Sphericity is met when which condition holds?

Sphericity is met when which condition holds?

• A. The variance-covariance matrix is a scalar multiple of the identity.
• B. The variances are unequal.
• C. The sample size is large.
• D. The data are non-parametric.

Answer: (A)

Sphericity means the spread of the variables is uniform in all directions, so all variances are equal and all covariances are zero. This happens exactly when the variance-covariance matrix is a scalar multiple of the identity, written as σ^2 times the identity matrix. In that case every diagonal element is the same variance and every off-diagonal element is zero, so the variables are uncorrelated and equally variable—precisely the condition for sphericity. If variances differ, or covariances are not zero, sphericity is violated. A large sample size doesn’t by itself enforce this structure, and whether data are non-parametric isn’t about the covariance matrix being proportional to the identity.

Sphericity means the spread of the variables is uniform in all directions, so all variances are equal and all covariances are zero. This happens exactly when the variance-covariance matrix is a scalar multiple of the identity, written as σ^2 times the identity matrix. In that case every diagonal element is the same variance and every off-diagonal element is zero, so the variables are uncorrelated and equally variable—precisely the condition for sphericity. If variances differ, or covariances are not zero, sphericity is violated. A large sample size doesn’t by itself enforce this structure, and whether data are non-parametric isn’t about the covariance matrix being proportional to the identity.