Same-sex marriage is supported by most Americans. And after last week’s landmark Supreme Court decision, it’s also the law of the land. But how it will play out in the presidential campaign is far from settled. While the 2016 Democratic presidential candidates were quick to embrace the Supreme Court’s decision, the Republican candidates — beyond saying that marriage should remain between a man and a woman — were split.

Some Republicans, including Marco Rubio, said the country must abide by “the law of the land.” Jeb Bush essentially agreed, calling for additional protections for those who object to the decision on religious grounds. The candidates further to the right, including Scott Walker and Ted Cruz, took a stronger stand, calling for a constitutional amendment to overturn the court’s ruling and allow for states to ban same-sex marriage. Cruz even said the Supreme Court decision was part of the “darkest 24 hours” in the nation’s history.

A look at public opinion on same-sex marriage and what drives party affiliation suggests that Cruz, Walker and the other candidates on the right may be risking the party’s appeal in the general election. The Republican Party’s opposition to same-sex marriage is one of the top positions that may have kept voters from identifying with and potentially voting for the GOP.

Polling generally suggests that same-sex marriage is not a top issue for most voters. A February CNN/ORC survey found that just 17 percent of Americans said the issue of gay marriage would be “extremely important” in choosing a candidate to support for president — the lowest of any of nine issues tested.

But digging deeper provides a different perspective. Beyond the importance voters place upon it directly, gay marriage may have symbolic power because of the messages it sends to voters about the parties.

I’ve taken a look at individual responses from two 2014 Pew Research Center surveys. I wanted to see which of 14 issues (ranging from abortion to gay marriage to size of government) provided the most information about a person’s party identification after controlling for demographic factors like age, education, income, race, and religious attendance.

More specifically, I ran a series of logistic regression analyses that tested each issue, along with the demographic categories I described above. The chart you see below reflects the regression coefficient associated with each issue variable. Here’s how to decode the numbers. Positive values mean that support for that issue predicts a greater likelihood of affiliating with the party. Support for gun rights, for example, is predictive of a greater likelihood of identifying as Republican. The larger the coefficient, the more influence the issue has. Negative values, conversely, predict less likelihood of supporting the party.

In the Republican column, the coefficient for gay marriage is large and negative, meaning that supporting it substantially reduces the likelihood that someone will identify as Republican. In fact, based on the regressions, the only variable more predictive of Republican identification is whether a person believes health care coverage is the government’s responsibility. Gay marriage is more important than classic “wedge issues” like guns or abortion in predicting whether someone identifies as a Republican.

Let’s be clear about what these results mean. What they say is that if I meet you at a picnic and you tell me that you support gay marriage, that gives me a lot of information about whether you’re a Republican — more than almost any other attitudinal question I might ask you.

It’s still difficult to prove that support for gay marriage causes people not to identify as Republican, as opposed to merely being correlated with it. Voters sometimes first choose a party they like and then adopt that party’s position on an issue. Gay marriage has been a fairly polarizing issue, so it is possible that some voters are merely choosing the position that lines up with their party. Still, at least our method is controlling for demographics. Views on gay marriage still seem to matter a lot even once you account for those.

But how can gay marriage matter so much even if voters say it isn’t so important? One theory is that it’s a gateway issue. Voters may look to a position on gay marriage to understand whether the Republican Party represents their values. If people are for gay marriage, then they may not consider voting for a Republican, even if they agree with the candidate on other issues. They may see the party as too out of touch with the times.

Many young people have probably run into situations like this with friends. As Ari Fleischer was quoted as saying in the Washington Post, “When a young voter sees a Republican coming, many of them roll their eyes and wonder why they can’t get with modern life.”

So if support for gay marriage makes someone less likely to identify as Republican, does that make them more likely to identify as a Democrat? It does, of course. But look carefully at the chart I showed you above. While gay marriage ranks as the second-most-important issue in predicting whether someone will identify as Republican, it falls to fifth on the list in predicting if someone will call themselves a Democrat.

This may mean that a lot of people who reject the Republican Party because of its opposition to gay marriage aren’t lining up with the Democrats either and instead are choosing to be independents. Republicans might not care about this, provided those independents were still voting Republican in the end. Many self-described independents are hidden partisans, indeed, usually voting for one party or the other even if they reject the party label. In the Pew surveys, many voters who answered the question about gay marriage were also asked about how they planned to vote in the 2014 midterm elections. It turns out that a person’s opinion on gay marriage was actually slightly more significant in explaining their 2014 vote choice than it was in explaining their party identification, after controlling for demographics.

That suggests that there are voters Republicans aren’t getting because of gay marriage, and it’s why Republicans who have their eye on the general election, like Bush, want to move on from the subject.

Other Republicans, for the moment, are spending more time trying to win the evangelical vote in the primaries. But candidates like Walker, Cruz, Bobby Jindal and Mike Huckabee, in their outspoken opposition to the Supreme Court’s decision, risk increasing the salience of gay marriage instead of letting the issue die down.

This means that gay marriage could hurt Republicans in 2016. But the good news for Republicans is that if same-sex marriage is a major issue in the 2016 election, it will likely be the last time. It’s difficult to imagine that we’ll still be talking about the politics of gay marriage on a national stage in five years, when the public as a whole is likely to be even more in favor of it.