Nonbinary variance-balanced designs–part I, optimality

Abstract

Let D(ν, b, k) be a class of designs with ν treatments and b blocks, each of size k. We show that (i) if a variance-balanced design exists in D(ν, b, k), then it is E-optimal. Let Δ(ν, b, k) ⊂ D(ν, b, k) containing all the variance-balanced designs. Then we prove that (ii) a design d Є’Δ(ν, b, k)is ER -optimal in Δ(ν, b, k) if its replication numbers are as uniform as possible, (iii) a design d in Δ(ν, b, k) is AR -optimal and DR -optimal in Δ(ν, b, k) if its replication numbers are all equal except one, and (iv) a design d in Δ(ν, b, k) has maximum average efficiency in Δ(ν, b, k) if its replications numbers are all equal but one. Average efficiencies of designs in three classes are computed and tabulated at the end.