Optimal Designs for the Two-Dimensional Interference Model

Recently, there have been some major advances in the theory of optimal designs for interference models when the block is arranged in one-dimensional layout. Relatively speaking, the study for two-dimensional interference model is quite limited partly due to technical difficulties. This article tries to fill this gap. Specifically, we set the tone by characterizing all possible universally optimal designs simultaneously through one linear equations system (LES) with respect to the proportions of block arrays. However, such a LES is not readily solvable due to the extremely large number of block arrays. This computational issue could be resolved by identifying a small subset of block arrays with the theoretical guarantee that any optimal design is supported by this subset. The nature of two-dimensional layout of the block has made this task very technically challenging, and we have theoretically derived such subset for any size of the treatment array and any number of treatments under comparison. This facilitates the development of the algorithm for deriving either approximate or exact designs. Supplementary materials for this article are available online.