My cat Mimi has stuck her butt outside the litter box and peed off and
on for a couple of years.  I have had her urine checked multiple
times, and it's always negative.  So since Peewee died, she has peed
outside the litter boxes every day!!  I figured they must still smell
like him to some degree, so I removed those boxes completely, and
replaced them with an Jumbo hooded litter box from Van Ness.  I also
switched to an unscented litter.  Finally - problem solved - no more
peeing outside the litter box!!  I think the key was the hood and the
extra-high sides.

-L.