We study ballot independence for election schemes. First, we formally define ballot independence as a cryptographic game and prove that ballot secrecy implies ballot independence. Secondly, we introduce a notion of controlled malleability and show that it is sufficient for ballot independence. We also show that non-malleable ballots are sufficient, but not necessary, for ballot independence. Thirdly, we prove that ballot independence is sufficient for ballot secrecy under practical assumptions. Our results show that ballot independence is necessary in election schemes satisfying ballot secrecy. Furthermore, our sufficient conditions will enable simpler proofs of ballot secrecy.