Relative efficiency of certain randomization procedures in an n×n array when spatial correlation is present

Abstract For n ≤ 5 and n ≠ 4 optimal designs are obtained in an n×n layout under a general class of spatial correlation functions. For n = 2 or 3, the optimal design is a randomized latin square, and for n = 5 it is the knight's move latin square. Although an optimal design does not exist for n = 4 or n>5, an optimal design does exist for n = 4 (the type G systematic latin square of Addelman, 1976) within the subclass of latin square designs. The optimality criterion used is related to, but not the same as, the A-optimality criterion. Here the objective function that is used is the model variance of a treatment contrast averaged over the randomization distribution. Some numerical efficiency comparisons are given for 4 ⩽ n ⩽ 13 for various randomization schemes. It is found that when the schemes are more efficient than a randomized latin square, the gains in efficiency are modest, usually much less than 10%.