Use of social media (type and frequency). Enroll in University (as a result of using SM to search). Remain at University (class/course activities using SM).
What is the best approach to use to determine the relationship?
It's perhaps worthwhile trying to acquire a dataset from a MOOC provider. For example, einztein.com is a social network dedicated to social learning. Although this doesn't give you access to specific university-based learning, it would allow you to measure things like engagement from students. If it has to be for students enrolled at a university, it is likely that Evgenia's suggestion is most feasible. Although, in theory, you could ask students to install a plug-in to measure their OSN activity and report back...it is unlikely many would be happy with that though!
I am not sure of the relevance between enrolling via social media and then using social media in a university course as students can do the former without the latter and the latter without the former. I would think it more relevant to study their use of social media once enrolled in university. I am currently working on a project with Professor Neil Selwyn who has done a great deal of work in this area. One of the case studies investigates social media use. I have provided below the links to the project. If it is not available yet it will be available early in the new year. Cheers, Kevin.