# Interpretation of coefficients in multilevel mixed effects negative binomial regression analysis

I am wondering how to correctly interpret and describe the coefficients (betas) in a multilevel mixed effects negative binomial regression analysis. I conducted this analysis in Stata with the menbreg command. I further standardised the independent variables. Let's assume the following regression output where restaurant visits of individuals are regressed on their financial income and age. Many of them did not visit restaurants at all, therefore the negative binomial regression analysis. Visits of different restaurant are nested in individuals.

Given that it is multilevel mixed effects negative binomial regression based on standardised predictors, how to correctly interpret the coefficients (betas) below? What does one unit change in the predictor variable mean in regards to the dependent variable? All advice would be highly appreciated.

Assuming that you have used the log link, and that the estimates given are on the log-odds scale:

• Every 1 standard deviation change in Income is associated with a 0.32 change in the log-ods of the outcome, leaving Age unchanged. Alternatively, every 1 standard deviation change in Income is associated with a $$100 \times (\exp{0.32} - 1 ) = 37.7$$ percent change in the outcome, leaving Age unchanged.

• Every 1 standard deviation change in Age is associated with a 0.22 change in the log-odds of the outcome, leaving Income unchanged. Alternatively, every 1 standard deviation change in Income is associated with a $$100 \times (\exp{0.22} - 1 ) = 24.6$$ percent change in the outcome, leaving Age unchanged.

Also, with most software, the fixed effects coefficients have an interpretation that is conditional on the random effects due to the nonlinear link function. That is, the estimates for Income and Age are for subjects visiting the same retaurant. It would be good to verify that the estimates from menbreg produce conditional estimates - if they are marginal estimates instead, then this interpretation does not apply.

• How do log odds work with a count outcome like restaurant visits? Aug 10, 2021 at 15:13
• I'm not sure what you mean. With any model that models count data with a log link (eg poisson regression) the raw estimates are on the log-odds scale. Actually there are some software that produces odds ratios by default, so it would be worth checking whether these estimates are on the log-odds scale, or whether they are odds-ratios (ie, the raw estimates have already been exponentiated) Aug 10, 2021 at 15:17
• Many thanks for your swift reply, Robert. I find your explanation very clear and helpful. I am not sure though if I completely understand the second part on whether the estimates are conditional or marginal. Do I understand correctly, that conditional estimates relate to the individual? What would be a marginal estimate? Also, I just checked the Stata manual but unfortunately did not find an answer to (1) if the are marginal or conditional estimates and (2) if log odds are the default ... stata.com/manuals13/memenbreg.pdf
– Lea
Aug 10, 2021 at 17:51
• You're welcome. Yes, in your case the conditional estimates refer to the grouping variable for which random intercepts are specified. Marginal estimates are produced by default from Generalised Estimating Equations. Some mixed model software, such as GLMMAdaptive in R can output both. As for the defaults in the Stata program you are using, you might consider asking the Statalist about that Aug 10, 2021 at 19:56
• Thank you for the explanation regarding conditional and marginal estimates. Yes, I will reach out to statalist regarding the defaults and as soon as I receive their answer I will share it here too for complete insight. Thank you very much for all your helpful advice, Robert! It is highly appreciated.
– Lea
Aug 11, 2021 at 7:31