Significance test for 13 dichotomous DVs?

I am studying unprofessional conduct. I have every case of misconduct over a 5-year period. I am curious if misconduct varies by education level.

As you can see, misconduct is broken into 13 mutually exclusive categories, with a binary coding scheme. Theoretically, all 13 charges could exist in the same case. The independent variable (education) is three level: high school degree, AD, BA. This is a complete sample from 1 out of 50 National Institutions for this five year period.

I know the population proportions and can perform a goodness of fit test. However, could someone explain to me how to test if there is a difference between the misconduct of each educational group? (i.e. does education impact the type of misconduct?)

I could determine the proportion of successes for each IV level, and contrast these? I could also run a Cochran's Q test for each level, but how would I use this to compare the IVs? Could I treat this like a survey and run a test for multiple responses?