Sample size for an equivalence trial: means

When to use it

Showing two formulations, devices or methods are interchangeable, for example a generic against a brand.

Not for

When only the new treatment being worse matters: use non-inferiority, which needs fewer participants.

Notation

$n$: Participants per group
$\delta$: Margin
$d$: Expected difference, new minus reference; higher is better
$\sigma$: Common standard deviation
$\alpha$: 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}{\sigma\sqrt{2/n}} - z_{1-\alpha}\right) + \Phi\!\left(\frac{\delta + d}{\sigma\sqrt{2/n}} - z_{1-\alpha}\right) - 1 \;\ge\; 1-\beta

Equivalence is shown when both one-sided tests reject: the difference is above −δ and below +δ. n is the smallest whole number whose power reaches the target; with d = 0 this is n = 2σ²(z₁₋α + z₁₋β/₂)²/δ² [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, in the outcome’s units

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

Standard deviation $\sigma$

Values: Above 0

From previous trials of the reference treatment.

Expected true difference $d$ Optional

Values: Smaller in size than the margin

Usually 0: the treatments are expected to be equally good. Assuming the new treatment is better shrinks the sample, and should only be done with evidence.

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 ±5

Margin 5, SD 10, no true difference, α = 0.05 for each test, 80% power: 200 × (1.645 + 1.282)² / 25 = 68.5, so 69 per group.

Group 1: 69 · Group 2: 69 · Total: 138

Protocol statement
To show equivalence within margins of ±5 with 80% power at a one-sided significance level of 0.05 for each, using two one-sided tests, and assuming a standard deviation of 10 and a true difference of 0, the required sample size is 69 per group (138 in total). Calculated with StatWisely (statwisely.com) using: n_per_group: solve normal-approximation 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

If the treatments truly differ by 1: 81 per group. Any true difference makes equivalence harder to show.

Group 1: 81 · Group 2: 81 · Total: 162

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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: means. Method document, engine version 1.0.0, reviewed 2026-10-02. https://statwisely.com/methods/equivalence-means/

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.