1. Latin Square Designs

2. 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

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,…, 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

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 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

11. 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

12. Using SPSS for a Latin Square experiment RowsCols Trts 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 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

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 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

22. 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

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

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

25. 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

26. 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

27. 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

28. 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

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 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

31. 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

32. 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

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 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

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 , , , …) 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  G  B  C  D  E  F  G  A  C  D  E  F  G  A  B  D  E  F  G  A  B  C  E  F  G  A  B  C  D  F  G  A  B  C  D  E  G  A  B  C  D  E  F 

37. Note: There exists at most (t –1) t x t Latin squares L 1, L 2, …, L t-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 L 1, L 2, L 3, L 4, L 5, L 6.

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 i = 1,2,…, t j = 1,2,…, t y ij(kl) = the observation in i th row and the j th column receiving the k th Latin treatment and the l th Greek treatment k = 1,2,…, t l = 1,2,…, t

42.  = overall mean  k = the effect of the k th Latin treatment  i = the effect of the i th row  ij(k) = random error  j = the effect of the j th column No interaction between rows, columns, Latin treatments and Greek treatments l = the effect of the l th Greek treatment

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 SourceS.S.d.f.M.S.F p-value LatinSS La t-1MS La MS La /MS E GreekSS Gr t-1MS Gr MS Gr /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-3)MS E TotalSS T t 2 - 1

45. The Anova Table for Example SourceS.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

46. Next topic: Incomplete Block designsIncomplete Block designs