How to Report Mauchly's Test of Sphericity in APA 7 (With Example)

Mauchly's test checks sphericity, the assumption of a repeated-measures ANOVA that the variances of the differences between every pair of conditions are equal. It only applies when the within-subjects factor has three or more levels. If it fails, the ANOVA's F test is too liberal, and you correct its degrees of freedom. This guide gives you the APA 7 format for both outcomes.

What you need before you write a single word

1. Mauchly's W.
2. The approximate χ² and its degrees of freedom (k(k − 1)/2 − 1 for k conditions; 2 for three conditions).
3. The exact p value.
4. If it is significant: Greenhouse-Geisser ε (or Huynh-Feldt ε) and the corrected degrees of freedom for the ANOVA.

The APA 7 format template

When sphericity holds:

Mauchly's test indicated that the assumption of sphericity was met, W = .XX, χ²(df) = X.XX, p = .XXX.

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When it is violated:

Mauchly's test indicated that the assumption of sphericity was violated, W = .XX, χ²(df) = X.XX, p = .XXX, so Greenhouse-Geisser corrected degrees of freedom are reported (ε = .XX).

Formatting rules reviewers actually check:

• No leading zero on W, p or ε (none can exceed 1).
• After a violation, the ANOVA's df are fractional: F(1.29, 14.18), not F(2, 22).
• Italicize W and p; χ² and ε are Greek and stay upright.

Worked examples

Sphericity met. Anxiety measured at three time points in 40 adults:

Mauchly's test indicated that the assumption of sphericity was met, W = .86, χ²(2) = 5.84, p = .054.

Sphericity violated. Words recalled under three study conditions:

Mauchly's test indicated that the assumption of sphericity was violated, W = .45, χ²(2) = 8.01, p = .018, so Greenhouse-Geisser corrected degrees of freedom are reported (ε = .64).

The ANOVA that follows then uses the corrected df: each original df multiplied by ε. The full write-up is in how to report a repeated-measures ANOVA.

Mistakes reviewers catch

• Reporting Mauchly's test for two conditions. With two levels there is only one difference, so sphericity always holds and the test is not computed.
• Correcting nothing after a violation. The ANOVA's df and p must come from the Greenhouse-Geisser (or Huynh-Feldt) row.
• Uncorrected whole-number df after saying sphericity was violated. Reviewers check that the df match the correction.
• Relying on it in small samples. Mauchly's test has little power with few participants; many researchers apply the correction whenever ε is below about .75.

Let KyroStat do the write-up for you

KyroStat checks sphericity for every within-subjects factor, applies the correction when it is needed, and writes the sentence and the corrected ANOVA from your data.

Frequently asked questions

Greenhouse-Geisser or Huynh-Feldt? A common rule: use Greenhouse-Geisser when ε is below .75, and Huynh-Feldt when it is above. Say which one you used.

Does a mixed ANOVA need Mauchly's test? Yes, for its within-subjects factor. See how to report a mixed ANOVA.

Where does it go in the Results? Right before the repeated-measures ANOVA, or in a short assumptions paragraph. See how to check statistical test assumptions.