Latin Square Designs

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Latin Square Designs
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• Latin Square Designs
• Selected Latin Squares
• 3 x 3 4 x 4
• A B C A B C D A B C D A B C D A B C D
• B C A B A D C B C D A B D A C B A D C
• C 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 6
• A B C D E A B C D E F
• B A E C D B F D C A E
• C D A E B C D E F B A
• D E B A C D A F E C B
• E C D B A E C A B F D
• F E B A D C

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A Latin Square
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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
•  

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• 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  
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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).

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• 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

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• The Model for a Latin Experiment

i 1,2,, t
j 1,2,, t
k 1,2,, t
yij(k) the observation in ith row and the jth
column receiving the kth treatment
m overall mean
tk the effect of the ith treatment
No interaction between rows, columns and
treatments
ri the effect of the ith row
gj the effect of the jth column
eij(k) random error
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• 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.

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• The Anova Table for a Latin Square Experiment

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• The Anova Table for Example

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Using SPSS for a Latin Square experiment
Trts
Rows
Cols
Y
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Select Analyze-gtGeneral Linear Model-gtUnivariate
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Select the dependent variable and the three
factors Rows, Cols, Treats
Select Model
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Identify a model that has only main effects for
Rows, Cols, Treats
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The ANOVA table produced by SPSS
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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

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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.

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• 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.

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• The weight gain after a fixed period is measured
  for each of the test animals and is tabulated
  below

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• The Anova Table for Example

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

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Graeco-Latin Square Designs
Mutually orthogonal Squares
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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
• Aa Be Cb Df Ec Fg Gd
• Bb Cf Dc Eg Fd Ga Ae
• Cc Dg Ed Fa Ge Ab Bf
• Dd Ea Fe Gb Af Bc Cg
• Ee Fb Gf Ac Bg Cd Da
• Ff Gc Ag Bd Ca De Eb
• Gg Ad Ba Ce Db Ef Fc

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Note

• At most (t 1) t x t Latin squares L1, L2, ,
  Lt-1 such that any pair are mutually orthogonal.

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

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• 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

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• The Model for a Greaco-Latin Experiment

j 1,2,, t
i 1,2,, t
k 1,2,, t
l 1,2,, t
yij(kl) the observation in ith row and the jth
column receiving the kth Latin treatment and the
lth Greek treatment
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m overall mean
tk the effect of the kth Latin treatment
ll the effect of the lth Greek treatment
ri the effect of the ith row
gj the effect of the jth column
eij(k) random error
No interaction between rows, columns, Latin
treatments and Greek treatments
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• 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.

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• The Anova Table for a
• Greaco-Latin Square Experiment

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• The Anova Table for Example