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 GraecoLatin square and
HyperGraecoLatin 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 GraecoLatin square designs
there are 3 nuisance factors, and for HyperGraecoLatin square designs
there are 4 nuisance factors.

Nuisance factors used as blocking variables

The nuisance factors are used as blocking variables.
 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.
 As with the Latin square design, a GraecoLatin 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.
 A HyperGraecoLatin 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:
 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.
 They allow experiments with a relatively small number
of runs.
The disadvantages are:
 The number of levels of each blocking variable must
equal the number of levels of the treatment factor.
 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^{31}fractional factorial design).

Summary of designs

Several useful designs are described in the table below.
Some Useful Latin Square, GraecoLatin Square and
HyperGraecoLatin Square Designs
Name of
Design

Number of
Factors
k

Number of
Runs
N


3by3 Latin Square

3

9

4by4 Latin Square

3

16

5by5 Latin Square

3

25




3by3 GraecoLatin Square

4

9

4by4 GraecoLatin Square

4

16

5by5 GraecoLatin Square

4

25




4by4 HyperGraecoLatin Square

5

16

5by5 HyperGraecoLatin 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
Y_{ijk}

denoting any observation for which
X_{1} = i, X_{2} =
j, X_{3} = k
X_{1} and X_{2} are
blocking factors
X_{3} is the primary factor

\( \mu \)

denoting the general location parameter

R_{i}

denoting the effect for block i

C_{j}

denoting the effect for block j

T_{k}

denoting the effect for treatment k

Models for GraecoLatin and HyperGraecoLatin 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
R_{i}:

\( \bar{Y}_{i}  \bar{Y} \)
\( \bar{Y}_{i} \) = average of all Y for which
X_{1} = i

Estimate for
C_{j}:

\( \bar{Y}_{j}  \bar{Y} \)
\( \bar{Y}_{j} \) = average of all Y for which
X_{2} = j

Estimate for
T_{k}:

\( \bar{Y}_{k}  \bar{Y} \)
\( \bar{Y}_{k} \) = average of all Y for which
X_{3} = k


Randomize as much as design allows

Designs for Latin squares with 3, 4, and 5level 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 5Level Factors

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

3Level Factors
X_{1}

X_{2}

X_{3}

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)
L_{1} = 3 levels of factor X_{1} (block)
L_{2} = 3 levels of factor X_{2} (block)
L_{3} = 3 levels of factor X_{3}
(primary)
N = L1 * L2 = 9 runs
This can alternatively be represented as

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

4Level Factors
X_{1}

X_{2}

X_{3}

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)
L_{1} = 4 levels of factor X_{1} (block)
L_{2} = 4 levels of factor X_{2} (block)
L_{3} = 4 levels of factor X_{3}
(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 5level factors (and 2 nuisance or blocking factors)

5Level Factors
X_{1}

X_{2}

X_{3}

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)
L_{1} = 5 levels of factor X_{1} (block)
L_{2} = 5 levels of factor X_{2} (block)
L_{3} = 5 levels of factor X_{3}
(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).
