I have mice in the shed since having chickens. They've never got into the food bins but I guess there's small spillages at times. Found a mouse sat stuffing itself at the chicken feeder once and it just ignored me and carried on eating!
5 chickens in the garden and no noticeable rodent population - we have the occasional fox - but in a pretty urban area. We keep our food inside and only put out what they can eat in a day to minimize waste and slugs - which seem to enjoy chowing down on chicken food. Bleeurch!
We've lived here 9 years and I've seen two rats, one of those was before we got the chickens, so we are not exactly infested. Unfortunately my neighbour saw one and freaked out so I've had to buy bait boxes now, but really I think there are no more rats than there would be without the chickens iyswim.
We have chickens and yes, there have been recent rat sightings by the neighbours. It isn't unusual for this part of the world when all the waterways are so full anyway, but I have no doubt that the rats have been responsible for the amazing rate the feed was disappearing from the feeder! I wouldn't say the chickens bring the rats but their free food encourages them to stay around, along with other neighbours feeding birds, another neighbour leaving their bin bags outs, just general normal stuff. We've all put poisoned bait down, and we have stopped using a feeder (doing more controlled feeding three times a day instead) and fingers crossed, we haven't seen them since. As a bonus, it's saving a lot of money on feed as it lasts longer thanks ratty!
If you see mice that's a good sign - it means there are no rats around.
If possible, try to raise your chicken hut onto concrete blocks or logs of wood or something - that discourages rats from nesting under your chicken hut and gaining access to it easily.
Keep food stored in a galvanised bin with a secure lid. They can get thought plastic. Think the bastards can even get through concrete
Try to put only enough food out for the chucks as they can eat in a day ... so there's none left in their feeder overnight. Even if you lock the feeder in the hut with the chickens overnight, rats will inevitably find their way in.
Do not throw more bread for the chucks than they can eat. Give them bread cut down into tiny bits so they can eat it OK and don't leave it for rats.
If you do see a rat, get something done about it straight away, don't procrastinate, because already by that stage (once they're making themselves visible) it means there is an infestation.