Note: For a criminal conviction the standard of proof is "beyond a reasonable doubt", while for civil cases, such as evictions, the standard of proof is based on "the preponderance of the evidence". Note that a PHA cannot simply allege that criminal activity has occurred, however. Some sort of evidence will be required.
For example, if the PHA claims that drug-related criminal activity has occurred, proof that illegal drugs were involved will be needed (typically testimony by a medical examiner or forensic laboratory), as well as some proof tying the sale, possession, use, manufacture etc. of the drugs to the tenant being evicted.