Which statement about the sphericity assumption in Mauchly's test is true?

Which statement about the sphericity assumption in Mauchly's test is true?

• A. The data must be normally distributed.
• B. The variance-covariance matrix must be a scalar multiple of the identity matrix for sphericity to hold.
• C. The variance-covariance matrix must be orthogonal.
• D. The mean vector must be zero.

Answer: (B)

Sphericity is about the pattern of relationships among the repeated measurements, focusing on the covariance structure. Under sphericity, the covariance matrix of the repeated-measures outcomes is a scalar multiple of the identity matrix, meaning all time points have the same variance and the covariances between different time points behave in a way that makes the F-tests in repeated-measures ANOVA valid. Mauchly's test specifically checks this form: H0 is that the covariance matrix equals λ times the identity. If the test supports this, sphericity holds and standard results are appropriate; if not, degrees of freedom corrections (like Greenhouse-Geisser) are used. Other statements don’t capture this covariance-structure focus. Normality of the data isn’t what Mauchly’s test assesses; orthogonality of the covariance matrix isn’t the criterion here, and the mean vector does not need to be zero for sphericity to hold.

Sphericity is about the pattern of relationships among the repeated measurements, focusing on the covariance structure. Under sphericity, the covariance matrix of the repeated-measures outcomes is a scalar multiple of the identity matrix, meaning all time points have the same variance and the covariances between different time points behave in a way that makes the F-tests in repeated-measures ANOVA valid. Mauchly's test specifically checks this form: H0 is that the covariance matrix equals λ times the identity. If the test supports this, sphericity holds and standard results are appropriate; if not, degrees of freedom corrections (like Greenhouse-Geisser) are used.

Other statements don’t capture this covariance-structure focus. Normality of the data isn’t what Mauchly’s test assesses; orthogonality of the covariance matrix isn’t the criterion here, and the mean vector does not need to be zero for sphericity to hold.