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    Method document

    Achieved power for a stepped-wedge trial (proportions)

    Calculates the power of a cross-sectional stepped-wedge cluster trial with a binary outcome, from the number of clusters, steps, cluster size, intracluster correlation, and the two proportions.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-stepped-wedge-proportions/

    When to use it

    • Assess a rollout with a yes/no outcome in which every cluster starts on control and crosses to the intervention at a randomised time.

    Not for

    • The binary outcome is analysed on the risk-difference scale with a normal approximation; very small or large proportions need more care.

    Notation

    CC
    Clusters, a multiple of the number of steps.
    SS
    Steps; there are S + 1 periods.
    mm
    Measurements per cluster in each period.
    ρ\rho
    Intracluster correlation.
    σ2\sigma^2
    Total variance of one measurement.
    p1,p2p_1, p_2
    Proportions under intervention and control; σ² = p̄(1 − p̄).

    Formulas

    GLS variance of the treatment effect

    Var⁡(θ^)=[(X⊤V−1X)−1]θθ,V=ρσ2J+(1−ρ)σ2mI\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

    Each period has its own term, so the secular trend is adjusted for rather than assumed away [1, 2].

    Normal power

    Power=Φ ⁣(∣Δ∣Var⁡(θ^)−z1−α/tails)\text{Power}=\Phi\!\left(\frac{|\Delta|}{\sqrt{\operatorname{Var}(\hat\theta)}}-z_{1-\alpha/\text{tails}}\right)

    With few clusters this can overstate power; the planned small-sample analysis should inform the design.

    Parameters

    Number of clustersCC

    Values: A multiple of the steps

    Clusters split equally across the steps, one group switching at each.

    StepsSS

    Values: Integer, at least 1

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

    Measurements per cluster per periodmm

    Values: Positive

    A fresh sample in each period (cross-sectional design).

    Intracluster correlationρ\rho

    Values: Between 0 and 1

    From earlier cluster data on the same outcome.

    Proportion under interventionp1p_1

    Values: Between 0 and 1

    Expected proportion once a cluster has crossed over.

    Proportion under controlp2p_2

    Values: Between 0 and 1

    Expected proportion before crossover.

    Significance levelα\alpha

    Values: Between 0 and 1

    The type I error rate for the specified test.

    Test directionss

    Values: 1 or 2

    One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Assumptions

    • A fresh sample in every period, equal cluster-period sizes, and a constant intracluster correlation over time.
    • One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Worked examples

    A worked input set

    With 8 clusters over 4 steps, 20 measurements per cluster-period, ρ = 0.05, and proportions of 0.30 and 0.45 at α = 0.05: σ² = 0.375 × 0.625 = 0.234.

    Sample size: 8

    Open this example in the calculator

    Validation

    • At the size the stepped-wedge calculator returns, this power reaches that calculator’s target, and one fewer falls short; it is checked against an independent calculation in R.

    How to cite

    StatWisely. Achieved power for a stepped-wedge trial (proportions). Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-stepped-wedge-proportions/

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

    References

    1. 1.Hussey MA, Hughes JP. Design and analysis of stepped wedge cluster randomized trials. Contemporary Clinical Trials. 2007;28(2):182–191.
    2. 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.