# When someone says residual deviance/df should ~ 1 for a Poisson model, how approximate is approximate?

I've often seen the advice for checking whether or not a Poisson model fit is over-dispersed involving dividing the residual deviance by the degrees of freedom. The resulting ratio should be "approximately 1".

The question is what range are we talking about for "approximate" - what is a ratio that should set off alarms to go consider alternative model forms?

• Not an answer to this interesting question, but what I will often do is run several models (e.g. Poissson, NB, maybe zero-inflated versions) and compare them - both on AIC-type measures and on predicted values. – Peter Flom - Reinstate Monica Sep 21 '12 at 10:36
• This link might be of interest. Specially the section "Criteria For Assessing Goodness Of Fit". – user10525 Sep 21 '12 at 14:57
• @Procrastinator The link is a perfect example of what I'm talking about: "Then, if our model fits the data well, the ratio of the Deviance to DF, Value/DF, should be about one. Large ratio values may indicate model misspecification or an over-dispersed response variable; ratios less than one may also indicate model misspecification or an under-dispersed response variable." What's the range of "about 1"? 0.99 to 1.01? 0.75 to 2? – Fomite Sep 21 '12 at 19:41
• r-bloggers.com/… also has some information about how to answer this question, though @StasK's response covers it well enough. – flies Dec 9 '16 at 16:02

10 is large... 1.01 is not. Since the variance of a $$\chi^2_k$$ is $$2k$$ (see Wikipedia), the standard deviation of a $$\chi^2_k$$ is $$\sqrt{2k}$$, and that of $$\chi^2_k/k$$ is $$\sqrt{2/k}$$. That's your measuring stick: for $$\chi^2_{100}$$, 1.01 is not large, but 2 is large (7 s.d.s away). For $$\chi^2_{10,000}$$, 1.01 is OK, but 1.1 is not (7 s.d.s away).
• "so $\chi^2_k/k$ has a standard deviation of $\sqrt{2/k}$" can you direct me to somewhere that demonstrates this please? – baxx Mar 24 '19 at 19:18