If your goal is to make them say "I was wrong," you will almost always lose. Pride is a powerful motivator. If your goal is to get your way or change their behavior, let them save face.
People lose arguments because they let the topic drift. If you’re arguing about the dishes, don't let them bring up "that thing you did three years ago." How to Win Every Argument
If they get loud, you get quieter. It makes them look irrational and gives you the moral high ground. If your goal is to make them say