I am using the count data models as I have a count dependent variable (0, 1, 2, 3, 4).

There is no overdispersion present and no assumption of the Poisson model is violated. Pearson goodness-of-fit and Deviane goodness-of-fit both suggest the usage of Poisson regression instead of NB regression.

However, I am still interested to compare the results of both regressions and see if there are some changes. Surprisingly, the results are identical.

My question would be: Why do the Poisson and Negative Binomial Regressions both give the same results? Is it because of no violations of Poisson model assumptions? What should be done in this case?

Poisson results:
dnewoccposition3 .0989237 .028555 3.46 0.001 .042957 .1548904
dnewoccposition4 .1154875 .0467075 2.47 0.013 .0239424 .2070326
dnewoccposition5 .1718835 .0493519 3.48 0.000 .0751555 .2686115
NB results:
dnewoccposition3 .0989237 .028555 3.46 0.001 .042957 .1548904
dnewoccposition4 .1154875 .0467075 2.47 0.013 .0239424 .2070326
dnewoccposition5 .1718835 .0493519 3.48 0.000 .0751555 .2686115

Many thanks in advance!

Best,
Mehrzad