How do you compute the four-fifths disparate-impact ratio for a hiring screen?

## The quantity For each group `g`, the **selection rate** is the fraction of that group's applicants who receive the favourable decision: ``` selection_rate(g) = selected(g) / applicants(g) ``` The **disparate-impact ratio** (also called the adverse-impact ratio) compares the least-selected group with the most-selected one: ``` DI = min_g selection_rate(g) / max_g selection_rate(g) ``` The **four-fifths rule** — a rule of thumb in the US federal Uniform Guidelines on Employee Selection Procedures — treats `DI < 0.8` as an indicator of adverse impact worth investigating. Worked example. A screen sees 400 applicants from group A and 120 from group B. It passes 200 of A (50%) and 36 of B (30%). `DI = 0.30 / 0.50 = 0.60`. That is well under 0.8, so the screen is flagged. ## What it is and is not **It is a rate of decisions, not a rate of errors.** The four-fifths check never looks at the label. It does not ask whether the passed candidates went on to succeed, or whether the screen's false-negative rate differs between groups. Two screens with identical accuracy profiles can have wildly different ratios if their thresholds sit at different points. **It is threshold-dependent.** Because it is computed on hard decisions, the same scoring model produces different ratios at different cutoffs. If the score distributions of two groups differ in shape, raising the bar can widen or narrow the ratio non-monotonically. Reporting a ratio without reporting the threshold it was computed at is incomplete. **It is a flag, not a verdict.** In its original legal setting a sub-0.8 ratio establishes a prima facie case; the employer may then argue the screen is job-related and consistent with business necessity, and a challenger may argue a less discriminatory alternative exists. A candidate who says "0.65, therefore illegal" has overclaimed. The engineering translation is: this number opens an investigation and a documentation obligation, it does not close one. **It is unstable on small pools.** The ratio is a quotient of two proportions. If one group contributed 40 applicants, its selection rate has a confidence interval tens of points wide, and so does the ratio. Report the ratio with an interval, or at least with the group counts, so a reader can see how much weight it bears. ## The connection to proxies The reason this check matters in a machine-learning setting is that **it can fail badly on a model whose training table never contained the protected attribute**. Selection rates are computed after the fact, from decisions the deployed model made, joined against attribute data held for audit. A hiring screen built on features like commute distance, university, or years of continuous employment can produce a 0.6 ratio purely through proxies — no attribute, no explicit rule, and yet unequal selection rates. That is also why the check has to be a *reporting* step rather than an assumption. Nothing in training tells you the ratio; you have to compute it on realistic applicant flow, at the threshold you actually ship, and recompute it whenever the threshold or the applicant mix changes. A screen that passed at launch can fail six months later because the composition of the applicant pool moved, with the model untouched. ## Practicalities that come up - **Which groups.** Compute the ratio for every group you have data for, not only the pair you expect to be a problem. Include an "unknown/declined to state" bucket in the counts so nobody can move the ratio by pushing people into it. - **Multi-stage funnels.** If the model is one gate among several, compute the ratio at your gate *and* end to end. A gate that looks acceptable in isolation can dominate the cumulative ratio when three such gates compose. - **Which group is the denominator.** The convention is the most-selected group, which makes the ratio at most 1. Some teams instead compare against the overall rate; either is defensible provided you say which you used. - **Rates, not counts.** A group can be a small share of hires and still have the highest selection rate. Comparing raw hire counts instead of rates is the most common arithmetic mistake on this question. ## What good sounds like Give the formula, give a two-line worked example, state the 0.8 threshold and where it comes from, then immediately qualify it: decision rates not error rates, threshold-dependent, a flag rather than proof, and shaky when a group's applicant count is small. Close on the proxy point — that a model with no protected attribute in it can still produce a failing ratio, which is precisely why the ratio has to be measured rather than reasoned about.