There's a whole lot of literature about multi-class extensions for ROC.
I have some presentations with illustrations how the calculation works at softclassval's home page (softclassval calculates sensitivities etc. if you have partial class memberships, also for multiple classes - but that is probably an overkill for your problem).
For sensitivity and specificity, the spelled out definitions lead to a very straightforward extension:
- sensitivity: what proportion of truly class $c$ cases are correctly recognized by the model?
- specificity: what proportion of cases truly not belonging to class $c$ are correctly recognized as not coming from class $c$?
If you think about medical diagnostics/epidemiology, the set up is always multinomial from a philosophical point of view: the normal/healthy/control group in fact is rather a "not this disease" group which may contain a whole lot of other diseases. Sometimes classes are mutually exclusive, more often they are not (having, say, a brain tumour does not mean that you cannot have hepatitis nor does it save you from breaking your arm)
- I use package ROCR to plot ROCs, but there are plenty alternatives in R (e.g. pROC - pROC's home page has a comparison of several R packages dealing with ROC generation in R).