Four blocking variables or factors

Balanced designs also can be obtained from some r-by-r Latin square designs by adding a set of r Greek letters in such a way that each pair of Latin & Greek letters appears in exactly one square of the design.

Graeco-Latin designs are not possible for all r but are fairly easy to obtain when r is odd.

Graeco-Latin square for odd r

A Graeco-Latin design for tables in which the factors & blocking variables have an odd number of levels can be constructed from two ordinary Latin squares with diagonal bands of Latin and Greek letters. For example, consider the following two 5x5 Latin squares.

   X=1   X=2    X=3    X=4    X=5      X=1   X=2    X=3    X=4    X=5  
Block 1 A B C D E   α β γ δ θ
Block 2 E A B C D   β γ δ θ α
Block 3 D E A B C   γ δ θ α β
Block 4 C D E A B   δ θ α β γ
Block 5 B C D E A   θ α β γ δ

Superimopsing these gives a Graeco-Latin square.

   X=1   X=2    X=3    X=4    X=5  
Block 1
Block 2
Block 3
Block 4
Block 5

Permutation and randomisation

The above is only one of many possible 5x5 Graeco-Latin squares. To generate one particular design, you should randomly permute the levels of all four variables.

Since the above design is for a randomised block experiment with 5 blocks of size 5, the final step should be:

Warning: Although Graeco-Latin squares are very occasionally appropriate, the restrictions of equal levels for all factors and blocks means that:

Graeco-Latin squares are rarely used in practice.