Sample size for testing one rate against a reference

When to use it

Comparing a cohort’s event rate with a published or historical benchmark.

Not for

A comparison with a concurrent control group. Use the rate ratio.

Notation

$n$: Number of participants
$\lambda_0,\ \lambda_1$: Reference rate and expected rate
$t$: Follow-up time per participant
$k$: Negative binomial dispersion: variance μ + kμ²; 0 is Poisson
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\tau$: Number of tails: 2 for a two-sided test, 1 for one-sided
$\rho$: 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

n = \left\lceil \frac{\left[z_{1-\alpha/\tau}\sqrt{\lambda_0(1 + k\lambda_0 t)} + z_{1-\beta}\sqrt{\lambda_1(1 + k\lambda_1 t)}\right]^{2}}{t\,(\lambda_1 - \lambda_0)^{2}} \right\rceil

The normal approximation to the count, with the null variance for the test and the alternative’s under the alternative. A participant’s count over t has variance μ(1 + kμ), so each rate’s variance is λ(1 + kλt)/t; at k = 0 these are the Poisson λ₀ and λ₁ [1].

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 [2, 3].

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 [4].

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 $\lambda_1$

Values: Above 0, events per unit of time

The rate you expect with the intervention, in the same time unit as the follow-up.

Reference rate $\lambda_2$

Values: Above 0, different from λ₁

From routine data or previous studies.

Follow-up per participant $t$

Values: Above 0, in the rates’ time unit

Doubling the follow-up buys roughly the same information as doubling the participants, for a Poisson outcome.

Dispersion (negative binomial k) $k$ Optional

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

Clinical counts usually vary between people more than a Poisson process allows; a Poisson plan then has too little power. Take k from earlier data.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

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.

One- or two-sided $\tau$

Values: 2 (two-sided) or 1 (one-sided)

Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

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 [5]; 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 [3].

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.
Counts are Poisson, or negative binomial with the stated dispersion, and the reference rate is known rather than estimated.
The analysis is a z test on the rate scale with the reference rate’s variance. A Wald test on the log scale has a little less power for a rise in the rate and a little more for a fall.
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 [2].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

A 20% higher rate

1.2 against a reference of 1 a year, one year each: n = [1.96 × 1 + 0.8416 × 1.095]² / 0.2² = 207.6, rounded up to 208.

Sample size: 208

Protocol statement
To detect a rate of 1.2 against a reference of 1 per unit of time with 80% power at a two-sided significance level of 0.05, using a one-sample test of a rate, and assuming follow-up of 1 per participant, the required sample size is 208 participants. Calculated with StatWisely (statwisely.com) using: n = [z_(1-α/2)√λ₀ + z_(1-β)√λ₁]²/[t(λ₁-λ₀)²]. These are planning estimates; confirm the final design with a qualified statistician.

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Two years of follow-up

Over 2 years: 104.

Sample size: 104

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Over-dispersed counts

With dispersion k = 0.5 over one year the variances become 1.5 and 1.92: n = [1.96 × 1.225 + 0.8416 × 1.386]² / 0.2² = 318.0, rounded up to 319.

Sample size: 319

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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 counts at the planned sizes, tested on the rate scale, gave power of 0.80 to 0.82 against a target of 0.80.

How to cite

StatWisely. Sample size for testing one rate against a reference. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/rates-one-sample/

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.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
3.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
4.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
5.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

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected rateλ1\lambda_1λ1​

Reference rateλ2\lambda_2λ2​

Follow-up per participantttt

Dispersion (negative binomial k)kkkOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

One- or two-sidedτ\tauτ

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

A 20% higher rate

Two years of follow-up

Over-dispersed counts

Validation

How to cite

References

Expected rate $\lambda_1$

Reference rate $\lambda_2$

Follow-up per participant $t$

Dispersion (negative binomial k) $k$ Optional

Significance level $\alpha$

Power $1-\beta$

One- or two-sided $\tau$

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