Sample size for an equivalence trial: proportions

When to use it

Showing two treatments or methods give the same success rate within a margin.

Not for

When only the new treatment being worse matters: use non-inferiority.

Notation

$n$: Participants per group
$\delta$: Margin, on the proportion scale
$p_1,\ p_2$: Expected success proportions with the new and the reference treatment
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$q$: Expected dropout

Formulas

Two one-sided tests (TOST)

\text{power}(n) = \Phi\!\left(\frac{\delta - d}{\sqrt{V/n}} - z_{1-\alpha}\right) + \Phi\!\left(\frac{\delta + d}{\sqrt{V/n}} - z_{1-\alpha}\right) - 1, \qquad d = p_1 - p_2, \quad V = p_1(1-p_1) + p_2(1-p_2)

Both Wald tests must reject; n is the smallest whole number whose power reaches the target [1, 2].

Dropout

n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Parameters

Equivalence margin $\delta$

Values: Above 0, on the proportion scale

The largest difference in either direction that would still count as equivalent, justified clinically and set in advance.

Success with the new treatment $p_1$

Values: Between 0 and 1

Usually the same as p₂.

Success with the reference treatment $p_2$

Values: Between 0 and 1

From previous trials of the reference treatment.

Significance level $\alpha$

Values: For each of the two one-sided tests; usually 0.05

Each one-sided test at α gives a 1 − 2α confidence interval inside the margins, a 90% interval at α = 0.05 [1].

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Assumptions

The two groups are independent and equal in size, and the reference treatment works in this trial as it did in the trials that established it (constancy).
The per-protocol and intention-to-treat analyses should agree; poor adherence makes groups look alike and can favour a false claim of non-inferiority.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Margins of ±10 points

80% success with both, margin 0.1, α = 0.05 for each test, 80% power: 275 per group.

Group 1: 275 · Group 2: 275 · Total: 550

Protocol statement
To show equivalence within margins of ±10 percentage points with 80% power at a one-sided significance level of 0.05 for each, using two one-sided tests, and assuming success proportions of 80% with the new treatment and 80% with the reference, the required sample size is 275 per group (550 in total). Calculated with StatWisely (statwisely.com) using: n_per_group: solve Wald-normal TOST power for bounds (-Δ,+Δ). These are planning estimates; confirm the final design with a qualified statistician.

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A small true difference

With 0.82 against 0.8: 311 per group.

Group 1: 311 · Group 2: 311 · Total: 622

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Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.

How to cite

StatWisely. Sample size for an equivalence trial: proportions. Method document, engine version 1.0.0, reviewed 2026-10-02. https://statwisely.com/methods/equivalence-proportions/

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

References

1.Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics. 1987;15(6):657–680.
2.Chow SC, Shao J, Wang H. Sample Size Calculations in Clinical Research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008.