# 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?

10 is large… 1.01 is not. Since the variance of a $$χ2k\chi^2_k$$ is $$2k2k$$ (see Wikipedia), the standard deviation of a $$χ2k\chi^2_k$$ is $$√2k\sqrt{2k}$$, and that of $$χ2k/k\chi^2_k/k$$ is $$√2/k\sqrt{2/k}$$. That’s your measuring stick: for $$χ2100\chi^2_{100}$$, 1.01 is not large, but 2 is large (7 s.d.s away). For $$χ210,000\chi^2_{10,000}$$, 1.01 is OK, but 1.1 is not (7 s.d.s away).