Construction of sequentially counterbalanced designs formed from two Latin squares

Sequentially counterbalanced designs formed from two addition Latin squares are discussed for the case of an odd number of treatments. The properties of the different types of Latin squares, i.e. sets of orthogonal squares, Williams squares and nearly-balanced squares, are investigated, and it is shown that only a limited number of different designs formed from two squares are available. The situation for cross-over designs for five treatments over five periods using ten different treatment sequences is examined in detail, and a general formula is derived to produce a sequentially counterbalanced design for any odd number of treatments. A simple method of producing further different designs from this sequentially counterbalanced design is described.