Achieved power for a repeated-measures ANOVA

When to use it

Assess the sensitivity of a study that measures the same participants at several time points or conditions and tests whether the mean changes across them.

Not for

This is the within-subject main effect for one group. Comparing how two groups change over time is a group-by-time interaction, which needs a different calculation.

Notation

$N$: Participants, each measured m times.
$m$: Number of measurements per participant.
$f$: Cohen’s f for the differences between the time-point means.
$\rho$: Correlation between repeated measurements.
$\varepsilon$: Sphericity correction; 1 when sphericity holds.

Formulas

Degrees of freedom and noncentrality

\nu_1=(m-1)\varepsilon, \qquad \nu_2=(N-1)(m-1)\varepsilon, \qquad \lambda=\frac{f^2 N m \varepsilon}{1-\rho}

A higher correlation between measurements removes more between-person variation, so it raises power [1, 2].

Power of the within-subject F test

\text{Power}=P\left(F_{\nu_1,\nu_2,\lambda} > F_{1-\alpha;\,\nu_1,\nu_2}\right)

The alternative is a noncentral F distribution under compound symmetry [1].

Parameters

Number of participants $N$

Values: Integer, at least 2

Enter the participants with complete measurements.

Measurements per participant $m$

Values: Integer, at least 2

Time points or conditions each participant is measured under.

Effect size (Cohen’s f) $f$

Values: Positive

The SD of the expected time-point means divided by the within-time SD; 0.10, 0.25, and 0.40 are small, medium, and large.

Correlation between measurements $\rho$

Values: At least 0 and below 1

From earlier data on the same outcome; 0.5 is a common planning value when none is available.

Sphericity (ε) $\varepsilon$ Optional

Values: Above 0, at most 1

1 assumes sphericity; the lower bound is 1/(m − 1). A lower value is more conservative.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Assumptions

Measurements are approximately normal with equal variances and a common correlation (compound symmetry); departures from sphericity are handled through ε.
F and chi-square tests reject only in the upper tail, so there is no one- or two-sided choice.

Worked examples

A worked input set

For 30 participants measured 3 times, f = 0.25, a correlation of 0.5 between measurements, sphericity, and α = 0.05: ν₁ = 2, ν₂ = 58, and λ = 0.25² × 30 × 3/(1 − 0.5) = 11.25.

Sample size: 30

Open this example in the calculator

Validation

Power is the exact noncentral-F tail probability with G*Power’s within-factors parameterization, evaluated independently in R. At the size the repeated-measures calculator returns, it reaches that calculator’s target power.

How to cite

StatWisely. Achieved power for a repeated-measures ANOVA. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-repeated-anova/

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

References

1.Faul F, Erdfelder E, Lang AG, Buchner A. G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods. 2007;39(2):175–191.
2.Greenhouse SW, Geisser S. On methods in the analysis of profile data. Psychometrika. 1959;24(2):95–112.