A largely ignored study from the National Bureau of Economic Research in Cambridge polling 2,516 Americans showed that when people are given anonymity they report far greater queer sexual experience (up 59%), increased self-identifying as non-hetero (up 65%), and much higher antigay prejudice, namely:
"Respondents were 67% more likely to express disapproval of an openly gay manager at work and 71% more likely to say it is okay to discriminate against lesbian, gay, or bisexual individuals. The results show non-heterosexuality and anti-gay sentiment are substantially underestimated in existing surveys, and the privacy afforded by current best practices is not always sufficient to eliminate bias."
Bloomberg News confirms what you always knew:
"The veiled survey had no effect on the answers of young people to questions about their sexual orientation, apparently because social norms don’t much discourage young people from revealing the truth.
"But among Christians and older people, the effect of the veiled approach was especially large, increasing their reports of non-heterosexuality and of same-sex experiences by more than 100 percent.
"In best practices, only a minority of Republicans (35 percent) said they would be unhappy with an LGBT manager. Under the veiled report, most Republicans (67 percent) said they would be unhappy."