Bartlett's test of sphericity is used to determine whether a variance-covariance matrix is proportional to an identity matrix. Which of the following best describes this test?

• A. It tests whether the variance-covariance matrix is proportional to an identity matrix, implying equal diagonal elements and zero off-diagonal elements.
• B. It tests whether the data follow a normal distribution.
• C. It tests whether the means of all groups are equal.
• D. It tests whether observations are independent.

Answer: (A)

Bartlett’s test of sphericity is about whether the variables in a set are related enough to justify multivariate methods. Specifically, it tests the null hypothesis that the observed variance-covariance (or correlation) matrix is proportional to an identity matrix. If the matrix is proportional to the identity, all variables have equal variances and zero covariances with each other, meaning there’s no systematic structure to exploit. A significant result means the matrix differs from that identity form, indicating correlations among variables and that techniques like factor analysis or principal components may be appropriate. It’s not about normality, group means, or independence of observations.