Extended semi-Latin squares for use in field and glasshouse trials

Semi-Latin squares have a long history of use in experimental design, and their properties have been extensively studied. They can be interpreted as a special case of latinized block designs. They are row–column designs in which each treatment occurs once in each row and once in each column, but each row–column intersection has k treatments, where k > 1. These intersections are called cells. In field and glasshouse trials, the layout is typically a rectangle, but now each row–column intersection has only one treatment. By renaming these columns as lines and keeping the previous columns, we incorporate lines as an extra blocking factor into semi-Latin squares, thereby obtaining examples of latinized row–column designs. We call these extended semi-Latin squares. We have found a few isolated examples of the use of such designs in practice, but they do not seem to have been named or studied before. We maintain the convention that each treatment occurs once in each row and once in each column (or minor generalizations of this). However, both the cells and the lines form systems of incomplete blocks. We seek designs which provide maximum information on treatment comparisons using only contrasts that are orthogonal to cells and orthogonal to lines. We investigate the construction and properties of efficient extended semi-Latin squares by using both combinatorics and computer optimization. We present cases showing general balance and note that often this property does not align with design optimality.