Chapter 15: Repeated Measures ANOVA
Interactive Self-Study Guide — KIN 610
Ovande Furtado Jr., PhD. 
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- Fully expand and interact with the Decision Trees and Flashcards when you encounter them.
- Use the APA Builder to practice writing your results.
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Start here: complete the pre-check, then work through each section.
1. Why Use Repeated Measures?
Between-subjects vs. within-subjects
| Feature | Between-Subjects | Repeated Measures |
|---|---|---|
| Each participant measured… | Once (one group only) | Multiple times (all conditions) |
| Individual differences… | Stay inside the error term | Removed from the error term |
| Required sample size | Larger | Smaller |
| Statistical power | Lower | Higher |
| Independence assumption applies to… | Every observation | Participants (not repeated observations) |
The core intuition
Imagine measuring grip strength at pre, mid, and post training.
- Participant A (athlete) starts at 48 kg and ends at 55 kg — a 7 kg gain.
- Participant B (older adult) starts at 21 kg and ends at 28 kg — also a 7 kg gain.
A between-subjects design treats the 27 kg baseline gap between A and B as error — noise that obscures the treatment effect. A repeated-measures design asks: relative to each person’s own baseline, how much did they change? The 27 kg gap disappears; only the consistent ~7 kg gain is analysed.
Result: smaller error → larger F-statistic → greater power to detect the time effect.
2. Partitioning the Variance
In a one-way repeated measures ANOVA, total variance splits into three parts:
\[SS_{\text{total}} = SS_{\text{between subjects}} + SS_{\text{time}} + SS_{\text{error}}\]
Stable individual differences
Carved out of error
Treatment variance
F numerator
Subject × Time interaction
F denominator
| Source | What it captures | Role in the F-ratio |
|---|---|---|
| \(SS_{\text{between subjects}}\) | Consistent individual baselines (some people always score higher) | Removed — not in the denominator |
| \(SS_{\text{time}}\) | How group means change across time points | Numerator (\(MS_{\text{time}}\)) |
| \(SS_{\text{error}}\) | How individually inconsistent each participant’s changes are | Denominator (\(MS_{\text{error}}\)) |
3. The Three Assumptions
Before trusting your SPSS output, verify these three conditions:
1. Independence of participants
Each person in the study must be independent of every other person — no related pairs, no athletes on the same team influencing each other. Note: the repeated observations within the same person are intentionally correlated; that is the point of the design.
2. Normality of difference scores
Compute the pairwise differences between every pair of time points (mid − pre, post − pre, post − mid) and verify those difference variables are approximately normally distributed using Shapiro-Wilk. With N ≥ 30, the Central Limit Theorem makes the test fairly robust to moderate non-normality.
3. Sphericity
The variances of all pairwise difference scores must be approximately equal:
\[\text{Var}(\text{mid} - \text{pre}) \approx \text{Var}(\text{post} - \text{pre}) \approx \text{Var}(\text{post} - \text{mid})\]
Tested automatically by SPSS with Mauchly’s W (covered in Section 4).
These are different assumptions. Homogeneity of variance (Levene’s test) compares raw-score variances across groups in between-subjects ANOVA. Sphericity compares variances of pairwise differences within a repeated-measures design. A dataset can have unequal SDs at pre, mid, and post and still satisfy sphericity if individual patterns of change are consistent.
4. Sphericity & the Correction Decision
When sphericity is violated, the standard degrees of freedom are too large, making the F-test anti-conservative — you would claim significance more often than warranted. SPSS offers corrected rows that shrink df by multiplying by an epsilon (ε) estimate:
\[df_{\text{corrected}} = df \times \hat{\varepsilon}\]
The F-statistic itself does not change; only the df used to look up the p-value changes.
⚡ Interactive Decision Tree
Click each button to navigate the decision — just as you would with your SPSS output:
Epsilon (ε) quick reference
| ε value | Interpretation | Action in SPSS |
|---|---|---|
| ε = 1.0 | Perfect sphericity | Sphericity Assumed row |
| .75 ≤ ε < 1.0 | Mild violation | Huynh-Feldt row |
| ε < .75 | Serious violation | Greenhouse-Geisser row |
| ε = 1/(k−1) | Maximum possible violation | Greenhouse-Geisser row |
5. Post Hoc Tests & Effect Size
When to run pairwise comparisons
A significant omnibus F tells you that somewhere across the time points there is a meaningful change — but not which pairs differ. Follow up with Bonferroni-corrected pairwise comparisons in SPSS (Analyze → General Linear Model → Repeated Measures → Options → Compare main effects).
Never run post hoc pairwise comparisons when the omnibus F is not significant. Doing so capitalises on chance and dramatically inflates family-wise Type I error (false positives).
Partial eta-squared (η²_p)
SPSS reports partial η² by default:
\[\eta^2_p = \frac{SS_{\text{time}}}{SS_{\text{time}} + SS_{\text{error}}}\]
Note that \(SS_{\text{between subjects}}\) is excluded from the denominator. This makes η²_p typically larger than full η² (which uses \(SS_{\text{total}}\) in the denominator). Always label it as partial in your write-up.
| η²_p value | Conventional interpretation |
|---|---|
| .01 | Small effect |
| .06 | Medium effect |
| .14 | Large effect |
6. Power Dynamics: Sample Size & G*Power
Repeated measures designs require substantially fewer participants to achieve the same statistical power as between-subjects designs.
Why? Because the denominator for the \(F\)-ratio — the error term — specifically excludes stable, between-subjects variance (individual baseline differences). By mathematically filtering out this “noise,” the “signal” (the treatment effect) becomes much easier to detect. Consequently, when planning your target sample size in software like **G*Power**, you will almost always find that repeated measures ANOVA demands a smaller \(N\) than a one-way ANOVA for the same effect size and alpha level.
7. Pitfall Prevention
When running and reporting repeated measures ANOVA, researchers frequently stumble into a few consistent statistical traps. Avoid these four specific pitfalls:
- Ignoring Mauchly’s test and deferring blindly to “Sphericity Assumed.” Always assess the assumption first; failing to verify sphericity actively inflates your Type I error rate.
- Treating partial \(\eta^2_p\) as though it were full variance (\(\eta^2\)). Remember that \(\eta^2_p\) only evaluates the effect size relative to its specific partition of error, which systematically overstates the variance compared to the overall total pie.
- Running post hoc tests after a non-significant omnibus \(F\). Meaningless fishing expeditions into non-significant data exponentially increase the likelihood of false positives.
- Confusing within-subjects error with between-subjects variance. Between-subjects variance refers to stable baseline differences across individuals (which RM-ANOVA removes), not the variance driving the actual within-subjects error term.
8. Key Terms — Flash Review
Click each card to flip it and see the definition.
9. APA Reporting Practice
The template
✍️ Interactive APA Builder
Fill in your study’s values and click Generate to see a complete APA paragraph:
Your study values
Full worked example (textbook data)
“A one-way repeated measures ANOVA was conducted to examine the effect of training time (pre, mid, post) on muscular strength (N = 30). Mauchly’s test of sphericity 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 (M difference = 2.02 kg, p < .001, 95% CI [1.34, 2.70]), from mid- to post-training (M difference = 3.36 kg, p < .001, 95% CI [2.22, 4.50]), and from pre- to post-training (M difference = 5.38 kg, p < .001, 95% CI [4.54, 6.22]).”
10. Final Review & Your Score
Two final questions, then reveal your score.