Sample size for non-inferiority of a rate ratio

When to use it

A trial of a simpler, safer or cheaper treatment whose aim is to show that it does not raise the rate of exacerbations, infections or falls by more than an acceptable amount.

Not for

Showing that the treatment lowers the rate. Use the rate-ratio tests.
A margin that has not been justified in advance from earlier trials and clinical judgement.

Notation

$n_1,\ n_2$: Participants in groups 1 (treatment) and 2 (comparator)
$\lambda_1,\ \lambda_2$: Expected event rates per unit of time in groups 1 and 2
$t$: Follow-up time per participant
$k$: Negative binomial dispersion: variance μ + kμ²; 0 is Poisson
$M$: Non-inferiority margin on the rate-ratio scale
$r_a$: Allocation ratio n₂ / n₁
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

Participants in group 1

n_1 = \left\lceil \frac{(z_{1-\alpha} + z_{1-\beta})^{2}\left[\dfrac{1}{\lambda_1 t} + k + \dfrac{1}{r_a}\left(\dfrac{1}{\lambda_2 t} + k\right)\right]}{\left(\ln \dfrac{\lambda_1}{\lambda_2} - \ln M\right)^{2}} \right\rceil, \qquad n_2 = \lceil r_a\, n_1 \rceil

The one-sided Wald test that the log rate ratio lies on the right side of ln M: below it for a margin above 1, above it for a margin below 1. Each log rate has variance (1/(λt) + k)/n at the expected rates [1, 2]. Equal true rates are the usual planning assumption.

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [3, 4].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [5].

All adjustments together

n_{\text{adj}} = \left\lceil n \mathrm{DE}\;F_t \right\rceil

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

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

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Expected rate with the treatment $\lambda_1$

Values: Above 0

Usually the comparator’s rate: the treatment is expected to be as good.

Rate with the comparator $\lambda_2$

Values: Above 0

From earlier trials of the comparator.

Non-inferiority margin $M$

Values: Above 0 and not 1; the expected ratio λ₁/λ₂ on its non-inferior side

Above 1, such as 1.2, for an event the treatment must not make much more common; below 1, such as 0.8, for one it must not make much rarer. A narrower margin needs many more participants.

Follow-up per participant $t$

Values: Above 0, in the rates’ time unit

Longer follow-up yields more events per participant; with over-dispersion it helps less.

Dispersion (negative binomial k) $k$ Optional

Values: 0 or more; 0 (the default) for Poisson counts

Take k from earlier trials of the comparator; a Poisson plan for over-dispersed counts is too small.

Allocation ratio $r_a$ Optional

Values: Above 0; 1 for equal groups

The planned size of group 2 divided by group 1.

Significance level $\alpha$

Values: One-sided; usually 0.025

One-sided, since only one direction is being ruled out. 0.025 matches a two-sided 0.05 test.

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.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [6]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [4].

Expected loss of whole clusters $\ell$ Optional

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

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

Participants are independent apart from any clustering specified, each followed for the same time, and each group’s rate is constant over it.
Counts are Poisson, or negative binomial with the same stated dispersion in both groups.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [3].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Equal rates, a margin of 1.2

Rates of 1 and 1 a year over one year, margin 1.2, one-sided α = 0.025, 80% power, Poisson counts: (1.96 + 0.84)² × (1 + 1) / (ln 1.2)² = 472.2, rounded up to 473 per group.

Group 1: 473 · Group 2: 473 · Total: 946

Protocol statement
To show non-inferiority within a rate-ratio margin of 1.2 with 80% power at a one-sided significance level of 0.025, using a one-sided test of the rate ratio against the margin, and assuming event rates of 1 with the treatment and 1 with the comparator per unit of time and follow-up of 1 per participant, the required sample size is 473 per group (946 in total). Calculated with StatWisely (statwisely.com) using: n₁ = (z_(1-α)+z_(1-β))² × (1/(λ₁t) + k + (1/(λ₂t) + k)/r) / (ln(λ₁/λ₂) - ln M)²; n₂ = r × n₁. These are planning estimates; confirm the final design with a qualified statistician.

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Over-dispersed counts

With dispersion k = 0.5 the variance term grows from 2 to 3: 708.4, so 709 per group.

Group 1: 709 · Group 2: 709 · Total: 1418

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Two years of follow-up

Following each participant for 2 years halves the Poisson variance: 237 per group.

Group 1: 237 · Group 2: 237 · Total: 474

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A treatment expected to be slightly better

Expecting 0.9 against 1 with k = 0.5, the distance from the margin grows to ln 1.2 − ln 0.9 = 0.288: 296 per group.

Group 1: 296 · Group 2: 296 · Total: 592

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A rate the treatment must keep up

For an event that should not become much rarer, a margin of 0.8: (1.96 + 0.84)² × 2 / (ln 0.8)² = 315.3, so 316 per group.

Group 1: 316 · Group 2: 316 · Total: 632

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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.
Simulated Poisson and negative binomial trials at the planned sizes, analysed with Wald tests in generalised linear models, gave power of 0.79 to 0.84 against a target of 0.80, with the dispersion known or estimated, and a type I error of 0.020 to 0.029 at the margin against a nominal 0.025.

How to cite

StatWisely. Sample size for non-inferiority of a rate ratio. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/rates-noninferiority/

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

References

1.Zhu H, Lakkis H. Sample size calculation for comparing two negative binomial rates. Statistics in Medicine. 2014;33(3):376–387.
2.Chow SC, Shao J, Wang H. Sample Size Calculations in Clinical Research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008.
3.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
4.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
5.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
6.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013

When to use it

Not for

Notation

Formulas

Participants in group 1

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected rate with the treatmentλ1\lambda_1λ1​

Rate with the comparatorλ2\lambda_2λ2​

Non-inferiority marginMMM

Follow-up per participantttt

Dispersion (negative binomial k)kkkOptional

Allocation ratiorar_ara​Optional

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Average cluster sizemmmOptional

Variation in cluster sizeCV\mathrm{CV}CVOptional

Expected loss of whole clustersℓ\ellℓOptional

Correction for few clustersFtF_tFt​Optional

Assumptions

Worked examples

Equal rates, a margin of 1.2

Over-dispersed counts

Two years of follow-up

A treatment expected to be slightly better

A rate the treatment must keep up

Validation

How to cite

References

Expected rate with the treatment $\lambda_1$

Rate with the comparator $\lambda_2$

Non-inferiority margin $M$

Follow-up per participant $t$

Dispersion (negative binomial k) $k$ Optional

Allocation ratio $r_a$ Optional

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Average cluster size $m$ Optional

Variation in cluster size $\mathrm{CV}$ Optional

Expected loss of whole clusters $\ell$ Optional

Correction for few clusters $F_t$ Optional