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Officials from the chief medical examiner’s office in New York City were furious when they heard that Marina Stajic, one of their longest-serving laboratory directors, had openly questioned whether they had sufficiently verified the reliability of a novel form of DNA testing being used in criminal cases.

Ms. Stajic was concerned that incorrect use of the DNA testing technique could lead to wrongful convictions, she said. But her bosses took her questioning in a different light.

“Hold me down,” Dr. Barbara Sampson, the city’s medical examiner, wrote in an internal email to a colleague in 2014, when she found out that Ms. Stajic had voted on a state panel to compel the office to release a study proving the technique’s validity. “She sucks,” a lawyer for the office wrote about Ms. Stajic, in another internal email.

Ms. Stajic, who was fired from the medical examiner’s office about six months later, sued in 2016, claiming she was pushed out in part because she had challenged the controversial DNA testing technique. On Monday, the city agreed to settle her case for $1 million.