Many cross-over designs with t 2 2 treatments and periods and n = bt subjects (b an integer 21) which incorporate balance for first-order carry-over effects are already known: single squares with any even value oft and some odd ones also; and sequentially balanced pairs of squares for any odd t 2 3 . We describe the construction of designs with b = 3, for t = 5,7,9, .. . . This permits use of sequentially balanced designs for any combination of b > 1 and t, except for b = t = 3, for which no such design exists.
Introduction
The usefulness of cross-over designs for clinical studies continues to attract much attention. The potential advantages of the cross-over design are clear; use of subjects as their own controls often results in a great improvement in imputed precision, when typically within-subjects variation is much smaller than between-subjects variation. Moreover, in one sense the treatment 'groups' are more closely balanced than could ever be produced by simple random allocation, or even any degree of stratification, in a parallel-groups study, in that they consist of exactly the same subjects. However, the latter argument overstates the case; the same subject may enter the various treatment periods in quite different states. In particular, it is the existence of carry-over (or residual) effects that casts doubt on the validity of the design. The nature of the carry-over to be anticipated remains a matter of controversy. 1-3 The practical utility of cross-over designs depends on two issues: can designs be found that cope effectively with the carryover problem, and for what kinds of studies is this so? The issue of carry-over is recognized to be most crucial for two-treatment designs, which nevertheless remain the commonest medical application.
One type of study in which cross-over designs have proved particularly useful is the screening study to evaluate efficacy of rinsed oral hygiene agents in volunteers. One design involves determination of bacterial counts in saliva specimens taken immediately before and at intervals up to 7 hours after a single rinsing. Another involves plaque regrowth over a 4-day period, in which oral hygiene is maintained solely by use of the test rinse, starting from a zero baseline produced by professional cleaning. Commonly, between 3 and 7 treatments are compared, often including a negative control (water, saline or a standard toothpaste slurry) and/or a positive one, against which new formulations may be assessed. The positive control is usually the highly potent antimicrobial, chlorhexidine gluconate, the usefulness of which for routine oral hygiene is limited by its severe potential for extrinsic staining. In the plaque regrowth design, there is the (perhaps