Block Designs: A Randomization Approach pp 229-279 | Cite as

# Resolvable Designs

## Abstract

Resolvable block designs, introduced in Section 6.0.3, are important in practice because it is often useful to be able to perform an experiment with replicates one or more at a time. The present chapter is devoted only to those among (α,α_{2}, …,α_{a})-resolvable block designs which are a-resolvable for α ≥ 1, according to the concepts discussed in Section 6.0.3. A 1-resolvable block design is simply called resolvable in the usual sense of Bose (1942a). Note that α-resolvable block designs are necessarily equireplicate. Now, recalling the general form of Definition 6.0.2, one can write as follows. A block design with parameters *v, b, r, k*is said to be a-resolvable if the blocks can be separated into a sets (superblocks) of *b*_{h} blocks each (*b*=∑ _{ h=1 } ^{ a } *b*_{ h }) such that the *h*th superblock contains every treatment exactly a times, for *h* = 1,2, …,a. Further, note that if the block design is proper, i.e., *k* = *k***1**_{b},, then *b*_{1} = *b*_{2} = … = *b*_{a}(= *b*_{0}, say). Then one gets *b* = *b*_{0}a, *r* = α*a*, *v*α = *kb*_{0} and *b*α = *rb*_{0}. An a-resolvable proper block design with parameters *v, b, r, k* is said to be affine a-resolvable if every two distinct blocks from the same superblock intersect in the same number, *q*_{1}, of treatments, whereas any two blocks from different superblocks intersect in the same number, *q*_{2}, of treatments. Here *q*_{1} = (α - 1)*k*/(*b*_{0} - 1) and *q*_{2} = *k*^{2}/*v* (see Section 6.0.3). A more general case can be seen in Definition 6.0.3.

## Keywords

Block Design Incidence Matrix Proper Design Efficiency Factor Association Scheme## Preview

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