Dear all,
I am running multilevel regressions on a binary dependent variable characterised by a majority of zero values (85%), which is in turn influenced by binary independent variables with extremely high percentages of zero values (>90%).
I know that if my dependent variable were a count variable I could have used zero-inflated poison or negative binomial model, but I am told that they would not be adequate to analyse binary variables.

Which models would you suggest me to keep this characteristic of the data under control?

Thank you very much for your help