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.
Highlights
Comparative experiments in agriculture often involve the estimation of treatment effects against a background of high plot variability
The analysis shows the number of model terms, the mean estimated degrees of freedom (EDF), the mean AIC and the mean standard error of pairwise treatment differences (SED) averaged over simulations combination, the table shows the number of marginal model terms, the mean EDF, the mean AIC and the mean SED, calculated as described in the previous example
The mean pairwise SED for H3.1 was about 0.249 compared with about 0.260 for B3.3, which suggests that the best fitting HGAM model gave an improvement in precision of about 4% compared with the best fitting multi-level blocks model
Summary
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. One commonly used method is the mixed model which assumes a mixture of fixed treatment effects and random block effects, see Pinheiro and Bates (2000) and Piepho et al (2003). This paper examines multi-level block designs for block and spatial effects in field experiments. We first make a simulation study of 3 replicates of 48 varieties in complete randomized blocks with four levels of nesting using simulation data based on a spatial correlation model from a study of 244 UK cereal variety trials by Patterson and Hunter (1983). Edmondson analysis so the assumption of fixed block effects at the design stage does not preclude the choice of an alternative model at the analysis stage. See Gilmour and Trinca (2006) Section 5.1.1. for further discussion of fixed versus random effects block models
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