Multi-level Block Designs for Comparative Experiments
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Multi-level Block Designs for Comparative Experiments Rodney N. Edmondson Complete replicate block designs are fully efficient for treatment effects and are the designs of choice for many agricultural field experiments. For experiments with a large number of treatments, however, they may not provide good control of variability over the whole experimental area. Nested incomplete block designs with a single level of nesting can then improve ‘within-block’ homogeneity for moderate sized experiments. For very large designs, however, a single level of nesting may not be adequate and this paper discusses multi-level nesting with hierarchies of nested blocks. Multi-level nested block designs provide a range of block sizes which can improve ‘within-block’ homogeneity over a range of scales of measurement. We discuss design and analysis of multi-level block designs for hierarchies of nested blocks including designs with crossed block factors. We describe an R language package for multi-level block design and we exemplify the design and analysis of multi-level block designs by a simulation study of block designs for cereal variety trials in the UK. Finally, we re-analyse a single large row-and-column field trial for 272 spring barley varieties in 16 rows and 34 columns assuming an additional set of multi-level nested column blocks superimposed on the existing design. For each example, a multi-level mixed blocks analysis is compared with a spatial analysis based on hierarchical generalized additive (HGAM) models. We discuss the combined analysis of random blocks and HGAM smoothers in the same model. Key Words: Block designs; Mixed models; GAM models; HGAM models; Hierarchical nesting; Nested blocks; Row-and-column blocks; D-optimality; Trend analysis.
1. INTRODUCTION Comparative experiments in agriculture often involve the estimation of treatment effects against a background of high plot variability. Effective control of plot variability is essential for good treatment estimation and the most common method of control is the randomized complete blocks design. Randomized complete blocks can be effective against a range of nuisance effects such as patchy fertility, row-and-column effects or even the residual effects of previous treatments. Randomized complete blocks, however, contain every treatment
R. N. Edmondson (B) Rana House 18 Church Street, Wellesbourne, Warwick CV35 9LS, UK (E-mail: [email protected]). © 2020 The Author(s) Journal of Agricultural, Biological, and Environmental Statistics https://doi.org/10.1007/s13253-020-00416-0
R. N. Edmondson
in every block and may be too large to give good control of variability for experiments with many treatments. It is common practice, therefore, to subdivide large replicate blocks into incomplete blocks for improved intra-block homogeneity, Bailey (1999, 2008), Dean et al. (2015). Sometimes agricultural field trials have crossed blocks and then there are many additional options for nesting and crossing of blocks within the same design, see, for example, Piepho et al
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