Derive closed form interpretation of negative binomial regression linear term coefficient Consider an experimental setup where any given individual is either exposed or unexposed to some treatment A. Some over-dispersed count variable is measured as the outcome. This can be modelled with a negative binomial regression as follows:
$log\big(E[\text{outcome}|\text{exposed to treatment A}]\big)=\beta_0+\beta_1I(\text{exposed to treatment A})$
The parameters $\beta_0,\beta_1$ have no closed form solution and are solved using some optimization technique.
However, it is always the case that:
$\beta_1=\log{\frac{\mu(\text{outcome}|\text{exposed to treatment A})}{\mu(\text{outcome}|\text{unexposed to treatment A})}}$
I cannot find any derivation of this result. Any help deriving this mathematically?

First, note that, as $I(\text{exposed to treatment A})= 0$ when the individual was not exposed to treatment A and $1$ when they were exposed to A, it follows that:
$$\log\big(E[\text{outcome}|\text{not exposed to A}]\big)=\beta_0$$
and
$$\log\big(E[\text{outcome}|\text{exposed to A}]\big)=\beta_0 + \beta_1$$
so
$$\log\big(E[\text{outcome}|\text{exposed to A}]\big) -\log\big(E[\text{outcome}|\text{not exposed to A}]\big) = \beta_1 $$
and, as $\log a - \log b = \log (a/b)$, we are a single algebraic step from the desired result.
