Assume that you have a Poisson model with overdispersion. Besides negative binomial models, what are other appropriate count-data modeling regression techniques?
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If you're willing to impose an upper bound on your counts, the beta-binomial works well. Its story is that the binomial probability for each of your count responses is drawn from a beta distribution, which is bounded between zero and one and is used to model binomial probabilities in a Bayesian context. There is also a negative beta binomial, which is more or less what it sounds like. At the bottom of any wikipedia page on a count distribution, there is a box with links to other distributions, including some fun count distributions. Click the 'show' button in the bluish box near the bottom of this page to open it. But beta binomial and negative binomial are both chosen for computational reasons as much as anything else. There's no reason Poisson processes have to have gamma distributed means and binomials have to have beta distributed means other than that someone figured out how to do the math for them by hand. If you're using a more sophisticated multilevel modeling package like BUGS, you can make any mixture distribution you want. You could have a count variable whose mean is drawn from a uniform distribution like a die roll or from anything else. So you can think about your problem, see if any of the existing methods are good enouh for your purposes, with the comfort of knowing you can always build your own if you need to. |
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