Selection Rate

Selection rate is the share of candidates who receive the positive outcome:

\[ \text{Selection Rate} = \frac{\text{Number Selected}}{\text{Number Considered}} \]

Twenty hires from two hundred applicants is 10%. Two hundred and fifty approvals from a thousand loan applications is 25%. In machine learning it is the rate at which a model predicts the positive class, which is the same quantity under a different name — and it is the quantity demographic parity compares across groups.

The definition is simple. The two hard parts are the denominator and the grouping.

Combining roles can invent a disparity

import numpy as np

rng = np.random.default_rng(0)
"""Two roles with very different hiring rates, and two groups that applied to
   them in different proportions. Within each role both groups have the same hiring probability; observed
   rates vary because hires are sampled."""
roles = {"warehouse (hires 60%)": 0.60, "engineering (hires 10%)": 0.10}
applicants = {"group A": {"warehouse (hires 60%)": 900, "engineering (hires 10%)": 100},
              "group B": {"warehouse (hires 60%)": 100, "engineering (hires 10%)": 900}}

totals = {}
for group, counts in applicants.items():
    hired = applied = 0
    for role, rate in roles.items():
        n = counts[role]
        h = rng.binomial(n, rate)
        print(f"  {group}  {role:<24} applied {n:>4}  hired {h:>4}  rate {h / n:.4f}")
        hired += h
        applied += n
    totals[group] = hired / applied
    print(f"  {group}  {'overall':<24} applied {applied:>4}  hired {hired:>4}  rate {hired / applied:.4f}\n")

low, high = min(totals.values()), max(totals.values())
print(f"impact ratio on the combined figures: {low / high:.4f}"
      f"   four-fifths rule: {'indicates adverse impact' if low / high < 0.8 else 'does not fire'}")
#   group A  warehouse (hires 60%)    applied  900  hired  520  rate 0.5778
#   group A  engineering (hires 10%)  applied  100  hired    5  rate 0.0500
#   group A  overall                  applied 1000  hired  525  rate 0.5250
#
#   group B  warehouse (hires 60%)    applied  100  hired   66  rate 0.6600
#   group B  engineering (hires 10%)  applied  900  hired   72  rate 0.0800
#   group B  overall                  applied 1000  hired  138  rate 0.1380
#
# impact ratio on the combined figures: 0.2629   four-fifths rule: indicates adverse impact

Group B is hired at a higher rate than group A in the warehouse role, 0.6600 against 0.5778, and a higher rate in engineering, 0.0800 against 0.0500. There is no role in which group A does better.

The combined impact ratio is 0.2629, which falls well below the four-fifths screening threshold. That combined result calls for examining the job populations and process; it alone does not establish a legal conclusion about within-role treatment.

Nothing is wrong with the arithmetic. Group A applied mostly to a role that hires 60% of applicants and group B applied mostly to one that hires 10%, so the combined rates measure which jobs each group applied for rather than how each group was treated. This is the standard shape of Simpson’s paradox, and it runs in both directions: the same mechanism can hide a real per-role disparity behind a healthy combined ratio.

The practical rule is that a selection rate is only interpretable within a population where everyone faced the same decision. Aggregating across roles, locations, seniority levels or hiring stages combines different decision populations. The total rate describes the overall pipeline but does not directly isolate differences in treatment within the same role or stage.

Which is not to say the combined figure is meaningless. If one group is systematically steered toward low-selectivity roles, that pattern is itself worth examining — but it is a question about the pipeline, not about the hiring decision, and the two need to be reported separately to be acted on.

Defining the denominator

“Number considered” is a choice that moves the rate substantially, and the choices are rarely written down.

• Everyone who applied, including incomplete and withdrawn applications — the largest denominator and the lowest rates.
• Everyone eligible, after minimum qualifications are applied — but the qualification screen is itself a selection step with its own rates to check.
• Everyone who reached this stage — the right denominator for finding where in a pipeline the groups diverge, and the one that requires stage-by-stage reporting.

Two teams using different denominators will report different rates for the same process, so the definition belongs in the report rather than in the code.

Comparing rates across groups

Under the US Uniform Guidelines the comparison is against the group with the highest selection rate — not against a designated majority group and not pairwise between any two groups you happen to pick. With three or more groups this matters: the ratio each group receives depends on which one tops the table.

And the ratio is a rule of thumb rather than a requirement. Falling below 0.80 yields evidence of adverse impact that calls for analysis; it is not a violation, and clearing it is not compliance. On small applicant pools the ratio is also extremely noisy — the four-fifths rule article shows it firing 61.7% of the time on twenty applicants per group when the two processes are identical.

Report counts, rates, the reference group and an interval. A rate without its denominator is not a measurement.

References

• 29 CFR 1607.4, Uniform Guidelines on Employee Selection Procedures — Information on impact