Precision in block experiments

SUMMARY A study has been made to facilitate the design of block experiments in circumstances where considerable balance can be introduced but not so much as to permit the adoption of a standard design. First, to provide a theoretical basis, the effect of merging treatments is considered. It is then shown that many standard designs owe their distinctive characteristics to features that can often be introduced in nonstandard cases. Finally, it is assumed that one treatment has been allocated provisionally to plots. Limits are found for the precision of estimation of its mean however the other treatments are allocated. Similar expressions are given for the case in which two treatments have been allocated and it is desired to estimate the difference betweeni their means. In the past many specific experimental designs have been studied and expressions published for their precision of estimation, both of treatment means and of differences betweeii them. At the other extreme, methods have been developed that facilitate the analysis of data from experiments that have hardly any restrictions on design at all; for example, Tocher (1952), Kuiper (1952), as developed by Corsten (1958), Nelder (1965a,b) and Wilkinson (1970). Sucli methods imply ways of determining precision in particular cases and the lemma of Freeman (1957) also is useful for that purpose. Sometimes, however, the designer of experiments is at neither extreme. He cannot secure the complete regularity of the publislhed designs, yet he can build in enoug,h pattern to obtain precise estimation of certain important effects. Classes of design useful in such circumstances lhave been reviewed previously (Pearce, 1963). The various formulae for standard errors have muclh in common, which suggests the existence of expressions more general than any now known. In this paper attention will be confined to block designs. The conclusions are in three parts. First in ? 3 some results will be presented relating to the effect of merging two treatments to form one. The results are of interest on their own account because treatments are sometimes specified conditionally; for example, 'trees will be irrigated whenever the soil moisture deficit exceeds 8 cm', and in unexpected circumstances may become merged unintentionally. The results are needed here, however, as a basis for what follows. Secondly, a number of relationships between treatments will be presented in ? 4, together with the consequent effects on precision. Thirdly, it will be supposed that one or two of the treatments have already been provisionally allocated to plots. In ? 5 limits of precision will be given that apply regardless of the allocation of the others.