Repeated-Measures ANOVA
Within-subjects ANOVA for pre/post or multi-condition designs — with Mauchly's sphericity test, corrections, partial eta-squared, and APA reporting.
What it is
A repeated-measures ANOVA (also called within-subjects ANOVA) tests whether an outcome changes across conditions or time points when the <em>same participants</em> provide data at every level. It is the within-subjects counterpart to a one-way ANOVA. Because each participant serves as their own control, individual differences are removed from the error term, which gives the test more statistical power than a between-groups design of the same size. Key outputs include the F statistic, degrees of freedom (df), and p-value for the within-subjects effect, and partial eta-squared (partial η²) as the effect size. Because the test assumes sphericity — equal variances of the differences between all pairs of conditions — Explonia runs Mauchly's test automatically. When sphericity is violated (Mauchly's p < .05), the degrees of freedom are corrected using either the Greenhouse–Geisser or Huynh–Feldt correction, making the test conservative enough to be valid. Corrected pairwise comparisons identify which specific conditions differ.
When to use it
- The same participants are measured at three or more time points (e.g., pre, mid, post).
- Each participant completes all conditions in a within-subjects experiment.
- You want to test change over time with greater power than a between-groups design.
- You need an effect size (partial η²) for your APA write-up alongside F and p.
When not to use it
- Different participants are in each group — use a between-groups comparison (independent t-test or one-way ANOVA) instead.
- You have only two repeated measurements and your outcome is continuous — a paired-samples t-test gives the same result with simpler reporting.
- Carry-over effects between conditions make within-subjects measurement inappropriate — consider a between-groups design.
Assumptions
- The same participants are measured at every condition or time point (within-subjects design).
- Sphericity — the variances of the differences between all pairs of conditions are equal; Mauchly's test checks it.
- When sphericity is violated (Mauchly's p < .05), the degrees of freedom are corrected (Greenhouse–Geisser or Huynh–Feldt).
- The outcome is continuous and approximately normally distributed within each condition.
- No carry-over effects between conditions that would make within-subjects measurement inappropriate.
Thresholds & cutoffs
| Criterion | Rule of thumb | Source |
|---|---|---|
| Mauchly's W | p < .05 → sphericity violated → apply a correction | Mauchly (1940) |
| Greenhouse–Geisser ε | ε < .75 → prefer Greenhouse–Geisser correction | Greenhouse & Geisser (1959) |
| Huynh–Feldt ε | ε ≥ .75 → Huynh–Feldt correction is less biased | Huynh & Feldt (1976) |
| Partial η² | .01 small, .06 medium, .14 large (rough guides) | Cohen (1988) |
| Significance (α) | p < .05 conventional; correct pairwise comparisons for multiple tests | Standard NHST convention |
How to report it in APA
Mauchly's test indicated a violation of sphericity, W = .81, p = .012; therefore, degrees of freedom were corrected using the Greenhouse–Geisser estimate (ε = .87). A repeated-measures ANOVA revealed a significant time effect, F(1.73, 206.93) = 24.17, p < .001, partial η² = .17. Corrected pairwise comparisons showed that scores at Time 2 were significantly higher than at Time 1 (p = .003) and Time 3 (p < .001).
How Explonia runs it
In Explonia, open your study, choose Repeated Measures under Analyses, and assign your within-subjects variables to the time-point or condition slots. Explonia fits the model, runs Mauchly's sphericity test automatically, and applies the Greenhouse–Geisser correction if sphericity is violated. It reports F, corrected df, p, and partial η² in a single table, along with corrected pairwise comparisons for every pair of conditions. The Article description box drafts an APA paragraph — including the Mauchly result and the correction used — that you can edit before copying. All analyses are available on every tier; the limit across tiers is the number of responses in your study.
Frequently asked questions
What is sphericity and why does it matter?
Sphericity means that the variances of the differences between all pairs of conditions are equal. Repeated-measures ANOVA assumes sphericity. When this assumption is violated, the F-test is too liberal — it produces too many false positives. Mauchly's test checks this assumption, and if it is significant (p < .05) you should apply a correction.
Which correction should I choose when sphericity is violated — Greenhouse–Geisser or Huynh–Feldt?
When the Greenhouse–Geisser epsilon (ε) is below .75, the Greenhouse–Geisser correction is generally preferred because it is more conservative. When ε is .75 or above, the Huynh–Feldt correction is less biased. Report whichever correction you apply, along with the ε value, in your write-up.
How do I report partial eta-squared in APA?
Report partial η² alongside F, degrees of freedom, and p. For example: F(1.73, 206.93) = 24.17, p < .001, partial η² = .17. Cohen's (1988) conventions label .01 small, .06 medium, and .14 large, though these are rough guides rather than field-specific standards.
What is the difference between repeated-measures ANOVA and a between-groups ANOVA?
In a repeated-measures (within-subjects) design, the same participants provide data at every condition or time point. In a between-groups design, different participants are assigned to each group. Repeated measures use each participant as their own control, which increases statistical power but requires the sphericity assumption.
Last updated: July 2026
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