# Negative Binomial Substitute for Poisson Applied to NYC Crime Data

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

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.