Hillary Clinton is the first woman to clinch a major US party's presidential nomination, and she's competing against Donald Trump, a man who is accused of being a sexual predator. So it's no surprise that sexism has been a major theme on the 2016 campaign trail — from Trump's blatantly misogynistic remarks, to more subtle sexism from pundits who criticize Hillary's "shouting" or tell her to "smile."

But will sexism or stereotypes about Clinton's gender actually affect who votes for her? That's a much more complicated question.

One experiment found that Clinton's gender could cost her as much as 24 points against Trump among male voters, and 8 points overall.

In late February of 2016, a Fairleigh Dickinson University PublicMind poll asked New Jersey voters who they would support if the general election were held today. But in order to test the effect of "gender-role threat," half of the respondents were primed with a question about how much they make compared with their spouse before they were asked whom they support in the election. The other half were asked the same question after they'd already answered questions about the election.

Some men might perceive the idea of a wife making more than them as a threat to their masculinity. Raising the question at all reminds people that traditional gender norms can be disrupted — and could "simulate the sorts of subtle gender-based attacks that can be expected when Clinton is a general election candidate," writes Dan Cassino, director of experimental research for the poll and a professor of political science at Fairleigh Dickinson in Madison, New Jersey.

Just asking that one gender prime question made men choose Trump over Clinton in shocking numbers. Men who weren't asked the gender prime question supported Clinton 49-33, but those who were primed supported Trump 50-42. That's a 24-point swing in support, which is simply massive.

Overall, Clinton was up 19 points among New Jersey voters who didn't get the gender prime question, but that lead dropped to 11 points among voters who did get the question. Women actually favored Clinton more after the gender priming question, shifting 12 points in her favor, but that wasn't nearly enough to overcome the drop in male support.

Respondents were also asked about Sanders, but the numbers barely budged for him. That suggests this is more about gender than party, Cassino said.

This is a striking finding, to be sure, one that suggests some Americans are less comfortable with the idea of a female president than we might hope. But will it translate at the ballot box?

Cassino told Vox it probably will. He used other data from the study to come up with a rough estimate of how much gender-role threat could hurt Clinton among people who weren't explicitly primed beforehand, and found that it could still make a 5 to 6 percent difference.

Jennifer Lawless, director of the Women in Politics Institute at American University, is more skeptical.

"The results you find in experiments often generate gender differences that don't wind up emerging in actual campaigns or when you survey voters," Lawless told Vox.

That's because gender is just one of many, many factors that play a role in voters' decisions. And of course, you can't segment the electorate so half of them gets a specific piece of information right before voting.

"When you look at the difference in support based on who got what question first, and then you think about how that might manifest in a real campaign, it's almost not transferable," Lawless said. "That's not to say it's not a real finding. It just winds up getting muted by the cacophony of a real-world campaign."

Party identification tends to matter a lot more than gender when it comes time to vote, Lawless said. Even gender priming is polarizing, which Cassino also noted — it makes liberal men express more liberal views and conservative men more conservative ones. And since Clinton herself has been such a recognizable and divisive figure in American politics for decades, Lawless said, it can be hard to untangle "sexism" from "Clintonism."

But, Cassino told Vox, party identification isn't always stable. Another study he ran in 2008 accidentally found that asking about Clinton before asking about demographic information even made men less likely to identify as Democrats.

"Independents" have a much smaller role in presidential elections than people think, Cassino said. But Democrats do also have higher "defection rates" than Republicans in presidential elections. That is, they're more likely to vote against their party. And gender-role threat is likely to increase this defection rate not just among "independent" men but also among some men who identify as Democrats.

Lawless acknowledged that there may be some gender backlash among certain demographics, and that gender-role threat could account for some of Clinton's poor performance with men in polls so far. It could cost her a point or two. But Lawless doubts it will make a big difference in November.