But you are setting the quota, so looking at the data of a respondent you also know which quota the person was screened out for. Of course, if you change quotas during field time, this can be an issue. So you might have to note down which quotas you have changed and when. Also, you should see which quota was triggered by the last question that was registered in the response table. Of course, this only works if you have simple quotas that depend only on one question.
However, you could think about including hidden equation type question where you store if and why someone should be screened out.
This should work for "normal" screenouts, but I don't think this would work for quotas though, as they need to check the database.
You might consider using the stats functions to create your "quota". You could take the stats within an equation question and calculate whether the quota is met or not and then write into the equation question, if the respondent should be screened out or not. Then set the quota on this equation type question.
Just a few ideas on how to approach this to make your life easier.
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