Optimal designs for crossover model with partial interactions

This paper studies the universally optimal designs for estimating total effects under crossover models with partial interactions. We provide necessary and sufficient conditions for a symmetric design to be universally optimal, based on which algorithms can be used to derive optimal symmetric designs under any form of the within-block covariance matrix. To cope with the computational complexity of algorithms when the experimental scale is too large, we provide the analytical form of optimal designs under the type-H covariance matrix. We find that for a fixed number of treatments, say (Formula presented.), the number of distinct treatments appearing in the support sequences increases with the increase of the number of periods, (Formula presented.), until (Formula presented.), in which case all (Formula presented.) treatments appear. The optimal design can be constructed from up to two representative sequences, within which each treatment appears in consecutive periods with equal or almost equal numbers of replications.