Repeated-measures ANOVA
The same people measured at three time points is not three paired t tests. Those comparisons share subjects, so they are not independent, and the resulting test carries a second assumption — sphericity — that most people never check.
Two conditions: paired t test. Three or more conditions on the same cases: repeated-measures ANOVA, in wide format, with Mauchly's test run first and the Greenhouse–Geisser correction applied when it rejects.
Why three paired t tests is the wrong answer
With three conditions you could run AB, AC and BC as three paired t tests. Two things go wrong.
First, the same problem as with independent groups: three tests at 5% each give a family-wise error rate near 14%, not 5%.
Second — and this is specific to repeated measures — the three tests are computed from the same subjects. The errors are correlated. That correlation is information: a subject who scores high at baseline tends to score high at follow-up. Repeated-measures ANOVA uses it by partitioning out the between-subject variation, which is why it is more powerful than an ordinary one-way ANOVA on the same numbers. Three separate t tests throw that structure away.
A worked example, with the assumption failing
Eighteen subjects measured under three conditions where the spread differs markedly between them — a realistic pattern when a later measurement is more variable than an earlier one.
| Step | Result | Interpretation |
|---|---|---|
| Condition means | 9.29, 9.51, 8.55 | Very similar |
| Repeated-measures F, uncorrected | F(2, 34) = 0.50, p = .612 | No effect, partial η² = .028 |
| Mauchly's test of sphericity | W = 0.358, χ²(2) = 16.44, p < .001 | Sphericity is violated |
| Greenhouse–Geisser ε | 0.609 | Substantial correction |
| Corrected test | F(1.22, 20.71) = 0.50, p = .524 | Still no effect |
Note what the correction did and did not do. It changed the degrees of freedom from 2 and 34 to 1.22 and 20.71 — a large reduction, because ε is only 0.609. It did not change the conclusion, because the effect was nowhere near significance to begin with.
That is the honest version of this example, and it is worth internalising: the correction is not a formality that always flips a result. It matters most when F sits near the boundary, which is exactly when you cannot afford to skip it.
What sphericity actually means
The F test in a repeated-measures design compares condition means against a pooled error term. That pooling is only valid if the error variability is comparable across the comparisons — formally, if the variance of the difference between any two conditions is the same for every pair.
When that fails, the pooled error term is wrong in a specific direction: the test behaves as though it has more degrees of freedom than it has, and the p value comes out too small. The Greenhouse–Geisser correction estimates how much the assumption is broken (ε, between 1 and 1/(k−1)) and shrinks the degrees of freedom accordingly.
Wide format, and why it matters
Repeated-measures data belong in wide format: one row per subject, one column per condition.
| subject | baseline | week_4 | week_12 |
|---|---|---|---|
| 1 | 12.4 | 10.1 | 8.7 |
| 2 | 14.0 | 12.8 | 11.9 |
| 3 | 11.7 | 9.9 | 9.2 |
Wide format makes the pairing explicit and makes listwise deletion visible — if one condition is missing for a subject, that subject drops out of the whole analysis, and you can see it happen. Long format, with a subject identifier and a condition column, hides that.
When the outcome is ordinal
Likert items and graded scales should not be pushed through a repeated-measures ANOVA. Use the Friedman test: it ranks within each subject and compares the rank totals across conditions, making no distributional assumption about the values themselves.
What to report
The F statistic with both degrees of freedom, the p value, partial η², and — always — whether the degrees of freedom were corrected. A corrected result presented without saying so is indistinguishable from an uncorrected one, and a reviewer cannot reconstruct it.
With correction: F(1.22, 20.71) = 0.50, p = .524, partial η² = .03, with the Greenhouse–Geisser correction applied after Mauchly's test indicated violated sphericity, W = .36, p < .001.
Frequently asked
Why not three paired t tests?
The comparisons are not independent, and the error rate compounds. Repeated-measures ANOVA also uses the between-subject variation, so it is more powerful.
What is sphericity?
Equal variances of the differences between every pair of conditions. When it fails, the uncorrected p value is too small.
Should I always apply Greenhouse–Geisser?
Apply it when Mauchly rejects. Applying it routinely is defensible but then it should be labelled as your primary analysis, not presented as an uncorrected one.
Wide or long format?
Wide: one row per subject, one column per condition.
Which test should I use? · t test vs ANOVA · Method notes · Verification record