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My question is regarding the interpretation of the percent of deviance explained (and other $R^2$ anaologs or pseudo $R^2$ values for GLMs.

Is this a meaningful summary statistic for models other than Gaussian? That is, is it at least as meaningful/useful as the $R^2$ statistic in the typical OLS regression model. (We'll work under the assumption that $R^2$ is useful summary measure in an OLS framework).

Of course, percent of deviance explained is equivalent to the $R^2$ value if the link is Gaussian.

fit1<-glm( rating~1,
           data=attitude,family = gaussian)
fit2<-glm( rating~complaints+privileges+learning+raises+critical+advance,
           data=attitude,family = gaussian)

(deviance(fit1)-deviance(fit2))/deviance(fit1)

[1] 0.732602

summary(lm( rating~complaints+privileges+learning+raises+critical+advance,
            data=attitude))

Multiple R-squared: 0.7326

Is there a similarly meaningful interpretation of the percent of deviance explained from models from other GLM families, like binomial (logit/probit) or poisson regression? Or, does the change in link function and the characteristics of the model/response affect the interpretation of this statistic?

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    $\begingroup$ Is this: ats.ucla.edu/stat/mult_pkg/faq/general/psuedo_rsquareds.htm helpful, or do you mean something else? $\endgroup$
    – gung - Reinstate Monica
    Commented Jan 4, 2012 at 20:04
  • $\begingroup$ Indeed. The deviance R^2 from my question is the same as McFadden's measure. My question is really about interpretation of the deviance R^2 for non-Gaussian models. With a normal link it corresponds to SSR/SST and thus has a clean and neat interpretation. Is this true of other families? Do the results depend on the link used and the characteristics of the response? $\endgroup$
    – Brett
    Commented Jan 4, 2012 at 20:47
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    $\begingroup$ Walking down another path, I stumbled onto a very clear answer to my question in by @probabilityislogic in the thread at: stats.stackexchange.com/questions/3559/… $\endgroup$
    – Brett
    Commented Jan 6, 2012 at 16:58
  • $\begingroup$ Could you re-post your comment as an answer to your own question if it covers everything you wanted? Then you can accept your answer, and this won't appear as an unanswered question. $\endgroup$
    – Corvus
    Commented Feb 2, 2013 at 12:06
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    $\begingroup$ The general deviance based R2, 1-residual deviance/null deviance, applies to any GLM with any family & link function. McFadden's was originally defined for logistic regression (where the log-likelihood of a saturated model=0) and in that case the above formula is correct, but the generically correct version would be 1-residual deviance/null deviance, see stats.stackexchange.com/questions/359906/… $\endgroup$
    – Tom Wenseleers
    Commented Jun 14, 2019 at 23:12

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