Interpretation of fixed effects in random intercepts Poisson model I've been trying to do some research online to see how to interpret the fixed effects estimates for a random intercept model fit using the Poisson distribution with a log link. Is it accurate that when interpreting the fixed estimates, we have to say they are conditional on random effects (i.e., they are subject-specific)? I've seen this on various articles, but some also leave out the conditional part and treat them as marginal effects. Can someone please explain? Thank you!
 A: If a mixed effects Poisson regression model uses a log link and includes a random intercept but no random slopes, then its fixed effects have a dual interpretation: they can be interpreted as both conditional and marginal fixed effects.
If, on the other hand, a mixed effects Poisson regression model uses a log link and includes a random intercept as well as one or more random slopes, then its fixed effects can only be interpreted as conditional fixed effects.
For further details, see the article Marginal or conditional regression models for correlated non‐normal data? by Muff, Held and Keller
(Methods in Ecology and EvolutionVolume 7, Issue 12, 2016):
https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.12623
The paragraph of most interest to you in this article is the following paragraph in the section Interpretation of the parameters:
"Another exception where conditional and marginal models are not incompatible are log‐linear models, such as Poisson regression, where all parameters except the intercept are the same for the marginal and conditional models (Zeger, Liang & Albert 1988; Neuhaus, Kalbfleisch & Hauck 1991), although this only holds when the respective conditional model includes a random intercept but no random coefficients (Grömping 1996)."
