Bradley Jones | Searcharxiv

arXiv2026

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

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.

stat.ME↗

arXiv2022

D- and A-optimal Screening Designs

In practice, optimal screening designs for arbitrary run sizes are traditionally generated using the D-criterion with factor settings fixed at +/- 1, even when considering continuous factors with levels in [-1, 1]. This paper identifies cases of undesirable estimation variance properties for such D-optimal designs and argues that generally A-optimal designs tend to push variances closer to their minimum possible value. New insights about the behavior of the criteria are found through a study of their respective coordinate-exchange formulas. The study confirms the existence of D-optimal designs comprised only of settings +/- 1 for both main effect and interaction models for blocked and un-blocked experiments. Scenarios are also identified for which arbitrary manipulation of a coordinate between [-1, 1] leads to infinitely many D-optimal designs each having different variance properties. For the same conditions, the A-criterion is shown to have a unique optimal coordinate value for improvement. We also compare Bayesian version of the A- and D-criteria in how they balance minimization of estimation variance and bias. Multiple examples of screening designs are considered for various models under Bayesian and non-Bayesian versions of the A- and D-criteria.

stat.ME↗