I'm on the school board of a small private school and I can (anecdotally) tell you the factors that determine a child's success there are the same as they'd be

I'm on the school board of a small private school and I can (anecdotally) tell you the factors that determine a child's success there are the same as they'd be at the public school - engaged parent(s). I can also tell you that the social environment at our school is dramatically different from that of public schools.