Sample size for comparing event rates across three or more groups

When to use it

A trial of several arms, or a cohort split by an unordered exposure, whose outcome is a count of events over follow-up.

Not for

Ordered groups such as doses, where a trend test is more powerful.
Two groups. Use the rate-ratio tests.

Notation

$n$: Participants in each group
$g$: Number of groups
$\lambda_j$: Expected event rate per unit of time in group j
$t$: Follow-up time per participant
$k$: Negative binomial dispersion: variance μ + kμ²; 0 is Poisson
$u_j$: Information per participant about group j’s log rate, 1/(1/(λⱼt) + k)
$\alpha$: 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
$\bar{\theta}$: The u-weighted mean of the log rates

Formulas

Noncentral chi-square power

\Pr\!\left(\chi'^{2}_{g-1,\Lambda} > \chi^{2}_{g-1,\,1-\alpha}\right) \ge 1-\beta, \qquad \Lambda = n \sum_{j} u_j\,(\ln \lambda_j - \bar{\theta})^{2}

The Wald (or, in large samples, likelihood-ratio) test that every group’s log rate is equal, on g − 1 degrees of freedom. Each log rate is estimated with variance 1/(n uⱼ), the Poisson 1/(nλⱼt) widened by the dispersion [1, 2].

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 in each group $\lambda_1, \dots, \lambda_g$

Values: At least three, each above 0, not all equal

Events per participant per unit of time: the smallest pattern of differences worth detecting.

Follow-up per participant $t$

Values: Above 0, in the rates’ time unit

Longer follow-up yields more events, and so more information, per participant.

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; take k from earlier data and try plausible values.

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.

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

The groups are independent and of equal size, every participant is followed for the same time, and each group’s rate is constant over it.
Counts are Poisson, or negative binomial with the stated dispersion, the same in every group.
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

Three arms, Poisson counts

Rates of 1, 1.25 and 1.5 a year over one year: Σuⱼ(ln λⱼ − θ̄)² = 0.0990, and 80% power on 2 degrees of freedom needs Λ = 9.635, so n = 9.635 / 0.0990 = 97.3, rounded up to 98 per group.

98 per group · Total: 294

Protocol statement
To detect rates of 1, 1.25, 1.5 per unit of time with 80% power at a significance level of 0.05, using a test that the event rate is the same in all 3 groups, and assuming follow-up of 1 per participant, the required sample size is 98 per group (294 in total, across 3 groups). Calculated with StatWisely (statwisely.com) using: n per group: noncentral χ² power on g-1 df, λ=n·Σuⱼ(ln λⱼ-θ̄)², uⱼ=1/(1/(λⱼt)+k). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Over-dispersed counts

With dispersion k = 0.5 each group carries less information: Σuⱼ(ln λⱼ − θ̄)² falls to 0.0617, so 157 per group.

157 per group · Total: 471

Open this example in the calculator

Two years of follow-up

Following each participant for 2 years doubles the Poisson information: 49 per group.

49 per group · Total: 147

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.
Simulated Poisson and negative binomial counts at the planned sizes, analysed with likelihood-ratio tests in generalised linear models, gave power of 0.80 to 0.82 against a target of 0.80.

How to cite

StatWisely. Sample size for comparing event rates across three or more groups. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/rates-several-groups/

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.Agresti A. Categorical Data Analysis. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2013.
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

Noncentral chi-square power

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected rate in each groupλ1,…,λg\lambda_1, \dots, \lambda_gλ1​,…,λg​

Follow-up per participantttt

Dispersion (negative binomial k)kkkOptional

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

Three arms, Poisson counts

Over-dispersed counts

Two years of follow-up

Validation

How to cite

References

Expected rate in each group $\lambda_1, \dots, \lambda_g$

Follow-up per participant $t$

Dispersion (negative binomial k) $k$ 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