I am aware of the several similar questions existing here (like this, this or this), but my question is slightly different and remains after going through most of those posts. Specifically, my question is about deciding between the two distributions based on a priori reasoning rather than a data-driven goodness of fit approach. dbwilson mentioned the differences in weighting approaches between the two, and Ver Hoef & Boveng (2007) provides an excellent comparison in an ecological context.

As I understand it, Quasi-poisson (hereafter QP) has a variance that changes linearly with its mean (multiplied by the estimated constant dispersion parameter), whereas Negative Binomial (hereafter NB) has a variance that increases quadratically with its mean. As a result, for low values of mean NB has a lower variance than QP, while for high values of mean QP has a lower variance than NB. Additionally, the weighting also increases linearly in QP, while it increases then soon reaches an asymptote for NB. This means that NB is highly sensitive at lower mean values (because the assumption of the distribution is low variance at low means) and less sensitive at higher mean values.

In the Ecology paper, the authors were studying hauled-out harbour seals, and this data is naturally highly overdispersed. Higher mean values (large groups of seals) are most common and actually what the scientists are interested in. Moreover, variations seen at low mean values are not much more impactful than those at high mean values (at least to the tested relationship). Hence, for their scenario QP is preferred over NB. They also showed some flaws with using NB in such a case.

On the other hand, for many other ecological scenarios like abundances of organisms (say, birds), lower mean values are more common and these are more impactful and should be weighted more than the variances at high mean values. e.g., a difference of 5 birds where mean is 2 should be weighted more than a difference of 5 birds where mean is 25. This would suggest that NB is the better option. However, the NB also assumes a quadratically increasing variance with the mean, which is improbable in such scenarios. e.g., differences of 20-25 birds at a site where mean is 25 are rare. This would suggest that NB is less appropriate.

This is mainly what is confusing me. Are my interpretations above regarding the two incorrect? Am I mixing up separate things? Why do the distinctions between the two in terms of variance and in terms of weighting procedures suggest opposite solutions? How can I intuitively imagine real ecological scenarios where each method might be appropriate?


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