Balanced Incomplete Block Designs

Abstract

The randomized complete block design is rightly seen as the norm to be aimed at in the designing of experiments. This is due to its simplicity, its capability of reducing error, and the ease with which missing values can be handled. We use more complicated designs in agriculture only when: (1) It is desired to remove further sources of variability (Latin Square, Graeco-Latin Square, Youden Square), (2) Such designs are demanded by the practical details of field operations (split-plots, criss-cross, etc.), and (3) There are too many treatments for reasonably homogenous replications to be laid out in the field, and each replication has to be divided into several smaller blocks (incomplete blocks, confounded factorials). Under (3), “confounding” is a well-known device used in cases where the treatments have a factorial structure. When faced with unstructured treatments, particularly with many cultivars of a crop, then we often resort to more general incomplete block designs (usually balanced). This discussion paper introduces these balanced, incomplete blocks designs in general terms. Plant breeding and selection programs require the screening of large numbers of genotypes (lines, varieties, accessions, etc.), particularly in their early stages. When the amount of available seed does not permit more than one plot for each entry, the “augmented designs” (R.G. Petersen, 1980) probably provide the most suitable solution although other methods for adjusting the treatment observations are available. Lattices are probably the best-known class of incomplete block designs and in this paper we will introduce the properties and the analysis of balanced lattices, deferring to a future paper the introduction to partially balanced lattices.