Repeated-Measures Analysis of Variance

Overview

Repeated-measures ANOVA compares a numerical outcome measured on the same participants at three or more occasions or conditions.

The method separates:

• stable between-participant differences;
• systematic differences across time or condition;
• residual within-participant variation.

Because measurements from one participant are correlated, ordinary one-way ANOVA is inappropriate.

This lecture presents the classical balanced one-factor repeated-measures ANOVA. Modern longitudinal studies often use linear mixed-effects models or generalised estimating equations because they handle incomplete data, unequal timing and flexible covariance structures more naturally [4], [5].

Why this matters

If 24 participants are measured at three visits, the dataset contains 72 rows but only 24 independent participants.

Treating all 72 observations as independent usually understates uncertainty.

Running three paired tests also fragments the research question and multiplies testing. Repeated-measures ANOVA provides one overall time test, followed by adjusted comparisons if required.

When to use this method

Use one-factor repeated-measures ANOVA when:

• one numerical outcome is measured at three or more occasions or conditions;
• the same independent participants contribute at every level;
• the design is balanced and complete or nearly so;
• the mean time or condition profile is the target;
• and covariance assumptions are sufficiently reasonable.

Prefer a mixed-effects model when:

• visits are missing;
• timing varies;
• participants have different numbers of observations;
• time is continuous;
• covariate-dependent trajectories are required;
• or a flexible covariance structure is needed.

Research question and variables

The outcome is systolic blood pressure.

The within-participant factor is time with three levels:

• baseline;
• month 3;
• month 6.

The estimand is the set of population mean differences across occasions.

For a study with treatment groups and time, a one-factor repeated-measures ANOVA is not enough. The primary treatment question usually concerns a group-by-time interaction or a prespecified time-specific contrast.

Data structure

Long repeated-measures structure

simulated data

participant_id	time	sbp_mmHg
P001	Baseline	153
P001	Month 3	148
P001	Month 6	153
P002	Baseline	166
P002	Month 3	155
P002	Month 6	160

Note: The complete dataset contains 72 rows from 24 participants.

Participant identifier and time must uniquely identify each record.

Statistical assumptions

Sphericity

Sphericity requires equal variances of all pairwise difference scores among the repeated levels.

With exactly two levels, sphericity is automatic. With three or more, violation can make the uncorrected F test too liberal.

The Greenhouse–Geisser correction multiplies numerator and denominator degrees of freedom by an estimated epsilon between $1/(k-1)$ and 1 [1].

The F value remains unchanged, but the corrected p-value is larger.

How the method works

The repeated-measures ANOVA partitions variation into:

• time;
• participants;
• time-by-participant residual.

For the Greenhouse–Geisser correction:

$$df_1^*=\epsilon df_1$$ $$df_2^*=\epsilon df_2$$

Worked example

Reproducible R analysis

Base R can represent the participant and participant-by-time error strata using aov() with an Error() term [3].

wide <- data.frame(
  participant_id = factor(sprintf("P%03d", 1:24)),
  Baseline = c(
    153,166,154,160,157,157,136,162,156,137,143,146,
    169,165,149,144,140,140,143,144,155,148,139,155
  ),
  `Month 3` = c(
    148,155,146,159,152,151,127,150,149,136,141,146,
    174,162,147,137,136,137,134,142,147,143,131,151
  ),
  `Month 6` = c(
    153,160,145,139,149,141,137,150,139,122,135,124,
    158,157,143,132,124,136,136,130,146,137,133,139
  ),
  check.names = FALSE
)

long <- reshape(
  wide,
  varying = c("Baseline", "Month 3", "Month 6"),
  v.names = "sbp_mmHg",
  timevar = "time",
  times = c("Baseline", "Month 3", "Month 6"),
  direction = "long"
)

long$time <- factor(
  long$time,
  levels = c("Baseline", "Month 3", "Month 6")
)
long <- long[order(long$participant_id, long$time), ]

# Base R represents participant and participant-by-time error strata [3]
fit <- aov(
  sbp_mmHg ~ time + Error(participant_id / time),
  data = long
)

summary(fit)

R ANOVA output

Error: participant_id
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals 23

Error: participant_id:time
          Df Sum Sq Mean Sq F value   Pr(>F)
time       2 1336.0   668.0   34.84 6.14e-10
Residuals 46  882.0    19.2

Greenhouse–Geisser epsilon in base R

S <- cov(wide[c("Baseline", "Month 3", "Month 6")])
k <- ncol(S)
Cmat <- diag(k) - matrix(1 / k, k, k)
S_centered <- Cmat %*% S %*% Cmat

epsilon_gg <- sum(diag(S_centered))^2 /
  ((k - 1) * sum(diag(S_centered %*% S_centered)))

F_time <- 34.8408
df1 <- 2
df2 <- 46

p_gg <- pf(
  F_time,
  df1 = epsilon_gg * df1,
  df2 = epsilon_gg * df2,
  lower.tail = FALSE
)

data.frame(
  epsilon_GG = epsilon_gg,
  corrected_df1 = epsilon_gg * df1,
  corrected_df2 = epsilon_gg * df2,
  F = F_time,
  p_value_GG = p_gg
)

Corrected output

 epsilon_GG corrected_df1 corrected_df2       F   p_value_GG
      0.631         1.262        29.017 34.8408 4.77e-07

Pairwise comparisons

p1 <- t.test(wide[["Month 3"]], wide$Baseline, paired = TRUE)
p2 <- t.test(wide[["Month 6"]], wide$Baseline, paired = TRUE)
p3 <- t.test(wide[["Month 6"]], wide[["Month 3"]], paired = TRUE)

raw_p <- c(p1$p.value, p2$p.value, p3$p.value)
holm_p <- p.adjust(raw_p, method = "holm")

pairwise_result <- data.frame(
  contrast = c(
    "Month 3 - Baseline",
    "Month 6 - Baseline",
    "Month 6 - Month 3"
  ),
  mean_difference = c(
    mean(wide[["Month 3"]] - wide$Baseline),
    mean(wide[["Month 6"]] - wide$Baseline),
    mean(wide[["Month 6"]] - wide[["Month 3"]])
  ),
  ci_lower = c(p1$conf.int[1], p2$conf.int[1], p3$conf.int[1]),
  ci_upper = c(p1$conf.int[2], p2$conf.int[2], p3$conf.int[2]),
  holm_p = holm_p
)

pairwise_result

Interpreting the output

Presenting the results

Greenhouse–Geisser-corrected time effect: $F(1.26,29.02)=34.84$, $p<0.001$, $\epsilon=0.63$.

Visualising the results

Suggested reporting

Why mixed-effects models are often preferable MMed extension

Classical repeated-measures ANOVA generally requires complete balanced data and a restrictive covariance assumption.

A linear mixed-effects model can:

• include participants with incomplete visits under stated missing-data assumptions;
• model unequal visit timing;
• include random effects;
• compare covariance structures;
• and estimate treatment-by-time contrasts.

The model should be chosen according to the estimand and study design rather than software convenience.

Common mistakes

Knowledge check

Why are 72 rows not 72 independent observations?

Answer and reasoning

They arise from 24 participants, each measured three times. Measurements within a participant are correlated.

What does sphericity concern?

Answer and reasoning

It concerns equality of variances of all pairwise difference scores among repeated levels.

What does the Greenhouse–Geisser correction change?

Answer and reasoning

It reduces the numerator and denominator degrees of freedom by epsilon, producing a more conservative p-value when sphericity is doubtful. The F value remains unchanged.

Why are pairwise visit comparisons adjusted?

Answer and reasoning

Three comparisons form a family. Holm adjustment limits family-wise type I error while identifying where differences occur.

When should a mixed model replace repeated-measures ANOVA?

Answer and reasoning

Common reasons include incomplete visits, unequal timing, flexible covariance needs, treatment-by-time modelling or participant-specific random effects.

Key take-home messages

References

1. Greenhouse SW, Geisser S. On methods in the analysis of profile data. Psychometrika. 1959;24:95–112. https://doi.org/10.1007/BF02289823
2. Geisser S, Greenhouse SW. An extension of Box’s results on the use of the F distribution in multivariate analysis. Ann Math Stat. 1958;29(3):885–891. https://doi.org/10.1214/aoms/1177706545
3. R Core Team. Fit an analysis of variance model. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/aov.html
4. Gueorguieva R, Krystal JH. Move over ANOVA: progress in analyzing repeated-measures data and its reflection in papers published in the Archives of General Psychiatry. Arch Gen Psychiatry. 2004;61(3):310–317. https://doi.org/10.1001/archpsyc.61.3.310
5. Twisk JWR. Applied longitudinal data analysis for epidemiology. 2nd ed. Cambridge: Cambridge University Press; 2013. https://doi.org/10.1017/CBO9781139342834