Substitutes for the Non-existent Square Lattice Designs for 36 Varieties
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Substitutes for the Non-existent Square Lattice Designs for 36 Varieties R. A. Bailey , Peter J. Cameron , L. H. Soicher, and E. R. Williams Square lattice designs are often used in trials of new varieties of various agricultural crops. However, there are no square lattice designs for 36 varieties in blocks of size six for four or more replicates. Here, we use three different approaches to construct designs for up to eight replicates. All the designs perform well in terms of giving a low average variance of variety contrasts. Supplementary materials accompanying this paper appear online. Key Words: A-optimality; Computer search; Resolvable block designs; Semi-Latin squares; Sylvester graph.
1. INTRODUCTION In variety-testing programmes, later-stage trials can involve multiple replications of up to 100 varieties: see Patterson et al. (1978). Even at a well-run testing centre, variation across the experimental area makes it desirable to group the plots (experimental units) into homogeneous blocks, usually too small to contain all the varieties. As R. A. Fisher wrote in a letter in 1938, “… on any given field agricultural operations, at least for centuries, have followed one of two directions”, so that variability among the plots is well captured by blocking in one or both of these directions, with no need for more complicated spatial correlations: see Fisher et al. (1990, p. 270). Thus, on land which has been farmed for centuries, or where plots cannot be conveniently arranged to allow blocking in two directions (rows and columns), it is reasonable to assume the following model for the yield Yω on plot ω: Yω = τV (ω) + β B(ω) + εω .
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An extended abstract for this paper is in Bailey and Cameron (2018). R. A. Bailey (B)· P. J. Cameron, School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (E-mail: [email protected]). L. H. Soicher, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK. E. R. Williams, Statistical Consultancy Unit, Australian National University, Canberra 2600, Australia. © 2020 The Author(s) Journal of Agricultural, Biological, and Environmental Statistics https://doi.org/10.1007/s13253-020-00388-1
R. A. Bailey et al.
Here, V (ω) denotes the variety planted on ω and B(ω) denotes the block containing ω. The variety constants τi are the unknown parameters of interest, and the block constants β j are unknown nuisance parameters. The quantities εω are independent identically distributed random variables with zero mean and common (unknown) variance σ 2 . Denote the number of varieties by v. For management reasons, it is often convenient if the blocks can themselves be grouped into replicates, in such a way that each variety occurs exactly once in each replicate. Such a block design is called resolvable. Let r be the number of replicates. Yates (1936, 1937) introduced square lattice designs for this purpose. In these, v = n 2 for some positive integer n, and each replicate consists of n bl
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