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I've read in several places that a negative-binomial model is a reasonable alternative to a Poisson regression when the latter shows overdispersion. However, none of the several sources I read said whether it is also an improvement over a Poisson that shows underdispersion.
So is a negative-binomial worth considering for count data where the variance doesn't equal the mean? Or more specifically, only when the variance is larger than the mean?
The variance of a variable $y$ with negative binomial distribution with expectation $\mathrm{E}(y) = \mu > 0$ and dispersion parameter $\theta > 0$ is $\mathrm{Var}(y) = \mu + \frac{1}{\theta} \cdot \mu^2$. Thus, the negative binomial distribution is always overdispersed with $\mathrm{Var}(y) > \mu$ and only reaches equidispersion for $\theta \rightarrow \infty$ (i.e., the Poisson distribution).
There are various ways how underdispersion can occur, e.g., through zero-inflation or zero-truncation, which both have corresponding regression models. There are also more flexible extensions of the Poisson that can accomodate underdispersion, e.g., Conwell-Maxwell-Poisson among others.
If you do not need a full likelihood model, a quasi-Poisson model with estimated dispersion parameter (which can be $< 1$) may be useful as well.
