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Underdispersion is when there is less variability than there 'ought' to be in the data. Eg, the variance of counts could be less than the mean, whereas the variance of a Poisson should equal the mean.

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What does the dispersion parameter mean in negative binomial regression?

If you are using glm.nb from the MASS package, the variance is $\mu+\mu^2/\theta$, so the variance is larger than $\mu$. Different packages may define the dispersion parameter differently. But, if it …
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