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Insights into the Relationship Between D- and A-optimal Designs

Andrew T. Karl, Bradley Jones

Abstract

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.

Insights into the Relationship Between D- and A-optimal Designs

Abstract

For a fixed linear-model basis, we show that the criterion factors into an inverse- scale term and a dimensionless sphericity factor that depends only on eigenvalue dispersion. This factor isolates exactly the part of not controlled by the determinant, explaining why designs that are exact or near ties in can differ materially in coefficient-variance, aliasing, and prediction-variance behavior. We illustrate the factorization on a published tie and on screening settings with infinitely many -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 -class and introduces sphericity profiles.
Paper Structure (16 sections, 24 equations, 4 figures, 4 tables)

This paper contains 16 sections, 24 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Scale and sphericity in A-optimality
  3. Notation and spectral conventions
  4. A sphericity index and a factorization of A
  5. Connection to JMP Evaluate Design efficiencies
  6. Geometric interpretation: overall scale versus roundness
  7. A published $D$ tie in screening designs
  8. Further screening example: infinitely many D-optimal designs
  9. Space-filling post-screening: MaxPro divided by sphericity
  10. From inverse D to MaxPro: a sphericity-weighted post-screen
  11. Illustration with an FFF candidate pool
  12. Extensions via Kiefer's Phi-class and sphericity profiles
  13. Kiefer's $\Phi$-class and scale/shape separation
  14. Sphericity profiles
  15. A brief note on worst-case prediction variance
  16. ...and 1 more sections

Figures (4)

  • Figure 1: One-step augmentation illustration.
  • Figure 2: Ordered normalized covariance spectra.
  • Figure 3: FFF pool illustration: extremes of MaxPro$/\mathcal{S}$.
  • Figure 4: Sphericity profiles $\mathcal{S}_r(C)$ as a function of $r$.