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I have a Poisson GLM in R that is overdispersed, so I fit a quasipoisson GLM, however the residual deviance nor the degree of freedom change. Can that happen? What does it mean in that case?

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Yes, it can and is perfectly fine. Actually in this case, the estimated coefficients should be the same as well as the residual deviance and the degrees of freedom! But have a look at the standard errors of the estimated coefficients. You will see larger standard errors when fitting a quasipoisson GLM. It means that you are fitting a more flexible model that allows an over-dispersion. Therefore, you have larger confidence intervals.

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