What does the dispersion parameter means in negative binomial regression?

I am completely new to the topic of negative binomial regression and am unsure about what the output of my regression exactly means. Before I decided to use the negative binomial regression, i did check for overdispersion with the dispersiontest (Mod.poisson) of my Poisson model, which resulted in a dispersion of 1985 with a p value smaller than 0.05, telling me the data is overdispersed. But when I perform a negative binomial regression, there is standing: "Dispersion parameter for Negative Binomial(0.6974) family taken to be 1". Doesn't that value (0.6974) mean that there is underdispersed as it's below 1, so the variance is 0,6974 times smaller than the mean? That seems contradictory to me ... ?

• What software are you using to fit the Poisson and Negative Binomial regression?
– Eli
Commented Sep 28, 2021 at 21:04
• I am using R Studio Commented Sep 29, 2021 at 0:45

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 is doing negative binomial regression, the variance will always be at least as large as the variance of the Poisson distribution.