What Is Repeated Measures ANOVA?
Repeated measures analysis of variance (ANOVA) tests whether the mean of a continuous outcome differs across three or more related conditions or time points. The same participants are measured repeatedly, so observations within a participant are correlated rather than independent.
Typical examples include measuring blood pressure before treatment, after four weeks, and after eight weeks; comparing reaction time under three experimental conditions; or recording the same students’ scores at several stages of a course.
When Should You Use It?
Use a one-way repeated measures ANOVA when:
- One categorical within-subject factor has three or more levels.
- The dependent variable is continuous.
- The same participants contribute a measurement at every level.
- You want to test an overall difference among the condition means.
For only two related measurements, a paired-samples t-test is sufficient. For designs containing both within-subject and between-subject factors, use a mixed ANOVA or a linear mixed-effects model.
Research Question and Hypotheses
Suppose a researcher measures concentration scores for the same participants under no music, instrumental music, and music with lyrics.
Null hypothesis (H0): all population means are equal.
μ1 = μ2 = μ3
Alternative hypothesis (H1): at least one condition mean differs.
The omnibus ANOVA indicates whether a difference exists somewhere, but it does not identify which pairs differ. Pairwise follow-up tests are needed for that question.
Key Assumptions
1. Continuous Outcome
The dependent variable should be measured on an interval or ratio scale.
2. Related Observations
Measurements must come from the same participants, matched units, or another genuinely linked design. Different participants in each group require an independent-groups ANOVA instead.
3. No Serious Outliers
Extreme values within a condition or extreme participant change patterns can distort means and variability. Inspect plots and verify unusual values before deciding how to handle them.
4. Approximate Normality
For each pair of conditions, the distribution of difference scores should be reasonably normal. ANOVA is often robust to modest departures in balanced data, but small samples require greater caution.
5. Sphericity
Sphericity means that the variances of all pairwise difference scores are equal. It is relevant when the within-subject factor has at least three levels. Mauchly’s test is commonly reported, although graphical assessment and subject-matter judgment also matter.
If sphericity is violated, apply a correction such as Greenhouse-Geisser or Huynh-Feldt. The correction reduces the degrees of freedom and produces a more conservative p-value; it does not change the calculated F statistic.
How the Test Separates Variability
Repeated measures ANOVA distinguishes systematic differences among conditions from participant-to-participant differences and residual error. Accounting for stable individual differences can provide more statistical power than treating repeated observations as independent.
The test statistic is:
F = MScondition / MSerror
Here, MS denotes a mean square: a sum of squares divided by its degrees of freedom. A large F value indicates that condition-related variability is large relative to unexplained within-participant variability.
Worked Example
Eight participants complete a task under three conditions. Their scores are:
| Participant | No Music | Instrumental | Lyrics |
|---|---|---|---|
| 1 | 72 | 80 | 69 |
| 2 | 68 | 75 | 65 |
| 3 | 75 | 82 | 71 |
| 4 | 70 | 78 | 67 |
| 5 | 74 | 81 | 70 |
| 6 | 69 | 76 | 66 |
| 7 | 77 | 84 | 73 |
| 8 | 71 | 79 | 68 |
The condition means are 72.00 for no music, 79.38 for instrumental music, and 68.63 for music with lyrics. The descriptive pattern suggests the highest concentration under instrumental music, but inferential testing is required before generalizing beyond this sample.
Analysis Workflow
- Confirm that participant identifiers correctly link repeated observations.
- Inspect descriptive statistics and individual trajectory plots.
- Check unusual observations and the plausibility of normal difference scores.
- Run the repeated measures ANOVA.
- Assess sphericity and use corrected results when necessary.
- If the omnibus test is significant, perform planned contrasts or adjusted pairwise comparisons.
- Report an effect size and confidence intervals where available.
Post Hoc Comparisons
Testing every pair without adjustment increases the probability of at least one false-positive result. Common procedures include Bonferroni-adjusted paired comparisons and Holm’s sequential correction. Planned contrasts specified before examining the results may answer focused questions more efficiently.
For the example, useful comparisons might be instrumental music versus no music, instrumental music versus lyrics, and no music versus lyrics.
Effect Size
A p-value addresses compatibility with the null hypothesis, not practical importance. Partial eta squared is frequently reported:
η2p = SScondition / (SScondition + SSerror)
Interpret effect size in the research context rather than relying only on universal labels. Generalized eta squared can be useful when comparing effects across different designs.
How to Report the Result
A concise report should include condition means and variability, the F statistic, degrees of freedom, p-value, effect size, the sphericity decision, and adjusted follow-up comparisons.
Concentration scores differed across the three listening conditions, F(corrected df) = value, p = value, partial η2 = value. Greenhouse-Geisser-corrected results were used because the sphericity assumption was not met. Holm-adjusted comparisons indicated that instrumental music produced higher scores than both other conditions.
Replace the placeholders with the actual software output. Do not report an uncorrected p-value when a correction is required.
Common Mistakes
- Analyzing repeated measurements as if they came from independent groups.
- Reporting only Mauchly’s test and ignoring the corrected ANOVA table.
- Running multiple paired t-tests without controlling multiplicity.
- Concluding that every condition differs after a significant omnibus test.
- Ignoring missing observations, which may cause complete-case deletion in traditional ANOVA.
- Interpreting statistical significance as proof of a large or useful effect.
Alternatives
Use the Friedman test when an ordinal outcome or severe distributional problems make a rank-based approach more appropriate. Consider a linear mixed-effects model when observations are missing, measurement times are unequal, the correlation structure is important, or participant-specific trajectories are of interest.
Summary
Repeated measures ANOVA compares three or more related means while accounting for the dependence created by measuring the same units repeatedly. A sound analysis checks the design and assumptions, applies a sphericity correction when needed, follows a significant omnibus test with multiplicity-controlled comparisons, and reports effect size alongside statistical significance.