Latin Square Designs. Selected Latin Squares 3 x 34 x 4 A B CA B C DA B C DA B C DA B C D B C AB A D...

Latin Square Designs

Latin Square Designs Selected Latin Squares

3 x 3 4 x 4A B C A B C D A B C D A B C D A B C DB C A B A D C B C D A B D A C B A D CC A B C D B A C D A B C A D B C D A B

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

5 x 5 6 x 6A B C D E A B C D E FB A E C D B F D C A EC D A E B C D E F B AD E B A C D A F E C BE C D B A E C A B F D

F E B A D C

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 DB C D AC D A BD 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:

Week 1 2 3 4 5 1 5.83 6.22 7.67 9.43 6.57 D P F C R 2 4.80 7.56 10.34 5.82 9.86 P D C R F

Drivers 3 7.43 11.29 7.01 10.48 9.27 F C R D P 4 6.60 9.54 11.11 10.84 15.05 R F D P C 5 11.24 6.34 11.30 12.58 16.04 C R P F D

The Model for a Latin Experiment

kijjikkijy

i = 1,2,…, t j = 1,2,…, t

yij(k) = the observation in ith row and the jth column receiving the kth treatment

= overall mean

k = the effect of the ith treatmenti = the effect of the ith row

ij(k) = random error

k = 1,2,…, t

j = the effect of the jth column

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 S.S. d.f. M.S. F p-value

Treat SSTr t-1 MSTr MSTr /MSE

Rows SSRow t-1 MSRow MSRow /MSE

Cols SSCol t-1 MSCol MSCol /MSE

Error SSE (t-1)(t-2) MSE

Total SST t2 - 1

The Anova Table for Example


Week 51.17887 4 12.79472 16.06 0.0001

Driver 69.44663 4 17.36166 21.79 0.0000

Car 70.90402 4 17.72601 22.24 0.0000

Error 9.56315 12 0.79693

Total 201.09267 24

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

Tests of Between-Subjects Effects

Dependent Variable: COST

191.530a 12 15.961 20.028 .000

2120.050 1 2120.050 2660.273 .000

69.447 4 17.362 21.786 .000

51.179 4 12.795 16.055 .000

70.904 4 17.726 22.243 .000

9.563 12 .797

2321.143 25

201.093 24

SourceCorrected Model

Intercept

DRIVER

WEEK

CAR

Error

Total

Corrected Total

Type IIISum of

Squares dfMean

Square F Sig.

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

The ANOVA table produced by SPSS

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 diet• C represents the low protein-beef Diet• D represents the low protein-cereal diet• E represents the low protein-pork 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:

Appetite Category 1 2 3 4 5 6 1 62.1 84.3 61.5 66.3 73.0 104.7 A B C D E F 2 86.2 91.9 69.2 64.5 80.8 83.9 B F D C A E

Initial 3 63.9 71.1 69.6 90.4 100.7 93.2 Weight C D E F B A

Category 4 68.9 77.2 97.3 72.1 81.7 114.7 D A F E C B 5 73.8 73.3 78.6 101.9 111.5 95.3 E C A B F D 6 101.8 83.8 110.6 87.9 93.5 103.8 F E B A D C



Inwt 1767.0836 5 353.41673 111.1 0.0000

App 2195.4331 5 439.08662 138.03 0.0000

Diet 4183.9132 5 836.78263 263.06 0.0000

Error 63.61999 20 3.181

Total 8210.0499 35

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


Inwt 1767.0836 5 353.41673 111.1 0.0000

App 2195.4331 5 439.08662 138.03 0.0000

Source 631.22173 2 315.61087 99.22 0.0000

Level 2611.2097 1 2611.2097 820.88 0.0000

SL 941.48172 2 470.74086 147.99 0.0000

Error 63.61999 20 3.181

Total 8210.0499 35

Experimental Design

Of interest: to compare t treatments (the treatment combinations of one or several factors)

The Completely Randomized Design

Treats1 2 3 … t

Experimental units randomly assigned to treatments

The Model for a CR Experiment

ijiijy

i = 1,2,…, t j = 1,2,…, n

yij = the observation in jth observation receiving the ith treatment

= overall mean

i = the effect of the ith treatmentij = random error

The Anova Table for a CR Experiment


Treat SSTr t-1 MST MST /MSE

Error SSE t(n-1) MSE

Randomized Block Design

Blocks

All treats appear once in each block

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

1

2

3

⁞

t

The Model for a RB Experiment

ijjiijy

i = 1,2,…, t j = 1,2,…, b

yij = the observation in jth block receiving the ith treatment

= overall mean

i = the effect of the ith treatment

ij = random error

j = the effect of the jth blockNo interaction between blocks and treatments

• A Randomized Block experiment is assumed to be a two-factor experiment.

• The factors are blocks and treatments.

• It is assumed that there is no interaction between blocks and treatments.

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

The Anova Table for a randomized Block Experiment


Treat SST t-1 MST MST /MSE

Block SSB n-1 MSB MSB /MSE

Error SSE (t-1)(b-1) MSE

The Latin square Design

All treats appear once in each row and each column

Columns

Row

s1

2

3

⁞

t2

3

t

1

13

2

The Model for a Latin Experiment

kijjikkijy

i = 1,2,…, t j = 1,2,…, t

yij(k) = the observation in ith row and the jth column receiving the kth treatment

= overall mean

k = the effect of the ith treatmenti = the effect of the ith row


k = 1,2,…, t


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 Anova Table for a Latin Square Experiment


Treat SSTr t-1 MSTr MSTr /MSE




Total SST t2 - 1

Graeco-Latin Square Designs

Mutually orthogonal Squares

DefinitionA Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters , , , …) 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 B C D E F GB C D E F G A

C D E F G A BD E F G A B CE F G A B C DF G A B C D EG A B C D E F

Note:

There exists at most (t –1) t x t Latin squares L1, L2, …, Lt-1 such that any pair are mutually orthogonal.

e.g. It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L1, L2, L3, L4, L5, L6 .

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:

P e r i o d 1 2 3 4 5 6 7

1 3 0 4 4 3 6 3 5 0 5 0 4 4 1 7 5 1 9 4 3 2 ( A ( B ( C ( D ( E ( F ( G 2 3 8 1 5 0 5 4 2 5 5 6 4 4 9 4 3 5 0 4 1 3 B ( C ( D ( E ( F ( G ( A 3 4 3 2 5 6 6 4 7 9 3 5 7 4 6 1 3 4 0 5 0 2 ( C ( D ( E ( F ( G ( A ( B

C o w s 4 4 4 2 3 7 2 5 3 6 3 6 6 4 9 5 4 2 5 5 0 7 ( D ( E ( F ( G ( A ( B ( C 5 4 9 6 4 4 9 4 9 3 3 4 5 5 0 9 4 8 1 3 8 0

( E ( F ( G ( A ( B ( C ( D

6 5 3 4 4 2 1 4 5 2 4 2 7 3 4 6 4 7 8 3 9 7

( F ( G ( A ( B ( C ( D ( E

7 5 4 3 3 8 6 4 3 5 4 8 5 4 0 6 5 5 4 4 1 0 ( G ( A ( B ( C ( D ( E ( F

The Model for a Greaco-Latin Experiment

klijjilkklijy

i = 1,2,…, t j = 1,2,…, t

yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment

k = 1,2,…, t l = 1,2,…, t

= overall mean

k = the effect of the kth Latin treatment

i = the effect of the ith row



No interaction between rows, columns, Latin treatments and Greek treatments

l = the effect of the lth Greek treatment

• 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 Anova Table for a Greaco-Latin Square Experiment


Latin SSLa t-1 MSLa MSLa /MSE

Greek SSGr t-1 MSGr MSGr /MSE




Total SST t2 - 1



Protein 160242.82 6 26707.1361 41.23 0.0000

Lysine 30718.24 6 5119.70748 7.9 0.0001

Cow 2124.24 6 354.04082 0.55 0.7676

Period 5831.96 6 971.9932 1.5 0.2204

Error 15544.41 24 647.68367

Total 214461.67 48

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Latin Square Designs. Selected Latin Squares 3 x 34 x 4 A B CA B C DA B C DA B C DA B C D B C AB A D...

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