Even before Hillary Clinton officially launched her bid for the White House last year, political observers speculated about whether she would be hurt by sexism. Would gender stereotypes lead voters to doubt Clinton’s ability handle a national security crisis? Would people focus on her appearance instead of her policy ideas? Would Americans be willing to cast ballots for a female commander in chief? There was intense debate about these questions during, and especially after, Clinton’s first primary battle eight years ago. The country now has an opportunity to ponder these questions in an entirely new way — a presidential election in which for the first time in U.S. history, a male candidate faces a female opponent.

Recent political science research has cast doubt on the notion that sexism would derail Clinton’s campaign, including here in the Monkey Cage. Numerous studies of nonpresidential races have found that women are just as likely as men to win. In most campaigns, the media do not treat female candidates differently than male candidates. And even if voters possess gender stereotypes, those attitudes rarely cost women votes, in part because party identification is so powerful.

But sexism can be subtle and difficult to detect. People who aren’t willing to vote for a woman may not be willing to say so in a survey, instead giving an answer that is more socially acceptable. This is known as “social desirability” bias. If social desirability obscures sexism, standard polling could underestimate the gender bias that Clinton might face on Election Day.

Here’s how we looked for hidden gender bias

Earlier this year, we designed two studies to determine whether hidden sexism — the kind that might be hard to detect in a traditional survey — could lower support for Clinton. Those studies were just published in the Special Elections Issue of PS: Political Science & Politics.

In our first study, a nationally representative survey of 810 voters (391 of whom were Democrats) conducted in January, we used an approach that has uncovered bias in other contexts. Instead of simply asking respondents how they planned to vote, we randomly assigned them into two groups. Half were asked to tell us which candidate they would say they supported if they wanted to make the best impression possible on another person. We told the other half to answer as if they wanted to make the worst impression.

We then compared how many of the Democrats named either Clinton or Sen. Bernie Sanders as the choice that makes the best impression to how many said that same candidate makes the worst impression. If many more people say a particular candidate gives a good impression than a bad impression, then there is social desirability in claiming to vote for that candidate. And if voting for that candidate is the “cool” or “politically correct” answer, presumably some will give that answer even if that’s not how they plan to vote.

Our first survey came during the Democratic primary

When we asked people which candidate choice they thought would make the best or worst impression, about the same percentage said that “Clinton” would make the best impression as said “Clinton” would make the worst impression.

So we did not see any social desirability bias when it came to voting for Clinton. Rather, Bernie Sanders emerged as “the cool candidate” most prone to social desirability bias — especially among young Democrats. If anything, this suggests primary polls underreported, not overreported, support for Clinton; presumably, some young Democrats said they’d vote for “Sanders” just to make a good impression.

Our second survey came as the general election loomed.

In another survey in late April, we randomly assigned 900 respondents to one of two groups. One group was asked whether they would vote for Donald Trump or Hillary Clinton in the general election. The second group was asked whom their neighbors would vote for.

The logic of this approach, pioneered by the Obama campaign in 2008 to detect racial bias, is that respondents who don’t want to vote for a woman might be reluctant to express opposition to Clinton because of social desirability. But they might be more willing to express their own biases under the guise of discussing their neighbors. If more people said that their neighbors wouldn’t vote for Clinton than said that about themselves, that would suggest hidden gender bias among voters.

But as you’ll see in the graph below, we find no such evidence of hidden sexism. The percentage of respondents saying that their neighbors would vote for Clinton is actually higher than the share saying they themselves would vote for Clinton. That’s exactly the opposite of the pattern we would expect from voters harboring hidden gender bias.

However, we do find evidence that “Won’t Vote” may be seen as the socially desirable response. More people said that they wouldn’t vote than said the same about their neighbors. But this was driven entirely by Republicans. Some Republicans felt more comfortable saying they wouldn’t vote than saying that they would defect to Clinton or even vote for Trump. These “won’t vote” respondents, however, divided about evenly in whether they said their neighbors would vote for Clinton or for Trump — suggesting that this group will break evenly between the two.

Trump has said he suffers from the opposite of social desirability bias, and that polls may underestimate his support. Trump may be correct that some of his supporters are telling pollsters they “won’t vote.” But some hidden voters are likely to cross party lines to vote for Clinton. Those who would vote for Trump are a small and, as Clinton’s lead increases, increasingly inconsequential number. But we find no evidence that social desirability will lead pollsters to overestimate support for the Clinton candidacy.

The polls, in other words, are probably accurate.

Finally, in the same April survey, we investigated more general support for female candidates for president and the relationship between those general attitudes and support for Clinton specifically. We again randomly assigned people to either indicate if they “were more or less likely to vote for a woman” or if their neighbors “were more or less likely to vote for a woman.”

In this last experiment, we found two reasons that sexism shouldn’t hurt Clinton at the ballot box. First, although some voters said they didn’t want to elect a woman as president, partisanship overrode that. Some Democrats may prefer a male president, but they are far more concerned about having a Republican in the Oval Office.

The second reason Clinton should not suffer at the ballot box because she is a woman is especially notable. The number of people who are more likely to vote for a candidate because she is a woman outnumber those who are less likely to vote for a woman. And note this survey was taken before the most recent allegations of Trump’s sexual predations. Enthusiasm for electing the United States’ first female president seems to, if you’ll pardon us, trump any gender bias that may exist in the American electorate.

Ryan L. Claassen is professor of political science at Kent State University. John Barry Ryan is associate professor of political science at Stony Brook University.