### On MSE-optimal crossover designs

Christoph Neumann, Joachim Kunert
2018 Annals of Statistics
In crossover designs, each subject receives a series of treatments one after the other. Most papers on optimal crossover designs consider an estimate which is corrected for carryover effects. We look at the estimate for direct effects of treatment, which is not corrected for carryover effects. If there are carryover effects, this estimate will be biased. We try to find a design that minimizes the mean square error, that is the sum of the squared bias and the variance. It turns out that the
more » ... s out that the designs which are optimal for the corrected estimate are highly efficient for the uncorrected estimate. MSC 2010 subject classifications: Primary 62K05; secondary 62K10. Keywords and phrases: Optimal Design, crossover Design, MSE-Optimality. 1 C. Neumann and J.Kunert/MSE-optimal crossover designs 2 A possible compromise might be analyzing in a model without carryover effects but choosing the design in such a way that the carryover effects have as little impact on the estimates as possible. David et al. (2001) showed that this approach can be quite useful, at least in agricultural studies. Compared to the vast literature on the optimality of designs in the model with carryover effects, there is only a very small number of papers on the choice of designs if the carryover effect is neglected. The most relevant paper for our work is Azaïs and Druilhet (1997) who present a bias-criterion, which is similar to the optimality criterion by Kiefer (1975) . We note that, apart from the disadvantage of having biased estimates, there is the advantage of a smaller variance of the estimators neglecting the carryover effects. The present paper considers an optimality criterion that gives a compromise between these two opposing attributes. This criterion is the well-known mean square error (MSE). Calculating the MSE We consider the set of crossover designs Ω t,n,p with t treatments, n units and p periods. If d ∈ Ω t,n,p is applied, then y ij , the j-th observation on unit i, arises from a model with additive carryover effects, i.e.