Achieved power for a relative risk

When to use it

Assess a cohort study or trial with a binary outcome at the numbers exposed and unexposed.

Not for

The implied exposed risk, RR × p₂, must stay below 1.

Notation

$n_1$: Exposed participants; unexposed are r n₁.
$p_1, p_2$: Proportions in the two groups.
$r$: Allocation ratio n₂/n₁.
$\bar p$: Weighted pooled proportion, (p₁ + r p₂)/(1 + r).
$\mathrm{RR}$: Relative risk; p₁ = RR·p₂.

Formulas

Power, inverting the sample-size formula

\text{Power}=\Phi\!\left(\frac{|p_1-p_2|\sqrt{n_1}-z_{1-\alpha/\text{tails}}\sqrt{(1+1/r)\bar p\bar q}}{\sqrt{p_1q_1+p_2q_2/r}}\right)+I_{\text{tails}=2}\,\Phi\!\left(\frac{-|p_1-p_2|\sqrt{n_1}-z_{1-\alpha/\text{tails}}\sqrt{(1+1/r)\bar p\bar q}}{\sqrt{p_1q_1+p_2q_2/r}}\right)

p₁ is the exposed risk implied by the relative risk. The Fleiss formula without continuity correction, as the relative-risk calculator uses [2].

Parameters

Number exposed $n_1$

Values: Integer, at least 2

Exposed participants; the unexposed follow from the allocation ratio.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Unexposed per exposed participant.

Risk in the unexposed $p_2$

Values: Between 0 and 1

Outcome risk without the exposure.

Relative risk $\mathrm{RR}$

Values: Positive, not 1, with RR × p₂ below 1

The smallest relative risk worth detecting.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

Participants are independent and followed for the same period in both groups [1].
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 100 exposed and 100 unexposed, a risk of 0.15 in the unexposed, and RR = 2 at α = 0.05: p₁ = 0.30.

Needed for analysis: 100 · To approach: 200

Open this example in the calculator

Validation

Power is the relative-risk sample-size formula solved for z_β, with the far tail added for a two-sided test. At the size that calculator returns it reaches the target, one fewer falls short, and it agrees with an independent calculation in R.

How to cite

StatWisely. Achieved power for a relative risk. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-relative-risk/

Cite the original references for the formulas your design uses as well.

References

1.Lwanga SK, Lemeshow S. Sample Size Determination in Health Studies: A Practical Manual. Geneva: World Health Organization; 1991.
2.Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2003.