On the efficiency of some supplemented (α1,α2,…,αR)-resolvable block designs
Introduction
Supplemented block designs are often used in plant-breeding experiments and in many other fields of research with control or standard treatments. There are situations in which an experimental material for certain treatments is limited. So, such treatments (supplementary treatments, say) are treated differently than the rest of treatments (basic treatments, say) in the experiments. Supplemented block designs are widely described in many papers, for instance in Caliński and Ceranka (1974), Puri et al. (1977), Nigam et al. (1981), Nigam and Puri (1982), Kageyama (1993), etc.
We assume the design for the basic treatments (called basic design) to be an (α1,α2,…,αR)-resolvable block design (cf. Caliński and Kageyama, 2000).
There are nested block (NB) designs (cf. Morgan, 1996; Caliński and Kageyama, 2000) in which superblocks are divided into blocks. The blocks have constant size but the number of units may vary from superblock to superblock. In the superblocks there are α1,α2,…,αR replicates of the basic treatments, respectively.
The basic design is assumed to be orthogonal supplemented at the superblock level by groups of the supplementary treatments in such a way as to be proper. This property and adequate randomization of experimental units give an orthogonal block structure of the final supplemented block design. Thus it may be analyzed using the techniques for multistratum experiments with orthogonal block structure (cf. Nelder, 1965).
In the paper we will be considering connected efficiency balanced (EB) (α1,α2,…,αR)-resolvable block designs in general as well as BIB (α1,α2,…,αR)-resolvable designs as the basic designs. In both cases, the final design will be a connected and partially efficiency balanced (PEB) (α1,α2,…,αR)-resolvable block design.
The definition of EB designs can be found for instance in Caliński and Kageyama (2000). Especially for PEB designs, anyone can dip into Nigam and Puri (1982) and for BIB designs into Cochran and Cox (1957). The definition of the orthogonal block structure of a design may be found in Nelder (1965).
Section snippets
The basic design
Let us consider as the basic design, say D1, any (α1,α2,…,αR)-resolvable block design with parameters which denote in turn: the number of basic treatments, the number of blocks, the number of treatment replications, the vector of block sizes and the number of observations (n1=v1r1). The blocks of D1 can be separated into R superblocks of blocks of sizes k(1),k(2),…,k(R) so thatwhere is the vector of ones of order x.
Examples of constructions
Example 1 Consider a supplemented block design with incidence matrix:Thus the , are incidence matrices of BIB designs with parameters:Hence, D1 with incidence matrix is 3-resolvable BIB design and its parameters are as follows:
Remarks on the analysis
Since units have to be randomized before they enter the experiment, randomization models with three or four strata are suitable. The model with three strata, say model I, is obtained when the two-step randomization (blocks→plots) is performed in the experiment, while the three-step randomization (superblocks→blocks→plots) leads to the model with four strata, say model II. In both models we have a zero stratum (0) generated by the vector of ones, intra-block stratum (1) and inter-block stratum
Acknowledgements
The work was partially supported by grant KBN 3 P06A 017 22.
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