On optimality properties of simple block designs in the approximate design theory
Consider an experiment with v treatments and b blocks. An exact block design of size n is a set of integers \(n_{ij}\in \{0,1,...,n\}\) with \(i=1,2,...,v\), and \(j=1,2,3,...,b\) such that their sum is n in which the experimenter observes \(n_{ij}\) times with the i-th treatment in the j- th block. This paper deals with approximate block designs which are sets of weights \(w_{ij}\in [0,1]\) with sum equal to 1, meaning thereby that \(w_{ij}\) fraction of all observations is to be taken on the ith treatment in the j-th block. In this paper optimality properties of approximate block designs are discussed under variations of (a) the class of competing designs, (b) the optimality criterion, (c) the parametric function to be investigated, and (d) the statistical model (interaction effects, additive effects). Optimality means here universal optimality, uniform optimality, and \(j_ p\)-optimality. The optimal designs turn out to be the product of their treatment and block marginals, and uniform designs whose support is specified well in advance. A geometric account of uniform optimality is also given. The classical balanced incomplete block designs (BIBD) are shown to be simultaneously \(j_ p\)-optimal for a maximal system of identifiable parameters.
scientific article; zbMATH DE number 3843018