Optimality and Contrasts in Block Designs with Unequal Treatment Replication

The computer construction of optimal or near-optimal experimental designs is common in practice. Search procedures are often based on the non-zero eigenvalues of the information matrix of the design. Minimising the average of the pairwise treatment variances can also be used as a search criterion. For equal treatment replication these approaches are equivalent to maximising the harmonic mean of the design's canonical efficiency factors, but differ when treatments are unequally replicated. This paper investigates the extent of these differences and discusses some apparent inconsistencies previously observed when comparing the optimality of equally and unequally replicated designs.