Construction of uniform-balanced cross-over designs for any odd number of treatments

Cross-over designs balanced for simple carry-over effects are commonly applied in clinical studies for the comparison of treatments for chronic conditions such as hypertension or asthma. Uniform-balanced cross-over designs have the desirable property that the treatment sequences are arranged so that, in the full design, each treatment is followed by every other treatment equally often. Such designs for an even number of treatments and the same number of sequences and periods are readily constructed using suitable cyclic Latin squares. For an odd number of treatments, pairs of squares may be combined to give uniform-balanced designs. Recently, computer search techniques have been used to find nearly-balanced Latin squares which may be combined in pairs or in sets of three to produce designs with the overall properties of uniformity and balance. In this paper, simple generating formulae are described which will give, for any odd number of treatments t > 3, uniform-balanced cross-over designs with p = t periods and n = kt treatment sequences for any k > or = 2. Tables of cross-over designs obtained from these simple formulae are presented for t < or = 15.