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From: Interpreting Residual and Null Deviance in GLM R we see that the residual and null deviances are calculated from the baseline log-likelihood value given by the saturated model.

It also notes that the smaller these deviances are, the better the fit the model is to the data. However, this effect can also be attained by just comparing the fitted and null model to a theoretical (perfect) model having a perfect fit to the data. Note that in this case likelihood-ratio tests are also unaffected.

So in what situations is the saturated model giving more information than the theoretical model with perfect fit? The only reason I can think of is given by the comment to this answer, but in this case isn't it more sensible to compare the log-likelihoods of the saturated model to a perfect model?

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  • $\begingroup$ Terminology differs across sub-fields and over time, but the way I have seen the term saturated model used is that it refers to all categorical variables with all possible interactions. In that case the saturated model is the theoretical model with perfect fit. However, occasionally I have seen it used without the condition that the variables are categorical. In that case there is a difference between the perfect. So it depends on the exact definition used for saturated. $\endgroup$ – Maarten Buis Feb 21 '17 at 8:36
  • $\begingroup$ @MaartenBuis, what field are you familiar with? $\endgroup$ – Alex Feb 21 '17 at 21:35

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