All the old wive's tales about a strong association between certain
items on the heap and rats are complete hogwash. As a ratcatcher
said to me, they are attracted by the warmth, protection and worms.
I compost everything (including turkey carcases) and there is NO
correlation between when I do so and when rats invade.

Sounds more like an old ratcatcher's tale to me. Assuming we are talking of
the brown rat, ratus norvegicus, then you are rather wide of the mark.
Probably the major factor in the worldwide success of the brown rat is that
it will eat such a wide variety of foods. Although it is omnivorous
preferred foods are those that are starch-rich or high protein in particular
grain or seeds. A proven favourite is apple peel and if you can it is better
to avoid putting it on a compost heap frequented by rats. As for access to
the heap the rat will have burrowed upwards from below. It may be worth
asking the Environmental Health Department of your local council if they
provide any rodent eradication free of charge.