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

TL;DR

This paper analyzes the mathematical relationship between D- and A-optimal experimental designs, revealing how A-criteria can be decomposed into scale and shape factors related to eigenvalue dispersion, which explains differences in design properties.

Contribution

It introduces a factorization of the A-criterion into an inverse-D scale and a sphericity factor, providing new insights into design optimality and variability.

Findings

A-criterion factors into inverse-D scale and sphericity based on eigenvalues.

Designs with similar D-optimality can differ significantly in other properties.

The approach connects to Kiefer's Phi-class and introduces sphericity profiles.

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 $\Phi$-class and introduces sphericity profiles.