The Multinomial model is not part of the standard family. The implementation is based on Chapter 6 in Fahrmeir et al. (2013).
multinomial()An object of class familyLMMsolver with the following components:
character string with the family name.
the link function.
the inverse of the link function.
function giving the deviance for each observation as a function of (y, mu, wt)
Fahrmeir, Ludwig, Thomas Kneib, Stefan Lang, Brian Marx, Regression models. Springer Berlin Heidelberg, 2013.