Compare means across three or more conditions measured on the same subjects. Results include F-statistic, sphericity test, effect size, and post-hoc comparisons in APA format.
Repeated measures ANOVA (also called within-subjects ANOVA) is a statistical method used to compare means across three or more related groups where the same subjects are measured under each condition. Unlike one-way ANOVA, which compares independent groups, repeated measures ANOVA accounts for the correlation between measurements taken on the same individuals, resulting in greater statistical power because individual differences are removed from the error term.
Use repeated measures ANOVA when the same participants are measured under three or more conditions or at three or more time points. Common scenarios include longitudinal studies tracking changes over time, within-subjects experiments where every participant experiences all conditions, and crossover clinical trials where patients receive multiple treatments in sequence.
| Feature | Repeated Measures ANOVA | One-Way ANOVA |
|---|---|---|
| Design | Within-subjects | Between-subjects |
| Subjects | Same subjects in all conditions | Different subjects per group |
| Error term | Removes individual differences | Includes individual differences |
| Statistical power | Higher | Lower |
| Special assumption | Sphericity | Homogeneity of variance |
| Sample size needed | Smaller | Larger |
A clinical psychologist measures anxiety scores (0–100) for 8 patients at baseline, after 4 weeks of therapy, and after 8 weeks of therapy.
Baseline (n = 8)
45, 52, 48, 55, 50, 47, 53, 49
M = 49.88, SD = 3.23
4 Weeks (n = 8)
58, 65, 62, 68, 63, 60, 66, 61
M = 62.88, SD = 3.23
8 Weeks (n = 8)
70, 78, 74, 80, 75, 72, 79, 73
M = 75.13, SD = 3.56
Results
F(2, 14) = 186.47, p < .001, η²p = .96
There was a significant effect of time on anxiety scores. The very large effect size indicates that time in therapy explained 96% of the within-subjects variance, showing substantial improvement over the treatment period.
Repeated measures ANOVA has four key assumptions:
1. Normality
The dependent variable should be approximately normally distributed at each level of the within-subjects factor. With moderate sample sizes (n ≥ 15), the F-test is robust to violations of normality. For severely non-normal data, consider the Friedman test as a non-parametric alternative.
2. Sphericity (Compound Symmetry)
Sphericity requires that the variances of the differences between all pairs of conditions are approximately equal. This is the repeated measures equivalent of homogeneity of variance. Mauchly's test checks this assumption. When violated, use the Greenhouse-Geisser (more conservative) or Huynh-Feldt correction to adjust degrees of freedom.
3. No Carryover Effects
The effect of one condition should not carry over to the next. Counterbalancing the order of conditions across participants helps minimize carryover. In longitudinal studies, this is inherently difficult to control.
4. Interval or Ratio Data
The dependent variable must be measured on a continuous scale. For ordinal repeated measures data, use the Friedman test instead.
Sphericity is a critical assumption unique to repeated measures ANOVA. When sphericity is violated, the standard F-test becomes liberal (produces too many false positives). The Greenhouse-Geisser (GG) correction adjusts for this by multiplying the numerator and denominator degrees of freedom by epsilon (ε), a value between 1/(k-1) and 1. When ε = 1, sphericity is perfectly met. As ε decreases, the correction becomes more severe, yielding larger (more conservative) p-values.
Report the F-statistic, degrees of freedom, p-value, and partial eta-squared. If the Greenhouse-Geisser correction was applied, report the corrected degrees of freedom and note it:
Without Correction (Sphericity Met)
A repeated measures ANOVA revealed a significant effect of time on anxiety scores, F(2, 14) = 186.47, p < .001, η²p = .96.
With Greenhouse-Geisser Correction
Mauchly's test indicated that the assumption of sphericity had been violated, χ²(2) = 8.45, p = .015. Therefore, a Greenhouse-Geisser correction was applied (ε = .62). There was a significant effect of time, F(1.24, 8.68) = 186.47, p < .001, η²p = .96.
Note: Always report Mauchly's test result and specify which correction was used if sphericity was violated. Report corrected degrees of freedom to two decimal places.
StatMate's repeated measures ANOVA calculations have been validated against R's ezANOVA() and SPSS GLM Repeated Measures output. The implementation partitions variance into between-conditions, between-subjects, and error components. Mauchly's test and Greenhouse-Geisser epsilon are computed from the centered covariance matrix. Bonferroni-corrected post-hoc tests use paired t-tests with adjusted alpha levels.
T-Test
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