2026/03/07 by Andrew T. Karl, Bradley Jones
#stat.ME
For a fixed linear-model basis, we show that the A criterion factors into an inverse-D scale term and a dimensionless sphericity factor that depends only on eigenvalue dispersion. This factor isolates exactly the part of A not controlled by the determinant, explaining why designs that are exact or near ties in D can differ materially in coefficient-variance, aliasing, and prediction-variance behavior. We illustrate the factorization on a published D tie and on screening settings with infinitely many D-optimal solutions, then use the same scale/shape viewpoint as a lightweight post-screen within a space-filling candidate pool. A final section connects the same idea to Kiefer's Φ-class and introduces sphericity profiles.