In my experience the predator that most frequently eats the head and leaves the rest is a Great Horned Owl. Do your birds roost outside the coop at night?
Pictures of your coop and run will help too.
We had rats one year, who beheaded three chickens overnight. It all depends on who can enter their coop at night. Unless the birds are just 'out there'.
Mary
I bought four hens not knowing they wouldn't coop up at night. Never thought of an owl. Must have scared one of them. She went in the coop for the first time tonight. Hoping the other two will start doing the same.