Repeated Measures ANOVA

What Is a Repeated Measures ANOVA?

A repeated measures ANOVA (Analysis of Variance) is used when the same participants are measured under three or more different conditions, or at three or more points in time. It is the extension of the paired samples t-test to situations involving more than two measurements. Where a one-way ANOVA compares separate groups of different people, a repeated measures ANOVA compares multiple measurements taken from the same people.

Why Do We Need It?

Imagine a cognitive psychologist who wants to find out how noise level affects reading comprehension. She asks 30 participants to read passages under three conditions: silence, moderate noise, and loud noise. Every participant experiences all three conditions. Because the same people appear in every condition, any differences in comprehension scores cannot be attributed to individual differences in reading ability — those are held constant across conditions. This is the central advantage of a repeated measures design: by removing the variability due to individual differences, it becomes easier to detect the effect of the experimental variable (noise level in this case).

Repeated measures designs also have a practical benefit: they require fewer participants. Instead of recruiting 90 people and assigning 30 to each condition, the researcher needs only 30 people who each complete all three conditions. This makes the design efficient and cost-effective.

How Does It Work?

Like the one-way ANOVA, the repeated measures version works by partitioning variability. The total variability in the data is broken into three components:

• Between-conditions variance: This captures the differences among the condition means. If the experimental manipulation has an effect, this component will be large.
• Between-subjects variance: This captures the stable differences between individual participants (for example, some people are simply better readers than others). Because the same people appear in every condition, this source of variation can be identified and removed from the error term.
• Residual (error) variance: This is what remains after accounting for condition effects and individual differences. It represents unexplained variation — the background noise of the data.

The test produces an F-ratio, calculated by dividing the between-conditions variance by the residual variance. Because the between-subjects variance has been removed from the error term, the residual is typically smaller than it would be in a between-subjects design, making the F-ratio larger and the test more sensitive. A p-value accompanies the F-ratio: if it falls below your significance level (usually 0.05), you conclude that at least one condition mean differs from the others.

What Does the Result Mean?

A significant result tells you that not all condition means are equal, but it does not specify which conditions differ. As with a one-way ANOVA, you would follow up with post-hoc pairwise comparisons (for example, Bonferroni-corrected paired t-tests) to pinpoint where the differences lie.

The Sphericity Assumption

The repeated measures ANOVA carries an important assumption that does not apply to the one-way ANOVA: sphericity. In simple terms, sphericity requires that the variances of the differences between all possible pairs of conditions are roughly equal. For example, if you have conditions A, B, and C, the spread of the A-minus-B difference scores should be similar to the spread of the A-minus-C and B-minus-C difference scores.

When sphericity is violated, the F-ratio can be inflated, leading to false positives. Mauchly's test is commonly used to check this assumption. If it is violated, corrections are available — the most common being the Greenhouse-Geisser and Huynh-Feldt corrections — which adjust the degrees of freedom downward to produce a more conservative (and more accurate) p-value.

Other Assumptions

• Normality: The scores in each condition (or, equivalently, the difference scores between conditions) should be approximately normally distributed. The test is reasonably robust to moderate violations with larger sample sizes.
• No extreme outliers: Outliers within any condition can distort means and inflate error, so it is wise to screen for them beforehand.

When to Use It

Use a repeated measures ANOVA whenever the same participants provide data under three or more conditions or at three or more time points. Common examples include tracking patient symptoms at multiple follow-up visits, comparing performance across several task difficulties, or measuring attitudes before, during, and after an intervention. If you have only two conditions with the same subjects, a paired samples t-test is sufficient. If different people appear in each condition, you need a one-way (between-subjects) ANOVA instead.

A Quick Example

A researcher asks 25 participants to rate the pleasantness of three different room temperatures: 18 °C, 22 °C, and 26 °C. Each participant rates all three temperatures on a scale from 1 to 10. The mean ratings are 5.1, 7.8, and 6.3 respectively. A repeated measures ANOVA yields F(2, 48) = 12.37 with a p-value of less than 0.001. The researcher concludes that perceived pleasantness differs significantly across temperatures. Post-hoc comparisons reveal that 22 °C is rated significantly more pleasant than both 18 °C and 26 °C, while the difference between 18 °C and 26 °C is not significant.

Repeated measures ANOVA is a versatile and efficient method for analysing data from within-subjects designs. By accounting for individual differences, it gives you greater statistical power and a clearer picture of how your experimental conditions affect the outcome.