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.