Pairwise orthogonal F-rectangle designs

Abstract

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r= pv row by c= qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with v n columns, ( v n −1)/( v−1) rows, v symbols, and of strength two.