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!

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)."

• Thanks for the answer! Would this also be the case with negative binomial models since those are also log-linear? Dec 19, 2020 at 6:06
• Yes, all mixed effects models with a log link and just random intercept(s) would have fixed effects which lend themselves to a dual interpretation: conditional and marginal. I added the plural for random intercept here because these mixed effects models can have crossed random factors (e.g., site and year, where each site is assessed once per year or multiple times per year on all/some years in the study) for which you allow random intercepts; or, they could have nested random factors (e.g., transect nested within site) for which you allow random intercepts. Dec 19, 2020 at 17:01
• Many people get caught out and use the dual interpretation (i.e., conditional and marginal) of fixed effects in mixed effects modes with a log link, random intercept(s) AND random slope(s). That is plain wrong, as only the conditional interpretation is applicable in the presence of random slope(s) in those models. Sadly, I know of at least one published paper where this type of interpretational mistake was made and went unflagged by the reviewers. Dec 19, 2020 at 17:09
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• Hi Isabella, not sure if you know anything about this, but if we have add a random effect (intercept) to the zero-inflation part of a zero-inflated Poisson model, is the interpretation of the parameter in the zero-inflation part conditional on the random effect? Dec 19, 2020 at 18:15