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!



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 Mar 4 '19 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 Mar 4 '19 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 Mar 4 '19 at 15:05

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