Relative Risk Calculator

Relative risk, RRR and ARR

The risk in each group is the number with the outcome divided by the number in the group. From the two risks:

• Relative risk (RR) = risk with treatment ÷ risk with control. 1 means no difference.
• Relative risk reduction (RRR) = 1 − RR. A relative risk of 0.75 is a 25% relative reduction.
• Absolute risk reduction (ARR) = risk with control − risk with treatment. It is a difference in percentage points.
• Number needed to treat (NNT) = 1 ÷ ARR, rounded up.

When treatment raises the risk, the same arithmetic gives the relative risk increase, the absolute risk increase and the number needed to harm.

Confidence intervals.

• For the relative risk, the calculator uses the log method (Katz): SE of ln(RR) = √(1/a − 1/n₁ + 1/c − 1/n₂), where a and c are the events and n₁ and n₂ the group sizes.
• For the absolute difference, it uses Newcombe's hybrid score interval, which behaves well with small groups and with risks near 0% or 100%.
• The P value comes from the two-proportion z test. When counts are small, Fisher's exact test replaces it.

Why the absolute numbers matter

A relative risk is the same at every baseline risk, but the benefit to a patient is not.

• REDUCE-IT: icosapent ethyl lowered major cardiovascular events from 22.0% to 17.2% over a median of 4.9 years. That is a relative risk of 0.78, an absolute reduction of 4.8 points, and an NNT of 21.
• The same relative reduction at a 2% baseline would remove about 0.4 percentage points, an NNT near 230.

Headlines often quote only the relative number ("58% lower risk"). Before you act on one, ask what the risk was to begin with. HypeCheck checks health claims like these against the study behind them.

Common mistakes

1. Quoting only the relative reduction. Always give the baseline risk and the absolute difference too.
2. Mixing up percent and percentage points. A fall from 1.9% to 0.8% is 1.1 percentage points, or 58%.
3. Comparing relative risks from studies with different follow-up. A 5-year relative risk and a 1-year relative risk aren't the same quantity.
4. Using a relative risk from a case-control study. Risks can't be estimated there; use the odds ratio.
5. Assuming the relative risk of an event is the reciprocal of the relative risk of no event. It isn't: an RR of 0.42 for eclampsia is an RR of about 1.01 for staying free of it.

References

1. Katz D, Baptista J, Azen SP, Pike MC. Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics. 1978;34:469-74.
2. Newcombe RG. Interval estimation for the difference between independent proportions: comparison of eleven methods. Stat Med. 1998;17:873-90.
3. Magpie Trial Collaborative Group. Lancet. 2002;359:1877-90.
4. Bhatt DL, Steg PG, Miller M, et al. Cardiovascular risk reduction with icosapent ethyl for hypertriglyceridemia. N Engl J Med. 2019;380:11-22.
5. Laupacis A, Sackett DL, Roberts RS. An assessment of clinically useful measures of the consequences of treatment. N Engl J Med. 1988;318:1728-33.