The basic Latin square is a very restricted design since it is only applicable to situations where the treatment and blocking factors all have the same number of levels. It is also the basis of some other pairwise orthogonal designs that can be applied in more general situations.

Multiple Latin squares

An r2 Latin square design can be repeated in multiple sets of blocks to allow the variable represented by the rows or columns to have more than r levels. For example, the following pairwise orthogonal design for three treatments uses two Latin squares to allow it to be used for 18 experimental units that are indexed with blocking variables with 3 and 6 levels.

      Column 1          Column 2         Column 3    
Row 1 A C B
Row 2 C B A
Row 3 B A C
Row 4 B C A
Row 5 A B C
Row 6 C A B

The design could similarly be used for two controlled factors with 3 levels, with 6 blocks of size 3.

Smaller number of factor levels

In the number of rows and columns in a basic Latin square design is a multiple of a, r = ab, then the rows, columns or 'letters' can be grouped to give levels for a factor with a levels.

For example, the diagram below initially shows a 6x6 Latin square.

   
  C1 C2 C3 C4 C5 C6
Block 1    A       D       C       E       F       B   
Block 2 F A D B E C
Block 3 D C A F B E
Block 4 B F E C A D
Block 5 C E B A D F
Block 6 E B F D C A

Select Grouped levels from the pop-up menu to form a pairwise orthogonal design for two factors with 3 levels each in six blocks of size 6. (The columns in the original design are grouped in pairs and the treatments are grouped as AD, BE and CF.)