In other words, instead of having a two class problem I am dealing with 4 classes and still would like to assess performance using AUC.
It seems you are looking for multi-class ROC analysis, which is a kind of multi-objective optimization covered in a tutorial at ICML'04. As in several multi-class problem, the idea is generally to carry out pairwise comparison (one class vs. all other classes, one class vs. another class, see (1) or the Elements of Statistical Learning), and there is a recent paper by Landgrebe and Duin on that topic, Approximating the multiclass ROC by pairwise analysis, Pattern Recognition Letters 2007 28: 1747-1758. Now, for visualization purpose, I've seen some papers some time ago, most of them turning around volume under the ROC surface (VUS) or Cobweb diagram.
I don't know, however, if there exists an R implementation of these methods, although I think the
stars() function might be used for cobweb plot. I just ran across a Matlab toolbox that seems to offer multi-class ROC analysis, PRSD Studio.
Other papers that may also be useful as a first start for visualization/computation:
- Visualisation of multi-class ROC surfaces
- A simplified extension of the Area under the ROC to the multiclass domain
1. Allwein, E.L., Schapire, R.E. and Singer, Y. (2000). Reducing multiclass to binary: A unifying approach for margin classifiers. Journal of Machine Learning Research, 1:113–141.
While the math is beyond me this general review article has some references you will likely be interested in, and has a brief description of multi-class ROC graphs.
An introduction to ROC analysis by Tom Fawcett Pattern Recognition Letters Volume 27, Issue 8, June 2006, Pages 861-874
Link to pdf as provided by gd047- thanks