Regression Analysis in Pay Discrimination Class Actions

A pay regression estimates how much of the difference in pay between two groups of employees remains after the legitimate factors that set pay, such as job, level, tenure, location and experience, are held constant. In a pay discrimination class action it is the principal common proof, and nearly every dispute about it concerns a handful of choices made before the model is run. This note explains what the model estimates, how courts have treated it, which choices move the result, how to read the output, and what it takes to attack or defend one. The practice it describes is on our pay equity and wage discrimination page.

What a pay regression estimates

The model is a single equation. The natural logarithm of each employee’s pay is written as a function of the factors the employer says determine pay, plus an indicator for the protected group, plus a residual:

ln(pay_i) = α + β·Group_i + γ₁X_1i + γ₂X_2i + … + γ_kX_ki + ε_i

The X terms are the controls: job family, grade or level, location, tenure, time in job, education, prior experience, full-time status, year. The coefficient β is the quantity the case turns on. It is the average difference in log pay between the protected group and everyone else after the controls are accounted for, and for small values it reads directly as a percentage: a coefficient of −0.030 means the group is paid about 3.0 percent less than otherwise similar employees. The residual ε is everything the model does not capture, and its spread is what produces the standard error of β, the figure that says whether the estimated difference could be chance.

Two features of the specification are standard rather than optional. Pay is modeled in logs because pay differences are proportional, a 5 percent gap means something different at $40,000 and at $240,000, and because log pay is far less skewed than dollars. And the model is estimated on employee-level data, not on averages by job or department, because averaging throws away the within-group variation the test depends on.

What the coefficient does not measure is intent. A regression can show that a disparity exists, that it is unlikely to be chance, and that the employer’s stated factors do not account for it. Whether the remainder reflects discrimination is an inference for the trier of fact from the whole record. The model narrows the question; it does not answer it.

How courts treat regression evidence

The Supreme Court settled the admissibility question in Bazemore v. Friday, 478 U.S. 385 (1986). A regression that omits some measurable variables may still prove a plaintiff’s case; whether the omissions matter goes to the weight of the evidence, not its admissibility, unless the model is so incomplete that it is irrelevant. The courts of appeals had already been applying that principle in pay cases. In Segar v. Smith, 738 F.2d 1249 (D.C. Cir. 1984), and Sobel v. Yeshiva University, 839 F.2d 18 (2d Cir. 1988), the courts held that an employer cannot rebut a regression merely by naming variables it leaves out; it has to show that the omitted factor would change the result. Ottaviani v. State University of New York at New Paltz, 875 F.2d 365 (2d Cir. 1989), is the counterpart: a defendant that produced its own regressions and showed the disparity was not statistically significant prevailed after a bench trial.

The class-action cases concern fit rather than admissibility. In Wal-Mart Stores, Inc. v. Dukes, 564 U.S. 338 (2011), the plaintiffs’ regressions were run at the regional and national level, but pay and promotion decisions were made by store managers exercising discretion; the Court held that the aggregate results could not establish a common question about decisions made store by store. Comcast Corp. v. Behrend, 569 U.S. 27 (2013), requires that a damages model measure only the damages attributable to the theory of liability on which the class is certified. Tyson Foods, Inc. v. Bouaphakeo, 577 U.S. 442 (2016), permits representative statistical evidence where each class member could have relied on the same evidence in an individual action.

District courts apply those cases to the specification itself. In Chen-Oster v. Goldman, Sachs & Co., 325 F.R.D. 55 (S.D.N.Y. 2018), a class of female associates and vice presidents was certified on regression evidence tied to firm-wide evaluation and promotion processes. In Moussouris v. Microsoft Corp., No. 2:15-cv-01483 (W.D. Wash. June 25, 2018), aff’d, No. 18-35791 (9th Cir. Dec. 24, 2019), certification was denied where the challenged pay and promotion decisions ran through managerial discretion the statistics did not isolate. The lesson of the two cases is the same: the regression has to be built at the level at which the employer actually made the decisions, and it has to test the practice the complaint identifies.

The specification decisions that drive the result

Which controls belong in the model

Every control in a pay regression is a claim that the factor is a legitimate determinant of pay. Job family, level, location, tenure, education and prior experience are usually uncontroversial. Three kinds of variables are not, because they may be products of the very decisions being challenged. Starting salary, if set by the same managers under the same policies, can carry the disparity into the model as an explanation of itself. Job assignment or grade, where the claim is that women or minorities were assigned to lower grades for the same work, does the same. Performance ratings, where ratings fed raises and the ratings process is challenged, do the same again. The literature calls these tainted variables.

The answer is not to omit them by reflex; it is to run the model both ways and say what each specification shows. If the disparity survives the inclusion of starting salary, the question is settled. If it disappears, the next question is whether starting salaries themselves differ by group after the legitimate factors are held constant, and that is a second regression, not an argument. Under the federal Equal Pay Act the same logic appears as a legal rule: in the Ninth Circuit, prior salary is not a “factor other than sex” that can justify a pay difference (Rizo v. Yovino, 950 F.3d 1217 (9th Cir. 2020) (en banc)), and California’s Labor Code § 1197.5 says prior salary cannot justify any disparity. In other circuits the treatment of prior salary varies, which is one more reason to show the result with and without it.

Pooled or separate

A workforce spread across facilities, business units or regions can be modeled in one pooled regression with fixed effects for each unit, or in a separate regression for each unit. The pooled model compares employees only with others in the same unit while estimating a single disparity across units; it has more statistical power and gives one answer. Separate models allow the disparity to differ by unit and lose power as the units get small. Which is right depends on where the decisions were made. If compensation was set centrally under one policy, the pooled model matches the employer’s conduct. If unit managers held the discretion, the separate results have to be shown, and a stratified test can combine them without assuming they are the same. A standard interaction test (a Chow test) asks whether the coefficients differ across units by more than chance, which is evidence on the question rather than an assumption about it. Dukes is, in statistical terms, a holding about this choice.

Functional form, time and standard errors

Three further choices recur. The first is functional form: whether tenure enters linearly or with a squared term, whether experience and level interact, whether the gap is allowed to differ by level. The second is time: a snapshot regression on one year’s pay, a separate regression for each year, or a stacked panel with year effects that uses every employee-year. The panel is usually preferable because it tracks the same employees through raises, and it also exposes the third choice: when the same employee appears in several years, the observations are not independent, and standard errors have to be clustered by employee or they will overstate significance. A report that stacks eight years of data and reports standard errors as if it had eight times the sample has made an error that a competent rebuttal will find.

Who is in the sample

Part-time employees, executives on separate pay plans, acquired workforces on legacy scales, employees with missing education or hire-date fields, and employees who appear in payroll but not in the HRIS all have to be accounted for. Each exclusion is a choice, and the report states the count excluded, the reason, and the result with and without the exclusion where it is material. The six questions to put to any statistical report include this one for a reason: a disparity estimated on the employees who happened to have complete records is a disparity among those employees, not among the class.

Reading the output

A regression table reports, for the group coefficient, four numbers that matter: the coefficient itself, its standard error, the test statistic (the coefficient divided by the standard error, often reported in “standard deviations”), and the resulting p-value or confidence interval. The illustration below is hypothetical.

Courts have treated a test statistic of roughly two to three standard deviations as the point at which chance becomes an unlikely explanation, following Hazelwood School District v. United States, 433 U.S. 299, 308 n.14 (1977), and Castaneda v. Partida, 430 U.S. 482, 496 n.17 (1977). That is a convention about chance, not a finding about size. A 0.4 percent gap can be significant in a workforce of 50,000 and a 6 percent gap insignificant in a department of 40, which is why a careful report presents the magnitude in percent and dollars beside the test statistic, and why the hub page’s note on what the statistics can and cannot show matters for both sides.

A supplementary decomposition, usually the Oaxaca-Blinder method, splits the raw gap into the part explained by differences in the controls (women have less tenure on average, say) and the part that is unexplained. It is useful for exposition and for California’s § 1197.5, which asks whether the employer’s factors account for the entire differential, but it is the same regression presented differently, not an independent test.

Diagnostics and robustness

A regression that reports one specification and one number invites the reply that a different, equally reasonable specification gives a different answer. The report should therefore show its sensitivity: the coefficient under alternative control sets, with and without the contested variables, year by year and pooled, with and without the sample exclusions. The standard diagnostics belong in the workpapers if not the report: heteroskedasticity-robust standard errors; influence statistics that identify whether a handful of highly paid employees drive the result; variance inflation factors where tenure, age and experience are nearly collinear; and a placebo test that reassigns the group indicator at random to confirm the model does not manufacture significance.

One old dispute deserves a sentence. The “reverse regression” approach, which regresses qualifications on pay rather than pay on qualifications, was advanced in the early 1980s as a fairer test; Arthur Goldberger showed that when qualifications are measured with error it is biased toward finding no discrimination, and courts have not adopted it. The direct regression described here remains the standard in both the economics literature and the Reference Manual on Scientific Evidence’s guide to multiple regression.

From regression to certification and damages

At class certification the regression is offered as evidence that a common question exists: whether the employer’s compensation practices produced a disparity across the class. Its fit to the liability theory is what the court examines. A regression that controls for the practice the complaint challenges has assumed away the case; one that ignores a legitimate factor the employer actually used has overstated it; and one estimated at the wrong level of the organization answers a question nobody asked. Our note on class certification describes how the economic questions line up with Rule 23.

At the damages stage the same coefficient becomes the engine of a class-wide calculation. The simplest approach applies the estimated percentage to each class member’s actual pay in each period inside the limitations window; a more individualized approach compares each member with matched comparators. Comcast requires that whichever model is used measure the damages the liability theory implies, and Dukes forbids a “Trial by Formula” that denies the employer the chance to raise individual defenses. The practical answer is a model that produces a defensible class-wide figure and can also be read member by member, so that individual issues can be addressed in a second stage without rebuilding it. The arithmetic of that step is on the hub’s class-wide damages section.

Attacking and defending a regression

The disputes between opposing experts are predictable enough to list. Most rebuttals, and most cross-examinations, turn on ten points:

1. How the comparison pool was defined, and whether the next most reasonable definition changes the result.
2. Which legitimate factors were omitted, and whether including them changes the result.
3. Which included factors are products of the challenged decisions.
4. Whether the level of aggregation matches the level at which decisions were made.
5. Functional form: logs, interactions, how tenure and experience enter.
6. How standard errors were computed, and whether repeated observations were clustered.
7. How many records were dropped, why, and whether the dropped records differ from the rest.
8. Whether the analysis is a snapshot or follows employees over time.
9. Whether significance and magnitude were both reported, and in substantive units.
10. Whether the analysis can be rerun from the data, code and variable definitions produced.

The last point is the one that decides the others. Replication requires the data as produced, the analysis dataset after cleaning, the code that moves from one to the other, a crosswalk of variable definitions, and a log of exclusions. Rule 26(a)(2)(B) requires an expert report to state the facts or data considered; in a regression case that means the files, not a description of them. When we are retained to review an opposing regression we begin by rerunning it exactly, then change one choice at a time, so that the report shows which choices drive the conclusion and by how much. The same discipline applies to our own reports. Five problems to look for in an opposing economic damages report covers the non-statistical failures that often accompany the statistical ones.

A worked illustration

The figures are invented to show the mechanics and do not describe any matter. A company employs 1,200 people in four job families at three locations. The raw difference in average pay between women and men is 8.4 percent. A regression on job family, level, location, tenure and its square, education, full-time status and year reduces the unexplained difference to 2.9 percent, with a standard error of 0.94 percent: 3.1 standard deviations, a gap unlikely to be chance. The employer’s expert adds starting salary as a control. The coefficient falls to 0.6 percent and is no longer significant. The gap has been explained, in the sense that employees who started at lower salaries are paid less now.

The third regression asks whether starting salaries differed by sex after the same legitimate factors are held constant. They did, by 4.1 percent, at 3.6 standard deviations. The two experts are not disagreeing about arithmetic. They disagree about whether starting salary is a legitimate factor or the mechanism of the disparity, which is a question about how the employer set starting pay, answered from the record. A report that shows all three results, and says what each one supports, has done the statistical work; the characterization is for counsel and the court.

Where to start

If you have received a regression report, send it to us for review with its workpapers; the first task is to rerun it. If you are evaluating a matter, start with a conflict check using party names only. For the data the analysis needs, see the hub’s what we request from counsel; for the individual Equal Pay Act measure that a class model scales up, see how a forensic economist measures a pay disparity.

Frequently asked questions

Does Bazemore mean any regression is admissible?

No. Bazemore holds that omitting some measurable variables goes to weight rather than admissibility. A model so incomplete that it cannot inform the question, or one that fails the reliability requirements of Rule 702, can still be excluded. The practical standard is whether the model controls for the major factors the employer actually used to set pay and tests the practice the complaint identifies.

What is a tainted variable?

A control variable that is itself a product of the decisions being challenged, such as a starting salary set by the same managers, a job grade assigned at hire, or a performance rating that fed the raises in dispute. Including it can absorb the disparity into the “explained” part of the model. The remedy is to show the result with and without it and to test whether the variable itself differs by group.

Should performance ratings be a control?

Only if the ratings process is independent of the challenged pay decisions, which is a factual question. Where ratings are not challenged and were made by different people under a separate process, they are a plausible control. Where the complaint alleges that ratings were part of the mechanism, the report runs the model both ways.

Can a regression prove intent?

No. It can establish a disparity, show that chance is an unlikely explanation, and show how much of the disparity the employer’s stated factors account for. In a disparate-impact claim intent is not an element; in a pattern-or-practice claim the trier of fact draws the inference from the statistics together with the rest of the evidence.

How does a litigation regression differ from a pay equity audit?

An audit uses the employer’s own factors, is usually conducted under privilege, and aims at forward-looking adjustments. A litigation regression is built to the statute’s comparison, confined to the limitations period, run under both sides’ specifications, and documented so the other side can rerun it. The models look alike; the standards they must meet do not.