation of y for a one unit change in x depends on the values of the random effects and covariates, but I am still confused by how it is subject specific in the log-odds scale (I read your response to the other question - I just didn't understand the notation and distinction between subject-specific and marginal in the log-odds scale) - could you explain this again in more plain language? Feb 27, 2020 at 23:48

Following @dimitris answer In the logistic multi-level models the odds ratio is a cluster-specific, which means it is based on the assumption of holding the random effect constant as well as other covaraites in the model. This is particularly problematic for a cluster-level variable as it has no variation within clusters. But you can transform the cluster-specific OR to population average OR or also called marginal OR by accounting for the shrinkage caused by the random effect using this equation

Population average OR = cluster-specific OR /sqrt(1+0.346*(random effect variance)

This will give a quite similar or very close OR compared to the GEE Check out this Austin & Merlo 2017 article

• Use of population averaged when no population is envisioned and you don't have the sampling weights for how subjects were drawn from the population is a common nomenclature problem that leads to confusion. You are speaking of sample-averaged effects. To the original Q I prefer to think of this as do you want to estimate group effects or do you want to also be able to estimate outcomes on individual subjects? For the former stick with "marginal" models (including gen. least squares and transition models) and for the latter use random effects. Jan 29, 2021 at 12:26
• In some instances you are interested in both the group effects and the effects on subjects, so the point is that you can get an approximation of the marginal effects from the random-effects models. Jan 29, 2021 at 14:54
• For nonlinear models that requires a lot of trouble. Models such as Markov longitudinal models that have no random effects are so much easier and faster. Jan 30, 2021 at 2:19