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I am modelling some count data and I suspect my data to be underdispersed. I intend to use the Poisson distribution so that I can use information criteria (BIC) for optimal variable selection. However, when I run the GLM using a Quasi Poisson distribution, it returns a dispersion parameter of 0.34. If I then run dispersiontest() from the AER package on the refitted simulated residuals, I obtain a dispersion of 0.32, however the p-value is 0.072, and thus insignificant. Is this underdispersion therefore relevant in my model that is used for prediction? And if so, how should I handle it? Is negative binomial an appropriate replacement?

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If you are only interested in the predictions, then a Poisson regression model and a quasiPoisson regression model gives identical predictions, whether there are under- or over-dispersion.

The estimated (under)-dispersion parameter is used for:

  • confidence intervals/standard errors for parameters

  • Calculating quasi-AIC and such ...

so if you are not using this, you an as well use regular Poisson models.

If you do model validation/selection/comparison via, say, cross-validation, the way you implement that will not be influenced by underdispersion.

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