Multi-level Block Designs for Comparative Experiments

Abstract

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.