A right-leaning fact-checker is fighting critics on the left who say its conclusion that a lot of noncitizens vote illegally is bunk.

The online battle of debunking and rebuttal is playing out as a much larger war has erupted between President Trump’s commission on election integrity and Democratic state leaders. They are refusing to provide the panel with public voter registration data. Left-wing groups are suing to stop the commission’s work, which could settle the noncitizen debate by collecting enough data.

In the fact-check standoff, there is Just Facts, a small New Jersey firm of conservative and libertarian policy analysts who promote what they say is solid independent research.

Just Facts President James D. Agresti issued a blockbuster report in June. Using previous research, polling data and Census Bureau figures, his team concluded that as few as 594,000 noncitizens or as many as 5.7 million voted in the 2008 presidential election. If accurate on the high side, it would vindicate Mr. Trump’s contention that a lot of illegal ballots were cast in his race for the White House last year with Democrat Hillary Clinton.

Challenging Mr. Agresti are fact-checkers PolitiFact and Snopes.com, which conservatives generally view as liberal; liberal college professors; and left-leaning news sites such as HuffPost.

When Mr. Agresti read their broadsides, he issued a rebuttal that essentially said they do not understand statistics. Scientific random samples, though small, can be used to estimate behaviors of larger population sectors as all polls do, he said.

PolitiFact wrote: “The number comes from a conclusion by Just Facts, a conservative/libertarian think tank. Just Facts’ numbers came from a study by Old Dominion University researchers. That study was based on a survey which showed that 38 people out 32,800 claimed to be noncitizens who had actually voted. Just Facts used data from the study and census estimates on the noncitizen population to come up with a national figure of noncitizen voters.

“But other researchers and political scientists have said the small number is not a reliable source of data on noncitizen voters nationwide. We rate this claim False.”

Mr. Agresti wrote in his rebuttal that his critics’ articles “are littered with sophomoric inconsistencies, irrationalities and outright falsehoods.”

The referenced survey is the Cooperative Congressional Election Study spearheaded by Harvard University. Its study, with polling by YouGov.com, is the database used by Old Dominion University policy analysts who issued the first extensive study of its kind on how many noncitizens vote illegally.

The left, whose researchers and politicians contend that few aliens vote, immediately scorned the product from Old Dominion.

‘Mathematically illiterate notions’

The liberal press declared the ODU study debunked even though the authors issued a paper defending their work. A group of liberal professors circulated a letter calling on colleagues to blacklist the study.

The debate is complex because it is intertwined in polling data, census numbers, actual recorded voter behavior and assumptions about how people answer surveys.

But there is a central fact: The Harvard study polled 32,800 people. Of those, 339 were self-described noncitizens, and of those, 38 said they definitely voted or were in a database that showed they voted.

The Harvard researchers said the sample is too small to draw conclusions. They deconstructed the 38 figure and eliminated all of them based on a comparison with voter databases and other factors.

PolitiFact, Snopes and liberals asserted that the Harvard researchers debunked ODU and Just Facts.

PolitiFact points to Brian Schaffner, a political science professor at the University of Massachusetts at Amherst and an author of the Harvard study. It quoted his Politico column: “Survey respondents occasionally select the wrong response by accident — perhaps because they are rushing through and not reading the questions carefully, because they do not fully understand the terminology being used, or because they simply click on the wrong box on the page.”

Mr. Agresti said the Old Dominion policy analysts took the issue of sampling into consideration by “weighting” the data to match Census Bureau numbers.

In addition, he said, 14 percent of self-declared noncitizens who said they were not registered to vote were in fact on voter rolls, according to databases.

Of those, 8 percent answered, “I definitely voted” in 2008, and of those, 82 percent said they voted for Barack Obama.

“Given the poor math skills of many adults, the arguments of Snopes and PolitiFact may sound convincing to some people, but these so-called fact-checkers are using mathematically illiterate notions instead of concrete, quantifiable facts,” Mr. Agresti wrote.

As background, Mr. Agresti points to a scientific survey by the marketing consultancy McLaughlin & Associates. In a poll of Hispanics for the California wine industry, 13 percent of noncitizens said they were registered to vote. Matched against the country’s noncitizen population, that means 800,000 to 2.2 million noncitizen Hispanics were registered to vote, Just Facts concluded.

About 20 million noncitizens, including 11.8 million Hispanic adults, are living in the U.S., the Census Bureau said in 2013.

Commission’s mission

Mr. Agresti said it is more likely that noncitizens said they were citizens in the Harvard survey, not the other way around, because technically they otherwise would be admitting to a crime.

“Moreover, even if Just Facts disregarded all noncitizens who said they voted and limited its analysis only to votes confirmed by voting records, the data would show that 590,000 to 3.9 million noncitizens voted in the 2008 election,” he said. “In accord with scientific research methods, Just Facts fully accounts for the sampling error in its study, and this is why the results span a wide range of 594,000 to 5.7 million illegal voters.”

PolitiFact said it conferred with three elections analysts outside the Harvard study who agreed with the debunkers.

Mr. Agresti said that while Politifact cast him as an ideologue conservative and critics as politically pure, it failed to disclose that two of the quoted analysts contributed money to the Obama campaign. The third wrote an op-ed urging millennials not to vote for a third party because it would help Donald Trump.

One way to settle the noncitizen debate could be the Presidential Advisory Commission on Election Integrity. The co-chairmen, Vice President Mike Pence and Kansas Secretary of State Kris Kobach, want the states to provide what is normally public voter registration data.

The commission would compare those lists with other data, such as Social Security rolls and Homeland Security Department files on noncitizens who hold green cards.

In 2013, the Social Security Administration found that 700,000 illegal immigrants were using Social Security numbers with fraudulent birth certificates and 1.8 million had cards that did not match their names.

Democrat-led states are stonewalling the commission, and a leader of this resistance is Virginia Gov. Terry McAuliffe.

He has vetoed a string of bills designed to weed out illegal voters. His appointees also broached the idea of making optional the citizenship question on voter registration forms but abandoned the idea amid strong opposition from Republicans.

Meanwhile, the Public Interest Legal Foundation, a conservative grass-roots group dedicated to eliminating voter fraud, released a report in May that found at least 5,500 noncitizens were registered to vote in Virginia this decade and at least 1,852 had voted and had cast more than 7,000 votes.

The foundation said the numbers are likely just part of a bigger figure because the 5,500 likely disclosed their noncitizenship status when acquiring driver’s licenses.

The Public Interest Legal Foundation said Mr. McAuliffe’s administration repeatedly has blocked its access to records, forcing the group to file lawsuits.