Repeated Measures ANOVA

Definition

Core Statement

Repeated Measures ANOVA is used to compare means when the same subjects are measured multiple times (e.g., pre-test, post-test, follow-up). It accounts for the correlation between measurements from the same individual, increasing statistical power.

Purpose

Test if means differ across repeated measurements on the same subjects.
Account for within-subject correlation (violates independence assumption of standard ANOVA).
More powerful than between-subjects designs (subjects serve as their own controls).

When to Use

Use Repeated Measures ANOVA When...

Same subjects measured at multiple time points or conditions.
You have a continuous dependent variable.
Sphericity assumption is met (or corrected).

Alternatives

Two time points only: Use paired Student's T-Test.
Sphericity violated: Use Greenhouse-Geisser correction or MANOVA.
Non-normal data: Use Friedman Test.

Theoretical Background

The Model

Similar to One-Way ANOVA, but includes a subject effect:

Y_{i j} = μ + α_{i} + π_{j} + ε_{i j}

Term	Meaning
$μ$	Grand mean
$α_{i}$	Effect of time/condition $i$
$π_{j}$	Subject effect (individual differences)
$ε_{i j}$	Random error

Sphericity Assumption

Critical Assumption

Sphericity means the variances of differences between all pairs of repeated measures are equal.

Test: Mauchly's Test of Sphericity.

If violated ( $p < 0.05$ ), use Greenhouse-Geisser or Huynh-Feldt correction.

Why Not Standard ANOVA?

Standard ANOVA assumes independence. Repeated measures from the same person are correlated, violating this. RM-ANOVA accounts for this correlation.

Assumptions

Continuous dependent variable.
Normality of residuals (or differences).
Sphericity: Variances of all pairwise differences are equal. (Test with Mauchly's Test).
No missing data (or handle appropriately with mixed models).

Limitations

Pitfalls

Sphericity violations are common. Always check and apply corrections if needed.
Missing data is problematic. RM-ANOVA requires complete data for all time points. Use Linear Mixed Models (LMM) for flexibility.
Carryover effects: If conditions are sequential, earlier conditions may influence later ones.

Python Implementation

import pandas as pd
from statsmodels.stats.anova import AnovaRM

# Example: Pain Scores at 3 Time Points
data = pd.DataFrame({
    'Subject': [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4],
    'Time': ['Pre', 'Mid', 'Post'] * 4,
    'Pain': [8, 6, 4, 7, 5, 3, 9, 7, 5, 6, 4, 2]
})

# Repeated Measures ANOVA
rm_anova = AnovaRM(data, depvar='Pain', subject='Subject', within=['Time'])
result = rm_anova.fit()
print(result.summary())

R Implementation

# Example Data (Wide Format)
df <- data.frame(
  Subject = 1:4,
  Pre = c(8, 7, 9, 6),
  Mid = c(6, 5, 7, 4),
  Post = c(4, 3, 5, 2)
)

# Convert to Long Format
library(tidyr)
df_long <- pivot_longer(df, cols = c(Pre, Mid, Post), 
                        names_to = "Time", values_to = "Pain")

# Repeated Measures ANOVA
library(rstatix)
res.aov <- anova_test(data = df_long, dv = Pain, wid = Subject, within = Time)
get_anova_table(res.aov)

# Check Sphericity (Mauchly's Test)
# If p < 0.05, apply Greenhouse-Geisser correction

# Alternative: ezANOVA
library(ez)
ezANOVA(data = df_long, dv = Pain, wid = Subject, within = Time, detailed = TRUE)

Interpretation Guide

Output	Interpretation
F = 12.5, p = 0.002	Time has a significant effect on Pain.
Mauchly's p < 0.05	Sphericity violated. Use corrected results (Greenhouse-Geisser).
Greenhouse-Geisser ε = 0.75	Moderate violation; correction applied.

One-Way ANOVA - Between-subjects version.
Mixed ANOVA (Between-Within) - Combines RM and between factors.
Friedman Test - Non-parametric alternative.
Linear Mixed Models (LMM) - More flexible; handles missing data.
Mauchly's Test of Sphericity