Risk Quantification & Public Health
From relative risk to number needed to treat
Relative risk and odds ratio
The 2 × 2 layout. Exposure goes in the rows and outcome in the columns. Cell a is exposed and diseased, b is exposed and not diseased, c is unexposed and diseased, and d is unexposed and not diseased. Every ratio below comes from these cells.
| Group | Disease | No disease | Risk in group |
|---|---|---|---|
| Exposed | a | b | a/(a + b) |
| Unexposed | c | d | c/(c + d) |
Relative risk (RR). RR is the risk in the exposed divided by the risk in the unexposed. It requires incidence, so it comes from cohort studies and RCTs. An RR above 1 means harm, below 1 means protection, and exactly 1 means no association. A vaccine with an RR of 0.5 halves the risk of disease in the vaccinated.
Odds ratio (OR). The OR equals ad/bc, the odds of exposure among cases divided by the odds of exposure among controls. Case-control studies start from disease status and cannot measure incidence, so they report the OR. In a case-control study of a rare cancer, an OR of 4.0 means roughly 4-fold higher odds of disease with exposure, approximating the relative risk.
Rare disease assumption. When the disease is rare, the OR approximately equals the RR. As the disease becomes common, the OR moves further from 1 than the RR and overestimates the effect size.
For an RR above 1, the OR is even higher. For an RR below 1, the OR is even lower. The OR always sits further from the null.
Absolute vs relative effects: ARR, RRR, NNT and NNH
Absolute risk reduction (ARR). ARR is the risk in controls minus the risk in the treated. If 8% of controls and 2% of treated patients have events, ARR = 0.08 − 0.02 = 0.06, or 6 percentage points.
Relative risk reduction (RRR). RRR = ARR / control risk = 1 − RR. The same data give RRR = 0.06/0.08 = 75%. The identical trial sounds far more impressive as a 75% RRR than as a 6% ARR, which is why drug marketing favors RRR. ARR and NNT convey the real-world impact.
Number needed to treat (NNT). NNT = 1/ARR. An ARR of 0.06 gives 1/0.06 ≈ 16.7, so about 17 patients must be treated to prevent one event. A lower NNT means a more effective therapy. Always round up to the next whole patient, because a fraction of a person cannot be treated. For example, stroke in 12% on placebo vs 4% on treatment gives ARR = 0.08 and NNT = 12.5, which rounds up to 13.
Attributable risk and NNH. Attributable risk (AR) is the risk in the exposed minus the risk in the unexposed, the excess caused by the exposure. For a drug's adverse effect it is also called the absolute risk increase. The number needed to harm (NNH) is 1/AR. A higher NNH means a safer drug, because more patients are exposed before one is harmed. Attributable risk percent, (RR − 1)/RR, is the fraction of disease in the exposed that is due to the exposure.
| Measure | Formula | Worked example |
|---|---|---|
| ARR | Control risk − treated risk | 20% vs 15% → 0.05 |
| RRR | ARR / control risk = 1 − RR | 0.05/0.20 = 25% |
| NNT | 1 / ARR (round up) | 10% → 5%: 1/0.05 = 20 |
| AR | Exposed risk − unexposed risk | Smokers 30% vs nonsmokers 5% → 25 percentage points |
| AR% | (RR − 1)/RR | Same smokers: RR 6 → 5/6 ≈ 83% |
| NNH | 1 / AR (absolute risk increase) | Higher value = safer drug |
Hazard ratios and confidence intervals
Hazard ratio (HR). The HR comes from survival (time-to-event) analysis. It is typically estimated with a Cox proportional hazards model, and survival curves are usually shown as Kaplan-Meier plots. It compares the instantaneous event rate between groups. An HR of 0.7 means a 30% lower hazard in the treatment arm at any given moment. An HR of 0.65 (95% CI 0.50–0.85) is a 35% relative reduction in the hazard. An HR of 1.5 (95% CI 1.2–1.9) is a 50% increase. Both are statistically significant because their intervals exclude 1.
Reading the confidence interval.
- Ratio measures (RR, OR, HR): the null value is 1, so a 95% CI that includes 1.0 is not statistically significant. An RR of 1.3 with a CI of 0.9–1.8 is not significant.
- Difference measures (ARR, AR): the null value is 0, so a CI that crosses 0 is not significant.
Prevention, outbreaks and reporting
| Level | Goal | Examples |
|---|---|---|
| Primary | Stop disease before it starts | Vaccines (HPV vaccine before sexual debut), seatbelts, smoking cessation |
| Secondary | Detect early, asymptomatic disease | Pap smear, mammography, blood pressure screening |
| Tertiary | Limit complications of established disease | Diabetic foot care, cardiac rehabilitation |
Attack rate. The attack rate is the proportion of people at risk who become ill during an outbreak, a form of incidence. Comparing attack rates among people who did and did not eat each food identifies the likely source.
Epidemic curves. A point-source outbreak, such as a single contaminated meal, produces one sharp peak. A propagated outbreak spreads from person to person and produces successive waves separated by roughly one incubation period.
Reportable diseases. Physicians must report notifiable diseases to the local or state health department for surveillance. Examples include HIV/AIDS, tuberculosis, hepatitis A, B and C, measles, syphilis, gonorrhea, chlamydia, salmonellosis, shigellosis and rabies. Others include pertussis, mumps, rubella and Lyme disease. Reporting is a legal duty that overrides patient confidentiality.