Salary data provided by Sen. Elizabeth Warren’s office to outlets such as the Huffington Post paint a misleading picture of employment in the Massachusetts Democratic senator’s office and fail to disprove the fact reported Tuesday that Warren paid men significantly more than women in 2016.

Warren’s office failed to respond to numerous emails from the Washington Free Beacon for the Tuesday report, which found that the median salary for men that worked for Warren in 2016 was $21,000 higher than the median salary for women.

Instead it reached out to the Huffington Post, a liberal publication owned by telecom giant Verizon, which on Wednesday published an article titled "No, Elizabeth Warren Doesn’t Pay Women Less Than Men" that said the Free Beacon "misleadingly accused" the Massachusetts senator of gender pay inequality by using "incomplete data."

The data that Warren’s office supplied to the Huffington Post, which was shared with the Free Beacon by reporter Laura Bassett, fails to disprove that Warren paid men significantly more than women in 2016 and is statistically flawed.

Warren’s office included every employee that has ever worked in the office, regardless of how long, in its calculations. It then took "the average salary that employee earned over the course of the time they worked for Warren," according to the Huffington Post, and then took the median of those averages for each gender. After jumping through those hoops, it found that the median salaries were nearly identical, with women coming out slightly on top $44,108 to $43,348.

The decision to look at numerous years rather than just a single fiscal year—the preferred method of both the Free Beacon and U.S. Census Bureau—was the first mistake made by Warren’s office, according to Mark Perry, a University of Michigan economics professor and scholar at the American Enterprise Institute.

"If you look at multiple years of salary data, there is too much turnover and changes in staff to make any valid comparisons of gender differences in income," Perry said.

After analyzing both sets of data, Perry pointed out that the high level of turnover at lower-paid positions is what pulled median salaries so far down in the analysis done by Warren’s office.

"Since there is much greater turnover at the lower-paid positions compared to the higher-paid positions, it would be difficult to design a statistical methodology to account for the turnover of younger workers/staffers," Perry said.

Here is a little more detail from Perry on how this method can distort the reality of pay in Warren’s office:

About two-thirds of the staffers in the spreadsheet provided are employees who made $50,000 or less, and that’s the group with the highest turnover. The fact that they are reporting medians so low—about $44,000—means that the lowest paid staffers are over-represented. With higher turnover, that low-paid group gets double-counted, and triple-counted. For example, if there are 20 low-paid staffers (the group most likely to quit if they find something better with more pay), and 15 of them quit in a given year, and are replaced with 15 new staff, there would be 35 staff members counted in that year for the lowest-paid group. And that would distort a meaningful comparison of salaries by gender. Your method avoids that issue, by only looking at full-time staff who worked the entire year and is a much better approach.

The problems become more apparent when you look at the employees who wind up included in the analysis when you include everybody.

One employee included in the analysis by Warren’s office, Ceferina Murrell, worked as a press assistant from July 17 to July 23, a six-day span that included both a Saturday and Sunday. She was paid $777.75 for her time.

In the data prepared by Warren’s office, however, Murrell is assigned a salary of $47,313—a number higher than the median for women—even though she actually took home just a tiny fraction of that number.

"We should compare actual earnings that have actually been paid," Perry notes.

Similarly egregious is the inclusion of Walter Cross, who at $27,948 is the lowest paid male in the data prepared by Warren’s office. Cross receives just a fraction of his pay from Warren’s office—he is a systems administrator who splits his time between four Senate offices and earns nearly $150,000 each year.

Also included was a $134,816 salary for a woman named Alexandra Swartsel, who has never appeared on Warren’s payroll documents. She most recently worked for the Senate Democratic Conference Committee in 2016, which was chaired by former Sen. Harry Reid (D., Nev.). It is unclear why Warren included Swartsel and her six-figure salary in the analysis. No other employees who were never paid by Warren were included.

None would have been included in the Free Beacon’s method, which was laid out in the original report and aims to create what in statistics is referred to as an "apples-to-apples" comparison. To the best of our ability, using publicly available official personnel and office expense account filings, we attempt to only compare full-time staffers that worked the full year.

Perry says the Free Beacon’s method meets the mark and is consistent with the numbers used on Equal Pay Day, which was a day that Warren recognized before this year.

"Equal Pay Day is based on the gender differences in median annual salaries of full-time workers in the most recent previous year," Perry explained. "So I think your pay gap analysis of Warren’s staff is totally valid (and preferred) by looking at full-time salaries of her staff in 2016, and including only those staff members who worked for the entire year. That is a valid methodology to try to do an ‘apples-to-apples’ comparison."

In addition to providing its own numbers to the Huffington Post, Warren’s office complained that the Free Beacon didn’t include former chief of staff Mindy Myers, who left the office on January 3, 2016, and adviser Kaaren Hinck, who left the office on February 21, 2016 (and was referred to by the Huffington Post as "Haaren Hinck"), in its 2016 calculations. It also complained that Lauren Miller, a new media director who was paid $69,375 by Warren’s office, actually made $120,000 if you include her work for Warren’s political operation.

As a creative exercise, the Free Beacon included salaries for both Hinck and Myers ($118,000 and $170,000, according to Warren's office), and upped Miller’s compensation to $120,000.

Without including Myers’s male replacement as chief of staff, the median male salary was still $16,766 higher than the median female salary, with women making 77.3 percent of what men made—a rate that remains lower than the national average in 2016.

Warren’s office has still not spoken with the Free Beacon regarding its 2016 pay gap.

In addition to the Huffington Post, Warren’s office also shared its data with the managing editor of Young Voices, who writes about criminal justice and feminist issues and had her piece published by the Daily Beast.

Warren is yet to publicly comment on the issue of gender pay inequality since the Free Beacon’s Equal Pay Day report.