Tiers, structure formulae and the analysis of complicated experiments

The use of several structure formulae has many advantages for specifying the models for comparative experiments. In many experiments, two formulae are sufficient to ensure that the analysis-of-variance table correctly represents the confounding relationships in the design. The first formula usually defines the systematic (or treatment) terms, and the second the error terms (or strata). This allows the correct error to be determined for each term and enables the full information to be obtained on terms that are estimated in more than one stratum. The formulae should be derived from sets of factors, based on the randomization, and termed tiers. A detailed discussion of data from a tasting trial of produce from a viticultural experiment shows that sometimes more than two tiers, and hence structure formulae, may be needed. A new algorithm is developed to determine the structure of such multiphase experiments, and to generate their correct analysis.