Achieved power for an odds ratio

When to use it

Assess a case-control study at the number of cases and controls available.

Not for

An odds ratio is not a risk ratio; for a cohort or trial use the relative-risk tab.

Notation

$n_1$: Cases; controls 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).
$\mathrm{OR}$: Odds ratio; p₁ = OR·p₂/(1 − p₂ + OR·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 exposure among cases implied by the odds ratio. The Fleiss formula without continuity correction, as the odds-ratio calculator uses [2].

Parameters

Number of cases $n_1$

Values: Integer, at least 2

Cases available; controls follow from the allocation ratio.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Controls per case.

Exposure among controls $p_2$

Values: Between 0 and 1

From a comparable population.

Odds ratio $\mathrm{OR}$

Values: Positive, not 1

The smallest odds ratio 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

Cases and controls are sampled independently, and exposure is measured the same way in both [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 cases and 200 controls, 20% exposure among controls, and OR = 2 at α = 0.05: p₁ = 0.333.

Needed for analysis: 100 · To approach: 300

Open this example in the calculator

Validation

Power is the odds-ratio 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 an odds ratio. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-odds-ratio/

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.