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I would please like to enquire if it's appropriate for me to compare the fit of a Poisson vs. a negative Binomial model for my data, given that the two models are nested, i.e. the negative Binomial and Poisson regression are the same model when the one additional parameter the negative Binomial model adds (alpha, which captures the overdispersion present) is zero.

Thank you for any insight. It's extremely appreciated.

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  • $\begingroup$ Let us know if you have further questions or need more explanation. If this answer or any other one solved your issue, please mark it as accepted :) $\endgroup$ Dec 18, 2020 at 10:06

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Yes: model selection criteria, such as the BIC, the AIC, or the minimum length criterion, are commonly used in the literature to compare models based on their goodness of fit (and regularized for their complexity, ie for their number of free parameters).

Here, since the negative Binomial has 2 parameters (instead of only 1 for a Poisson distribution), it is going to be more penalized by the AIC and the BIC than the Poisson distribution.

However, the validity of these criteria rely on some strong assumptions, that you will need to verify and justify. For instance, using the BIC requires that your data are i.i.d., that you have enough of them, that you correctly obtained your Maximum Likelihood Estimators of the parameters of the models.

An interesting reference is Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological methods & research, 33(2), 261-304.

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    $\begingroup$ I was always under the impression that comparing models with different likelihoods with AIC and BIC is somewhat wrong since the likelihood functions are different. Whether you are high or low in one function when compared to another is somewhat meaningless (as opposed to higher or lower in the same likelihood function with different covariates). Do I have a wrong picture here? $\endgroup$ Dec 10, 2020 at 14:46
  • $\begingroup$ Comparing different likelihoods is precisely the point of model comparison ! I think the confusion might come from your definition of likelihood: a likelihood is not some kind of function that is unique to a model and cannot be used for another one. When comparing models $\mathcal{M}_1$ and $\mathcal{M}_2$ based on data $\mathcal{D}$, you compare their respective likelihoods $p(\mathcal{D}|\mathcal{M}_1)$ and $p(\mathcal{D}|\mathcal{M}_2)$. The probability $p$ can be used for different models. $\endgroup$ Dec 10, 2020 at 15:14
  • $\begingroup$ (+1) Worth nothing that software may drop terms in the log-likelihood that are constant across a particular class of models. $\endgroup$ Dec 10, 2020 at 19:30

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