The Post's Michelle Ye Hee Lee explains why White House press secretary Sean Spicer's claims on Jan. 24 about voter fraud in the presidential election don't add up. (Bastien Inzaurralde/The Washington Post)

Update: Here is a letter signed by many political scientists supporting the evidence in this post against the research that Trump has used to defend his incorrect claims about voter fraud.

Yet again, Donald Trump is claiming that the reason that he lost the popular vote was because of millions of illegal votes. He is threatening a mass investigation. When pressed for evidence to back up Trump’s beliefs, Sean Spicer, the White House Press Secretary, claimed that there was a study “out of Pew in 2008 that showed 14 percent of people who have voted were not citizens.”

Mr. Spicer is mixing up his studies – the Pew research that he is referring to says nothing of the sort. There is a study by political scientists Jesse Richman, Gulshan Chattha, and David Earnest that purports to use data from a large national survey — the Cooperative Congressional Election Study (CCES) — to show that some 14% of non-citizens are registered to vote. President Trump has previously referred to this study in public as evidence of illegal voting. Unfortunately for President Trump and Mr. Spicer, the study is wrong.

Richman, Chattha and Earnest’s study was summarized at The Monkey Cage and provoked three rebuttals (here, here, and here) as well as a response from the authors.

After this exchange, we published a peer-reviewed piece arguing that this study is wrong and that there is absolutely no evidence from the data that non-citizens voted in recent presidential elections.

We argue that the findings in the Richman et al. article can be entirely explained by measurement error. Specifically, survey respondents occasionally select the incorrect response to a question merely by accident.

In 2012, we re-interviewed 19,000 respondents who had originally taken the CCES survey in 2010. We asked about a respondent’s citizenship status in both 2010 and 2012. A very large fraction (99.25 percent) of respondents indicated that they were citizens in both waves of the survey. Only 85 respondents said they were non-citizens in both waves.

But the remaining 56 respondents actually changed their response between 2010 and 2012 — including 20 who responded that they were citizens in 2010 but non-citizens in 2012, a highly unrealistic change.

Thus, it appears as though about 0.1-0.3 percent of respondents are citizens who incorrectly identify themselves as non-citizens in the survey. With a sample size of 19,000, even this low rate of error can result in a number of responses that appear notable when they are not. The mistake that Richman and his colleagues made was to isolate this small portion of the sample and extrapolate from it as if it were representative of some larger population.

Given the extremely low rate of voting among purported “non-citizens” described in the Richman et al. article, it is almost certainly the case that all of the non-citizen voters that they report are actually citizen voters who simply clicked the wrong box on the survey.

Indeed, especially telling is this: Of the 85 respondents who said they were non-citizens in both 2010 and 2012, there was not a single voter. In other words, among the group of respondents who we can actually be confident are non-citizens, none voted.

Thus the best estimate of the percentage of non-citizens who vote is zero.

Unfortunately, Richman et al. failed to account for the fact that even very low rates of measurement error on the citizenship question would bias their estimates of voting among this small group of respondents.

Even more unfortunate is the fact that the Trump White House is now using this debunked result to validate his claims about a rigged election.

Stephen Ansolabehere is professor of government at Harvard University.

Samantha Luks is a managing director at YouGov.

Brian Schaffner is professor of political science at the University of Massachusetts-Amherst.

An earlier version of this post was published on Oct 19, 2016.