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I've been self-studying GLMs and I have some questions regarding the deviance in the context of GLMs. In Generalized Additive Models An Introduction with R, the author defines the deviance of a model as

$$ 2\big(l(\hat{\beta}_{max}) - l(\hat{\beta})\big)\phi $$

where $\phi$ is the scale parameter, $\hat{\beta}_{max}$ denotes the MLE of the saturated model and $\hat{\beta}$ is the MLE of the fitted model. However, I have also seen the deviance defined simply as

$$ 2\big(l(\hat{\beta}_{max}) - l(\hat{\beta})\big), $$

without the scale parameter, this is the definition used on Wikipedia. Which definition is the correct one? And what definition of deviance is used in the summary output in R for GLMs?

Any help is much appreciated!

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  • $\begingroup$ It is worth noting that single-parameter GLMs assume $\phi = 1$, which may explain the difference in definition you have seen. For example. logistic regression, which is probably the best known GLM, is also a single-parameter GLM. $\endgroup$
    – Frans Rodenburg
    Commented Jun 24 at 12:00
  • $\begingroup$ @FransRodenburg I'd argue OLS is the best known GLM. But I take your point :) $\endgroup$
    – ischmidt20
    Commented Jun 24 at 18:34
  • $\begingroup$ The first definition is the correct definition in the context of GLMs. You seem to have taken the second formula from the Wikipedia page for "deviance", which is not quite correct. The authors of that page seem to have forgotten about the possibility of a scale (dispersion) parameter. The normal deviance they give for example doesn't match their definition. $\endgroup$
    – Gordon Smyth
    Commented Jun 26 at 11:11

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