Resolved designs viewed as sets of partitions

This chapter explains two incomplete-block designs. In both designs all the treatments occur in the same number of blocks. Such a design is called equireplicate. The two most important statistical properties of incomplete-block designs are balance and resolvability. A design is balanced if every pair of distinct treatments concurs in the same number of blocks. A design is resolvable if the set of blocks can be partitioned into superblocks each of which is complete in the sense that all treatments occur in it once. In the context of statistical design, orthogonality is the most important relation between a pair of partitions on the same set. The first non-balanced incomplete-block designs used for experiments were the designs now called square lattices, introduced by Yates. A compact way of showing partitions on the same set is to write the elements of the set as a row, with one row beneath it for each partition.