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I have some crime data where the crime rate for a given year in a given town is reported as number of police reports per thousands of people in this town, eg $5/6$ for five reported crimes in a town of 6000 people. In the nature of things there will be some high-variance measurements from small towns and some less-variable measurements from larger towns.

If it was only the number of police reports it would be count data, and I'd figure that the sensible choice would be a Poisson distribution. Is there a similar sort of theoretically founded choice for this count-per-something situation?

Below is a histogram of the rates and the log of these rates; from these it's kind of tempting to call the data log-normal but I have no theoretical basis for saying so. Further below is a very long string you can copy to get a vector of these rates in R.

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rates <- c(15.4, 11.6, 10.9, 10.9, 10.5, 10, 9.8, 9.8, 9.5, 9, 9, 8.9, 8.8, 8.8, 8.7, 8.4, 8.2, 8.2, 8.2, 8, 7.9, 7.9, 7.8, 7.7, 7.6, 7.6, 7.6, 7.5, 7.5, 7.4, 7.4, 7.3, 7, 7, 6.9, 6.9, 6.8, 6.8, 6.8, 6.8, 6.7, 6.7, 6.6, 6.6, 6.5, 6.5, 6.5, 6.4, 6.4, 6.4, 6.3, 6.3, 6.3, 6.3, 6.2, 6.2, 6.2, 6.1, 6.1, 6.1, 6.1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5.9, 5.9, 5.9, 5.9, 5.9, 5.8, 5.8, 5.8, 5.8, 5.8, 5.7, 5.7, 5.7, 5.6, 5.6, 5.6, 5.5, 5.5, 5.5, 5.5, 5.5, 5.4, 5.4, 5.4, 5.4, 5.4, 5.4, 5.4, 5.4, 5.3, 5.3, 5.3, 5.3, 5.3, 5.3, 5.2, 5.2, 5.2, 5.2, 5.2, 5.2, 5.1, 5.1, 5.1, 5, 5, 5, 5, 4.9, 4.9, 4.9, 4.9, 4.9, 4.8, 4.8, 4.7, 4.7, 4.7, 4.6, 4.6, 4.6, 4.6, 4.5, 4.5, 4.5, 4.5, 4.5, 4.4, 4.4, 4.4, 4.4, 4.3, 4.3, 4.3, 4.3, 4.3, 4.3, 4.3, 4.3, 4.2, 4.2, 4.2, 4.2, 4.2, 4.2, 4.2, 4.2, 4.2, 4.2, 4.1, 4.1, 4.1, 4.1, 4, 4, 4, 4, 4, 4, 4, 4, 3.9, 3.9, 3.9, 3.9, 3.9, 3.9, 3.9, 3.9, 3.8, 3.8, 3.8, 3.8, 3.8, 3.8, 3.7, 3.7, 3.7, 3.7, 3.7, 3.7, 3.7, 3.7, 3.7, 3.6, 3.6, 3.6, 3.6, 3.6, 3.6, 3.6, 3.6, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.5, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4, 3.3, 3.3, 3.3, 3.3, 3.3, 3.2, 3.2, 3.2, 3.2, 3.2, 3.2, 3.2, 3.2, 3.2, 3.2, 3.1, 3.1, 3.1, 3.1, 3.1, 3.1, 3.1, 3.1, 3.1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.9, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.8, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.7, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.6, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.4, 2.4, 2.4, 2.4, 2.4, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.3, 2.2, 2.2, 2.2, 2.2, 2.2, 2.2, 2.2, 2.2, 2.1, 2.1, 2, 2, 2, 2, 2, 2, 2, 1.8, 1.8, 1.6, 1.5, 1.5, 1.4, 1.4)
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    $\begingroup$ Modelling the log of the crime rate is common. When there are zeroes, a popular alternative is to use a Poisson model predicting the counts, but use the population as an offset term. $\endgroup$ – Andy W Sep 29 '16 at 17:06
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    $\begingroup$ This is relevant: stats.stackexchange.com/questions/142338/… $\endgroup$ – kjetil b halvorsen Sep 29 '16 at 17:47
  • $\begingroup$ @AndyW That's interesting what you say about the offset Poisson, how would I do it that practice? $\endgroup$ – einar Sep 30 '16 at 19:21
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    $\begingroup$ Pretty much all statistical software allows you to include an exposure term for Poisson models (which is equivalent to an offset). Depending on the software, this will either be the population counts, or the log of the population counts. See this discussion for how an offset term in a Poisson regression can help you interpret the model as rates. $\endgroup$ – Andy W Sep 30 '16 at 20:04
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Sometimes the negative binomial is used too.

The most interesting option is, however, the Hurdle Models!

They work as a Poisson, but the probability of getting a 0 has its own parameter.

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