If I remember correctly from all of my reading on this family, the real Laura and Lazy, Lousy, Liza Jane never were exactly the best of friends. Rose was sort of a rebel as a teen and definitely an independent thinker; for whatever reason, she left home and attended school near EJ's residence in Louisiana, I think.
EJ eventually became the pariah/black sheep of the entire family, as she convinced Pa and Ma Wilder to invest in rice farming which ended up losing them their entire life savings. Bad move. But she was also quite a modern woman, taking up her own claim in the Dakota Territory, moving to Washington D.C. later to work for the Department of the Interior, marrying at the age of 42, remarrying after she became a widow, and eventually divorcing her second husband. Quite a life for a nineteenth-century woman!
What bugs me about the TV show is how they depict EJ. I mean, I love Lucy Lee Flippen, but she's nothing like the real EJ. And although there's nothing wrong with Flippen's appearance in real life, she didn't resemble EJ in the least. EJ was more of a classic beauty. She was also (according to the books) quite nasty to Laura; it would have been interesting to see that play out on-screen.