The teams that go down at the end are usually the ones that panic and start making desperate coaching changes or tactics that are completely off the wall. That destroys any confidence the squad had built up, even if it was fragile. If you want to try to predict, look less at their position in the table and more at consistency over the last five games - a team that's been losing consistently is in worse shape than one that's alternated wins and losses, even if they have the same number of points overall.
One thing not many people mention: the schedule matters a lot. Teams that face the top 4 clustered in the final rounds are basically doomed, while whoever has easier games in their last matches can pull themselves out of nowhere. It's worth checking who still plays who before you bet on a prediction.
About big teams going down... it's rare, but not impossible. What I see is that even a big team playing badly has the financial structure to make emergency signings in January if needed. The smaller clubs don't have that flexibility, so they're the ones who end up going down. But then it goes back to what I said: there are still rounds to play, and football is merciless with teams that lose their heads.