Some E and MV-optimal designs for the two-way elimination of heterogeneity

Abstract

SummaryIt is well known that in experimental settings where v treatments are being tested in b blocks of size k, a group divisible design having parameters ~2=~1+1 and whose corresponding C-matrix has maximal trace is both E and MV-optimal among all possible competing designs. In this paper, we show that under certain conditions, the E and MVoptimal group divisible block designs mentioned in the previous sentence can be used to construct E and MV-optimal row-column designs to handle experimental situations in which heterogeneity is to be eliminated in two directions and where v treatments are being tested in b columns and k rows. Examples are given to illustrate how the results obtained can be applied.

Some sufficient conditions for the E- and MV-optimality of block designs having blocks of unequal size

“…We begin by stating a result which can be proven using techniques analogous to those used in Constantine [4] or Jacroux [9]. …”

On the E-optimality of complete designs under an interference model

“…Only the problem of determining E-optimal designs under the model with block effects as the only nuisance parameters over the class of designs is partially solved, see e.g. Constantine, 1981, Jacroux, 1982, 1983, Srivastav and Shankar, 2003. However, there is no characterization of E-optimal designs under an interference model.…”

Optimal row–column design for three treatments

“…These designs are all subject to strict combinatorial requirements, with the consequence that in the universe of all possible row-column experiments (v, p, q), they comprise a very small fraction. Optimality progress outside of these combinatorially rarefied settings has been at best sporadic, with most known results being for the E-criterion; see Jacroux (1985Jacroux ( , 1986Jacroux ( , 1987Jacroux ( , 1990, Bagchi and van Berkum (1991), Singh and Gupta (1991), Das (1993), Bagchi (1996), and Parvu and Morgan (2005). It has long been known that optimal, equireplicate block designs (v treatments in b blocks of size k) can be arranged into optimal rowcolumn designs (of size b × k) whenever b is a multiple of v; the underlying technical result was formalized by Magda (1980).…”