Repeated Measures ANOVA Calculator

Sphericity is about the DIFFERENCES, not the conditions

It is usually stated as “equal variances and equal covariances”, which is compound symmetry and is stronger than what is actually required. The real condition is simpler and easier to check mentally.

Every pairwise difference between conditions must have the same variance. Condition 1 minus condition 2, condition 1 minus condition 3, condition 2 minus condition 3: all of those difference scores need a common variance.

Which is why two conditions can never violate it. There is one difference score, so there is nothing for it to differ from — ε is exactly 1 by construction, and the analysis is a paired t-test wearing an F.

It fails most often when the conditions are ordered in time. Measurements taken close together correlate more than measurements far apart, so the difference between adjacent timepoints is less variable than the difference between the first and the last.

When it fails the F test is liberal, not conservative. The p-value is too small, so the error is in the direction that produces findings rather than the one that hides them — which is the reason this correction is not optional.

Epsilon measures the violation, and multiplies both degrees of freedom

Greenhouse-Geisser and Huynh-Feldt do not change F. They change the reference distribution it is compared against, by shrinking the degrees of freedom.

ε runs from 1 (perfect sphericity) down to 1/(k−1), which is the worst possible case and is the “lower bound” row in the table. Both degrees of freedom are multiplied by it.

On the shipped preset ε = 0.431738, so df go from (3, 21) to (1.2952, 9.0665). Fractional degrees of freedom are correct here, not a rounding error, and the F distribution is perfectly well defined at them.

Greenhouse-Geisser is conservative and Huynh-Feldt is not. GG underestimates ε when the true value is near 1; HF corrects that and can overshoot past 1, which is why it is capped. The common rule is GG below 0.75 and HF above.

The cost of correcting when sphericity holds is tiny. The tool’s second preset has ε = 0.960540, and correcting moves p from 1.23×10⁻⁵ to 1.74×10⁻⁵. That asymmetry — large gain when it matters, negligible cost when it does not — is the argument for always reporting the corrected value.

Mauchly’s test is not a good gate

The standard workflow is: run Mauchly, correct only if it is significant. That workflow has a problem at both ends of the sample-size range.

At small n Mauchly is underpowered. It fails to detect real violations exactly when the design is small, which is exactly when the violation does the most damage to the F test.

At large n it is oversensitive. It flags trivial departures from sphericity that would move the p-value in the fourth decimal, prompting a correction nobody needed.

It also assumes multivariate normality, so a significant Mauchly can be reporting a distributional problem rather than a sphericity one.

Which is why the modern recommendation is to skip the gate. Report the corrected p-value unconditionally; when sphericity holds the correction costs almost nothing, and when it does not the correction is the only defensible number. The tool prints Mauchly because it is asked for, and prints all four p-values so the decision is visible either way.

Removing subject variation is the entire point

The source table has a row that a between-subjects ANOVA does not: subjects. Its size is the whole argument for the design.

On the fourth preset the subject row is 99.25% of the total variation. Subject means run from about 12 to about 48 while the conditions do not differ at all.

A between-subjects design would have left all of that in the error term, where it would have buried any condition effect under variation that has nothing to do with the manipulation.

Which is why a repeated-measures design needs far fewer participants for the same power — often by a factor of several, depending on how correlated the measurements are.

The costs are order effects and dropout. Every subject experiences every condition, so practice, fatigue and carryover all become confounds unless the order is counterbalanced; and one missing measurement removes the whole subject from a complete-case analysis.

When to use something else entirely

The sphericity correction patches a model rather than replacing it, and for some designs the replacement is better.

A mixed-effects model needs no sphericity assumption at all. It models the covariance structure explicitly instead of correcting for its failure, and it handles missing data without dropping subjects.

MANOVA treats the conditions as a multivariate outcome and also avoids the assumption, at the cost of needing more subjects than conditions and usually less power when sphericity does roughly hold.

The Friedman test is the rank-based alternative, which assumes nothing about the distribution and discards the magnitudes entirely — a real trade rather than a strictly safer option.

For exactly two conditions, use a paired t-test and say so. It is the same test — F is exactly t squared, verified to 1.4×10⁻¹⁴ on the tool’s third preset — and the t is more readable.

Reporting a repeated measures ANOVA

Four things, and the second is the one most often missing.

Report the corrected degrees of freedom, not the nominal ones. “F(1.30, 9.07) = 3.55, p = 0.085” tells a reader a correction was applied and how large it was; “F(3, 21) = 3.55, p = 0.085” is internally inconsistent.

Name the correction and give epsilon. Greenhouse-Geisser and Huynh-Feldt give different p-values, and ε is the one number that says how far from sphericity the data sat.

Give an effect size. Partial eta squared is conventional here; omega squared is less biased and the tool prints both.

And say how order was handled. Counterbalanced, randomised or fixed are different designs, and a fixed order makes practice and fatigue indistinguishable from the condition effect.

Sources and methodology

References for the repeated-measures F and its sphericity corrections.

• Greenhouse & Geisser, “On Methods in the Analysis of Profile Data”, Psychometrika 24(2), 1959
• Huynh & Feldt, “Estimation of the Box Correction…”, Journal of Educational Statistics 1(1), 1976
• Mauchly, “Significance Test for Sphericity of a Normal n-Variate Distribution”, Annals of Mathematical Statistics 11(2), 1940
• Friedman Test Calculator — the rank-based alternative
• ANOVA Calculator — the between-subjects case
• How we calculate — methodology

Method. Epsilon is computed from the actual covariance matrix across conditions, double-centred, rather than from a compound-symmetry shortcut — so it reflects the specific violation in your data rather than a summary of it. Mauchly’s W is computed on k−1 orthonormal contrasts with eigenvalues from a cyclic Jacobi rotation, which is why W comes out exactly 1 for two conditions rather than approximately. The two-condition identity is asserted rather than assumed: F equals the paired t squared to 1.4×10⁻¹⁴ on the third preset, and ε is exactly 1. Corrected p-values are computed by scaling BOTH degrees of freedom, which is the correction as defined rather than the common shortcut of scaling only the numerator. Ragged rows, a single subject and a single condition all return no result. That engine is verified on every change against 96 assertions. The count and the per-case breakdown are published on the formula verification page.

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