WASHINGTON — President Donald Trump and his allies have long insisted that Russian's 2016 propaganda campaign on social media had no impact on the presidential election.

A new statistical analysis says it may well have.

The study, by researchers at the University of Tennessee, Knoxville, does not prove that Russian interference swung the election to Trump. But it demonstrates that Trump's gains in popularity during the 2016 campaign correlated closely with high levels of social media activity by the Russian trolls and bots of the Internet Research Agency, a key weapon in the Russian attack.

"Our results show that the weeks when Russian trolls were accumulating likes and retweets on Twitter, that activity reliably foreshadowed gains for Trump in the opinion polls," wrote Damian Ruck, the study's lead researcher, in an article explaining his findings.

The study found that every 25,000 re-tweets by accounts connected to the IRA predicted a 1 percent increase in opinion polls for Trump.

In an interview with NBC News, Ruck said the research suggests that Russian trolls helped shift U.S public opinion in Trump's favor. As to whether it affected the outcome of the election: "The answer is that we still don't know, but we can't rule it out."

Given that the election turned on 75,000 votes in Michigan, Wisconsin and Pennsylvania, "it is a prospect that should be taken seriously," Ruck wrote, adding that more study was needed in those swing states.

He points out that 13 percent of voters didn't make their final choice until the last week before the election.

Ruck said the correlation between troll activity and Trump's popularity remained true even when controlling for Trump's own Twitter activity and other variables.

"It turns out that the activity of Russian Twitter trolls was a better predictor of Donald Trump's polling numbers than his own Twitter activity," he wrote.

Ruck was among a group of researchers who won a Defense Department grant this year to study Russian disinformation campaigns in Georgia, Ukraine, and Belarus through March 2024.

The researchers found to their surprise that the Russian propaganda did not drive down Democrat Hillary Clinton's popularity, even though much of it was aimed at discrediting her. One factor may have been that the target audience was mostly right-wing media consumers who already disliked Clinton.

The study noted that Trump's own policy proposals occasionally seemed to dovetail with Russian propaganda.

For example, Russian propaganda exploited the December 2015 shooting in San Bernardino, California "to stoke fears about Muslim immigration," the study says.

Shortly afterward, Trump announced his support of a ban on Muslims entering the United States.

The president's son-in-law and top adviser, Jared Kushner, in April derided the Russian propaganda effort as "a couple of Facebook ads."

In fact, the Internet Research Agency, a troll farm led by an oligarch close to Putin, conducted "a vast, coordinated campaign that was incredibly successful at pushing out and amplifying its messages," according to cybersecurity firm Symantec, which conducted an in-depth analysts of nearly 4,000 Twitter accounts involved in what U.S. intelligence agencies assess was a Russian-government–sponsored propaganda operation designed in part to help Trump get elected president.

The IRA's disinformation campaign was described in the final report by special counsel Robert Mueller, and in his indictment of multiple Russian nationals. U.S intelligence agencies assessed that the social media manipulation was part of a Russian intelligence operation designed to undermine American democracy by exacerbating divisions, hurt Clinton and help Trump. Experts say the U.S. government and American social media companies have yet to develop a strategy to prevent such manipulation from happening again. Just last week, a different cybersecurity firm exposed an Iranian effort to manipulate U.S. social media with fake accounts.

The IRA's basic strategy, Symantec found, was to use a small core of Twitter accounts to push out new content. And they harnessed a wider pool of automated accounts to amplify those messages.

"We also see that 91 percent of first retweeters of IRA tweets were non-IRA bots," the Tennessee study says, "which suggests that propaganda spread into networks of real U.S. citizens."

The Trump administration has not developed a strategy to deal with the social media propaganda that is already cropping up in connection with the 2020 election, and Trump continues to downplay the issue of foreign election interference. Last week, he sparked a furor when he seemed to joke about it during a joint appearance with Russian President Vladimir Putin, saying, in a mocking tone, "Don't meddle in the election."

The White House did not immediately respond to a request for comment.