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When you run a Poisson regression and afterwards a negative binomial regression because you observed over-dispersion using Poisson, you do not loose a DF with the neg. binomial model. I would expect loss of 1 degree of freedomF because of the dispersion parameter.

Thank you!

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It does cause the loss of a degree of freedom so I'm not sure why you think it does not. Consider the following basic example for negative-binomial regression using the MASS package in R:

library("MASS")
data("quine", package = "MASS")
po <- glm(Days ~ ., data = quine, family = poisson)
nb <- glm.nb(Days ~ ., data = quine)

The number of estimated coefficients is the same

length(coef(po))
## [1] 7
length(coef(nb))
## [1] 7

But the degrees of freedom (df) are increased by 1 due to the extra $\theta$ parameter:

logLik(po)
## 'log Lik.' -1142.592 (df=7)
logLik(nb)
## 'log Lik.' -546.5755 (df=8)

And the extra dispersion parameter has the effect that standard errors are larger (much larger in this case) while the coefficient estimates remain rather similar:

cbind("Coef (Pois)" = coef(po), "Coef (NB)" = coef(nb),
  "SE (Pois)" = sqrt(diag(vcov(po))), "SE (NB)" = sqrt(diag(vcov(nb))))
##             Coef (Pois)   Coef (NB)  SE (Pois)   SE (NB)
## (Intercept)   2.7153802  2.89458002 0.06468292 0.2284246
## EthN         -0.5336043 -0.56937170 0.04188300 0.1533334
## SexM          0.1615966  0.08232026 0.04253447 0.1599150
## AgeF1        -0.3339014 -0.44842815 0.07009331 0.2397466
## AgeF2         0.2578284  0.08808014 0.06241921 0.2361930
## AgeF3         0.4276938  0.35690095 0.06768619 0.2483244
## LrnSL         0.3489430  0.29210914 0.05204305 0.1864747
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