Document 3B


EXPERIMENTAL DESIGN PROBLEM EXAMPLES

These examples are intended to illustrate the general principles for fitting subsets of treatments into sets of
blocks, the sizes of the blocks being determined so as to provide homogeneous plots within a block. In
most cases it will be assumed that

 (i) the set of treatments is pre-defined, possibly with factorial structure (complete or incomplete),
 (ii) the experiment is to include 2, 3 or 4 replicates of the treatment set,
 (iii) each replicate is to be divided into two or more blocks of the same or similar size.

 The initial set of problems have been generated from the problems that (should) have been solved for
 various on-farm experiments at Poza Rica and Chalco in early 1990. Other problems from the past, or for
 the future, or purely hypothetical, will be added as they are suggested to me. (Such suggestions will be
 welcome!).


 1. Principles

 1.1 Treatments Without Structure

 Ideally, each pair of treatments should occur together (in a block) equally frequently, or at worst, with
 frequencies differing by at most one. Particularly with relatively small numbers of treatments (e.g.twelve),
 this is not always possible to achieve. So we need some "rules" for the division into blocks in each replicate
 such that the resulting design will come as near to the equal pair-wise occurrence as possible.

 Where each of several replicates is split into blocks the splits in the different replicates should be as
 different as possible. This is to be interpreted in the sense that, for the first two replicates, each block group
 of treatments in the first replicate should be split as equally as possible between the blocks of the second
 replicate. For further replicates the division of treatments into blocks should be as different as possible (in
 the same sense) to each of the divisions in previous blocks

 1.2 Complete Factorial Structure

 In each replicate the division into blocks should be such that for each factor in each block the levels of that
 factor should occur as equally as possible. This requirement is intended to maximize the information about
 each factor main effect. Further, for each pair of factors for which the two-factor interaction is likely to be
 important, all combinations of levels from those two factors should occur as evenly as possible in each
 block. Where there are still arbitrary choices to be made the equal occurrence in each block of all
 combinations of levels of three factors should be aimed at.

 1.3 Other Structure

 For incomplete factorial structures the general principle about equal occurrence in each block of the levels
 of each factor and of the combinations of levels of pairs of factors still applies. When the set of treatments
 includes some non-factorial structure, the important treatment contrasts should be identified. For each
 contrast each block should contain a similar balance between the groups of treatments compared in the
contrast.