Hi Maria, thanks for your feedback! You are right in that the LR test is appropriate for nested models. The LR test is useful for testing the validity of the null hypothesis that the constrained model (the one with less number of parameters) has the same goodness-of-fit as the complex model.

Recollect that the NB(1 or 2) model has a dispersion parameter alpha which the Poisson model lacks. That makes the NB model the 'complex model' and the Poisson model the constrained model, thereby making the LR test suitable for use. That being said, there are still some technicial difficultities in using the LR test for comparing Counts based models. If you are interested in knowing the details, please refer to Chapter 3, / Sec 3.4) Regression Analysis of Count Data by Cameron and Trivedi.

Good luck!