I am trying to measure the performance of a student in five categories. I give the student a test of 100 problems. Each problem requires the student to use knowledge from one or more of the categories in order to solve the problem. I know the score for each problem, and I know which categories were required to solve the problem, but I don't know the weighting of the categories for each problem. For example, problem #1 requires knowledge of category A and B. The student scores a 90% on that problem. This could mean he scored 0% on A and 90% on B, 90% on A and 0% on B, or anything in between.

I do have a random mix of problem that might look something like this

Prob Categories

1 A C

2 A B E

3 D

4 B C D

...

Does anyone know how to tackle this type of problem?

I don't even know if I can say anything meaningful from the data at all?

Thanks,

Greg