Neighbor balance and evenness of distribution of treatment replications in row‐column designs

Abstract

Row‐column designs allow error control in field experiments by blocking in two dimensions. While this strategy can capture spatial heterogeneity aligned with blocks and account for effects due to the farming operations along rows and columns, it suffers from the occasional clustered occurrence of several replications of the same treatment. This property of classical row‐column designs has hampered their more widespread use in practice. A further issue of practical importance is the degree of neighbor balance of a design, that is, the frequency of adjacencies of pairs of treatments. This paper proposes two design strategies to simultaneously optimize the evenness of spatial distribution of treatment replication as well as neighbor balance. Three examples are given to illustrate the proposed methods and demonstrate that both approaches yield comparable and satisfactory results.

Error variance bias in neighbour balance and evenness of distribution designs

“…It should be emphasised, however, that using NB & ED designs and associated row‐column blocking does not preclude the adoption of more sophisticated analysis methods such as spatial models (see Rodriguez‐Alvarez et al . and references therein) and/or extra blocking structures as discussed by Piepho, Michel & Williams ().…”

“…Piepho, Michel & Williams () described two methods for the construction of NB & ED designs; here we will use the direct optimisation method. In summary, consideration is given to minimising the number of pairwise treatment neighbours in one direction, namely adjacent long thin plot boundaries.…”

“…Further batches of 200 spatial designs for 30 treatments and 38 treatments have been generated with CycDesigN using the LV spatial efficiency factor as a criterion (Williams & Piepho ). CycDesigN has been used to construct 200 resolvable row‐column designs for 30 treatments and five replicates of 5×6 plots where the replicates are latinised by columns; the transpose of this design layout thus produces the desired 6×25 array of plots. The package has also generated a further 200 resolvable LV spatial designs for 30 treatments. The direct optimisation method described in section 3.4 of Piepho, Michel & Williams () has been used to produce 200 NB & ED designs for each of the 30 treatment and 38 treatment situations. The procedure starts with an efficient non‐resolvable row‐column design and adopts a convenient grouping of the 25 columns.…”

“…This is a measure assessing the evenness of the spatial distribution of replications of a treatment. The larger its value, the more even the distribution (Piepho, Michel & Williams ).…”

“…Recently, there has been interest in the construction and use of neighbour balance and evenness of distribution (NB & ED) designs (Piepho, Michel & Williams ). These are row–column designs, where, as with other types of spatial design, randomisation has been restricted in order to address certain balance and spread criteria.…”