I know that quasi-Poisson gives more weight to large values, whereas Negative Binomial gives more weight to small values (see source). Small values are more frequent in my dependent variable, so it seems to me that I should choose Negative Binomial, because it would fit more data points more accurately. Does this justification make sense?
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$\begingroup$ Read your source carefully, your claim is wrong. The weights (as applied in iterative estimation) will be different, accordingly direction (more vs less) depends on the "scale" parameter in the NB model and can go either way. $\endgroup$– AdamOCommented Jun 4, 2021 at 17:57
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No, the justification does not make sense. The NB model is guaranteed to fit points better. This is because the predictions from Quasipoisson models are exactly the same as the Poisson model, but the NB is one parameter bigger. As Tukey and Akaike and others have noticed: more parameters = better fit. If the goal is prediction, you can use some model validation methods, like split sample validation, to see whether the NB model performs better in an external, independent dataset. If for inference, there are other posts on this topic. See, for instance, here: How to deal with overdispersion in Poisson regression: quasi-likelihood, negative binomial GLM, or subject-level random effect?
