On MSE-Optimal Circular Crossover Designs

In crossover designs, each subject receives a series of treatments, one after the other in p consecutive periods. There is concern that the measurement of a subject at a given period might be influenced not only by the direct effect of the current treatment but also by a carryover effect of the treatment applied in the preceding period. Sometimes, the periods of a crossover design are arranged in a circular structure. Before the first period of the experiment itself, there is a run-in period, in which each subject receives the treatment it will receive again in the last period. No measurements are taken during the run-in period. We consider the estimate for direct effects of treatments which is not corrected for carryover effects. If there are carryover effects, this uncorrected estimate will be biased. In that situation, the quality of the estimate can be measured by the mean square error, the sum of the squared bias and the variance. We determine MSE-optimal designs, that is, designs for which the mean square error is as small as possible. Since the optimal design will in general depend on the size of the carryover effects, we also determine the efficiency of some designs compared to the locally optimal design. It turns out that circular neighbour-balanced designs are highly efficient.