Donald Trump and Kris Kobach in November 2016. (Carolyn Kaster/AP)

After Matthew Dunlap, one of the members of President Trump’s disbanded voting fraud panel, released documents from the commission showing that it had failed to turn up any evidence of widespread voter fraud last week, the panel’s vice chairman, Kansas Secretary of State Kris Kobach, made his case for the commission’s existence.

One of the foremost proponents of stricter voter identification laws, Kobach, who is running in the primary Tuesday for the Republican nomination for the state’s governorship, has been undeterred since a federal judge struck down a restrictive voting law for which he had advocated in the state.

And in a statement sent to The Washington Post, Kobach accused Dunlap of being “willfully blind to the voter fraud in front of his nose,” pointing to studies from two conservative groups about the supposed voter fraud about which he has been so vocal: a database from the Heritage Foundation that found 983 convictions in state, local and federal elections dating back decades; and a study from the Government Accountability Institute, a nonprofit founded by Stephen K. Bannon and another Breitbart editor, that purported to find 8,400 instances of double voting in the 2016 election.

Kobach’s response was included in the reports of outlets such as CNN, the Associated Press and HuffPost. But election experts interviewed by The Post said that the two studies made for a flawed portrait of the issue of voter fraud. Examining them provides a window into the ways in which statistics are massaged and studies are selectively deployed in the push to address the supposed mass scourge of voter fraud with stricter voter identification laws. Though a handful of people vote illegally, either intentionally or unintentionally, every year, election experts say that there is no evidence that voter fraud is a widespread issue of any statistical significance.

Myrna Pérez, a deputy director at New York University’s Brennan Center and an expert on voting rights, pointed out that Heritage Foundation’s database catalogued a range of misconduct that included actions like vote buying or ballot-altering by elected officials — a far cry from the type of fraud alleged by President Trump that informed the commission’s creation.

“It’s important to distinguish between insiders perpetrating fraudulent votes and individual voters,” Pérez said. “The responses we see to this meme of fraud frequently put the burden on the voter.”

A study the Brennan Center did of the Heritage Foundation’s database showed that 105 cases came from the past five years and 488, about half, from the past 10 years. The cases date back as far as 1948, to the election when Harry S. Truman defeated Thomas Dewey.

“Over the period considered by Heritage, there have been over 3 billion votes cast in federal elections alone, and many more when you include the state and local elections also covered in the database,” the Brennan Center’s study reads. “The number of cases in the database represent a minuscule portion of the overall number of votes cast during this time span. In reviewing decades of cases and billions of votes cast, the Heritage Foundation has identified just 10 cases involving in-person impersonation fraud at the polls (fewer than the number of members on the President’s Commission).”

In a statement, the Heritage Foundation said that the database was only a “representative sampling.”

“We routinely update the database with new cases,” it said in the statement distributed by spokeswoman Breanna Deutsch. “Heritage has gathered cases from nearly every state, proving that election integrity is an issue in need of serious attention, not polarizing rhetoric.”

Experts had even harsher words for the Government Accountability Institute study, which was released in July 2017, about two months after Trump had formed the commission. The 36-page report, a copy of which can be found on the White House’s website, claims to have identified 8,400 cases of double voting in the 2016 election, based on data on 75 million voters from 21 states. The nonprofit said it compared lists of people who voted in different states and identified duplicates by matching their names, birthdays and a portion of their social security numbers. Virtual DBS, the firm contracted to crunch the data for the report, said that its methodology would result “in virtually zero false positives.”

But people who work regularly with election data said they thought it was significantly flawed.

“The report was amateurishly done,” said Paul Gronke, a political-science professor at Reed College and the director of the school’s nonpartisan Early Voting Information Center. “They were sloppy in their use of terms, and lack of precision is a big problem when you are accusing individuals or groups of committing a felony.”

The GAI declined to comment on Tuesday morning, through a spokeswoman.

Max Hailperin, a professor emeritus of mathematics, computer science and statistics at Gustavus Adolphus College in Minnesota who has worked with government officials on the state’s electoral system for years, said that the study suffered from a glaring vulnerability that can result from large data sets.

“Whenever you test a large volume of data for some rare occurrence, you wind up with two things that happen at low rates: one is finding what you’re looking for; and the other is finding artifacts in the data that mimic what you’re looking for,” he said.

Gronke pointed out that even if the 8,400 instances were taken as true, “those are beyond minuscule.”

“You need a scientific calculator to display that many significant digits,” he said. “Yet this is what a presidential commission focuses on, when we have the reality of foreign governments interfering in our election process?”

Hailperin explained how rare instances of human error in a large set of numbers — millions of data points that grow exponentially when multiplied against one another — could manifest as what might appear to be a significant number of matches, if not examined closely. Small mistakes, like people signing the wrong line of a voting roster, or errors as paper sign-ins are computerized later, are a part of the voting system, he said, and out of 75 million voters, the errors could add up.

“So there’s some very small rate of error that results in some small number of false appearances of double-voting,” Hailperin said. “There’s a lot of sloppiness in the records, and that’s unfortunate. But sloppy records are not the same thing as rampant fraud.”

For a study to be credible, it would need to account for this by calculating a rate of which these type of false positives were occurring and then adjusting the results accordingly, he said.

“You would have to randomly sample some of those and go out and do the investigation and get the on the ground truth,” he said.

Hailperin said he had a word for these type of studies about voting fraud: vigilante matching.

“This is very much of a genre,” he said. “When other vigilante-matching projects akin to this one have been followed up on by law-enforcement authorities, generally what they find is that a very, very small proportion of these apparent red flags actually turn out to be anything real.”

Dunlap, too, criticized Kobach for sharing the numbers, saying in a statement that “as usual, Secretary Kobach offers no proof, just allegations.”

“That two people have the same name and are registered in two states is a fact; it does not necessarily follow that they are the same person, committing a crime,” he wrote.

In an interview, Kobach defended the studies, saying that GAI’s report had informed his belief that more than 100,000 people participated in double voting in the 2016 election. Responding to criticism, he compared the data gathered by the Heritage Foundation to drunken-driving statistics.

“On any given day there may be a million cars on the highway but there could be 10 drunk drivers. You wouldn’t say that drunken driving isn’t a problem,” Kobach said. “The point is that there are enough fraudulent votes to recognize that it is a problem.”

He said that his political beliefs about stricter voter ID laws were based on facts, saying that his office had convicted 15 people for voter fraud in Kansas since it was given that authority to do so in 2015.

“Convictions only represent the tip of the iceberg,” he said. “I don’t know how many convictions a person has to see before he acknowledges that voting fraud is a reality.”

Federal judge Julie Robinson, who heard a case against Kobach, disagreed. Robinson overturned the law in Kansas that required proof of citizenship to register to vote in June, saying that the issue of noncitizen registrations could be “largely explained by administrative error, confusion, or mistake.” She had dismissed Kobach’s arguments about the “tip of the iceberg.”

“Instead, the Court draws the more obvious conclusion that there is no iceberg,” she wrote, “only an icicle, largely created by confusion and administrative error.”

This story was updated to include a statement from the Heritage Foundation. 

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