Chapter 15: Analysis of Variance With Repeated Measures
Student Resources
I use the 4 “P’s” framework to help you learn the material in this chapter: Prepare, Practice, Participate, and Perform. To increase the chances to succeed in this course, I strongly encourage you to complete all four “P’s” for each chapter.
1 Prepare
1.1 Chapter Overview
This chapter introduces one-way repeated measures ANOVA—the natural extension of the paired t-test to designs with three or more within-subject conditions or time points. You will learn how repeated measures designs exploit the correlation among observations on the same participants to remove individual differences from the error term, dramatically increasing statistical power. Key topics include the sphericity assumption, Mauchly’s test, the Greenhouse-Geisser and Huynh-Feldt corrections, Bonferroni-corrected post hoc comparisons, and effect size reporting with partial eta-squared and partial omega-squared.
1.2 Multimedia Resources
The following table provides access to video and slide resources for this chapter. Click the links to open them in an overlay for better viewing on all devices.
| Resource | Description | Link |
|---|---|---|
| Long Video Overview | A detailed video explaining repeated measures ANOVA, the sphericity assumption, corrections, effect sizes, and APA reporting in movement science research. | 🔗 Watch Video |
| 🆕 Interactive Self-Study Guide | NEW: An interactive, comprehensive module to be completed before attending class. Covers core concepts to prepare you for the lecture. | 🔗 View Guide |
| Slide Deck HTML | Interactive HTML slides for class. During class, the instructor controls the presentation; after class, review at your own pace. | 🔗 Open Slides |
| Slide Deck PDF | PDF version of the slide deck for download and offline viewing. | 🔗 Download PDF |
You can earn extra credit points by completing the Interactive Self-Study Guide before attending the lecture.
Instructions: 1. Click the link in the table above to start the guide. 2. Complete the activity (you may retake it as many times as you like). 3. Once satisfied with your result, take a screenshot of the Final Score Card. 4. Submit the screenshot to Canvas.
Important: Your screenshot must clearly show both your score and your full name (ensure you enter your name at the start of the activity). See Canvas for point details.
1.3 Read the Chapter
Read (Weir & Vincent, 2021, p. Ch.12) and (Furtado, 2026, p. Ch.15) to understand how to analyze within-subject data using repeated measures ANOVA.
To succeed in this course, you must read the textbook chapters assigned for each topic. This is the only way to learn the material in depth.
Once done, proceed to the next section to practice what you learned.
2 Practice
Practicing what you learned in the chapter is essential to mastering the material. Below are some resources to help you practice the material in this chapter.
2.1 Frequently Asked Questions
A repeated measures design (also called a within-subjects design) is one in which the same participants provide data under every level of the independent variable — for example, at three time points (pre, mid, post) or under three experimental conditions in a single session. It is used when the research question concerns change over time or across conditions within the same individuals, and when it is feasible and appropriate to measure each participant multiple times. Common movement science applications include tracking strength gains across a training program, measuring VO₂max before, during, and after an intervention, or recording movement error across practice blocks. Repeated measures ANOVA is appropriate when there are three or more levels of the within-subject factor; with only two levels, the paired t-test is equivalent.
In a between-subjects design, total variability is divided into two sources: variability due to the treatment and variability within groups (which includes all individual differences — genetics, training history, baseline fitness, etc.). These individual differences typically account for a large portion of the within-group variance, inflating the error term and reducing the F-ratio. In a repeated measures design, each participant serves as their own control across all conditions, so their consistent baseline level cancels out. The statistical result is that between-subjects variance is partitioned out and removed from the error term entirely, leaving a much smaller denominator for the F-ratio:
\[SS_{\text{total}} = SS_{\text{between subjects}} + SS_{\text{time}} + SS_{\text{error}}\]
A smaller error term means a larger F for the same treatment effect, which translates directly into greater statistical power. In practice, the same effect can often be detected with roughly half the number of participants required by an equivalent between-subjects design.
Sphericity requires that the variances of the differences between all pairs of time points are approximately equal. For a design with three time points (pre, mid, post), sphericity requires:
\[\text{Var}(\text{mid} - \text{pre}) \approx \text{Var}(\text{post} - \text{pre}) \approx \text{Var}(\text{post} - \text{mid})\]
Put plainly: the consistency of change should be similar across all pairs of time points. Sphericity matters because the standard repeated measures F-test uses degrees of freedom that assume this homogeneity. When sphericity is violated, the degrees of freedom are too large, the F-test becomes anticonservative, and Type I error is inflated — you reject H₀ more often than the stated α level. Sphericity has no counterpart in between-subjects ANOVA (which requires homogeneity of variance of raw scores, not differences) and is the most important and distinctive assumption to check in repeated measures designs.
Mauchly’s W is automatically reported by SPSS when you run a repeated measures ANOVA. It ranges from 0 (complete non-sphericity) to 1 (perfect sphericity).
- If p > .05: The sphericity assumption is not rejected — use the “Sphericity Assumed” row in the SPSS output.
- If p < .05: Sphericity is violated — apply a degrees-of-freedom correction.
SPSS also reports epsilon (ε) estimates alongside Mauchly’s W. Epsilon quantifies how severe the sphericity violation is: ε = 1.0 means perfect sphericity; values far below 1.0 indicate serious violations. Always inspect epsilon alongside the p-value — in large samples, Mauchly’s test may flag trivial violations as significant (ε ≥ .90 suggests approximate sphericity even if p < .05).
Both corrections adjust the degrees of freedom downward when sphericity is violated. The choice depends on the Greenhouse-Geisser epsilon (ε_GG):
- ε_GG ≥ .75: Use the Huynh-Feldt (HF) correction. The GG correction is overly conservative in this range, and HF provides a better balance of Type I error control and statistical power.
- ε_GG < .75: Use the Greenhouse-Geisser (GG) correction. The violation is serious enough that GG’s more conservative adjustment is warranted.
- Mauchly’s p > .05: Use the Sphericity Assumed row — no correction is needed.
Always report which correction was applied and the epsilon value, so readers can evaluate your analytic decision.
Post hoc tests are only appropriate after a significant omnibus within-subjects F-test. Running pairwise comparisons following a non-significant overall result inflates Type I error. The recommended approach is Bonferroni-corrected pairwise comparisons, which adjust the significance threshold for the number of comparisons made. In SPSS, these are requested under Estimated Marginal Means → Options → Bonferroni in the Repeated Measures dialog. SPSS reports each comparison as a mean difference, standard error, adjusted p-value, and 95% confidence interval, allowing you to determine not only whether each pair differs significantly but also the direction and magnitude of the difference.
Partial eta-squared is the effect size SPSS reports by default for repeated measures ANOVA:
\[\eta^2_p = \frac{SS_{\text{time}}}{SS_{\text{time}} + SS_{\text{error}}}\]
Note that \(SS_{\text{between subjects}}\) — often very large in human performance data — is excluded from the denominator. This is why η²_p is typically larger than full η² (which divides \(SS_{\text{time}}\) by total SS including between-subjects variance). η²_p answers the question “how large is the time effect relative to within-person error?” rather than “relative to everything?” Cohen’s (1988) benchmarks are commonly applied: small = .01, medium = .06, large = .14. Always label your effect size clearly as partial eta-squared to avoid misleading comparisons with full η² values from between-subjects studies.
Just as full η² overestimates the population effect in between-subjects ANOVA, partial eta-squared overestimates it in repeated measures designs, particularly with small samples. Partial omega-squared provides a less biased estimate:
\[\omega^2_p = \frac{(k - 1)(MS_{\text{time}} - MS_{\text{error}})}{(k - 1) \cdot MS_{\text{time}} + (n - k + 1) \cdot MS_{\text{error}}}\]
where \(k\) is the number of time points and \(n\) is the number of participants. SPSS does not compute ω²_p directly, but it can be calculated from the ANOVA source table values. Reporting ω²_p alongside η²_p is strongly recommended, especially in small-to-moderate samples, to give readers a realistic picture of the population effect.
A complete APA-style write-up must include:
- Mauchly’s test — W statistic, df, and p-value (to show sphericity was evaluated)
- Correction applied — name (GG or HF) and epsilon value, if sphericity was violated
- Omnibus F-statistic — with corrected df, exact p-value, η²_p, and ideally ω²_p
- Descriptive statistics — M and SD for each time point
- Bonferroni-corrected pairwise comparisons — mean differences, SEs, adjusted p-values, and 95% CIs
Example:
“A one-way repeated measures ANOVA was conducted to examine the effect of training time (pre, mid, post) on muscular strength. Mauchly’s test indicated that the sphericity assumption was not violated, W(2) = .932, p = .054. The within-subjects effect of time was statistically significant, F(2, 58) = 116.0, p < .001, η²_p = .80, ω²_p = .88. Post hoc Bonferroni-corrected pairwise comparisons indicated that strength increased significantly from pre- to mid-training (mean difference = 2.02 kg, p < .001, 95% CI [1.34, 2.70]), from mid- to post-training (mean difference = 3.36 kg, p < .001, 95% CI [2.22, 4.50]), and from pre- to post-training (mean difference = 5.38 kg, p < .001, 95% CI [4.54, 6.22]).”
- Ignoring Mauchly’s test: Always check sphericity before reading the F-table row. Using “Sphericity Assumed” when it is violated inflates Type I error.
- Confusing η²_p with full η²: Partial eta-squared excludes between-subjects variance from the denominator and will be larger than full η². Always label it as partial and do not compare it directly with η² values from between-subjects designs.
- Running post hoc tests after a non-significant omnibus F: This capitalizes on chance and inflates Type I error. Only run pairwise comparisons after a significant overall F.
- Confusing within-subjects error with between-subjects error: The error term in repeated measures ANOVA (MS_error = MS_subjects×time) reflects individual inconsistency in responses across time points — not total within-group variability. This distinction is critical for correctly interpreting the SPSS source table.
2.2 Test your Knowledge
Take this low-stakes quiz to test your knowledge of the material in this chapter. This quiz is for practice only and will help you identify areas where you may need additional review.
3 Participate
This section includes activities and discussions that will be completed during class time. Your active participation is essential for deepening your understanding of the material.
During class, we will: - Identify research scenarios appropriate for one-way repeated measures ANOVA versus between-subjects ANOVA or paired t-test - Interpret Mauchly’s test output and select the appropriate F-table row in SPSS - Apply the Greenhouse-Geisser or Huynh-Feldt correction based on epsilon values - Conduct and interpret Bonferroni-corrected pairwise comparisons following a significant omnibus F - Calculate partial eta-squared and partial omega-squared from SPSS source table values - Interpret line plots and spaghetti plots to visualize within-person trajectories - Practice writing a complete APA-style repeated measures ANOVA report
4 Perform
4.1 Apply Your Learning
Now that you’ve prepared, practiced, and participated, it’s time to demonstrate your mastery of the material through assignments and assessments.
I strongly encourage you to complete the previous “Ps” (Prepare, Practice, Participate) before attempting any assignments or assessments associated with this chapter.