Evan Vucci/Associated Press President-elect Donald Trump took to Twitter on Sunday to complain that illegal voters cost him the popular vote in the 2016 election.

"In addition to winning the Electoral College in a landslide, I won the popular vote if you deduct the millions of people who voted illegally," Trump said.

However, a political scientist who's likely behind Trump's assertion that "millions" of people illegally voted said "there is no way" they could explain how democratic candidate Hillary Clinton won the popular vote with a margin of more than 2.2 million votes (and growing).

Trump's baseless tweet made waves a day after Clinton's campaign said it would join vote recount petition efforts initiated by independent candidate Jill Stein. Stein has cited Russia's alleged hacking of Democratic Party email servers as one reason to recount votes in the swing states Michigan, Pennsylvania, and Wisconsin, and her campaign has raised more than $5 million to pay for legal and filing fees.

Marc Elias, the Clinton campaign's legal counsel, wrote in a Medium post on Saturday that while there's no "actionable evidence" of voting machine hacking or "attempts to alter the voting technology," her campaign intends "to participate in order to ensure the process proceeds in a manner that is fair to all sides."

Trump has since called Stein and Clinton's recount petitions a "scam".

However, if millions of illegal ballots were cast, as Trump believes, then the validity of his election as president would be questionable, given that he narrowly won the Electoral College by 107,000 votes.

Fortunately for Trump, voting experts say his claim is implausible.

Where Trump's illegal 'millions' come from

Voters cast their ballots during early voting at the Beatties Ford Library in Charlotte, North Carolina, U.S. on October 20, 2016. Chris Keane/Reuters

Business Insider asked the Trump campaign multiple times to verify the source of his claim that "millions" voted illegally, but we did not receive a response.

However, the figure almost certainly comes from a contentious September 2014 study of non-citizen voters — people who are legally prohibited from voting, but presumably do so anyway — that Trump has cited at rallies throughout his campaign, including one on October 22, 2016, in Cleveland, Ohio.

The study, which was peer-reviewed and published in the journal Electoral Studies, leaned on self-reported data from a 100-question, internet-based survey run by the Cooperative Congressional Election Studies (CCES). (The organization is funded by National Science Foundation grants and is led by researchers at Harvard University and Stanford University.)

In 2008 and 2010, the CCES surveys asked respondents about their citizenship status, and if they were registered to vote.

Jesse Richman, a political scientist at Old Dominion University, and two colleagues used data from these and other questions to ask how many non-citizens — people not born in the US, or have not yet endured the country's grueling naturalization process — might have voted in and possibly affected the outcome of elections.

The team focused on survey responses from after the 2008 presidential election, which was one of the largest voter turnouts in history. The researchers estimated that a maximum of 25.1% of non-citizens were registered to vote, and up to 11.3% illegally cast a ballot. Focusing on a population of 19.4 million non-citizens that year, Richman and his colleagues wrote "the number of non-citizen voters [...] could range from just over 38,000 at the very minimum to nearly 2.8 million at the maximum."

With such a wide and unreliable range, Richman and his team made an "adjusted" rate of verified voting at 6.4% of all non-citizens voting. The researchers called this their "best guess" of the actual voting rate, based on the CCES survey data and other studies of voting behavior.

"The adjusted estimate of 6.4 percent for 2008 is quite substantial, and would be associated with 1.2 million non-citizen votes cast in 2008 if the weighted CCES sample is fully representative of the non-citizen population," the authors wrote, noting that many votes could influence an election.

But the numbers differ wildly from other analyses, which indicate that non-citizen voting is extraordinarily rare; in fact, one national investigative reporting project determined that only 56 non-citizen were cast from 2000 through 2012.

Thus, Richman and his team — even though they noted their preliminary study's major limitations — met heated criticism from other researchers as popular websites like Breitbart, which has published articles with white nationalist themes, promoted the research to conservative audiences.

John Ahlquist and Scott Gehlbach, two researchers from the University of Wisconsin-Madison, tore into the study and its conclusions at the Washington Post's Monkey Cage blog. They cited the confusing nature of the CCES survey questions, extremely small sample size (less than 100 self-reported non-citizens in 2008, out of 32,800 subjects), and other steep limitations of the analysis.

Michael Tesler, a Brown University political scientist, argued at the same blog that respondents whom the CCES followed over the years showed signs of error — namely, reporting being citizens one year and non-citizens 2 years later, possibly at a rate of 71%.

While Richman and his colleagues responded to these and other criticisms, three researchers criticized his team's work in the same journal in July 2015. Their rebuttal says the initial study was "almost certainly flawed" and provided "a biased estimate of the rate at which non-citizens voted in recent elections", ultimately concluding "that the rate of non-citizen voting in the United States is likely [zero]."

Despite these criticisms, Richman says he stands by the work.

"I believe that there has surely been some non-citizen participation in the 2016 election," Richman told Business Insider in an email. "The 2016 CCES will hopefully include follow-up questions to verify the citizenship status of self-reported non-citizens. This should help get at the issues raised by our more perceptive critics."

Why Richman says Trump's claim is not plausible

If there was any independent political science researcher who could back up Trump's victimizing claims, it'd be Richman.

But he doesn't.

"I don't think that participation has been anywhere close to the level required to account for the entirety of Clinton's nation-wide popular vote margin," Richman told Business Insider.

Richman pointed us to a post he wrote at his university-hosted blog, which extrapolates his team's work in 2014 to the 2016 election.

In the post, Richman lays out why it's not plausible that non-citizen voter fraud could explain how Trump lost the popular vote by millions of ballots:

"The basic assumptions on which the extrapolation is based are that 6.4 percent of non-citizens voted, and that of the non-citizens who voted, 80 percent voted for Clinton and 20 percent voted for Trump. [...]

"As of this writing Trump is 2,235,663 votes behind Clinton in the popular vote. [...]

"If the assumptions stated above concerning non-citizen turnout are correct, could non-citizen turnout account for Clinton's popular vote margin? There is no way it could have. 6.4 percent turnout among the roughly 20.3 million non-citizen adults in the US would add only 778,524 votes to Clinton's popular vote margin. This is little more than a third of the total margin."

Richman went on to write that, while it's plausible non-citizens contributed to Clinton's margin of victory, it's "not at all" plausible that non-citizen voters could explain his loss.

"Even if we assume that 90 percent voted for Clinton and only 10 percent for Trump, a more than fourteen percent turnout would be necessary to account for Clinton's popular vote margin," Richman wrote. "This is much higher than the estimates we offered. Again, it seems too high to be plausible."

Business Insider sent Richman's post to Trump's campaign, asking for comment, but we haven't heard back.

Update (December 5, 2016): After this post was published, readers pointed out that Judicial Watch — a conservative legal interest group (which fueled controversy over Clinton's private email server) — hosts a PDF of Richman's full study, and that Richman links to the group's file in his Nov. 28 blog post at Old Dominion University.

However, in a series of emails to Business Insider, Richman asserted his academic independence and lack of any affiliation t0 the group.

"I have never had any ties with Judicial Watch, nor do I have any such ties today. I have never received any funding or other support from Judicial Watch, nor am I soliciting any funding from them," Richman told Business Insider in an email. "I linked to Judicial Watch merely because they posted a PDF that is outside of the journal paywall which makes it more feasible for people to read the study."