Construction of weak universal optimal block designs with various correlation structures and block sizes

• We construct weak universal optimal block designs for various correlation structures.
• Our results substantially generalize former research in optimal block design.
• Each design is proved precisely and confirmed by computation in MATLAB.
• Our methods can be widely used in optimal design with correlated observations.

Balanced block designs and in particular balanced incomplete block designs (BIBDs) have been in widespread use in agricultural, ecological, pharmaceutical, and industrial research for many years. However, this “balance” characteristic is only true for BIBDs with uncorrelated observations. With correlated observations within each block, the order of the observations matters and this order impacts variance and hence any notion of “balance”. Thus, there is need of research into construction of these useful designs when any one of a number of different correlation structures might exist within each block of units.In this paper, based on Balanced Incomplete Block Designs (BIBDs), we construct weak universal optimal block designs for the nearest neighbor correlation structure with common block sizes, for the hub correlation structure with any block size, and for circulant correlation with odd block size. We also approximate weak universal optimal block designs for the block-structured correlation.