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

## Graeco-Latin square designs

These designs handle 3 nuisance factors Graeco-Latin squares, as described on the previous page, are efficient designs to study the effect of one treatment factor in the presence of 3 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 3-, 4-, and 5-level factors are given on this page. These designs show what the treatment combinations would 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 Graeco-Latin square design be randomly selected from those available, then randomize the run order.

Graeco-Latin Square Designs for 3-, 4-, and 5-Level Factors
Designs for 3-level factors
 X1 X2 X3 X4 row blocking factor column blocking factor blocking factor treatment factor 1 1 1 1 1 2 2 2 1 3 3 3 2 1 2 3 2 2 3 1 2 3 1 2 3 1 3 2 3 2 1 3 3 3 2 1
with
k = 4 factors (3 blocking factors and 1 primary factor)
L1 = 3 levels of factor X1 (block)
L2 = 3 levels of factor X2 (block)
L3 = 3 levels of factor X3 (primary)
L4 = 3 levels of factor X4 (primary)
N = L1 * L2 = 9 runs
This can alternatively be represented as (A, B, and C represent the treatment factor and 1, 2, and 3 represent the blocking factor):

 A1 B2 C3 C2 A3 B1 B3 C1 A2
Designs for 4-level factors
 X1 X2 X3 X4 row blocking factor column blocking factor blocking factor treatment factor 1 1 1 1 1 2 2 2 1 3 3 3 1 4 4 4 2 1 2 4 2 2 1 3 2 3 4 2 2 4 3 1 3 1 3 2 3 2 4 1 3 3 1 4 3 4 2 3 4 1 4 3 4 2 3 4 4 3 2 1 4 4 1 2
with
k = 4 factors (3 blocking factors and 1 primary factor)
L1 = 3 levels of factor X1 (block)
L2 = 3 levels of factor X2 (block)
L3 = 3 levels of factor X3 (primary)
L4 = 3 levels of factor X4 (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 factor):

 A1 B2 C3 D4 D2 C1 B4 A3 B3 A4 D1 C2 C4 D3 A2 B1
Designs for 5-level factors
 X1 X2 X3 X4 row blocking factor column blocking factor blocking factor treatment factor 1 1 1 1 1 2 2 2 1 3 3 3 1 4 4 4 1 5 5 5 2 1 2 3 2 2 3 4 2 3 4 5 2 4 5 1 2 5 1 2 3 1 3 5 3 2 4 1 3 3 5 2 3 4 1 3 3 5 2 4 4 1 4 2 4 2 5 3 4 3 1 4 4 4 2 5 4 5 3 1 5 1 5 4 5 2 1 5 5 3 2 1 5 4 3 2 5 5 4 3
with
k = 4 factors (3 blocking factors and 1 primary factor)
L1 = 3 levels of factor X1 (block)
L2 = 3 levels of factor X2 (block)
L3 = 3 levels of factor X3 (primary)
L4 = 3 levels of factor X4 (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 factor):

 A1 B2 C3 D4 E5 C2 D3 E4 A5 B1 E3 A4 B5 C1 D2 B4 C5 D1 E2 A3 D5 E1 A2 B3 C4
Further information More designs are given in Box, Hunter, and Hunter (1978).