Pay Equity Analysis: The Specification Choices That Change the Answer

Two competent analysts, given the same payroll extract and asked the same question — “is there a pay gap?” — can return materially different answers, and both can be defensible. The divergence is rarely in the statistics. It is in three specification choices made before the model is fit: the model form, the control set, and how the adjusted and unadjusted gaps are presented alongside each other. This piece walks through each choice and the reporting risks that follow.

Model form: what question is the regression answering?

The industry default is OLS on log pay, with group indicators: log(pay) ~ group + controls. That specification estimates an average conditional gap, in percentage terms, at the mean of the controls. Three properties of that default deserve explicit acknowledgement:

• It is a conditional mean statement. A pay distribution can be equitable at the mean and unequal at the top decile, or the reverse. Quantile regression answers a different question — the gap at the median, at the 75th, at the 90th percentile — and organisations whose inequity lives in bonus-heavy senior bands will miss it entirely at the mean.
• It assumes the gap is constant across levels. An interaction term between group and level tests whether the gap widens with seniority. Omitting the interaction is itself a specification choice — one that forces a single number onto a structure that may not have one.
• The log transform changes the estimand. Coefficients are percentage effects, not currency effects; on skewed pay distributions the two tell different stories, and currency effects are often the ones leadership and legal actually care about.

None of this argues against the default. It argues that the default should be chosen, stated, and stress-tested — not inherited.

Control selection: the choice that decides the answer

Controls define the comparison. “Pay gap controlling for X” means: among employees alike on X, what gap remains? The legitimacy of a control hinges on whether it is a legitimate determinant of pay or a conduit of the inequity being measured — and several standard controls sit uncomfortably in between.

• Clearly legitimate: location, job family/function, hours, relevant experience. These are the comparison frame.
• Contested: job level. If promotions are themselves inequitably distributed, controlling for level “adjusts away” a gap that flows through blocked advancement. The adjusted number then answers “are people at the same level paid alike?” while silently setting aside “do people reach the same level alike?” Both questions matter. A level-controlled model that is reported as the pay gap has answered only the first.
• Contested: performance ratings. Ratings carry their own documented inter-rater and group biases. Conditioning on them controls for the inequity as well as the merit.
• Usually indefensible: salary history. Current trajectory compounds prior pay; conditioning on it absorbs the cumulative effect you are trying to measure.

The defensible practice is not to find the one true control set; there isn’t one. It is to run a control ladder: unadjusted, then market controls (location, function), then structural controls (level, tenure), then performance — and report the whole ladder. A gap that survives to the top rung is a within-job, within-level phenomenon. A gap that disappears when level enters has told you where to look: the promotion process, not the pay-setting process. The ladder converts a fragile point estimate into a diagnostic.

Adjusted and unadjusted: the reporting trap

The two numbers answer different questions. The unadjusted gap measures representation-weighted outcomes — who ends up where in the pay structure. The adjusted gap measures like-for-like pay treatment inside defined cells. Neither is “the real one.”

The reporting risk is asymmetry of presentation. Placed side by side without scaffolding, audiences select the congenial number: advocates cite the unadjusted figure, defenders cite the adjusted one, and both are now quoting your analysis out of context. The discipline is to present each number with the question it answers, in the same breath — and to refuse to publish either alone.

A second risk is false precision in small cells. Once controls slice the population, some comparison cells hold very few employees. A gap estimated on a triple-digit population reads differently from one estimated on n=14; the estimates are not comparable in reliability even when reported at the same decimal precision. Cell sizes and interval widths belong on the page next to the estimates, and very small cells should be pooled or suppressed rather than printed.

A third risk is legal-context blindness. Pay equity analyses routinely run under privilege for good reason; the specification decisions above should be documented at the time they are made, with rationale, because the same model may later be examined in a context where “why this control set?” is a question with consequences. An undocumented specification looks like a chosen one retrofitted to a preferred answer — including when it wasn’t.

What the analysis cannot tell you

A regression gap, adjusted or not, is a description of a pay distribution, not proof of discrimination or of its absence. Residual gaps are consistent with discrimination and also with unmeasured legitimate factors; null results are consistent with equity and also with underpowered cells. The analysis bounds the question; it does not close it. Remediation modelling — what it would cost to close a residual gap, and on whom the adjustment would fall — is a separate exercise with its own specification choices, and deserves the same documented-ladder treatment.

The data always wins over the narrative, but in pay equity the specification decides which data you see. Choose it openly, ladder it, and publish both gaps with their questions attached. The audience that is only ever shown one number will eventually be shown the other by someone else.