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?

  • 2
    $\begingroup$ 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. $\endgroup$
    – Peter Flom
    Sep 21, 2012 at 10:36
  • $\begingroup$ This link might be of interest. Specially the section "Criteria For Assessing Goodness Of Fit". $\endgroup$
    – user10525
    Sep 21, 2012 at 14:57
  • $\begingroup$ @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? $\endgroup$
    – Fomite
    Sep 21, 2012 at 19:41
  • $\begingroup$ r-bloggers.com/… also has some information about how to answer this question, though @StasK's response covers it well enough. $\endgroup$
    – flies
    Dec 9, 2016 at 16:02

2 Answers 2


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).

  • 1
    $\begingroup$ "so $\chi^2_k/k$ has a standard deviation of $\sqrt{2/k}$" can you direct me to somewhere that demonstrates this please? $\endgroup$
    – baxx
    Mar 24, 2019 at 19:18
  • $\begingroup$ amazon.com/…. Sorry to be an asshole, but that's a reference distribution in statistical inference; if you don't understand it, you should not be working with generalized linear models such as Poisson. $\endgroup$
    – StasK
    Apr 5, 2019 at 17:07
  • 7
    $\begingroup$ For future reference you can, instead of the prefix / apology about being an asshole thing, just state the information and a reference. It would probably save you typing, and make you appear less of an asshole, which might be a novel experience. $\endgroup$
    – baxx
    Apr 5, 2019 at 17:11
  • $\begingroup$ See edit and the wikipedia reference. I have volunteered a few hundred answers over a few years, so I admit it is a bit difficult for me to have a really novel experience. $\endgroup$
    – StasK
    Apr 5, 2019 at 17:28

Asymptotically the deviance should be chi-square distributed with mean equal to the degrees of freedom. So divide it by its degrees of freedom & you should get about 1 if the data is not over-dispersed. To get a proper test just look up the deviance in chi-square tables - but note (a) that the chi square distribution is an approximation & (b) that a high value can indicate other kinds of lack of fit (which is perhaps why 'around 1' is considered good enough for government work).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.