Abstract
Summary A general method of analysis is given for a design with a set of treatments added to a p-way orthogonal classification. If there is any grouping within the treatments, the treatment sum of squares may be partitioned; the partitioning is assisted by expressing this sum of squares as a quadratic form in the estimated treatment parameters. When the treatments fall into two groups, possibly with unequal replication, the complete partition of the treatment sum of squares is derived. Designs on three-way classifications are considered in some detail. The more useful ones are mostly on single Latin squares, and new ways of adding various numbers of treatments to 5 times 5 and 6times6 Latin squares are described, examples of possible designs being given. Lack of balance of the treatments with respect to one of the orthogonal classifications may be compensated for in another classification so that some designs are better balanced with three classifications than with one or two.
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