In count data models with dummies, what exactly means “on average”?

Question

There are some questions + answers out here that explain how to interpret coefficients from count data regressions (e.g. negative binomial), both as incidence rate ratios or marginal effects. Bottom line: Unlike in linear models, the coefficients are modelled at the mean of the other variables, such that the obtained effect is on average.

But there is one thing I can't get my head around: When I use dummy variables, how exactly is on average defined? Suppose I have

$$y = b_0 + b_1 X + b_2 D$$

with $D$ being a dummy and $b_1$ is the parameter I am interested in. Suppose also $D=1$ in exactly half of all observations. What is the mean observation now, at which $b_1$ is evaluated? Mean of $X$ and $0.5 D$?

Answer 1

I would say that in case where $D = \{0,1\}$ we have the outcome adjusted or not by $b_2$, in more formal way $D = 0$ is the reference level.

In case where you have more levels like $D = \{0,1,2,3\}$ and you code it on dummies then you have the difference between the mean (without taking D into consideration) and a specific level.

Please keep in mind that it also depends on the way you code the dummies, you can for example point one level as a reference and than you have a difference between this specific level and all the others.