Use of goodness of fit statistics in Poisson GLM

Question

I'm using JMP Pro 15 to run a generalized linear model with a Poisson distribution and log link function. I was told to use Poisson because I have count data with values ranging from a min of 1 to a max of 118 with most falling in the 5 to 50 range. The model shows 2 goodness of fit statistics (pearson, deviance) that both have p-values less than 0.05. Does this mean that the model results are invalid and I need to reanalyze using a different distribution? This blog below seems to indicate that the results of the deviance test alone are somewhat unreliable.

https://thestatsgeek.com/2014/04/26/deviance-goodness-of-fit-test-for-poisson-regression/

Mark_Bailey · Answer

The Poisson distribution uses a single parameter for the mean and the variance. As such, it might under-fit the data. The over-dispersion parameter allows for more variance than the mean.

IK1 · Answer

I did not originally, but just tried it. The overdispersion value is 2.5197 (for Pearson). I've never used this option before so not sure how to interpret it.

Mark_Bailey · Answer

Did you invoke the over-dispersion option in the launch dialog when you fit the Poisson distribution?