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I am trying to fit the number of crimes in a city with some enviromental variables (aka my features). I'm using a Poisson/Negative Binomial model since I have count data. The problems are:

  1. selecting the features which REALLY fit (I have sometimes big collinearity)
  2. select the smallest subset of significative features (p-values)
  3. understand the relative importance between them to fit the number of crimes

Possible solutions to the problems:

  1. Variance inflation factor (VIF). Is it ok? Sometimes interactions could be ignored. I mean: what if I have to test whenever two variabiles are important, but collinear? (e.g. population density and employment density)
  2. Stepwise (backward/foward/with Cross-Validation) steps to reduce the AIC. However, this does not mean that the resulting variables are significative (p-values). Should I care at all about the features' p-values?
  3. Standardize the input data, then looking at the Beta coefficient to see the importance of them. Is it really the best way?

Should I use something else? Keep in mind that my primary goal is not the $R^2$. I want to see and to be able to explain how some variables could explain the crime.

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Firstly, Variance Inflation Factor is inversely proportional to the $R^2$ value. So checking this might help , but multi-collinearity is a thing to look at.

Secondly , instead of $R^2$ , adjusted $R^2$ is a better way to validate a model.

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