Latin Square Designs

Latin Square Designs

Selected Latin Squares

3 x 3 4 x 4

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

5 x 5

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

6 x 6

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

A Latin Square

Definition

• A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column. Example - 4 x 4 Latin Square.

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

In a Latin square You have three factors: • Treatments (

t

) (letters A, B, C, …) • Rows (

t

) • Columns (

t

) The number of treatments = the number of rows = the number of colums =

t

.

The row-column treatments are represented by cells in a

t

x

t

array.

The treatments are assigned to row-column combinations using a Latin-square arrangement

Example

A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars. • The brands are all comparable in purchase price. • The company wants to carry out a study that will enable them to compare the brands with respect to operating costs. • For this purpose they select five drivers (Rows). • In addition the study will be carried out over a five week period (Columns = weeks).

• Each week a driver is assigned to a car using randomization and a Latin Square Design.

• The average cost per mile is recorded at the end of each week and is tabulated below: Drivers 1 2 3 4 5 1 5.83 D 4.80 P 7.43 F 6.60 R 11.24 C 2 6.22 P 7.56 D 11.29 C 9.54 F 6.34 R Week 3 7.67 F 10.34 C 7.01 R 11.11 D 11.30 P 4 9.43 C 5.82 R 10.48 D 10.84 P 12.58 F 5 6.57 R 9.86 F 9.27 P 15.05 C 16.04 D

The Model for a Latin Experiment

y ij

   

k

 

i

 

j

 

ij i

= 1,2,…,

t j

= 1,2,…,

t k

= 1,2,…,

t y ij(k)

= the observation in

i

th row and the

j

th  column receiving the

k

th = overall mean 

k



i



j

= the effect of the

i

th = the effect of the

i

th = the effect of the

j

th 

ij(k)

= random error treatment row column treatment No interaction between rows, columns and treatments

• A

Latin Square

experiment is assumed to be a

three

-factor experiment. • The factors are

rows

,

columns

and

treatments

. • It is assumed that there is

no interaction

between rows, columns and treatments. • The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Latin Square Experiment

Source Treat Rows Cols Error Total S.S.

SS Tr SS Row SS Col SS E SS T d.f.

t-1 t-1 t-1 (t-1)(t-2) t 2 - 1 M.S.

MS Tr MS Row MS Col MS E F MS Tr /MS E MS Row /MS E MS Col /MS E p-value

The Anova Table for Example

Source Week Driver Car Error Total S.S.

51.17887

69.44663

70.90402

9.56315

201.09267

d.f.

4 4 4 12 24 M.S.

12.79472

17.36166

17.72601

0.79693

F 16.06

21.79

22.24

p-value 0.0001

0.0000

0.0000

Using SPSS for a Latin Square experiment

Rows Cols Trts

Y

Select

Analyze->General Linear Model->Univariate

Select the dependent variable and the three factors – Rows, Cols, Treats Select

Model

Identify a model that has only main effects for Rows, Cols, Treats

The

ANOVA

table produced by

SPSS Tests of Between-Subjects Effects

Dependent Variable: COST Source Corrected Model Intercept DRIVER WEEK CAR Error Total Corrected Total Type III Sum of Squares 191.530

a 2120.050

69.447

51.179

70.904

9.563

2321.143

201.093

df 12 1 4 4 4 12 25 24 Mean Square 15.961

2120.050

17.362

12.795

17.726

.797

a. R Squared = .952 (Adjusted R Squared = .905) F 20.028

2660.273

21.786

16.055

22.243

Sig.

.000

.000

.000

.000

.000

Example 2

In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). There are a total of t = 3 X 2 = 6 treatment combinations of the two factors.

• Beef -High Protein • Cereal-High Protein • Pork-High Protein • Beef -Low Protein • Cereal-Low Protein and • Pork-Low Protein

In this example we will consider using a

Latin Square

design Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories. • A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories. • A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

In the latin square the letter • A represents the

high protein-cereal

diet • B represents the

high protein-pork

• C represents the

low protein-beef

diet Diet • D represents the

low protein-cereal

• E represents the

low protein-pork

diet diet and • F represents the

high protein-beef

diet.

The weight gain after a fixed period is measured for each of the test animals and is tabulated below: Initial Weight Category 5 6 3 4 1 2 1 62.1 A 86.2 B 63.9 C 68.9 D 73.8 E 101.8 F 2 Appetite Category 3 4 84.3 B 91.9 F 71.1 D 77.2 A 73.3 C 83.8 E 61.5 C 69.2 D 69.6 E 97.3 F 78.6 A 110.6 B 66.3 D 64.5 C 90.4 F 72.1 E 101.9 B 87.9 A 5 73.0 E 80.8 A 100.7 B 81.7 C 111.5 F 93.5 D 6 104.7 F 83.9 E 93.2 A 114.7 B 95.3 D 103.8 C

The Anova Table for Example

Source Inwt App Diet Error Total S.S.

1767.0836

2195.4331

4183.9132

63.61999

8210.0499

d.f.

5 5 5 20 35 M.S.

353.41673

439.08662

836.78263

3.181

F 111.1

138.03

263.06

p-value 0.0000

0.0000

0.0000

Diet SS partioned into main effects for Source and Level of Protein

Source Inwt App Source Level SL Error Total S.S.

1767.0836

2195.4331

631.22173

2611.2097

941.48172

63.61999

8210.0499

d.f.

5 5 2 1 2 20 35 M.S.

353.41673

439.08662

315.61087

2611.2097

470.74086

3.181

F 111.1

138.03

99.22

820.88

147.99

p-value 0.0000

0.0000

0.0000

0.0000

0.0000

Graeco-Latin Square Designs

Mutually orthogonal Squares

Definition

A

Greaco-Latin

square consists of two latin squares (one using the letters A, B, C, … the other using greek letters a , b , c , …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called

mutually orthogonal

.

Example:

a 7 x 7 Greaco-Latin Square A a B  C b D f E c B b C c D d E  F f G  C D E F G A f  a b c d D E F G A B  c d f  a E F G A B C  a b c d  F d G  A f B  C a D b F  G a A b B c C d D  E f G d A  B f C  D a E b F c

Note:

At most (

t

–1)

t

x

t

Latin squares

L

1 ,

L

2 , …,

L

t-1 that any pair are

mutually orthogonal

.

such It is possible that there exists a set of

six

7 x 7

mutually orthogonal

Latin squares

L

1 ,

L

2 ,

L

3 ,

L

4 ,

L

5 ,

L

6 .

The Greaco-Latin Square Design - An Example

A researcher is interested in determining the effect of two factors • • the percentage of

Lysine

in the diet and percentage of

Protein

in the diet have on

Milk Production

in cows. Previous similar experiments suggest that

interaction

between the two factors is negligible.

For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (

Lysine

and

Protein

). Seven levels of each factor is selected • 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and 0.6(G)% for

Lysine

and • 2(a), 4(b), 6(c), 8(d), 10(e), 12(f) and 14(g)% for

Protein

). • Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods .

A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (

Lysine

and

Protein

) to a period and a cow. The data is tabulated on below:

The Model for a Greaco-Latin Experiment

y ij

   

k

 

l

 

i

 

j

 

ij i

= 1,2,…,

t j

= 1,2,…,

t k

= 1,2,…,

t l

= 1,2,…,

t y ij(kl)

= the observation in

i

th row and the

j

th column receiving the

k

th and the

l

th Latin treatment Greek treatment

 = overall mean 

k

= the effect of the

k

th Latin treatment 

l

= the effect of the

l

th Greek treatment 

i

= the effect of the

i

th row 

j

= the effect of the

j

th column 

ij(k)

= random error No interaction between rows, columns, Latin treatments and Greek treatments

• A

Greaco-Latin Square

experiment is assumed to be a

four

-factor experiment. • The factors are

rows

,

columns

,

Latin treatments

and

Greek treatments

. • It is assumed that there is

no interaction

between rows, columns, Latin treatments and Greek treatments.

• The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Greaco-Latin Square Experiment

Source Latin Greek Rows Cols Error Total S.S.

SS La SS Gr SS Row SS Col SS E SS T d.f.

t-1 t-1 t-1 t-1 (t-1)(t-3) t 2 - 1 M.S.

MS La MS Gr MS Row MS Col MS E F MS La /MS E MS Gr /MS E MS Row /MS E MS Col /MS E p-value

The Anova Table for Example

Source Protein Lysine Cow Period Error Total S.S.

160242.82

30718.24

2124.24

5831.96

15544.41

214461.67

d.f.

6 6 6 6 24 48 M.S.

26707.1361

5119.70748

354.04082

971.9932

647.68367

F 41.23

7.9

0.55

1.5

p-value 0.0000

0.0001

0.7676

0.2204