While there has been plenty of commentary on this theme so far in the campaign, it turns out there is comparatively little hard evidence on how gender affects elections. The differences between men and women are not as pronounced as those between, say, white and black voters, which political scientists have studied in more depth.

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Women are slightly more likely to vote for Democrats, while men are slightly more likely to vote Republican. In 2012, for example, Mitt Romney won 54 percent of ballots cast by male voters, while 56 percent of women voted for President Obama, according to Gallup. It was the widest margin between the genders recorded in any presidential election since Gallup began compiling figures in 1952.

Yet, there is no reliable forecast for an election like the one that would put Trump up against Clinton, a match that could be unprecedented in terms of the politics of gender. A woman has never been a major party's presidential nominee before, and no candidate of either sex has ever run against a candidate like Trump. His scabrous comments about women could help both him and Clinton, by motivating female voters to turn out in opposition at the polls or by uniting men drawn to his rhetorical style in support of his candidacy.

"We're definitely in uncharted territory," Sanbonmatsu said.

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There is some preliminary data on this unusual campaign thanks to the American National Election Studies, a project that has been polling Americans on their preferences regarding candidates for decades. Recently, the project published a pilot study for this year's election, based on an online poll conducted in January.

Participants in the poll were asked how much discrimination there is against several different groups, including both men and women. Overall, 38 percent of men and 23 percent of women said there was at least "a moderate amount" of discrimination against men.

Men's perceptions of discrimination might amount to a kind of "resentment" at women's advances in society, said Leonie Huddy, a political psychologist at Stony Brook University who has studied the results of the poll.

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"It is a feeling that men are aggrieved, that things aren't right for them," she said. (She noted that women who perceive discrimination against men may be empathizing with the frustrations of their sons, brothers or spouses.)

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Participants in the poll were also asked to rate their feelings toward politicians on a 100-point scale. On this subjective scale, a score of 0 would represent very negative feelings toward a candidate, a score of 100 would represent very positive feelings, and 50 points would indicate indifference.

In an analysis, Huddy and her colleague Jennifer Lawless of American University found that men who said there was discrimination against men had somewhat colder feelings toward Clinton than toward male Democratic figures, such as her primary rival Bernie Sanders and President Obama.

The analysis took into account other factors, such as respondents' partisanship, education, race and political ideology, among other demographic factors. In other words, regardless of their party and political orientation, men concerned about men's status in society viewed Clinton more negatively.

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On the other side, men who said that men are discriminated against had more positive views of Trump and his competitor, Sen. Ted Cruz (R-Tex.). Their attitudes toward Sen. Marco Rubio (R-Fla.), who recently dropped out of the presidential race, were not meaningfully different.

The analysis predicted that the rare man who said there was "a great deal" of discrimination against men would give Trump a subjective score averaging about 13 points higher than a similar man who said there was no discrimination against men at all.

The figure was essentially the same for Cruz — 14 points in the senator's favor. Cruz, though, did best among respondents who opposed feminist public policies, such as mandatory equal pay or parental leave and increased spending on child care. Trump's supporters were ambivalent on these questions.

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Clinton fared about 12 points worse among men who perceived a great deal of discrimination against men, compared with otherwise similar men who perceived none at all.

"That could hurt her," Huddy said. The differences in how Sanders and Obama would be rated by these men were not significant in a statistical sense.

At the same time, Huddy added, Clinton might make up for any losses among male voters with greater enthusiasm among women. Meanwhile factors, such as race, partisanship and ideology, were more important in respondents' feelings toward the candidates, according to Huddy and Lawless's analysis.

"It's not just men and women," Huddy said. "It's the kind of view you have of the world."