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

5.3.3.2.1.

Latin square and related designs

Latin square (and related) designs are efficient designs to block from 2 to 4 nuisance factors

Latin square designs, and the related Graeco-Latin square and Hyper-Graeco-Latin square designs, are a special type of comparative design.

There is a single factor of primary interest, typically called the treatment factor, and several nuisance factors. For Latin square designs there are 2 nuisance factors, for Graeco-Latin square designs there are 3 nuisance factors, and for Hyper-Graeco-Latin square designs there are 4 nuisance factors.

Nuisance factors used as blocking variables

The nuisance factors are used as blocking variables.

1. For Latin square designs, the 2 nuisance factors are divided into a tabular grid with the property that each row and each column receive each treatment exactly once.

2. As with the Latin square design, a Graeco-Latin square design is a kxk tabular grid in which k is the number of levels of the treatment factor. However, it uses 3 blocking variables instead of the 2 used by the standard Latin square design.

3. A Hyper-Graeco-Latin square design is also a kxk tabular grid with k denoting the number of levels of the treatment factor. However, it uses 4 blocking variables instead of the 2 used by the standard Latin square design.

Advantages and disadvantages of Latin square designs

The advantages of Latin square designs are:

1. They handle the case when we have several nuisance factors and we either cannot combine them into a single factor or we wish to keep them separate.

2. They allow experiments with a relatively small number of runs.

The disadvantages are:

1. The number of levels of each blocking variable must equal the number of levels of the treatment factor.

2. The Latin square model assumes that there are no interactions between the blocking variables or between the treatment variable and the blocking variable.

Note that Latin square designs are equivalent to specific fractional factorial designs (e.g., the 4x4 Latin square design is equivalent to a 4^3-1fractional factorial design).

Summary of designs

Several useful designs are described in the table below.

Some Useful Latin Square, Graeco-Latin Square and Hyper-Graeco-Latin Square Designs

Name of Design	Number of Factors k	Number of Runs N
3-by-3 Latin Square	3	9
4-by-4 Latin Square	3	16
5-by-5 Latin Square	3	25
3-by-3 Graeco-Latin Square	4	9
4-by-4 Graeco-Latin Square	4	16
5-by-5 Graeco-Latin Square	4	25
4-by-4 Hyper-Graeco-Latin Square	5	16
5-by-5 Hyper-Graeco-Latin Square	5	25

Model for Latin Square and Related Designs

Latin square design model and estimates for effect levels

The model for a response for a latin square design is

\( Y_{ijk} = \mu + R_{i} + C_{j} + T_{k} + \mbox{random error} \)

with

Yijk	denoting any observation for which X1 = i, X2 = j, X3 = k X1 and X2 are blocking factors X3 is the primary factor
\( \mu \)	denoting the general location parameter
Ri	denoting the effect for block i
Cj	denoting the effect for block j
Tk	denoting the effect for treatment k

Models for Graeco-Latin and Hyper-Graeco-Latin squares are the obvious extensions of the Latin square model, with additional blocking variables added.

Estimates for Latin Square Designs

Estimates

Estimate for \( \mu \):	\( \bar{Y} \) = the average of all the data
Estimate for Ri:	\( \bar{Y}_{i} - \bar{Y} \) \( \bar{Y}_{i} \) = average of all Y for which X1 = i
Estimate for Cj:	\( \bar{Y}_{j} - \bar{Y} \) \( \bar{Y}_{j} \) = average of all Y for which X2 = j
Estimate for Tk:	\( \bar{Y}_{k} - \bar{Y} \) \( \bar{Y}_{k} \) = average of all Y for which X3 = k

Randomize as much as design allows

Designs for Latin squares with 3-, 4-, and 5-level factors are given next. 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 Latin square design be randomly selected from those available, then randomize the run order.

Latin Square Designs for 3-, 4-, and 5-Level Factors

Designs for 3-level factors (and 2 nuisance or blocking factors)

3-Level Factors

X1	X2	X3
row blocking factor	column blocking factor	treatment factor
1	1	1
1	2	2
1	3	3
2	1	3
2	2	1
2	3	2
3	1	2
3	2	3
3	3	1

with

k = 3 factors (2 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)
N = L1 * L2 = 9 runs

This can alternatively be represented as

A	B	C
C	A	B
B	C	A

Designs for 4-level factors (and 2 nuisance or blocking factors)

4-Level Factors

X1	X2	X3
row blocking factor	column blocking factor	treatment factor
1	1	1
1	2	2
1	3	4
1	4	3
2	1	4
2	2	3
2	3	1
2	4	2
3	1	2
3	2	4
3	3	3
3	4	1
4	1	3
4	2	1
4	3	2
4	4	4

with

k = 3 factors (2 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)
N = L1 * L2 = 16 runs

This can alternatively be represented as

A	B	D	C
D	C	A	B
B	D	C	A
C	A	B	D

Designs for 5-level factors (and 2 nuisance or blocking factors)

5-Level Factors

X1	X2	X3
row blocking factor	column blocking factor	treatment factor
1	1	1
1	2	2
1	3	3
1	4	4
1	5	5
2	1	3
2	2	4
2	3	5
2	4	1
2	5	2
3	1	5
3	2	1
3	3	2
3	4	3
3	5	4
4	1	2
4	2	3
4	3	4
4	4	5
4	5	1
5	1	4
5	2	5
5	3	1
5	4	2
5	5	3

with

k = 3 factors (2 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)
N = L1 * L2 = 25 runs

This can alternatively be represented as

A	B	C	D	E
C	D	E	A	B
E	A	B	C	D
B	C	D	E	A
D	E	A	B	C

Further information

More details on Latin square designs can be found in Box, Hunter, and Hunter (1978).