Test for analyzing significance of trend?

I am working on a dataset where a new administrator joins a school, and I want to see if it affects the number of students at the school. There are also characteristics of the administrator (age, gender, years of experience) and types of students (split by grade and core subject). I have the number of each type of student for the 3 years before and the 3 years after the administrator joins the school.

I know I'll need to run many tests to determine which administrator characteristics are significant, which student types are most affected, etc, but what has me stumped is how to use the 3 years before and 3 years after data. We don't know how quickly students may leave. I was thinking of using a paired t-test with the 9 year pairs tht could exist (-3 years and 1 years, -2 years and 1 years, etc) but that doesn't seem most efficient. Any suggestions for how to tackle this?