In March 2016 alone, a Trump rally in Chicago was called off after violent clashes broke out, while an anti-Trump protester was punched at a rally in North Carolina and another was punched and kicked at a rally in Arizona. The following month, several Trump supporters were assaulted at a California rally.

Mr. Trump himself repeatedly seemed to endorse attacks on his detractors, too.

“Maybe he should have been roughed up,” he said of one protester who was reportedly punched and kicked in November 2015. “I’d like to punch him in the face, I’ll tell ya,” he said of another a few months later. He even offered to pay legal fees for his supporters if they became too aggressive.

The supporters also often aimed offensive and violent rhetoric at Mrs. Clinton, suggesting she be killed.

To determine whether those words and news reports corresponded with an actual shift in violence, the researchers compiled a list of 31 Trump rallies and 38 Clinton rallies held in cities with assault data available online.