Sample size for a stepped-wedge trial: binary outcome

When to use it

A rollout evaluated on a binary outcome, such as the share of patients receiving recommended care.

Not for

Settings where a parallel cluster trial is feasible.

Notation

$I$: Number of clusters
$S$: Number of steps; there are S + 1 periods
$m$: Measurements per cluster in each period
$\rho$: Intracluster correlation coefficient (ICC)
$\sigma^{2}$: Total variance of the outcome
$\theta$: Treatment effect: a difference in means or proportions
$X_{ij}$: 1 if cluster i is on the intervention in period j, else 0
$p_1,\ p_2$: Proportions under control and intervention; σ² = p̄(1 − p̄)
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$q$: Expected dropout

Formulas

Variance of the treatment effect

\operatorname{Var}(\hat{\theta}) = \left[(X^{\top} V^{-1} X)^{-1}\right]_{\theta\theta}, \qquad V = \rho\sigma^{2} J + \frac{(1-\rho)\sigma^{2}}{m} I

The Hussey and Hughes model: cluster-period means share a cluster effect, and every period has its own term so the secular trend is estimated rather than assumed away [1]. The calculator solves this generalised least squares problem directly.

Clusters, or cluster size, with enough power

\Phi\!\left(\frac{|\theta|}{\sqrt{\operatorname{Var}(\hat{\theta})}} - z_{1-\alpha/2}\right) \ge 1-\beta

Clusters are added a sequence at a time, one cluster per step, until the power is reached; or, with the clusters fixed, the measurements per cluster-period are increased instead. The answer is reported as measurements, I × (S + 1) × m.

Dropout

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

Applied last, to the participants actually recruited.

Parameters

Proportion under control $p_1$

Values: Between 0 and 1

Before the intervention is rolled out.

Proportion under the intervention $p_2$

Values: Between 0 and 1, different from p₁

The variance used is that of the average proportion.

Number of steps $S$

Values: A whole number of at least 1

Times at which a group of clusters crosses to the intervention.

Measurements per cluster-period $m$

Values: Above 0

New participants measured in each cluster in each period.

Intracluster correlation (ICC) $\rho$

Values: From 0 to 1

From comparable cluster trials or surveys [3].

What to solve for $\cdot$

Values: Clusters, or cluster size for a fixed number of clusters

Solve for the cluster size when the clusters are fixed, such as every hospital in a region.

Number of clusters $I$ Optional

Values: A multiple of the number of steps

When solving for cluster size; it must divide evenly between the steps.

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.

Assumptions

A cross-sectional design: different people are measured in each period. A cohort followed throughout needs a model of repeated measurements on each participant.
The intervention effect is immediate and constant once a cluster crosses over, and the secular trend is the same in all clusters [2].
The calculation uses the normal approximation, which can overstate power when there are few clusters; with fewer than about 10–15, plan the analysis with a small-sample correction and allow for it here.
The binary outcome is handled with the variance of the average proportion, a linear approximation.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

30% to 45%

0.3 rising to 0.45, 4 steps, 20 per cluster-period, ICC 0.05: 12 clusters, 1200 measurements.

Sample size: 1200

Protocol statement
To detect a change from 30% to 45% with 80% power at a two-sided significance level of 0.05, using a stepped-wedge cluster-randomised trial with 4 steps, and assuming an intracluster correlation of 0.05, the required sample size is 12 clusters over 5 periods, with 20 measurements per cluster in each period, 1200 measurements in total. Calculated with StatWisely (statwisely.com) using: clusters: solve GLS Var(θ̂) from (X'V⁻¹X)⁻¹ with one term per period; V=σ_c²J+(σ_e²/m)I. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for missing outcomes

Expecting 10% of measurements to be missing: 1334 to collect.

Needed for analysis: 1200 · To recruit: 1334

Protocol statement
To detect a change from 30% to 45% with 80% power at a two-sided significance level of 0.05, using a stepped-wedge cluster-randomised trial with 4 steps, and assuming an intracluster correlation of 0.05, the required sample size is 12 clusters over 5 periods, with 20 measurements per cluster in each period, 1200 measurements in total. Allowing for 10% missing measurements, 1334 measurements should be collected. Calculated with StatWisely (statwisely.com) using: clusters: solve GLS Var(θ̂) from (X'V⁻¹X)⁻¹ with one term per period; V=σ_c²J+(σ_e²/m)I. These are planning estimates; confirm the final design with a qualified statistician.

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.

How to cite

StatWisely. Sample size for a stepped-wedge trial: binary outcome. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/stepped-wedge-proportions/

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

References

1.Hussey MA, Hughes JP. Design and analysis of stepped wedge cluster randomized trials. Contemporary Clinical Trials. 2007;28(2):182–191.
2.Hemming K, Haines TP, Chilton PJ, Girling AJ, Lilford RJ. The stepped wedge cluster randomised trial: rationale, design, analysis, and reporting. BMJ. 2015;350:h391.
3.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