lme4 and deviance

Dear list,

I'm trying to understand why the deviance function returns -2 log likelihood in lme4 and not the "true" deviance as with lm().

The scaled deviance is defined as the difference of -2 log likelihood between the model of interest and the saturated model but what is the saturated mixed model? For balanced data, some authors (e.g. Hoffman 2014, Longitudinal Analysis: Modeling Within-Person Fluctuation and Change) define the saturated mixed model as a model with saturated means (one fixed effect for each time point) and unstructured variance but such a model is not available for unbalanced data.

And if we can compute the scaled deviance, what is the deviance?

Thanks in advance for any hint.

Nad?ge Jacot