There are a few things that do change how they react in situations, like what outfit they wear, or if they will strip in public. But almost none of the really affect the story, other than Lin, and the contest.
I thought it was 10, but I heard you need a 14 or higher in Lin's stats (Obedience must be at least 20, and her Blowjobs, Vaginal, and Anal must be at least 14). Aside from that one event, the changes for them having the higher stats are pretty minor, the most important of them being exhibitionism and sluttiness.