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The only thing that three political scientists wanted to do was send mailers to thousands of Montana voters as part of a study of nonpartisan elections. What could possibly go wrong?

A lot, judging from the outrage and a state investigation. It has also raised thorny questions about political science field research, which isn’t uncommon, and its ability to affect an election.

The experiment, by the political scientists Adam Bonica and Jonathan Rodden of Stanford University and Kyle Dropp of Dartmouth College, sent mail to 100,000 Montana registered voters about two elections for the state’s supreme court. The Montana mailer, labeled “2014 Montana General Election Voter Information Guide,” featured the official state seal. It also placed the four judicial candidates on an ideological spectrum that included Barack Obama and Mitt Romney as reference points.

The mailers were designed to test whether voters who received information placing candidates on an ideological range would be more likely to vote. While political scientists have studied partisan races and voter turnout, less is known about nonpartisan races. Mr. Bonica, who had developed ideological scores for candidates and campaign donors, created a similar measure for state supreme court justices. The ideological scores for the Montana nonpartisan candidates were based in large part on the partisan candidates their donors had also given to. (The Upshot has written about Crowdpac, a start-up that Mr. Bonica is involved in.)

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Part of the offending mailer.

By putting the four candidates on an ideological range, the experiment also raised the ire of Montana officials, who expressed concern about the injection of partisanship into an officially nonpartisan race. “For a university to join in the flood of outside groups coming into Montana is not doing a service to our democracy,” said Senator Jon Tester, a Montana Democrat, in a telephone interview. He said the experiment was done “to influence voters” and described the mailer’s language as “the kind of stuff I get from the Montana Republican Party or Democratic Party.”

The use of the state seal also upset officials because it cannot be used on campaign literature. Groups wishing to use it must obtain the permission of the Secretary of State. Montana officials have begun an investigation into the mailers’ display of the official seal. Montana’s commissioner of political practices, Jonathan Motl, has asked Stanford and Dartmouth to disavow the mailers.

Stanford and Darmouth have jointly sent an open letter to voters apologizing for the mailer, and both are now investigating the project.

Senator Tester also said he would examine if any federal money had been used to pay for the study. (It was funded by a grant from the Hewlett Foundation and from Stanford itself, although the university has not detailed the source of that money).

The sharp reaction has veered into questions of whether the researchers were out to favor one candidate or another or whether the experiment was some kind of test of Crowdpac’s business.

Political scientists conduct field experiments to assess voter turnout in every election cycle. There is a large and growing body of work about the most effective methods for reaching voters and mobilizing them to vote, and field experiments are common.

Harvard political scientists have examined the effect of independent groups on ballot initiatives, for example, and like the Stanford-Dartmouth study, did so during a real election. The political scientists send different messages to different voters to test their effectiveness, a practice also used by political operatives. As John W. Patty, professor of political science at Washington University in St. Louis, put it, aside from using the state seal, “nothing that the researchers did would be inadmissible if they had just done it on their own as citizens.”

The size of this experiment — with mailers sent to nearly 15 percent of Montana’s registered voters — might be perceived as a large enough sample to directly sway the outcome. But nearly all field experiments conducted during elections affect the outcome in some way.

Melissa Michelson, a political science professor at Menlo College in California, said in a telephone interview that it could “poison the well” for future field experiments by making it harder to get institutional approval.

“Most of the time, experiments are nonpartisan or done in conjunction with an organization or campaign,” she said. The Montana mailers were not, she said, and “they made it look too official” by using the seal. Ms. Michelson, who wrote about the controversy on a political science blog, said adding ideological measures to the mailers was likely to raise the hackles of voters in the West. “Out here, that’s just not how we do judicial elections.”

Nonpartisan elections of judges have been seen as a way of reducing the influence of politics in the judicial system. But there is research that shows that voters are able to figure out partisan affiliations in such elections, although they err more often than in a partisan contest. Other studies have found that judges have similar characteristics whether selected in partisan or nonpartisan contests, weakening the argument that nonpartisan elections result in jurists of higher quality.

It’s well established that nonpartisan judicial elections are associated with lower turnout and lower voter information than partisan contests. Montana fits this pattern: In 2010, a state supreme court race drew about 47,000 fewer votes than a statewide partisan race for Congress in a state with fewer than 700,000 registered voters. The Stanford-Dartmouth experiment was aimed at seeing whether that gap could be narrowed by providing randomly selected voters with additional details about the candidates.

“We don’t have a lot of information about campaigning in these kinds of races until about 2002,” said Melinda Gann Hall, a political science professor at Michigan State University who studies state judicial elections. Since voters often take their cues from party affiliation, the lack of that information on nonpartisan ballots can make any independent information, whether advertisements or voter guides, more important, she said.

The controversy may have the result of raising the profile of the state supreme court contests, even possibly increasing voter turnout in Montana.

But someone else will have to study that.