Resolvable designs with large blocks

TL;DR

This paper investigates the optimality of resolvable block designs with two blocks per replicate, focusing on block concurrences and their role in achieving various optimality criteria, including E-optimality.

Contribution

It provides new conditions for optimality, characterizes E-optimal designs, and links optimal solutions to balanced arrays and affine-like structures.

Findings

Equalizing block concurrences is often optimal.

Sufficient conditions for strong optimalities are established.

E-optimal designs correspond to balanced arrays and affine-like structures.

Abstract

Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.