1. Randomized Block Design Blocks All treats appear once in each block

2. Latin Square Designs

3. The Latin square Design All treats appear once in each row and each column Columns Rows

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 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 AF E B A D C

5. 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

6. 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

7. 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).

8. 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:

9. 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

10. 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.

11. 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

12. 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

13. Using SPSS for a Latin Square experiment RowsCols Trts Y

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

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

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

17. The ANOVA table produced by SPSS

18. 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

19. 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.

20. 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.

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

22. 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

23. 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

24. Graeco-Latin Square Designs Mutually orthogonal Squares

25. 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 

26. Note: At most (t –1) t x t Latin squares L 1, L 2, …, L t-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 L 1, L 2, L 3, L 4, L 5, L 6.

27. 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.

28. 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(  ), 4(  ), 6(  ), 8(  ), 10(  ), 12(  ) and 14(  )% for Protein ). Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods.

29. 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:

30. 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

31.  = 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

32. 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.

33. 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

34. 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

35. Incomplete Block Designs

36. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size t. In each block we randomly assign the t treatments to the t experimental units in each block. The ability to detect treatment to treatment differences is dependent on the within block variability.

37. Comments The within block variability generally increases with block size. The larger the block size the larger the within block variability. For a larger number of treatments, t, it may not be appropriate or feasible to require the block size, k, to be equal to the number of treatments. If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

38. Comments regarding Incomplete block designs When two treatments appear together in the same block it is possible to estimate the difference in treatments effects. The treatment difference is estimable. If two treatments do not appear together in the same block it not be possible to estimate the difference in treatments effects. The treatment difference may not be estimable.

39. Example Consider the block design with 6 treatments and 6 blocks of size two. The treatments differences (1 vs 2, 1 vs 3, 2 vs 3, 4 vs 5, 4 vs 6, 5 vs 6) are estimable. If one of the treatments is in the group {1,2,3} and the other treatment is in the group {4,5,6}, the treatment difference is not estimable. 1212 2323 1313 4545 5656 4646

40. Definitions Two treatments i and i* are said to be connected if there is a sequence of treatments i 0 = i, i 1, i 2, … i M = i* such that each successive pair of treatments (i j and i j+1 ) appear in the same block In this case the treatment difference is estimable. An incomplete design is said to be connected if all treatment pairs i and i* are connected. In this case all treatment differences are estimable.

41. Example Consider the block design with 5 treatments and 5 blocks of size two. This incomplete block design is connected. All treatment differences are estimable. Some treatment differences are estimated with a higher precision than others. 1212 2323 1313 4545 1414

42. Definition An incomplete design is said to be a Balanced Incomplete Block Design. 1.if all treatments appear in exactly r blocks. This ensures that each treatment is estimated with the same precision The value of is the same for each treatment pair. 2.if all treatment pairs i and i* appear together in exactly blocks. This ensures that each treatment difference is estimated with the same precision. The value of is the same for each treatment pair.

43. Some Identities Let b = the number of blocks. t = the number of treatments k = the block size r = the number of times a treatment appears in the experiment. = the number of times a pair of treatment appears together in the same block 1.bk = rt Both sides of this equation are found by counting the total number of experimental units in the experiment. 2.r(k-1) = (t – 1) Both sides of this equation are found by counting the total number of experimental units that appear with a specific treatment in the experiment.

44. BIB Design A Balanced Incomplete Block Design (b = 15, k = 4, t = 6, r = 10, = 6)

45. An Example A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal. For this purpose: subjects will be asked to taste and compare these cereals scoring them on a scale of 0 - 100. For practical reasons it is decided that each subject should be asked to taste and compare at most four of the six cereals. For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal.

46. The design and the data is tabulated below:

47. Analysis for the Incomplete Block Design Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10, = 6 denotes summation over all blocks j containing treatment i.

48. Anova Table for Incomplete Block Designs Sums of Squares  y ij 2 = 234382  B j 2 /k = 213188  Q i 2 = 181388.88 Anova Sums of Squares SS total =  y ij 2 –G 2 /bk = 27640.6 SS Blocks =  B j 2 /k – G 2 /bk = 6446.6 SS Tr = (  Q i 2 )/(r – 1) = 20154.319 SS Error = SS total - SS Blocks - SS Tr = 1039.6806

49. Anova Table for Incomplete Block Designs

50. Designs for Estimating Carry-over (or Residual) Effects of Treatments

51. The Cross-over or Simple Reversal Design An Example A clinical psychologist wanted to test two drugs, A and B, which are intended to increase reaction time to a certain stimulus. He has decided to use n = 8 subjects selected at random and randomly divided into two groups of four. –The first group will receive drug A first then B, while –the second group will receive drug B first then A.

52. To conduct the trial he administered a drug to the individual, waited 15 minutes for absorption, applied the stimulus and then measured reaction time. The data and the design is tabulated below:

53. The Switch-back or Double Reversal Design An Example A following study was interested in the effect of concentrate type on the daily production of fat-corrected milk (FCM). Two concentrates were used: –A - high fat; and –B - low fat. Five test animals were then selected for each of the two sequence groups –( A-B-A and B-A-B) in a switch-back design.

54. The data and the design is tabulated below: One animal in the first group developed mastitis and was removed from the study.

55. The Incomplete Block Switch-back Design An Example An insurance company was interested in buying a quantity of word processing machines for use by secretaries in the stenographic pool. The selection was narrowed down to three models (A, B, and C). A study was to be carried out, where the time to process a test document would be determined for a group of secretaries on each of the word processing models. For various reasons the company decided to use an incomplete block switch back design using n = 6 secretaries from the secretarial pool.

56. The data and the design is tabulated below: BIB incomplete block design with t = 3 treatments – A, B and block size k = 2. ABAB ACAC BCBC

57. The Latin Square Change-Over (or Round Robin) Design Selected Latin Squares Change-Over Designs (Balanced for Residual Effects) Period = RowsColumns = Subjects

58. Four Treatments

59. An Example An experimental psychologist wanted to determine the effect of three new drugs (A, B and C) on the time for laboratory rats to work their way through a maze. A sample of n= 12 test animals were used in the experiment. It was decided to use a Latin square Change-Over experimental design.

60. The data and the design is tabulated below:

61. Orthogonal Linear Contrasts This is a technique for partitioning ANOVA sum of squares into individual degrees of freedom

62. Definition Let x 1, x 2,..., x p denote p numerical quantities computed from the data. These could be statistics or the raw observations. A linear combination of x 1, x 2,..., x p is defined to be a quantity,L,computed in the following manner: L = c 1 x 1 + c 2 x 2 +... + c p x p where the coefficients c 1, c 2,..., c p are predetermined numerical values:

63. Definition If the coefficients c 1, c 2,..., c p satisfy: c 1 + c 2 +... + c p = 0, Then the linear combination L = c 1 x 1 + c 2 x 2 +... + c p x p is called a linear contrast.

64. Examples 1. 2. 3.L = x 1 - 4 x 2 + 6x 3 - 4 x 4 + x 5 = (1)x 1 + (-4)x 2 + (6)x 3 + (-4)x 4 + (1)x 5 A linear combination A linear contrast

65. Definition Let A = a 1 x 1 + a 2 x 2 +... + a p x p and B= b 1 x 1 + b 2 x 2 +... + b p x p be two linear contrasts of the quantities x 1, x 2,..., x p. Then A and B are c called Orthogonal Linear Contrasts if in addition to: a 1 + a 2 +... + a p = 0 and b 1 + b 2 +... + b p = 0, it is also true that: a 1 b 1 + a 2 b 2 +... + a p b p = 0..

66. Example Let Note:

67. Definition Let A = a 1 x 1 + a 2 x 2 +... + a p x p, B= b 1 x 1 + b 2 x 2 +... + b p x p,..., and L= l 1 x 1 + l 2 x 2 +... + l p x p be a set linear contrasts of the quantities x 1, x 2,..., x p. Then the set is called a set of Mutually Orthogonal Linear Contrasts if each linear contrast in the set is orthogonal to any other linear contrast..

68. Theorem: The maximum number of linear contrasts in a set of Mutually Orthogonal Linear Contrasts of the quantities x 1, x 2,..., x p is p - 1. p - 1 is called the degrees of freedom (d.f.) for comparing quantities x 1, x 2,..., x p.

69. Comments 1.Linear contrasts are making comparisons amongst the p values x 1, x 2,..., x p 2.Orthogonal Linear Contrasts are making independent comparisons amongst the p values x 1, x 2,..., x p. 3.The number of independent comparisons amongst the p values x 1, x 2,..., x p is p – 1.

70. Definition denotes a linear contrast of the p means If each mean,, is calculated from n observations then: The Sum of Squares for testing the Linear Contrast L, is defined to be:

71. the degrees of freedom (df) for testing the Linear Contrast L, is defined to be the F-ratio for testing the Linear Contrast L, is defined to be:

72. Theorem: Let L 1, L 2,..., L p-1 denote p-1 mutually orthogonal Linear contrasts for comparing the p means. Then the Sum of Squares for comparing the p means based on p – 1 degrees of freedom, SS Between, satisfies:

73. Comment Defining a set of Orthogonal Linear Contrasts for comparing the p means allows the researcher to "break apart" the Sum of Squares for comparing the p means, SS Between, and make individual tests of each the Linear Contrast.

74. The Diet-Weight Gain example The sum of Squares for comparing the 6 means is given in the Anova Table:

75. Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) : (A comparison of the High protein diets with Low protein diets) (A comparison of the Beef source of protein with the Pork source of protein)

76. (A comparison of the Meat (Beef - Pork) source of protein with the Cereal source of protein) (A comparison representing interaction between Level of protein and Source of protein for the Meat source of Protein) (A comparison representing interaction between Level of protein with the Cereal source of Protein)

77. The Anova Table for Testing these contrasts is given below: The Mutually Orthogonal contrasts that are eventually selected should be determine prior to observing the data and should be determined by the objectives of the experiment

78. Another Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) : (A comparison of the High protein diets with Low protein diets) (A comparison of the Beef source of protein with the Pork source of protein)

79. (A comparison of the high and low protein diets for the Beef source of protein) (A comparison of the high and low protein diets for the Cereal source of protein) (A comparison of the high and low protein diets for the Pork source of protein)

80. The Anova Table for Testing these contrasts is given below:

81. Orthogonal Linear Contrasts Polynomial Regression

82. Orthogonal Linear Contrasts for Polynomial Regression

84. Example In this example we are measuring the “Life” of an electronic component and how it depends on the temperature on activation

85. The Anova Table SourceSSdfMSF Treat6604165.023.57 Linear187.501187.5026.79 Quadratic433.931433.9361.99 Cubic0.0010.000.00 Quartic38.57138.575.51 Error70107.00 Total73014 L = 25.00Q 2 = -45.00C = 0.00Q 4 = 30.00

86. The Anova Tables for Determining degree of polynomial Testing for effect of the factor

87. Testing for departure from Linear

88. Testing for departure from Quadratic

90. Post-hoc Tests Multiple Comparison Tests

91. Suppose we have p means An F-test has revealed that there are significant differences amongst the p means We want to perform an analysis to determine precisely where the differences exist.

92. Tukey’s Multiple Comparison Test

93. Let Tukey's Critical Differences Two means are declared significant if they differ by more than this amount. denote the standard error of each = the tabled value for Tukey’s studentized range p = no. of means, = df for Error

94. Scheffe’s Multiple Comparison Test

95. Scheffe's Critical Differences (for Linear contrasts) A linear contrast is declared significant if it exceeds this amount. = the tabled value for F distribution (p -1 = df for comparing p means, = df for Error)

96. Scheffe's Critical Differences (for comparing two means) Two means are declared significant if they differ by more than this amount.