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You wouldn't necessarily see them (and it doesn't have to be a them, even one can do a lot of damage). They are masters of getting into and out of spaces you would swear they couldn't (their joints are only barely attached--the original contortionists). On top of that, they are SMART! When we had pet rats, we used to give them puzzle boxes full of treats. The longest it ever took them to figure out the most complicated one was a day. Oh, and they LOVE eggs the way I love Cadbury Creme Eggs.
I would expect them to avoid the broodies - too much trouble if there are unguarded opportunities. And the poor barn cat might discourage them a little, but a full grown rat is bigger than you'd expect and very well armed. Claws on all four paws and a pair of incredibly sharp, very long teeth, make all but the most intrepid cat think twice. Because they are smart, traps can be really hit or miss. I hesitate to use poison, just because where that poison ends up can be hard to control--our neighbor's dog ate a poisoned squirrel and it was only the very expensive intervention of their vet that kept the dog from joining the squirrel across the rainbow bridge. Also, they can chew through ANYTHING, including hardware cloth.
You might try soaking wads of cotton in peppermint oil (the actual oil, not flavoring), and stuffing them into the corners of the coop. It works to deter mice, and I think it would work on rats as well, although I'm not sure how the chickens would feel about it.