E- and R-optimality of block designs for treatment-control comparisons

TL;DR

This paper characterizes E- and R-optimal block designs for treatment-control comparisons, providing theoretical insights and practical design classes that optimize statistical efficiency and confidence region volume.

Contribution

It introduces a comprehensive characterization of E-optimal approximate designs and identifies classes of R-optimal designs, extending optimal design theory for treatment-control studies.

Findings

All E-optimal approximate designs are characterized by simple linear constraints.

A class of E-optimal exact designs for unequal block sizes is identified.

A-optimal designs are also R-optimal, indicating strong performance for confidence regions.

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs characterized by simple linear constraints. Employing this characterization, we obtain a class of E-optimal exact designs for treatment-control comparisons for unequal block sizes. In the studied model, we justify the use of E-optimality by providing a statistical interpretation for all E-optimal approximate designs and for the known classes of E-optimal exact designs. Moreover, we consider the R-optimality criterion, which minimizes the volume of the rectangular confidence region based on the Bonferroni confidence intervals. We show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This further reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.