# Requirements for negative binomial regression

In my study, I experiment with fixed and mixed effects negative binomial regression to my data (in R) as the response variable is a count variable. Normally I should use Poisson regression but (and here I have doubts whether I have done it right) I checked overdispersion (variance > mean), theta parameter > 1, there is no zero-inflation). Are there any other requirements that I should check before deciding to switch from Poisson to negative binomial regression?

• The marginal distribution of the response in a Poisson regression is not Poisson, so checking for over-dispersion of the raw response does not appear to address the issue of whether the conditional variance is similar to the conditional mean, or larger/smaller than it. Typically even if the response were (conditionally) under-dispersed, the marginal distribution would still be over-dispersed. Commented Feb 27 at 16:19

Switching from Poisson to negative binomial regression is a common step when dealing with count data that exhibit overdispersion, meaning the variance exceeds the mean, which violates the Poisson assumption that the mean and variance are equal. You've correctly identified overdispersion and checked the theta parameter (indicating dispersion in the negative binomial model) and zero-inflation as key factors. Here are additional considerations to ensure you're making an informed choice in selecting negative binomial regression over Poisson:

Beyond checking for overdispersion, it's crucial to inspect the residuals of your Poisson model. If the residuals show patterns or are not randomly distributed around zero, this could indicate that the Poisson model is not fitting the data well, supporting a switch to a model that can handle overdispersion, like the negative binomial.

You've mentioned there's no zero-inflation, which is good. However, it's important to ensure this determination is based on a robust analysis. Zero-inflation occurs when there's an excess of zeros in the data than what the Poisson or negative binomial distribution would predict. Since you've ruled this out, it strengthens the case for using the negative binomial model if overdispersion is present.

Use statistical tests or criteria like AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion) to compare the fit of the Poisson and negative binomial models to your data. Lower values of AIC or BIC suggest a better fit. This quantitative comparison can provide a more objective basis for choosing between models.

Ensure that the relationship between your covariates and the response variable fits the assumptions of the negative binomial model. For instance, the link function used (usually log) should appropriately model the relationship between the covariates and the expected count.

The negative binomial distribution is more flexible than the Poisson, allowing for a varying mean and variance. If the theoretical distribution of your data suggests variability beyond what a Poisson process would generate, this theoretical justification supports using a negative binomial model.

Robustness Checks: Conduct sensitivity analyses to check the robustness of your model's conclusions. This can involve varying model specifications or subsets of data to ensure that your findings are not overly sensitive to specific choices made during modeling.

By systematically going through these steps, you'll be better positioned to decide whether the negative binomial regression model is the most appropriate for your data, ensuring that your analysis is both robust and well-justified.