This paper addresses the problems of determination and construction of universally optimal designs in two-dimensional blocks of size $p \times 2$, assuming within-block observations are correlated. Generalized least-squares estimation of treatment contrasts is considered in four fixed block-effects models: (I) with fixed row and column effects, (II) with the row effects only, (III) with the column effects only, and (IV) with neither row nor column effects. For a general dependence structure and $p = 2$, optimal designs for Model I are found to coincide with the least-squares optimal designs. For general p, Models I-IV, and the within-block correlation pattern described by the doubly geometric process, interesting nonbinary block patterns are found for the universally optimal designs. Only for Model IV for small, positive correlations do binary blocks turn out to be best, though binarity with respect to rows or columns is often required. Regardless of the model, the conditions frequently coincide with those for optimal nested row-column designs with uncorrelated errors: one class of these designs is found to be optimal for at least some values of the correlation parameters under all four models, and others are found to be optimal for particular models. The exact form of the blocks for a universally optimum design is found to be quite sensitive to the blocksize and to the magnitude of the correlations under both Models III and IV.
"Universally optimal designs with blocksize $p\times 2$ and correlated observations." Ann. Statist. 25 (3) 1189 - 1207, June 1997. https://doi.org/10.1214/aos/1069362744