D- and A-Optimal Screening Designs

Abstract

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 article 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 gained 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 unblocked 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 how Bayesian versions of the A- and D-criteria 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.

KEYWORDS:

• Bayesian optimal design
• Blocking
• Coordinate exchange algorithm
• Factorial experiments
• Minimum aliasing

Supplementary Materials

A PDF file includes proofs of all results and additional details of the optimization of the continuous coordinate exchange formula for $\mathcal{A}$-criteria. Tables for the optimal designs from Sections 4.2 are also included in the PDF, as well as additional results for the quadratic screening designs from Section 4.3. The optimal main effect designs for $(n,k)=(11,10)$ and (19, 18) that include non-integer coordinates are given as txt files. R files and jsl scripts are included to perform algorithmic optimization of the described criteria.

Acknowledgments

We would like to thank the editor, associate editor, and two anonymous referees for their valuable feedback that improved this article.