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Latin Square Designs

Latin Square Designs. 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

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Latin Square Designs

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  1. Latin Square Designs

  2. Latin Square Designs Selected Latin Squares 3 x 34 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 56 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

  3. A Latin Square

  4. 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

  5. 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

  6. 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).

  7. 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:

  8. 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

  9. 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.

  10. The Anova Table for a Latin Square Experiment

  11. The Anova Table for Example

  12. Using SPSS for a Latin Square experiment Trts Rows Cols Y

  13. Select Analyze->General Linear Model->Univariate

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

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

  16. The ANOVA table produced by SPSS

  17. Example 2 In this Experiment thewe 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

  18. 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.

  19. 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.

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

  21. The Anova Table for Example

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

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

  24. The Completely Randomized Design Treats 1 2 3 … t Experimental units randomly assigned to treatments

  25. The Model for a CR Experiment i = 1,2,…, t j = 1,2,…, n yij = the observation in jth observation receiving the ith treatment m = overall mean ti = the effect of the ith treatment eij = random error

  26. The Anova Table for a CR Experiment

  27. Randomized Block Design Blocks 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ t t t t t t t t t All treats appear once in each block

  28. The Model for a RB Experiment i = 1,2,…, t j = 1,2,…, b yij = the observation in jth block receiving the ith treatment m= overall mean ti= the effect of the ith treatment No interaction between blocks and treatments bj= the effect of the jth block eij = random error

  29. 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.

  30. The Anova Table for a randomized Block Experiment

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

  32. 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

  33. 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.

  34. The Anova Table for a Latin Square Experiment

  35. Graeco-Latin Square Designs Mutually orthogonal Squares

  36. 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

  37. Note: There exists at most (t –1) t x t Latin squaresL1, 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 .

  38. 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.

  39. 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.

  40. 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:

  41. 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

  42. 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

  43. 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.

  44. The Anova Table for a Greaco-Latin Square Experiment

  45. The Anova Table for Example

  46. Next topic: Incomplete Block designs

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