Multi-level Block Designs for Comparative Experiments

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.