Which statement about sphericity is true?

Which statement about sphericity is true?

• A. The variances of the differences between all pairs of treatments are equal
• B. The variances of the treatment scores must be increasing with time
• C. Means must be equal across all conditions
• D. Data must be non-normal

Answer: (A)

Sphericity refers to the relationship between how the repeated-measures differences behave across all pairs of conditions. Specifically, it means that if you take the difference scores for every pair of treatment levels (for each participant), the variances of those difference scores should be the same across all pairs. With k conditions, there are k(k−1)/2 such pairwise differences, and sphericity says their variances are equal. This property helps ensure the F tests in a repeated-measures ANOVA have the correct distribution under the null. If sphericity holds, the within-subjects test is straightforward to interpret. If it doesn’t, the test can be biased, leading to inflated or deflated Type I error rates, which is why corrections like Greenhouse-Geisser or Huynh-Feldt are used to adjust the degrees of freedom. For example, with three conditions, you’d look at the variances of the three pairwise difference scores (condition 1 minus 2, 1 minus 3, and 2 minus 3). Those three variances should be roughly equal for sphericity to hold. The other statements—variance in treatment scores increasing over time, means needing to be equal across conditions, or data needing to be non-normal—do not describe sphericity.

Sphericity refers to the relationship between how the repeated-measures differences behave across all pairs of conditions. Specifically, it means that if you take the difference scores for every pair of treatment levels (for each participant), the variances of those difference scores should be the same across all pairs. With k conditions, there are k(k−1)/2 such pairwise differences, and sphericity says their variances are equal. This property helps ensure the F tests in a repeated-measures ANOVA have the correct distribution under the null.

If sphericity holds, the within-subjects test is straightforward to interpret. If it doesn’t, the test can be biased, leading to inflated or deflated Type I error rates, which is why corrections like Greenhouse-Geisser or Huynh-Feldt are used to adjust the degrees of freedom.

For example, with three conditions, you’d look at the variances of the three pairwise difference scores (condition 1 minus 2, 1 minus 3, and 2 minus 3). Those three variances should be roughly equal for sphericity to hold. The other statements—variance in treatment scores increasing over time, means needing to be equal across conditions, or data needing to be non-normal—do not describe sphericity.