Method document

Sample size for several arms against a shared control: proportions

How many participants are needed in each active arm and the control when several treatments are each compared with one control on a yes/no outcome, with the significance level shared across the comparisons. Covers Dunnett’s and Bonferroni’s corrections, a larger control arm, clustering and dropout.

Engine version 1.0.0 · Reviewed 1 October 2026

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https://statwisely.com/methods/proportions-multi-arm/

When to use it

A multi-arm trial with a binary outcome in which each treatment is compared with the same control.

Not for

Comparing every arm with every other. Use a chi-square test of independence on the whole table.

Notation

$K$: Number of active arms
$n$: Participants per active arm
$c_r$: Size of the control relative to each active arm
$p_1,\ p_2$: Expected proportions in the control and in each active arm
$h$: Standardised difference between the proportions
$u$: Critical value after the multiplicity correction
$\alpha$: 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

Standardised difference

h = \frac{|p_1 - p_2|}{\sqrt{\left[p_1(1-p_1) + p_2(1-p_2)\right]/2}}

The difference in proportions divided by the average binomial standard deviation, so the same calculation as for means applies.

Participants per active arm

n = \left\lceil \left(1 + \frac{1}{c_r}\right)\left(\frac{u + z_{1-\beta}}{h}\right)^{2} \right\rceil, \qquad n_{\text{control}} = \lceil c_r\, n \rceil

With u Dunnett’s critical value, which allows for the comparisons sharing the control, or Bonferroni’s [1]. Power is that of a single comparison.

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

Number of active arms $K$

Values: A whole number of at least 2

Not counting the control.

Expected proportion in the control $p_1$

Values: Between 0 and 1

From routine data or previous trials.

Expected proportion in each active arm $p_2$

Values: Between 0 and 1, different from p₁

The proportion worth detecting for each treatment.

Multiplicity correction $u$

Values: Dunnett, Bonferroni or none

Dunnett’s is designed for comparisons with a shared control.

Control allocation $c_r$ Optional

Values: Above 0

About √K is close to optimal.

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 [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

Arms are independent, with the same expected proportion in every active arm.
Power is per comparison, and the normal approximation holds in every arm.
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

Three treatments, Dunnett’s correction

3 arms, 30% in the control against 45%: h = 0.15 / 0.478 = 0.314; Dunnett’s critical value is about 2.35, giving 208 in every arm and the control, 832 in total.

Control: 208 · Each active arm: 208 · Total: 832

Protocol statement
To detect a change from 30% in the control to 45% in each active arm with 80% power for each comparison at a two-sided significance level of 0.05, using comparisons of each of 3 active arms with a shared control, with Dunnett’s correction, the required sample size is 208 in each of the 3 active arms and 208 in the control arm (832 in total). Calculated with StatWisely (statwisely.com) using: n_arm = (1 + 1/c)(u + z_(1-beta))^2 / d^2, where u is the dunnett critical value for 3 comparisons with a shared control. These are planning estimates; confirm the final design with a qualified statistician.

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Bonferroni instead

Bonferroni’s critical value is about 2.40: 214 per arm, 856 in total.

Control: 214 · Each active arm: 214 · Total: 856

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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 several arms against a shared control: proportions. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/proportions-multi-arm/

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

References

1.Dunnett CW. A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association. 1955;50(272):1096–1121.
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