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If crowds of Massachusetts residents came into New Hampshire on Election Day as part of widespread voter fraud – a claim made by President Donald Trump’s administration and others – they managed to do so without creating any spikes in voter turnout and without creating any unusual changes in town-by-town support for Kelly Ayotte.

That’s the conclusion of a study from a trio of Dartmouth researchers, whose work follows their previous studies that failed to find evidence of any voter fraud during the 2016 presidential election in six states.

“Because of these results and a total lack of photographic evidence of buses infiltrating New Hampshire on Election Day 2016, we believe that Trump’s claims about a tainted election in New Hampshire are at best unsupported and at worst an intentional mistruth,” says the report from two professors and a postdoctoral fellow.

President Trump has said more than once that “thousands” of Massachusetts voters were bused into New Hampshire on Nov. 8 to vote illegally, costing him the state and also contributing to former governor Maggie Hassan’s very tight win over incumbent Kelly Ayotte for a U.S. Senate seat. Trump most recently made the claim to a group of senators Feb. 9.

David Cottrell, a postdoctoral fellow in the Program in Quantitative Social Science at Dartmouth and one of three authors of the just-released study, noted that analyzing such a claim is difficult because of its lack of specificity.

“The fraud hypotheses are often very vague and ambiguous. Determining exactly what people mean is difficult, and there are always different theories that come out later. It could go endlessly – that’s kind of the nature of these theories,” he said.

To analyze the claim, the group chose some effects that it thought would be visible if large numbers of voters were crossing over from Massachusetts, and looked at electoral data.

The factors are:

Turnout surges: The researchers compared 2012 and 2016 Election Day turnout by town and saw “no particular surges that would suggest thousands of out-of-staters voting there illegally” in any communities, including those closer to the Massachusetts border and thus more likely to be visited by any buses.

“The results are completely consistent with a legitimate election,” the researchers wrote.

Ayotte support: The authors supposed that Democratic voters bused into New Hampshire would cast ballots in the hotly contested race for U.S. Senate.

When they compared Republican Kelly Ayotte’s percentages in each town in 2016 vs. 2010, the last time she ran, and compared that to changes in voter turnout in 2016, they found “a slight positive relationship” between more voters and more Ayotte support. In other words, towns that saw a larger surge in voting turnout had higher increase in Ayotte support.

“That’s the opposite of what you might expect to see if anti-Ayotte ringers had been bused in to vote in New Hampshire,” the researchers wrote.

Massachusetts driver’s licenses as ID: “We find no relationship between the number of Massachusetts drivers licenses used for same-day New Hampshire voter registration and any increase or decrease in votes for Ayotte between 2010 and 2016,” they wrote.

Ayotte support close to Massachusetts: “If buses ferrying out-of-state voters avoided traveling far into New Hampshire, towns near the New Hampshire border should be relatively anti-Ayotte. ... In fact, New Hampshire towns near the Massachusetts border tended to favor Ayotte, not oppose her,” they wrote.

The analysis was done by Cottrell; Michael Herron, a professor of government at Dartmouth; and Sean Westwood, an assistant professor of government at Dartmouth.

(David Brooks can be reached at 369-3313 or dbrooks@cmonitor.com or on Twitter @GraniteGeek.)