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...
-
Upload
shanna-stafford -
Category
Documents
-
view
219 -
download
1
Transcript of 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...
![Page 1: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/1.jpg)
Latin Square Designs
![Page 2: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/2.jpg)
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
![Page 3: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/3.jpg)
A Latin Square
![Page 4: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/4.jpg)
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
![Page 5: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/5.jpg)
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
![Page 6: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/6.jpg)
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).
![Page 7: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/7.jpg)
• 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
![Page 8: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/8.jpg)
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
![Page 9: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/9.jpg)
• 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.
![Page 10: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/10.jpg)
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
![Page 11: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/11.jpg)
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
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
![Page 12: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/12.jpg)
Using SPSS for a Latin Square experiment
Rows Cols Trts Y
![Page 13: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/13.jpg)
Select Analyze->General Linear Model->Univariate
![Page 14: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/14.jpg)
Select the dependent variable and the three factors – Rows, Cols, Treats
Select Model
![Page 15: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/15.jpg)
Identify a model that has only main effects for Rows, Cols, Treats
![Page 16: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/16.jpg)
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
![Page 17: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/17.jpg)
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
![Page 18: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/18.jpg)
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.
![Page 19: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/19.jpg)
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.
![Page 20: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/20.jpg)
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
![Page 21: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/21.jpg)
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
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
![Page 22: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/22.jpg)
Diet SS partioned into main effects for Source and Level of Protein
Source S.S. d.f. M.S. F p-value
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
![Page 23: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/23.jpg)
Experimental Design
Of interest: to compare t treatments (the treatment combinations of one or several factors)
![Page 24: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/24.jpg)
The Completely Randomized Design
Treats1 2 3 … t
Experimental units randomly assigned to treatments
![Page 25: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/25.jpg)
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
![Page 26: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/26.jpg)
The Anova Table for a CR Experiment
Source S.S. d.f. M.S. F p-value
Treat SSTr t-1 MST MST /MSE
Error SSE t(n-1) MSE
![Page 27: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/27.jpg)
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
![Page 28: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/28.jpg)
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
![Page 29: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/29.jpg)
• 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.
![Page 30: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/30.jpg)
The Anova Table for a randomized Block Experiment
Source S.S. d.f. M.S. F p-value
Treat SST t-1 MST MST /MSE
Block SSB n-1 MSB MSB /MSE
Error SSE (t-1)(b-1) MSE
![Page 31: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/31.jpg)
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
![Page 32: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/32.jpg)
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
![Page 33: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/33.jpg)
• 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.
![Page 34: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/34.jpg)
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
![Page 35: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/35.jpg)
Graeco-Latin Square Designs
Mutually orthogonal Squares
![Page 36: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/36.jpg)
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
![Page 37: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/37.jpg)
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 .
![Page 38: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/38.jpg)
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.
![Page 39: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/39.jpg)
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.
![Page 40: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/40.jpg)
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
![Page 41: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/41.jpg)
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
![Page 42: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/42.jpg)
= overall mean
k = the effect of the kth Latin treatment
i = the effect of the ith row
ij(k) = random error
j = the effect of the jth column
No interaction between rows, columns, Latin treatments and Greek treatments
l = the effect of the lth Greek treatment
![Page 43: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/43.jpg)
• 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.
![Page 44: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/44.jpg)
The Anova Table for a Greaco-Latin Square Experiment
Source S.S. d.f. M.S. F p-value
Latin SSLa t-1 MSLa MSLa /MSE
Greek SSGr t-1 MSGr MSGr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-3) MSE
Total SST t2 - 1
![Page 45: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/45.jpg)
The Anova Table for Example
Source S.S. d.f. M.S. F p-value
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
![Page 46: 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 CB C D AB D A CB A D C C A BC D B AC D A BC A D BC.](https://reader033.fdocuments.us/reader033/viewer/2022051215/56649e205503460f94b0b0b0/html5/thumbnails/46.jpg)
Next topic: Incomplete Block designs