Module S06: Repeated Measures ANOVA - Statistics Visualizer

01

The Within-Subjects Design

Same People, Multiple Measures

S1

Subject 1

Time 1

Time 2

Time 3

Each subject is measured at ALL time points

Repeated measures ANOVA is used when the SAME subjects are measured multiple times - at different time points, under different conditions, or with different stimuli. It's the extension of the paired t-test to 3+ conditions.

Same people, multiple conditions

Within vs Between Subjects

Between-Subjects (One-Way)

Different people in each group
Group A: S1, S2, S3
Group B: S4, S5, S6
Group C: S7, S8, S9

Within-Subjects (RM)

Same people in all conditions
Cond A: S1, S2, S3
Cond B: S1, S2, S3
Cond C: S1, S2, S3

Between-subjects: different people in each group, individual differences add noise. Within-subjects: same people in all conditions, individual differences are controlled for.

Within-subjects controls individual differences

Advantages

More Statistical Power

Individual differences removed from error term → smaller error variance → larger F-values → easier to detect effects

Fewer Participants Needed

Each person contributes data to all conditions. 20 subjects × 3 conditions = 60 data points, but only 20 people to recruit.

Controls for Individual Differences

Fast vs slow responders, high vs low baseline - these differences don't contaminate the comparison.

Repeated measures designs are more powerful because between-subject variability is removed. The same person serves as their own control, so you're comparing apples to apples.

Much more powerful than between-subjects

02

Sphericity Assumption

The Key Assumption

Sphericity

The variances of the differences between all pairs of conditions should be equal.

In simpler terms: If you compute the difference scores between each pair of conditions, those difference scores should have similar variability.

Example with 3 time points:
• Var(Time1 - Time2) ≈ Var(Time1 - Time3) ≈ Var(Time2 - Time3)

Sphericity is like homogeneity of variance for repeated measures. It's automatically satisfied with 2 conditions (like paired t-test) but must be checked with 3+ conditions.

Equal variance of difference scores

Mauchly's Test

Mauchly's Test of Sphericity

H₀: Sphericity is met
If p < .05: Sphericity violated → use corrections
If p ≥ .05: Sphericity OK (or underpowered)

Example output:
Mauchly's W = .72, χ²(2) = 8.4, p = .015

Since p < .05, sphericity is violated. Use corrected F-test.

Mauchly's test checks sphericity. If significant (p < .05), sphericity is violated and you need to use corrected degrees of freedom for the F-test.

Significant Mauchly's test = violation

What to Do When Violated

Greenhouse-Geisser (ε_GG)

More conservative. Use when ε < .75 or when in doubt. Multiply df by ε to get corrected df.

Huynh-Feldt (ε_HF)

Less conservative. Use when ε ≥ .75. Slightly more powerful than G-G when violation is mild.

Epsilon (ε)

Ranges from 1/(k-1) to 1.0. ε = 1 means perfect sphericity. Lower ε = worse violation. SPSS/R report this automatically.

Epsilon (ε) quantifies how badly sphericity is violated. The corrections reduce degrees of freedom, making the test more conservative and honest about uncertainty.

Use G-G when ε < .75, H-F when ε ≥ .75

03

Reading the Output

Typical Output

Effect of Time on Test Scores (n=20, 3 time points)

Source	SS	df	MS	F	p
Time	400	2	200	10.0	.001
Error(Time)	760	38	20	-	-

Mauchly's W = .65, p = .02 → use correction
G-G ε = .74 → F(1.48, 28.1) = 10.0, p = .002

The output shows the within-subjects factor (Time). Notice Error(Time) - this is the error term specific to the repeated factor. When sphericity is violated, use corrected df.

df changes with G-G/H-F correction

Follow-Up Comparisons

After significant F, which conditions differ?

Paired t-tests with Bonferroni correction:
• Time 1 vs Time 2: p = .15
• Time 1 vs Time 3: p = .001*
• Time 2 vs Time 3: p = .03*

* significant at Bonferroni-adjusted α = .05/3 = .017

After finding a significant overall effect, run pairwise comparisons to see which specific conditions differ. Use Bonferroni or similar correction for multiple comparisons.

Bonferroni corrects for multiple comparisons

APA Style

Example Write-Up:

"A one-way repeated measures ANOVA revealed a significant effect of time on test scores, F(1.48, 28.1) = 10.0, p = .002, η²_p = .34. Greenhouse-Geisser correction was applied due to violation of sphericity (Mauchly's W = .65, p = .02). Bonferroni-corrected pairwise comparisons showed significant improvement from Time 1 to Time 3 (p = .001) and from Time 2 to Time 3 (p = .03), but not from Time 1 to Time 2 (p = .15)."

Report: the design, corrected df if applicable, F-value, p-value, effect size (partial η²), and pairwise comparison results.

Report sphericity test and any corrections used

04

Mixed Designs

Between + Within

Mixed ANOVA

At least one between-subjects factor AND at least one within-subjects factor

Example:
Between: Treatment group (Drug vs Placebo)
Within: Time (Pre, Post, Follow-up)

Tests: Main effect of Group, Main effect of Time, Group × Time interaction

Mixed designs combine between-subjects and within-subjects factors. The interaction is often key: does the pattern over time differ between groups?

Tests both between and within factors

Treatment Over Time

Significant Group × Time interaction: Drug group improves more over time

The interaction shows that the drug group improves over time while the placebo group stays flat. This Group × Time interaction is often the primary hypothesis in treatment studies.

Interaction = different patterns over time

★

Real-World Examples

Apply your knowledge to practical scenarios

01

Learning Across Practice Sessions

20 participants complete a task at 4 time points: baseline, after 1 week, 2 weeks, and 4 weeks of practice. Outcome is accuracy (%).

Q Why is this a repeated measures design? +

A The SAME 20 participants are measured at ALL 4 time points. Each person contributes 4 data points. This controls for individual differences in baseline ability.

Q Mauchly's test gives W = .45, p = .003. What should you do? +

A Sphericity is violated (p < .05). Use Greenhouse-Geisser or Huynh-Feldt corrected F-test. If ε < .75, use G-G; if ε ≥ .75, use H-F.

Q The G-G corrected F(1.89, 35.9) = 15.2, p < .001. What can you conclude? +

A There's a significant effect of practice time on accuracy. Performance differs across the 4 time points. Follow-up pairwise comparisons will show which specific time points differ.

Key Insights

Repeated measures controls for individual differences
Always check sphericity with 3+ levels
Report corrected df when sphericity is violated

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Repeated Measures ANOVA Quiz

Test your understanding • Passing score: 70%

Repeated Measures ANOVA

Within-Subjects Design

Same Subjects

Reduced Error