Method document

Sample size for several arms against a shared control

How many participants are needed in each active arm and in the control when several treatments are each compared with one control on a continuous outcome, with the significance level shared across the comparisons. Covers Dunnett’s and Bonferroni’s corrections, a larger control arm, a baseline covariate, clustering and dropout.

Engine version 1.0.0 · Reviewed 1 October 2026

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

When to use it

A multi-arm trial in which each of several treatments or doses is compared with the same control.

Not for

Comparing every group with every other, or testing whether any group differs. Use one-way ANOVA.

Notation

$K$: Number of active arms (comparisons with the control)
$n$: Participants per active arm
$c_r$: Size of the control relative to each active arm
$d$: Standardised effect of each treatment against the control
$u$: 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
$\tau$: Number of tails: 2 for a two-sided test, 1 for one-sided
$r_b$: Correlation between the outcome and a baseline covariate
$\rho$: 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 per active arm

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

Each comparison uses one active arm and the shared control, so its variance carries 1/n and 1/(c_r n). Power is that of a single comparison, as protocols usually state; the chance of detecting every true effect is lower.

The critical value u

u_{\text{Dunnett}}:\ \Pr\!\left(\max_{j} |T_j| \le u\right) = 1-\alpha, \quad \operatorname{corr}(T_j, T_{j'}) = \frac{1}{1 + c_r}; \qquad u_{\text{Bonferroni}} = t_{\nu,\,1-\alpha/(K\tau)}

Dunnett’s value accounts for the comparisons sharing the control, which correlates them, so it is smaller than Bonferroni’s and needs fewer participants [1]. Both use t on the trial’s degrees of freedom, iterated with n; “no correction” tests each comparison at the full α.

A larger control arm

c_r^{*} \approx \sqrt{K}

The control is used in every comparison, so giving it about √K times each active arm is close to the most efficient allocation.

Adjusting for a baseline covariate

n' = \left\lceil n\,(1 - r_b^{2}) \right\rceil + 1

Analysing with a baseline measurement as a covariate (ANCOVA) removes the share r_b² of the outcome’s variance the test would otherwise have to overcome. One participant per group is added back for the degree of freedom the covariate uses [3].

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 [4, 5].

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 [6].

All adjustments together

n_{\text{adj}} = \left\lceil n (1 - r_b^{2})\,\mathrm{DE}\;F_t \right\rceil + 1

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: three treatments against one control is K = 3.

Difference to detect $\Delta$

Values: Not zero, in the measurement’s units

The smallest difference that would matter, clinically or practically. Not the difference you hope for: a study sized for an optimistic effect is underpowered for a real but smaller one.

Standard deviation $\sigma$

Values: Above 0, in the measurement’s units

From a comparable published study, a pooled SD from several studies, or pilot data. The sample size grows with its square, so check a larger value too.

Cohen’s d, instead of the two above $d$ Optional

Values: Above 0

The difference in standard-deviation units. Cohen’s benchmarks of 0.2, 0.5 and 0.8 for small, medium and large are a last resort; a difference and SD from your own field are better [2].

Multiplicity correction $u$

Values: Dunnett, Bonferroni or none

Dunnett’s is designed for comparisons with a shared control and is the usual choice. No correction is only defensible when each comparison answers a separate question.

Control allocation $c_r$ Optional

Values: Above 0; 1 for a control the size of each arm

The control’s size relative to each active arm. 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.

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.

Baseline correlation $r_b$ Optional

Values: Between −1 and 1

The correlation between the outcome and its baseline measurement, from previous data. Only use it if the analysis will adjust for the baseline.

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 [7]; 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 [5].

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 standard deviation in every arm and the same effect assumed for each treatment.
Power is per comparison. To detect all K effects with a stated probability, more participants are needed.
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 [4].
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 against one control, d = 0.5, α = 0.05 two-sided, 80% power per comparison: Dunnett’s critical value is about 2.36, giving 82 in every arm and the control, 328 in total.

Control: 82 · Each active arm: 82 · Total: 328

Protocol statement
To detect a difference of 5 from the control 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, and assuming a standard deviation of 10, the required sample size is 82 in each of the 3 active arms and 82 in the control arm (328 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 ignores the shared control, so its critical value is larger, about 2.41: 85 per arm, 340 in total.

Control: 85 · Each active arm: 85 · Total: 340

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A larger control

With the control 1.7 times each arm, close to √3: 66 per active arm and 113 in the control, 311 in total, fewer than with equal arms.

Control: 113 · Each active arm: 66 · Total: 311

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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. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/means-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.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
3.Borm GF, Fransen J, Lemmens WAJG. A simple sample size formula for analysis of covariance in randomized clinical trials. Journal of Clinical Epidemiology. 2007;60(12):1234–1238.
4.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
5.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
6.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
7.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 per active arm

The critical value u

A larger control arm

Adjusting for a baseline covariate

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Number of active armsKKK

Difference to detectΔ\DeltaΔ

Standard deviationσ\sigmaσ

Cohen’s d, instead of the two abovedddOptional

Multiplicity correctionuuu

Control allocationcrc_rcr​Optional

Significance levelα\alphaα

Power1−β1-\beta1−β

One- or two-sidedτ\tauτ

Expected dropoutqqqOptional

Baseline correlationrbr_brb​Optional

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 treatments, Dunnett’s correction

Bonferroni instead

A larger control

Validation

How to cite

References

Number of active arms $K$

Difference to detect $\Delta$

Standard deviation $\sigma$

Cohen’s d, instead of the two above $d$ Optional

Multiplicity correction $u$

Control allocation $c_r$ Optional

Significance level $\alpha$

Power $1-\beta$

One- or two-sided $\tau$

Expected dropout $q$ Optional

Baseline correlation $r_b$ 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