Incomplete-block designs for experiments were first developed by Yates at Rothamsted Experimental Station. He produced a remarkable collection of designs for individual experiments. Two of them are shown, with the data from the experiment, in Example 4.3 on page 97 and Exercise 5.9 on page 141. This type of design poses two questions for statisticians: (i) what is the best way of choosing subsets of the treatments to allocate to the blocks, given the resource constraints? (ii) how should the data from the experiment be analysed?

Designs with partial balance help statisticians to answer both of these questions. The designs were formally introduced by Bose and Nair in 1939. The fundamental underlying concept is the association scheme, which was defined in its own right by Bose and Shimamoto in 1952. Theorem 5.2 on page 114 shows the importance of association schemes: the pattern of variances matches the pattern of concurrences.

Many experiments have more than one system of blocks. These can have complicated inter-relationships, like the examples in Section 7.1, which are all taken from real experiments. The general structure is called an orthogonal block structure. Although these were introduced independently of partially balanced incomplete-block designs, they too are association schemes. Thus association schemes play an important role in the design of experiments.