Should I consider data integrity issues when choosing a file compression algorithm?

Is there a relation between a data compression format (zip, gzip, bzip2, etc.) and it's ability to ease in recovering broken data?

I want to store some backups (like tar.gzs) but then I started wondering what could happen if the archive somehow becomes corrupt, like when changing a single bit randomly. Will the entire file become useless or are there mechanisms to minimize damage?

Most information I find is purely about storage efficiency and they never seem to mention possible corruptions...

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