Substantive Interpretation of Negative Binomial I am trying to interpret the output from a negative binomial regression.
Online, I read that we can exponentiate the coefficients to get substantively significant values.
However, I know that this is not the same as calculating the marginal effects.
What I am unable to find is what the difference is between marginal effects and exponentiated coefficients? 
Which should I use to substantively interpret my results? And which should I include in a paper - a table with the marginal effects or a table with exponentiated coefficients?
Thank you so much for your help,
Sky
 A: Let us understand why both marginal effects and exponentiated coefficients are about.
By using the Negative Binomial Regression, with $Y_i$ your dependent variable and $X=(x_1,...,x_p)$ the vector of explanatory variables. You assume (among other things) that:
$$E(Y_i|X) = e^{X_i \beta}$$
Consider a $dx_1$ increase in variable x1. You get:
$$E(Y_i|X+dx_1) = e^{X_i \beta} e^{dx_1 \beta_1} =  e^{dx_1 \beta_1} E(Y_i|X)$$
With $dx_1=1$, you get that:
$$E(Y_i|(x_1+1,x_2,...,x_p)) =   e^{\beta_1} E(Y_i|X)$$
$$\frac{E(Y_i|(x_1+1,x_2,...,x_p))- E(Y_i|X)}{ E(Y_i|X)} =   e^{\beta_1}$$
Hence, the exponentiated coefficient for variable $x_1$ gives you the percentgage increase due to an increase in 1 point of variable $x_1$. If $x_1$ is in year/month/day, it is the increase due to an increase in 1 year / 1 month / 1 day. If 1 unit of your variable has a meaning, this provides you immediately a substantive meaning. What is more, it is independent on your sample (as opposed to what comes below).
In comparison, marginal effects are additive effects.
The marginal effect of a $dx_1$ increase at value $X_0 $ is:
$$E(Y_i|X_0+dx_1)-E(Y_i|X_0) = (e^{dx_1 \beta_1}-1)E(Y_i|X_0) = (e^{dx_1 \beta_1}-1)e^{X_0 \beta}$$
This is what we are used to compute in the linear framework. Yet, there is a disadvantage: even if you take $dx_1=1$ as above, it depends on where it is computed (ie. $X_0$). Should you compute on a white woman aged 56 with 2 children, or on a black man aged 89 with 1 child, or so on... ? I think that conventions are to compute it at the fictious person with the mean for all variables, or with the median for all variables, but it remains conventions.
Hence, I would recommend to report exponentiated coefficients unless there is a specific requirement for the marginal effects. And I would make sure that a 1-point increase in my variable of interest can be easily understable (for instance, income is not in 1,000 euros, but in euros if I am interested in small income changes).
