Sample size for a repeated-measures ANOVA

When to use it

One group measured on three or more occasions, testing whether the mean changes over time or across conditions.

Not for

Comparing how groups change over time (a group-by-time interaction). Use the longitudinal designs.
Only two occasions. Use the paired t-test.

Notation

$N$: Number of participants
$M$: Number of occasions measured
$f$: Cohen’s f for the change across occasions
$\rho_m$: Correlation between occasions
$\varepsilon$: Sphericity correction, from 1/(M − 1) to 1
$\lambda$: Noncentrality of the F distribution
$\alpha$: 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

The smallest N with enough power

\Pr\!\left(F'_{\nu_1,\nu_2,\lambda} > F_{\nu_1,\nu_2,1-\alpha}\right) \ge 1-\beta, \qquad \nu_1 = (M-1)\varepsilon, \quad \nu_2 = (N-1)(M-1)\varepsilon, \quad \lambda = \frac{f^{2}\,N\,M\,\varepsilon}{1 - \rho_m}

The within-subject F test, powered on the noncentral F distribution [1]. A stronger correlation between occasions removes more between-person variation, so it raises λ and lowers N; a sphericity correction ε below 1 lowers both degrees of freedom and λ, as the Greenhouse–Geisser analysis does [2].

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 \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

Cohen’s f $f$

Values: Above 0

The SD of the occasion means divided by the within-occasion SD. Benchmarks of 0.10, 0.25 and 0.40 are a last resort [3].

Number of occasions $M$

Values: A whole number of at least 2

Every participant is measured on every occasion.

Correlation between occasions $\rho_m$

Values: At least 0 and below 1

How strongly a person’s measurements track each other, from previous data. Repeated measurements of the same person are often correlated 0.5 or more; assuming too strong a correlation undersizes the study.

Sphericity correction $\varepsilon$ Optional

Values: Above 0 and at most 1; 1 assumes sphericity

Below 1 when occasions further apart are less correlated than adjacent ones, as is common over time. 0.75 is a cautious choice when unsure.

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 [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

Participants are independent of one another apart from any clustering specified, and complete every occasion.
The occasions share one correlation (compound symmetry), relaxed by the sphericity correction ε.
Power is for the overall within-subject test, not for a particular contrast such as a linear trend.
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 occasions

f = 0.25, 3 occasions correlated 0.5, sphericity assumed; α = 0.05, 80% power: 28 participants.

Sample size: 28

Protocol statement
To detect a change across occasions of f = 0.25 with 80% power at a significance level of 0.05, using a repeated-measures ANOVA over 3 occasions, and assuming a correlation of 0.5 between occasions, the required sample size is 28 participants. Calculated with StatWisely (statwisely.com) using: n: solve noncentral-F power for the within-subject effect; df₁=(m-1)ε, df₂=(N-1)(m-1)ε, λ=f²Nmε/(1-ρ). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for non-sphericity

The same design with ε = 0.75: 34 participants.

Sample size: 34

Open this example in the calculator

More strongly correlated occasions

With a correlation of 0.8: only 12 participants, because each person serves as a closer control for themselves.

Sample size: 12

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 repeated-measures ANOVA. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/means-repeated-anova/

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

References

1.Muller KE, Barton CN. Approximate power for repeated-measures ANOVA lacking sphericity. Journal of the American Statistical Association. 1989;84(406):549–555.
2.Greenhouse SW, Geisser S. On methods in the analysis of profile data. Psychometrika. 1959;24(2):95–112.
3.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
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

The smallest N with enough power

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Cohen’s ffff

Number of occasionsMMM

Correlation between occasionsρm\rho_mρm​

Sphericity correctionε\varepsilonεOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

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 occasions

Allowing for non-sphericity

More strongly correlated occasions

Validation

How to cite

References

Cohen’s f $f$

Number of occasions $M$

Correlation between occasions $\rho_m$

Sphericity correction $\varepsilon$ Optional

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ 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