Negative-Binomial vs Poisson for Count Data Say that I was modelling the number of upvotes for a Facebook post under two treatments. What would be a more ideal distribution for modelling these counts: negative-binomial or a Poisson?
What would be the trade-offs here?
 A: For $Y \sim$ Poisson$(\lambda)$ we have $\text{E}(Y) = \lambda$ and $\text{Var}(Y) = \lambda$, so the variance should at least be well-approximated by the mean.  Often in real applications we find that the variance for count data is larger than the mean.
If on the other hand $Y \sim$ negative binomial$(r, p)$ we have $\text{E}(Y) = rp / (1 - p)$ which is less than $\text{Var}(Y) = rp / (1 - p)^2$, so the negative binomial is sometimes used to account for what's called "overdispersion."  The only real downside is that the model is slightly less parsimonious, but that arguably isn't much of a downside.
A: Aside from looking at the mean and variance of the response, you also need to understand if it makes sense for the data to come from a Poisson or a Negative Binomial distribution.
Your setup clearly suits Poisson regression in principle.
For Negative-Binomial distribution, you will need to have a data setup where you say "how many up-votes this post gets before the first down vote", and that will conform to a negative binomial. But as it stands, you cannot setup your problem in a way such that a NB distribution can make intuitive sense here.
