Sample size for a non-inferiority trial: proportions

When to use it

A non-inferiority trial with a yes/no outcome such as cure or response.

Not for

Showing the two are interchangeable in both directions: use equivalence for proportions.

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

Participants per group

n = \left\lceil \frac{(z_{1-\alpha} + z_{1-\beta})^{2}\left[p_1(1-p_1) + p_2(1-p_2)\right]}{(\delta + p_1 - p_2)^{2}} \right\rceil

The one-sided Wald test that p₁ − p₂ lies above −δ [1, 2].

Dropout

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

Applied last, to the participants actually recruited.

Parameters

Non-inferiority margin $\delta$

Values: Above 0, on the proportion scale

The largest loss of effect that would still be acceptable, justified clinically and from the reference treatment’s known benefit, 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: One-sided; usually 0.025

One-sided, since only inferiority is being ruled out; 0.025 matches a two-sided 0.05 test, as regulators usually expect.

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

A 10-point margin

80% success expected with both, margin 0.1, one-sided α = 0.025, 80% power: 7.849 × 0.32 / 0.01 = 251.2, so 252 per group.

Group 1: 252 · Group 2: 252 · Total: 504

Protocol statement
To show non-inferiority within a margin of 10 percentage points with 80% power at a one-sided significance level of 0.025, using a test of the difference against the margin, and assuming success proportions of 80% with the new treatment and 80% with the reference, the required sample size is 252 per group (504 in total). Calculated with StatWisely (statwisely.com) using: Wald approximation: n_per_group=[z_(1-α)+z_(1-β)]²(p₁q₁+p₂q₂)/(Δ+p₁-p₂)². These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A 5-point margin

Halving the margin to 0.05 quadruples the sample: 1005 per group.

Group 1: 1005 · Group 2: 1005 · Total: 2010

Open this example in the calculator

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 a non-inferiority trial: proportions. Method document, engine version 1.0.0, reviewed 2026-10-02. https://statwisely.com/methods/noninferiority-proportions/

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

References

1.Blackwelder WC. “Proving the null hypothesis” in clinical trials. Controlled Clinical Trials. 1982;3(4):345–353.
2.Chow SC, Shao J, Wang H. Sample Size Calculations in Clinical Research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008.