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I'm reading these questions and answers (http://study.sagepub.com/sites/default/files/chapter4.pdf) and am confused about 4.2.4 - 4.2.6

I agree that the Poisson model developed earlier is not a good fit to the true crime data (see Figure 4.5). The question then asks in 4.2.5 what is wrong with the Poisson assumptions and again I agree that we cannot assume crimes to be either independent or identically distributed.

However, it then asks for a new model in 4.2.7 and apparently we can use negative binomial? I don't see why? This is usually used for counting the number of trials required to get a certain number of successes? I don't understand how to map Poisson into Negative Binomial - can someone please explain? Ideally in a way that makes the equation (4.20) understandable!

Thanks

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The negative binomial distribution has various derivations and parametrizations. The one used here is the one were you assume that rates for observational units follow a gamma distribution $$\lambda_i \sim \text{Gamma}(1/\kappa, 1/\kappa)$$ (or alternatively one uses $\theta:=1/\kappa$) and that the counts per individual follow $$Y_i \sim \text{Poisson}(\mu \lambda_i).$$ I.e. counts for the same unit are correlated - e.g. indicating that the event rate can differ between them = one location just tends to have fewer events, while another one has more.

This distribution has mean $\mu$ and variance $\mu(1+\kappa\mu)$ - i.e. it's overdispersed relative to a Poisson distribution (more zeros, more high values). In the limit as $\kappa$ goes to zero, it becomes the Poisson. This is the parametrization typically used for negative binomial regression.

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  • $\begingroup$ Hi, do you have a recommended resource to read up on this? I am only familiar with negative binomial counting number of attempts to get required number of successes. I googled around and found a few things on negative binomial appearing as a special case of poisson where the poisson parameter is drawn from a gamma. However it wasn't well explained. I'd like to understand the equivalence between NB and Poisson a lot deeper/better.... $\endgroup$
    – user11128
    Commented Mar 4, 2019 at 14:26
  • $\begingroup$ Additionally, I don't see how the poisson distribution which counts the number of events, x>=0, can be modelled by NB which has a minimum number of events to meet the required number of successes, x>=r. $\endgroup$
    – user11128
    Commented Mar 4, 2019 at 14:32
  • $\begingroup$ Searching for negative binomial regression should bring up plenty of results (one of mine would be dx.doi.org/10.1002/sim.7549. For what is discussed here, you would actually do not use the parameterization of events needed until x successes, but rather a NegBin(mean rate, dispersion parameter) version of the distribution. However, if I remember correctly, I believe you can actually find a number of successes and probabilitiy that corresponds to a particular mean rate and dispersion parameter, it is just very tedious and impossible to remember, so nobody works with that version. $\endgroup$
    – Björn
    Commented Mar 4, 2019 at 15:05

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