The negative binomial regression model (nbreg command) is similar to a Poisson regression, only that the variance is allowed to be greater than what is assumed in a Poisson model. This extra variance is the overdispersion. If not accounted for, overdispersion leads to deflated standard errors which in turn may lead to errenous inference.

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Practical example

Let us first do a simple check to see what the situation looks like regarding overdispersion for our outcome children. Of course, this will not take any x-variable into consideration.

sum children, detail

The variance is considerably higher than the mean, which suggests that overdispersion might be an issue. Accordingly, it is a good idea to try out a negative binomial regression model.

Thus, we will re-run the multiple regression model that we specified for Poisson regression earlier, but now with the nbreg command:

nbreg children siblings sex ib1.educ if pop_poisson==1, irr

The output is very similar to the one we got for the Poisson regression. Additionally, we are presented with the results from the log-transformed overdispersion parameter (/lnalpha), as well as the untransformed estimate (alpha).

Note that we also get a LR test presented below the table, which compares this model to a Poisson model. The fact that the p-value (Prob >= chibar2) is below 0.05 (0.000) suggests that this model fits the data better than the traditional Poisson.