Measures of Risk: RR, OR, ARR, RRR & NNT
A board-focused walkthrough of the five core risk measures — RR and OR (measures of association, chosen by study design) and ARR, RRR, and NNT (measures of clinical impact) — built from the 2×2 table, with worked case-control and RCT vignettes plus confidence-interval interpretation.
The Two Families of Risk Measures
Measures of risk split into two families. Measures of association — relative risk (RR) and odds ratio (OR) — answer how strongly is an exposure linked to an outcome? Measures of impact — absolute risk reduction (ARR), relative risk reduction (RRR), and number needed to treat (NNT) — answer how much does a treatment actually help a real patient? Everything is built from a single 2×2 table: rows = exposed/treatment vs unexposed/control, columns = disease present vs absent.
Study design dictates the measure. A cohort study or RCT follows patients forward, so you can measure incidence and compute an RR. A case-control study starts from disease status and works backward, so incidence is unknowable and you can only compute an OR. The boards test the arithmetic and the interpretation — especially the classic trap where an impressive relative reduction conceals a trivial absolute benefit.
Label the 2×2: a = exposed + disease, b = exposed, no disease, c = unexposed + disease, d = unexposed, no disease.
- RR = [a/(a+b)] ÷ [c/(c+d)] — risk in exposed ÷ risk in unexposed; cohort/RCT. RR = 1 no effect, > 1 harmful, < 1 protective.
- OR = ad/bc (cross-product) — case-control; ≈ RR when the disease is rare (OR always sits farther from 1 than RR, exaggerating the effect as the outcome becomes common).
- ARR = CER − EER (control event rate − experimental event rate) = absolute drop in risk.
- RRR = ARR/CER = 1 − RR = fraction of baseline risk removed.
- NNT = 1/ARR (ARR as a decimal, then round UP); lower NNT = better therapy.
- NNH = 1/AR (attributable risk = risk_exposed − risk_unexposed); higher NNH = safer.
- NNT uses ARR, never RRR — the #1 calculation trap.
Side-by-Side Comparison
| Measure | Formula | Study design | What it tells you |
|---|---|---|---|
| RR | [a/(a+b)] ÷ [c/(c+d)] | Cohort / RCT | How many times the risk in exposed vs unexposed |
| OR | ad / bc | Case-control | Odds of exposure; ≈ RR if disease rare |
| ARR | CER − EER | RCT | Absolute risk removed by treatment |
| RRR | ARR/CER = 1 − RR | RCT | Fraction of baseline risk removed |
| NNT | 1 / ARR | RCT | Patients treated to prevent 1 event |
| NNH | 1 / AR | Cohort / RCT | Patients exposed to cause 1 harm |
Vignette: A case-control study enrolls 50 patients with pancreatic cancer and 50 cancer-free controls, then asks about prior smoking. 40 of 50 cases and 20 of 50 controls were smokers.
Q: Which measure of association applies, and what is its value?
Because subjects were selected by disease status, incidence is unknown → you cannot compute RR. Use the odds ratio.
- a (case, smoker) = 40, c (case, non-smoker) = 10
- b (control, smoker) = 20, d (control, non-smoker) = 30
- OR = ad/bc = (40 × 30) / (20 × 10) = 1200/200 = 6
Interpretation: cases had 6× the odds of prior smoking. Then report the OR with its 95% CI: if that interval excludes 1.0, the association is statistically significant.
Vignette: A 5-year RCT of a new statin reports myocardial infarction in 8% of placebo and 4% of statin patients. The drug ad boasts a "50% reduction in heart attacks."
Q: Calculate ARR, RRR, and NNT — and name the caveat.
- CER = 0.08, EER = 0.04
- ARR = 0.08 − 0.04 = 0.04 (4%)
- RRR = 0.04 / 0.08 = 0.50 (50%) ← the ad's figure
- NNT = 1 / 0.04 = 25 → treat 25 patients for 5 years to prevent 1 MI
Caveat: the 50% RRR sounds huge but is relative. The identical 50% RRR applied to a 1% baseline would give ARR 0.5% and NNT 200 — same relative number, far smaller real benefit. Always demand the ARR/NNT to judge true clinical impact.
- Cohort → RR; Case-Control → OR (Case-Control and Odds-Ratio pair up).
- OR = ad/bc — "cross-multiply" the corners of the 2×2.
- Rare disease → OR ≈ RR; as the outcome gets common, OR overstates the effect.
- NNT = 1 / ARR — never 1/RRR (the classic distractor answer).
- Lower NNT = better drug; Higher NNH = safer drug.
- Round NNT UP — you can't treat a fraction of a patient.
- CI crosses the null (1 for RR/OR, 0 for ARR) → NOT statistically significant.
Reading Significance Off the Confidence Interval
On the boards, significance is read straight off the 95% confidence interval, not just the point estimate. For a ratio measure (RR or OR) the null value is 1.0: if the CI includes 1.0, the result is not statistically significant (p > 0.05) — the true effect could be nothing. For a difference measure (ARR or risk difference) the null is 0, so significance requires the CI to exclude 0. A narrower CI (from a larger sample) means a more precise estimate.
Watch the classic item stem: "RR 1.8 (95% CI 0.9–3.2)" — the point estimate hints at harm, but because the interval crosses 1.0 you must conclude no significant association. A forest plot displays exactly this: each study's estimate with its CI, and whether it crosses the vertical null line.
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