Factorial experimentation in second-order Latin cubes

A Second-order Latin cube of size n x n x n can be used as the design for an experiment in three space dimensions, with the three sets of layers used as three sets of blocks. The n2 treatments are then orthogonal to the main effects X, Y and Z of the blocking systems. Particular interest attaches to second-order Latin cubes whose treatments are an n x n factorial set, with the main effects A and B of the treatment factors orthogonal to the interactions XY, XZ and YZbetween pairs of blocking systems. This note describes such designs where components of the interaction AB are each totally confounded with one of XY, XZ and YZ. Cubes with n = 4 are then described where components of A, B and AB are each partially confounded. Finally, a defective design with n = 4 is described, to illustrate the need for care in composing designs for three dimensions.