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When would one use a negative binomial regression and when would one use Poisson regression with respect to the mean and variance?

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I'm going to give contradictory advice to Shawn (+1) to encourage some discussion on this matter. Ultimately, I recommend almost never using negative binomial models over Poisson.

Poisson regression can be reframed a quasi-maximum likelihood (QMLE) model. That means as long as the relationship between the predictors and the outcome mean is correctly specified, the QMLE estimate of the coefficients is consistent, whether or not the outcome is actually Poisson distributed (including whether or not the outcome is over- or under-dispersed). The usual standard errors from the Poisson model are incorrect, though; you can easily use robust standard errors or bootstrap standard errors instead, which are valid and do not require strong assumptions about the form of the outcome distribution.

In contrast, the negative binomial (NB) model requires strong assumptions. If the outcome distribution is not negative binomial (e.g., it does not specifically follow the NB distribution or the dispersion is not accurately captured by the implied mean-variance relationship of the NB model, including if there is under-dispersion) the estimate isn't even consistent. Not only can NB not handle under-dispersion, it can't even handle correct specification of the mean vector if any other part of the model is incorrectly specified. Note, though, if the dispersion parameter is fixed at some value (any value), NB regression becomes a QMLE estimator and becomes consistent as long as the mean is correctly specified; again, you need to use robust or bootstrap standard errors for valid inference.

This is all described in Wooldridge (2010, Ch 8). Note that the usual negative binomial model is call NegBin II in this text.

In practice, there may be many reasons why you can't rely on robustness properties Poisson regression has because of its QMLE status, including when using it in a mixed effects model or when computing the conditional outcome density. In that case, it is a good idea to use an estimator that has a chance at being consistent, such as the negative binomial or other count models that are generalizations of Poisson like the generalized Poisson or Conway–Maxwell–Poisson models.

Reference

Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed). Cambridge, MA: MIT Press.

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    $\begingroup$ This is an interesting perspective and thanks for sharing (+1 to you as well sir). I haven't heard the arguments you have raised before, so I'll be curious to hear what others have to say about it. $\endgroup$
    – Shawn Hemelstrand
    Commented Sep 2 at 8:12
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    $\begingroup$ Could you please elaborate how the QMLE property helps for the consistency? $\endgroup$
    – Ggjj11
    Commented Sep 2 at 9:37
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Poisson regression can only be used when there is equidispersion, meaning the mean and variance are equal for the conditional distribution of the response. As soon as you enter into territories like under- or overdispersion, you need a model that addresses this. Typically overdispersion is a lot more common (when the variance is "excessive" compared to what the model assumes), and negative binomial regressions are at least one way to overcome this issue. It does this by using a "mixture" distribution, which combines a Poisson and gamma distribution to achieve a proper fit to the data. Some discussion of this point can be found in the initial pages of Hilbe's text on negative binomial regression (listed below).

Notably, Poisson and negative binomial regression are not the only solutions to count modeling. Sometimes you will have an overabundance of zeroes (where zero-inflated models are useful), extremely skewed distributions (where PIG models tend to sometimes do better), or very complicated distributional aspects that require hand-wiring the regression (where GAMLSS can be really useful). So which method you use will depend largely on the issues you are trying to address with your data.

I'll also add that by default one should almost always use negative binomial models over the Poisson because the assumption of equidispersion is rarely actually met, but you anyway have to check which dispersion issues are present in the model first before deciding on which to use.

Reference

Hilbe, J. M. (2011). Negative binomial regression (2nd ed). Cambridge University Press.

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    $\begingroup$ The opening sentence is (sorry) quite wrong. It confuses fitting a Poisson distribution (for which ideally variance and mean are equal) with fitting an outcome given predictors. For the latter, it's conditional distributions, not marginal distributions, that are relevant. As with plain regression for which $y = Xb$ is the main idea, so also for Poisson regression $y = \exp(Xb)$ is the main idea and all else is secondary. As in @Noah's answer, Poisson regression can work well in many problems -- even for outcome variables that are continuous -- although using appropriate SEs is important. $\endgroup$
    – Nick Cox
    Commented Sep 2 at 9:08
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    $\begingroup$ I've seen confusion between what is assumed about marginal distributions and what is assumed about conditional distributions hundreds of times both here and elsewhere, so it is vital to be explicit. I still think your first sentence exaggerates greatly. Just as conditional normal distributions are the least important ideal condition in plain regressions, so also conditional Poisson distributions are not essential for successful Poisson regressions. The outcome doesn't even have to be discrete. $\endgroup$
    – Nick Cox
    Commented Sep 2 at 9:45
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    $\begingroup$ In many ways Poisson regression is an unfortunate standard term, although I too use it in deference to convention. Note: I prefer ideal condition to assumption. Anscombe made this point tacitly in 1961! $\endgroup$
    – Nick Cox
    Commented Sep 2 at 9:45
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    $\begingroup$ See also blog.stata.com/2011/08/22/… The Stata context and detail are immaterial to the main point being made that is relevant here. $\endgroup$
    – Nick Cox
    Commented Sep 2 at 9:48
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    $\begingroup$ Thanks for your comments in reply. It's my impression that most textbooks are way behind what is known by people who've looked at this carefully, such as Jeff Wooldridge in econometrics, as cited by @Noah. His own introductory text doesn't puff Poisson regression enough in my view. In many ways the point is best approached from a generalized linear model perspective and is just that a logarithmic link is often a good idea for positive or even non-negative outcomes. $\endgroup$
    – Nick Cox
    Commented Sep 2 at 10:53

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