5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?
5.3.3.2. Randomized block designs

## Hyper-Graeco-Latin square designs

These designs handle 4 nuisance factors Hyper-Graeco-Latin squares, as described earlier, are efficient designs to study the effect of one treatment factor in the presence of 4 nuisance factors. They are restricted, however, to the case in which all the factors have the same number of levels.
Randomize as much as design allows Designs for 4- and 5-level factors are given on this page. These designs show what the treatment combinations should be for each run. When using any of these designs, be sure to randomize the treatment units and trial order, as much as the design allows.

For example, one recommendation is that a hyper-Graeco-Latin square design be randomly selected from those available, then randomize the run order.

Hyper-Graeco-Latin Square Designs for 4- and 5-Level Factors
Designs for 4-level factors (there are no 3-level factor Hyper-Graeco Latin square designs)
 X1 X2 X3 X4 X5 row blocking factor column blocking factor blocking factor blocking factor treatment factor 1 1 1 1 1 1 2 2 2 2 1 3 3 3 3 1 4 4 4 4 2 1 4 2 3 2 2 3 1 4 2 3 2 4 1 2 4 1 3 2 3 1 2 3 4 3 2 1 4 3 3 3 4 1 2 3 4 3 2 1 4 1 3 4 2 4 2 4 3 1 4 3 1 2 4 4 4 2 1 3
with
k = 5 factors (4 blocking factors and 1 primary factor)
L1 = 4 levels of factor X1 (block)
L2 = 4 levels of factor X2 (block)
L3 = 4 levels of factor X3 (primary)
L4 = 4 levels of factor X4 (primary)
L5 = 4 levels of factor X5 (primary)
N = L1 * L2 = 16 runs
This can alternatively be represented as (A, B, C, and D represent the treatment factor and 1, 2, 3, and 4 represent the blocking factors):

 A11 B22 C33 D44 C42 D31 A24 B13 D23 C14 B41 A32 B34 A43 D12 C21
Designs for 5-level factors
 X1 X2 X3 X4 X5 row blocking factor column blocking factor blocking factor blocking factor treatment factor 1 1 1 1 1 1 2 2 2 2 1 3 3 3 3 1 4 4 4 4 1 5 5 5 5 2 1 2 3 4 2 2 3 4 5 2 3 4 5 1 2 4 5 1 2 2 5 1 2 3 3 1 3 5 2 3 2 4 1 3 3 3 5 2 4 3 4 1 3 5 3 5 2 4 1 4 1 4 2 5 4 2 5 3 1 4 3 1 4 2 4 4 2 5 3 4 5 3 1 4 5 1 5 4 3 5 2 1 5 4 5 3 2 1 5 5 4 3 2 1 5 5 4 3 2
with
k = 5 factors (4 blocking factors and 1 primary factor)
L1 = 5 levels of factor X1 (block)
L2 = 5 levels of factor X2 (block)
L3 = 5 levels of factor X3 (primary)
L4 = 5 levels of factor X4 (primary)
L5 = 5 levels of factor X5 (primary)
N = L1 * L2 = 25 runs
This can alternatively be represented as (A, B, C, D, and E represent the treatment factor and 1, 2, 3, 4, and 5 represent the blocking factors):

 A11 B22 C33 D44 E55 D23 E34 A45 B51 C12 B35 C41 D52 E13 A24 E42 A53 B14 C25 D31 C54 D15 E21 A32 B43
Further information More designs are given in Box, Hunter, and Hunter (1978).