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If you have overdispersed observed abundance of multiple species including zero inflation the negative binomial distribution seems to be a reasonable choice. But if some species occur much more frequently than others, is there any reason to consider estimating species specific dispersion parameters?

I am not quite sure if I understand the logic behind using an overdispersion parameter. I suppose if the overdispersion is caused for example by too small sampling areas (i.e. small probability to observe the true abundance of a species) I would argue that the overdispersion parameter is equal for all species. But if the species ecological properties also affect the overdispersion I would argue for a species specific overdispersion parameter.

Is there any strategy to find out what decision should be made in such a case?

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The negative binomial dispersion parameter is not related to the species frequency or to the sampling area so generally speaking you do not need to worry about trying to estimate a different dispersion parameter for each species or group. A negative binomial generalized linear model with constant dispersion is easily able to accommodate different species frequencies and different sampling areas simply by varying the offsets or the linear model parameters. Indeed, the extra noise and uncertainty introduced into the model by trying to estimate lots of different dispersion parameters will usually overwhelm any possible gain from different dispersions.

The negative binomial dispersion is related to species clumping. If one species lives in larger family units than another species, then it is quite possible that would lead to a larger dispersion for that species. Similarly, if one species is more social than another and tends to group together, that might also lead to greater dispersion. However such clumping might also lead to quasi-Poisson type overdispersion instead of negative binomial dispersion. If you're not sure what type of overdispersion you have, I'd suggest that worrying about species-specific dispersion shouldn't be the top priority. Unless you are comparing radically different types of species, variation in dispersion are likely to be of much smaller magnitude than variation in frequency. Try simple models first.

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