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Latin Square Designs

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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 D A B D C A BD A B CD C B AD C B A 5 x 56 x 6 A B C D EA B C D E F B A E C DB F D C A E C D A E BC D E F B A D E B A CD A F E C B E C D B AE 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,…, tj = 1,2,…, t y ij(k) = the observation in i th row and the j th column receiving the k th treatment = overall mean k = the effect of the i th treatment i = the effect of the i th row ij(k) = random error k = 1,2,…, t j = the effect of the j th column No interaction between rows, columns and treatments

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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 SourceS.S.d.f.M.S.F p-value TreatSS Tr t-1MS Tr MS Tr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-2)MS E TotalSS T t 2 - 1

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The Anova Table for Example SourceS.S.d.f.M.S.F p-value Week 51.17887412.7947216.060.0001 Driver 69.44663417.3616621.790.0000 Car 70.90402417.7260122.240.0000 Error 9.56315120.79693 Total 201.0926724

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Using SPSS for a Latin Square experiment RowsCols Trts Y

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Select Analyze->General Linear Model->Univariate

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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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Latin Square -17 Advantages and Disadvantages Advantages: Allows for control of two extraneous sources of variation. Analysis is quite simple. Disadvantages: Requires t 2 experimental units to study t treatments. Best suited for t in range: 5 t 10. The effect of each treatment on the response must be approximately the same across rows and columns. Implementation problems. Missing data causes major analysis problems.

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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 SourceS.S.d.f.M.S.F p-value Inwt1767.08365353.41673111.10.0000 App2195.43315439.08662138.030.0000 Diet4183.91325836.78263263.060.0000 Error63.61999203.181 Total 8210.049935

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Diet SS partioned into main effects for Source and Level of Protein SourceS.S.d.f.M.S.F p-value Inwt1767.08365353.41673111.10.0000 App2195.43315439.08662138.030.0000 Source631.221732315.6108799.220.0000 Level2611.20971 820.880.0000 SL941.481722470.74086147.990.0000 Error63.61999203.181 Total 8210.049935

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

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The Completely Randomized Design Treats 123…t Experimental units randomly assigned to treatments

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The Model for a CR Experiment i = 1,2,…, tj = 1,2,…, n y ij = the observation in j th observation receiving the i th treatment = overall mean i = the effect of the i th treatment ij = random error

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The Anova Table for a CR Experiment SourceS.S.d.f.M.S.Fp-value TreatSS Tr t-1MS T MS T /MS E ErrorSS E t(n-1)MS E

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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

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The Model for a RB Experiment i = 1,2,…, tj = 1,2,…, b y ij = the observation in j th block receiving the i th treatment = overall mean i = the effect of the i th treatment ij = random error j = the effect of the j th block No interaction between blocks and treatments

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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.

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The Anova Table for a randomized Block Experiment SourceS.S.d.f.M.S.Fp-value TreatSS T t-1MS T MS T /MS E BlockSS B n-1MS B MS B /MS E ErrorSS E (t-1)(b-1)MS E

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The Latin square Design All treats appear once in each row and each column Columns Rows 1 2 3 t 2 3 t 1 13 2

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The Model for a Latin Experiment i = 1,2,…, tj = 1,2,…, t y ij(k) = the observation in i th row and the j th column receiving the k th treatment = overall mean k = the effect of the i th treatment i = the effect of the i th row i j(k ) = random error k = 1,2,…, t j = the effect of the j th column No interaction between rows, columns and treatments

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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 SourceS.S.d.f.M.S.F p-value TreatSS Tr t-1MS Tr MS Tr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-2)MS E TotalSS T t 2 - 1

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