Repeated-Measures ANOVA

Within-subjects analysis of variance with sphericity checks and corrections

Introduction

Repeated-measures ANOVA (rmANOVA) handles designs where the same subjects are measured at multiple time points or conditions. By treating subject as a blocking factor, it controls between-subject variability and gains power over between-subjects analysis. The main complication is the sphericity assumption on the covariance of the repeated measures.

Prerequisites

One-way ANOVA, dependence structure in repeated measures.

Theory

For one within-subjects factor with \(k\) levels measured on \(n\) subjects, the model is

\[Y_{ij} = \mu + \alpha_i + \pi_j + \varepsilon_{ij},\]

where \(\pi_j\) is a random subject effect and \(\alpha_i\) is the fixed effect of level \(i\).

Sphericity assumes the variances of all pairwise differences between repeated measurements are equal. Mauchly’s test assesses this assumption.

If sphericity is violated:

• Greenhouse-Geisser correction: adjusts degrees of freedom by multiplying by \(\hat{\varepsilon}_{\text{GG}} \in [1/(k-1), 1]\).
• Huynh-Feldt correction: less conservative for mild violations.

Modern alternative: fit a linear mixed-effects model (lmer) with an unstructured or compound-symmetric covariance – more flexible, handles missing data.

Assumptions

• Normality of within-subject residuals.
• Sphericity (or apply a correction).
• Subjects are independent; measurements within a subject are not.

R Implementation

library(afex); library(emmeans); library(rstatix)
set.seed(2026)

# 20 subjects measured at 4 time points
n_subj <- 20
times  <- 4
subject <- rep(1:n_subj, each = times)
time    <- rep(1:times, n_subj)

# Simulate with a time effect and subject-specific intercepts
df <- data.frame(
  subject = factor(subject),
  time    = factor(time),
  y       = rnorm(n_subj * times, mean = 50 + 2 * time, sd = 6) +
            rep(rnorm(n_subj, 0, 4), each = times)
)

# Using afex::aov_car
fit <- aov_car(y ~ time + Error(subject/time), data = df)
fit

# rstatix pipeline
df |> anova_test(dv = y, wid = subject, within = time)

# Post-hoc with Bonferroni
emm <- emmeans(fit, ~ time)
pairs(emm, adjust = "bonferroni")

Output & Results

Anova Table (Type 3 tests)

Response: y
   Effect         df  MSE       F  ges  p.value
1    time  2.61, 49.5 28.7  12.39  0.19  <.001 ***

Mauchly Tests for Sphericity:
 Effect   W       p
  time  0.87  0.78   # sphericity holds

Greenhouse-Geisser eps: 0.87

Large within-subject effect of time (F(2.61, 49.5) = 12.39, p < 0.001, generalised \(\eta^2 = 0.19\)); sphericity holds so no correction needed.

Interpretation

“A one-way repeated-measures ANOVA revealed a significant effect of time (F(2.61, 49.5) = 12.39, p < 0.001, generalised \(\eta^2 = 0.19\); Greenhouse-Geisser correction applied). Bonferroni post-hoc comparisons identified significant increases from time 1 to time 3 and from time 2 to time 4.”

Practical Tips

• Check Mauchly’s test; when rejected, apply Greenhouse-Geisser or Huynh-Feldt.
• For complex designs (multiple within factors, between x within interactions), a linear mixed model is preferred.
• Missing data in rmANOVA causes the entire subject to be dropped (listwise); mixed models handle it gracefully.
• Report generalised \(\eta^2\) for repeated-measures designs (partial \(\eta^2\) is upwardly biased).
• For non-normal residuals, use Friedman or an aligned-rank transform (ARTool::art).