Bigheaded people have big ones (4)
I believe the answer is:
egos
'bigheaded people have big ones' is the definition.
(I've seen this in another clue)
This is the entire clue.
(Other definitions for egos that I've seen before include "feelings of self-importance" , "Self-images" , "People's personal pride - the parts of them that are conscious" , "Lots of self-esteem" .)