# Different estimates for over dispersion using Pearson or Deviance statistics in Poisson model

I have a question about what is the best way to estimate the amount of over dispersion in a Poisson model.

I am unclear what to use as a measure for over/under dispersion as there seems to be a lack on consensus online. The Pearson Chi Squared statistic/Degrees of freedom or the Deviance Statistics/Degrees of freedom seem to be the most commonly recommended estimates, however nearly all sources I can find state to use one or the other, not identifying any advantages/disadvantages to either:

Source 1: The dispersion parameter can be estimated by the Pearson or deviance statistic divided by its degrees of freedom

Source 2: Thus the φˆpearson can be either larger or smaller than the φˆdeviance. These two estimators are approximately the same only when the values of yi are close to the predicted values, i.e. the GLM has a good fit to the data

Source 3: McCullagh and Nelder (1989) recommend where X2 is the usual Pearson goodness-of-fit statistic, If the model holds, then X2/(N - p) is a consistent estimate for σ2 in the asymptotic sequence N → ∞ for fixed ni's. The deviance-based estimate G2/(N - p) does not have this consistency property and should not be used.

This is the only source that recommends one over the other, however its source is very outdated.

I could find many more sources but there is never a concise argument as to which one should be used. The reason this is an issue for me is that I am currently trying to fit a Poisson model where Pearson Chi Sq/DF = 2.0806 (indicating over dispersion). However the Deviance statistic/DF = 0.4659 (indicating under dispersion).

The conclusions are entirely different depending on what estimate I use, and if I adjust the covariance matrix for these estimates, it would make the p-values bigger/smaller depending which one I chose. The issue is very similar to this post here: Very different scale parameter estimates in Poisson regression which was left unresolved.

My questions are:

1) Is one estimate of over dispersion better than another in certain scenarios + do you have any sources for this?

2) Are these statistics more likely to be different due to a lack of model fit?