When should I use repeated measures ANOVA?

Use repeated measures ANOVA when the same subjects are measured under three or more conditions or time points. Examples: comparing blood pressure before treatment, 1 week after, and 4 weeks after; or comparing scores when the same students experience all three teaching methods.

Sphericity is the assumption that the variances of the differences between all pairs of conditions are equal. It is tested using Mauchly's test. When violated (p < .05), apply the Greenhouse-Geisser or Huynh-Feldt correction to adjust degrees of freedom.

What is the Greenhouse-Geisser correction?

When sphericity is violated, the Greenhouse-Geisser correction multiplies the degrees of freedom by epsilon to produce more conservative F-tests. Epsilon near 1 indicates sphericity is met; near 1/(k-1) indicates severe violation. The corrected df yields a new, more accurate p-value.

How do I report repeated measures ANOVA in APA format?

Report the F-statistic, degrees of freedom (corrected if sphericity is violated), p-value, and partial eta-squared. Note if Greenhouse-Geisser correction was applied. Example: 'F(1.42, 9.94) = 35.12, p < .001, partial-eta-squared = .83 (Greenhouse-Geisser corrected).'

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同じ被験者で測定された3条件以上の平均値を比較します。結果にはF統計量、球面性の検定、効果量、多重比較がAPA形式で含まれます。

What is Repeated Measures ANOVA?

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.

When to Use Repeated Measures ANOVA

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.

Repeated Measures ANOVA vs. One-Way ANOVA

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

Worked Example: Therapy Progress Over Three Time Points

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, η²_p = .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.

Assumptions of Repeated Measures ANOVA

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.

Understanding Sphericity and the Greenhouse-Geisser Correction

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.

How to Report Repeated Measures ANOVA in APA Format

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, η²_p = .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, η²_p = .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.

Common Mistakes to Avoid

Using one-way ANOVA instead of repeated measures: When the same subjects are measured multiple times, treating them as independent groups ignores within-subject correlation and inflates the error term, reducing power.
Ignoring sphericity: Failing to check and correct for sphericity violations leads to inflated Type I error rates. Always report Mauchly's test and apply corrections when needed.
Unequal numbers of observations: All subjects must have data for all conditions. Missing data requires special handling (e.g., mixed-effects models or imputation).
Not counterbalancing: In within-subjects designs, order effects can confound results. Counterbalance the order of conditions when possible.
Ignoring effect size: A significant F-test alone does not convey practical importance. Always report partial eta-squared alongside the p-value.

Calculation Accuracy

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.

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