Is it currently possible to run a multinomial logistic regression with random subject and item effects, in R? I have a set of data in which participants get one of two types of items. I want to calculate if their responses are more likely to belong to one of X categories when they get one type vs. the other.
I am hoping to also include random subject and item effects.
Is it possible to run a multinomial logistic regression, including these random subject and item effects, in R?
 A: Yes it is possible.  This can be done with R packages for mixed effects regression such as "lme4" (see "glmer" function).  However it is not straightforward to accommodate the the multinomial nature of the dependent variable with "lme4" (it works best for binary variables).  Eventually you could use packages for choices modelling such as mlogit.  Your problem could be described as following: The (X) categories would be different choice options (e.g., product A, B, C, etc.) and the participants have to choose one of them. You could add a random effect to the choice of one category, for example by assuming that "preferences" for the categories are normally distributed over your sample. Your two items could be seen as an experimental manipulation (participants being allocated either to item #1 or #2). Again, it is possible to add a random effect to the type of items.
\begin{align}
P_{ntj} &= \frac{\exp(V_{ntj})}{(∑_j \exp(V_{ntj} ))}  \\
V_{ntj} &= β_{0n} + β_{1n} {\rm ITEM}_{ntj}  \\
β_{0n} &= μ_0 + ω_{0n} ~ N(0, σ_{0n})  \\
β_{1n} &= μ_1 + ω_{1n} ~ N(0, σ_{1n})  
\end{align}
Re "mlogit", you would be looking for something called mixed logit, random parameters logit, or even logit kernel model. Remark: Possible to use Probit specification instead. 
