I am looking for an appropriate measure of the "explained variance" of a Poisson GLM (using a log-link function).

I have found a number of different resources (both on this site and elsewhere) that discuss a number of different pseudo-$R^2$ measures, but nearly every site mentions the measures in relation to a logit-link function, and they don't discuss whether the pseudo-$R^2$ measures are appropriate for other link functions, such as log-link for my Poission distribution GLM.

For example, here are a few of the sites I've found:

Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)?



My question is: Are any of the methods discussed at those links (in particular, the FAQ on the UCLA page) appropriate for a Poission GLM (using a log-link function)? Is any particular method more appropriate and/or standardly used than any other method?

Some background:

This is for a research paper in which I am using a Poission GLM to analyze neural data. I am using the deviances of the models (calculated assuming a Poission distribution) to compare two models: One model (A) which includes 5 parameters that were left out of the other model (B). My interest (and the focus of the paper) is to show that that 5 parameters statistically improve the model fit. However, one of the reviewers would like an indication of how well both models fit the data.

If I were using OLS to fit my data the reviewer is effectively asking for the $R^2$ value for both the model with the 5 parameters and w/o the 5 parameters, to indicate how well either model explains the variance. It seems like a reasonable request to me. Lets say that, hypothetically, model B has an $R^2$ of 0.05 and model A has an $R^2$ of 0.25: even though that may be a statistically significant improvement, neither model does a good job of explaining the data. Alternatively, if model B has an $R^2$ of 0.5 and model A has an $R^2$ of 0.7, that could be interpreted in a very different way. I'm looking for the most appropriate measure that can be applied in a similar way to my GLM.

  • $\begingroup$ Why wouldn't a BIC work or a test of the difference in the log-likelihoods, particularly since one is a nested version of the other? $\endgroup$
    – user78229
    Commented Nov 6, 2015 at 13:13
  • $\begingroup$ This is a bit late for my purposes (the paper was published online this past Wednesday), but for the record: I am using the difference in the log-likelihoods as the primary measure, but a reviewer wanted a measure of "explained variance", so in the interest of appeasing the reviewers, I tried to come up with something. What I ended up with was something like what nukimov suggested below. $\endgroup$ Commented Nov 7, 2015 at 2:19

1 Answer 1


McCullagh and Nelder 1989 (page 34) give for the deviance function $D$ for the Poisson distribution:

$$ D = 2 \sum\left(y \log\left(\frac{y}{\mu} \right) - (y-\mu)\right) $$ (sign error in formula now corrected)

where y represents your data and $\mu$ your modelled output. I use this function to estimate the explained deviance $ED$ of a GLM with Poisson distribution like this:

$$ ED = 1 - \frac{D}{\text{total deviance}} $$

where total deviance is given by the same equation for $D$ but using the mean of $y$ (a single number, i.e., $\mathrm{mean}(y)$) instead of the array of modelled estimates $\mu$.

I do not know if this is 100% correct, it sounds logical for me and seems to work as you would expect an estimate of the explained deviance to work (it gives you 1 if you use $\mu = y$, etc).

  • 1
    $\begingroup$ I used the deviance function as the primary measure for the paper, using exactly the equation you provided above. However, a reviewer wanted a measure of "explained variance", so in the interest of appeasing the reviewers, I tried to come up with something. What I ended up with was: $$ pseudoR^2_M = \frac{ln(\Gamma_M) - ln(\Gamma_{Null})} {ln(\Gamma_{Sat}) - ln(\Gamma_{Null})} $$ $ln(\Gamma_{Sat})$ is the log-likelihood of a saturated model, $ln(\Gamma_{Null})$ is the log-likelihood of the null model, and $ln(\Gamma_{M})$ is the log-likelihood of the model in question. $\endgroup$ Commented Nov 7, 2015 at 2:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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