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.

翻译：针对固定的线性模型基，我们证明了A准则可分解为一个逆D尺度项和一个仅依赖于特征值离散度的无量纲球度因子。该因子精确分离了A准则中不受行列式控制的部分，从而解释了为何在D准则上完全或近似等价的设计，在系数方差、别名效应和预测方差行为上可能存在实质性差异。我们通过一个已发表的D等价案例及具有无穷多D-最优解的筛选场景演示了该分解，并运用相同的尺度/形状视角作为空间填充候选池中的轻量级后筛选工具。最后部分将同一思路与Kiefer的Φ类准则相关联，并引入了球度剖面概念。