D- and A-optimal Screening Designs

TL;DR

This paper compares D- and A-optimal screening designs, revealing limitations of D-optimal designs with fixed factor levels and highlighting the advantages of A-optimal designs in minimizing estimation variance.

Contribution

The study provides new insights into the behavior of D- and A-optimal criteria, including conditions for optimality and variance properties, and compares Bayesian and non-Bayesian approaches.

Findings

D-optimal designs often use fixed +/- 1 settings for factors.

A-optimal designs tend to minimize variances more effectively.

Multiple examples demonstrate differences between criteria under various models.

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 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.