Repeated Measures ANOVA#

The repeated measures ANOVA (rm-ANOVA) tests whether the means of three or more related measurements (e.g., at different time points) differ significantly. It is the extension of the paired t-test to more than two measurement points.

When to Use#

Use the rm-ANOVA when you want to:

Compare three or more related measurements from the same subjects
The dependent variable is metric (continuous)
The data are approximately normally distributed
The sphericity assumption is met (or corrected for)

Typical use cases:

Measurements at multiple time points (before, during, after)
Reaction times under different experimental conditions
Performance measurements across a semester

Assumptions#

Related measurements (the same subjects are measured multiple times)
Metric scale of the dependent variable
Normal distribution of the dependent variable at each measurement point
Sphericity (Mauchly's test) – if violated: Greenhouse-Geisser or Huynh-Feldt correction
No extreme outliers in the differences

Sphericity and Mauchly's Test#

Sphericity is a specific assumption of the rm-ANOVA. It requires that the variances of all pairwise differences between measurement points are equal. This assumption is tested using Mauchly's test:

If Mauchly's test is not significant ( $p > .05$ ): Sphericity can be assumed.
If Mauchly's test is significant ( $p \leq .05$ $p \leq .05$ ): Sphericity is violated. In this case, corrected F-values should be used:
- Greenhouse-Geisser correction ( $\varepsilon_{GG}$ ): More conservative, recommended when $\varepsilon < 0.75$
- Huynh-Feldt correction ( $\varepsilon_{HF}$ ): Less conservative, recommended when $\varepsilon \geq 0.75$

Note: For severe violations of the assumptions, the Friedman test is the appropriate nonparametric alternative.

Formula#

The test statistic of the rm-ANOVA:

F = \frac{MS_{condition}}{MS_{error}}

The sums of squares are decomposed into:

SS_{total} = SS_{condition} + SS_{subjects} + SS_{error}

where:

MS_{condition} = \frac{SS_{condition}}{k - 1}

MS_{error} = \frac{SS_{error}}{(k - 1)(n - 1)}

with:

$k$ = number of measurement points/conditions
$n$ = number of subjects

The degrees of freedom are $df_1 = k - 1$ and $df_2 = (k - 1)(n - 1)$ .

When sphericity is violated, the degrees of freedom are multiplied by the correction factor $\varepsilon$ :

df_1^{corr} = \varepsilon \cdot (k - 1), \quad df_2^{corr} = \varepsilon \cdot (k - 1)(n - 1)

Example#

Practical Example: Stress Levels During Therapy

A psychologist investigates whether patients' stress levels change over the course of therapy. She measures stress levels (using a standardized questionnaire) in 40 patients at four time points:

T1: Before therapy begins
T2: After 4 weeks
T3: After 8 weeks
T4: After 12 weeks (end of therapy)

Since the same patients are measured at all time points, these are repeated measures data. The rm-ANOVA tests whether the mean stress level changes significantly across the four time points.

Mauchly's test: Check sphericity ( $p = .02$ → sphericity violated)
Apply Greenhouse-Geisser correction ( $\varepsilon_{GG} = 0.68$ )
If significant: Pairwise comparisons with Bonferroni correction

Effect Size#

Partial eta-squared ( $\eta_p^2$ ) as a measure of effect size:

\eta_p^2 = \frac{SS_{condition}}{SS_{condition} + SS_{error}}

Effect Size	η²_p
Small	0.01
Medium	0.06
Large	0.14

Tip: When the omnibus test is significant, post-hoc comparisons (e.g., with Bonferroni correction) are required to determine which measurement points differ significantly from each other.