Design of Experiments chapter 4 - 7.pptx

This is a class note prepared from Book Montgomery. It introduce different design of experiments also it includes examples.

1. Design and Analysis of Experiments (Stat 2103) March, 2018 1

2. 4. Blocking 2

3. Introduction Blocking A block is a group of homogeneous experimental units Maximize the variation among blocks in order to minimize the variation within blocks Reasons for blocking To remove block to block variation from the experimental error (increase precision) Treatment comparisons are more uniform Increase the information by allowing the researcher to sample a wider range of conditions 3

4. Randomized Block Design A  nuisance  factor is a factor that has some effect on the response, but is of no interest to the experimenter; however, the variability it transmits to the response needs to be minimized or explained.  Blocking  is a technique for dealing with  nuisance   factors. If the nuisance variable is  known  and  controllable , we use  blocking  and control it by including a blocking factor in our experiment. 4

5. Randomized Block Design If you have a nuisance factor that is  known  but  uncontrollable , sometimes we can use  analysis of covariance  to measure and remove the effect of the nuisance factor from the analysis. Many times there are nuisance factors that are  unknown  and  uncontrollable  (sometimes called a  “lurking” variable) , use  randomization  to balance out their impact. Randomization is our insurance against a systematic bias due to a nuisance factor. 5

6. Randomized Block Design Criteria for blocking Proximity or known patterns of variation in the field gradients due to fertility, soil type animals (experimental units) in a pen (block) Time planting, harvesting Management of experimental tasks individuals collecting data runs in the laboratory 6

7. Randomized Block Design Criteria for blocking Physical characteristics height, maturity Natural groupings branches (experimental units) on a tree (block) 7

8. Randomized Block Design Demonstration ( The Hardness Testing Example The Hardness Testing Example) We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester Assignment of the tips to an experimental unit ; that is, a test coupon Structure of a completely randomized experiment The test coupons are a source of nuisance variability Alternatively, the experimenter may want to test the tips across coupons of various hardness levels The need for blocking 8

9. Randomized Block Design Demonstration ( The Hardness Testing Example The Hardness Testing Example) To conduct this experiment as a RCBD, assign all 4 tips to each coupon Each coupon is called a “ block ”; that is, it’s a more homogenous experimental unit on which to test the tips Variability between blocks can be large, variability within a block should be relatively small In general, a block is a specific level of the nuisance factor 9

10. Randomized Block Design Demonstration ( The Hardness Testing Example The Hardness Testing Example) A complete replicate of the basic experiment is conducted in each block A block represents a restriction on randomization All runs within a block are randomized 10

11. Randomized Block Design Demonstration ( The Hardness Testing Example The Hardness Testing Example) Suppose that we use b = 4 blocks: Notice the two-way structure of the experiment We are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks) 11

12. Randomized Complete Block Design Let y ij be the response for the ith treatment in the jth block. The standard model for an RCBD has a grand mean, a treatment effect, a block effect, and experimental error y ij = µ + + β j + ε ij This standard model says that treatments and blocks are additive, treatments have the same effect in every block and blocks only serve to shift the mean response up or down.   12

13. Randomized Complete Block Design The quantities are 13

14. Randomized Complete Block Design (Sum of Squares) ANOVA Partitioning of Total Sum of Squares 14

15. Randomized Complete Block Design (Sum of Squares) ANOVA Partitioning of Total Sum of Squares 15

16. Randomized Complete Block Design (Sum of Squares) ANOVA Table 16

17. Randomized Complete Block Design (Example) A hardness testing machine operates by pressing a tip into a metal test “coupon.” The hardness of the coupon can be determined from the depth of the resulting depression. Four tip types are being tested to see if they produce significantly different readings. However, the coupons might differ slightly in their hardness (for example, if they are taken from ingots produced in different heats). 17

18. Randomized Complete Block Design (Example) 18 Coupon (Block) Type of Tip 1 2 3 4 y i . yi. 2 1 9.3 9.4 9.6 10 38.3 1466.89 2 9.4 9.3 9.8 9.9 38.4 1474.56 3 9.2 9.4 9.5 9.7 37.8 1428.84 4 9.7 9.6 10 10.2 39.5 1560.25 y .j 37.6 37.7 38.9 39.8 y .. =154 ∑y i. 2 = 5930.54 y .j 2 1413.76 1421.29 1513.21 1584.04 ∑y .j 2 = 1483.075

19. Randomized Complete Block Design (Example) The hypothesis H : All tips give the same mean reading Ha : At least two tips give different mean readings. Decision: Reject , the mean measurement hardness from the four tips is not same 19 Source SS df MS F P-Value Treatment (Tip) 0.385 3 0.1283 14.4375 0.0009 Coupon (block) 0.825 3 0.2750 30.9375 Error 0.08 9 0.0089 Total 1.29 15

20. Randomized Complete Block Design (Example) Hardness as Completely randomized design Compare CRD and RCBD ? Which reduces noise? 20 Source SS df MS F Treatment (Tip) 0.385 3 0.1283 1.70 Error 0.905 12 0.0754 Total 1.29 15

21. RCBD or CRD ? Which is better, a RCBD or a CRD? Can check using “Relative Efficiency” which compares the variance of the estimate of the ith treatment mean under the two different experiment designs: Efficiency is calculated as the number of observations that would be required if the experiment had been conducted as a CRD without any blocking . 21

22. RCBD or CRD ? If the blocking was not helpful, then the relative efficiency equals 1. The larger the relative efficiency is, the more efficient the blocking was at reducing the error variance. The value can be interpreted as the ratio where n is the number of experimental units that would have to be assigned to each treatment if a CRD had been performed instead of a RCBD. 22

23. RCBD or CRD ? Example (Hardness test) This implies that it would have taken more than 8.47 times as many experimental units/treatment to get the same MSE as we got using the coupon as blocks. We would have needed approximately 34 (≈ 8.47*4) coupons per treatment in a CRD experiment testing the four types of tips. 23

24. Multiple Comparison RCBD When a significant result is found, determine where the difference lies Multiple comparison discussed in chapter 3 works Minor modification Replace number of replicates ( n ) in CRD by number of blocks ( b ) Replace error degrees of freedom ( a(n-1 ) ) in CRD by (a-1)(b-1) in RCBD. Refer : Scheffe , Tukey , LSD, Duncan, Bonferroni 24

25. Latin Square Design Latin square designs are used to simultaneously control (or eliminate) two sources of nuisance variability . It is a design in which each treatment occurs once and only once in each row and column The number of rows, columns and treatments are equal The total number of observations is p x p Example: Machines are to be tested to see whether they differ significantly in their ability to produce a manufactured part. Different operators and different time periods in the work day are known to have an effect on production. 25

26. Latin Square Design Examples of Latin square design 26

27. Latin Square Design (ANOVA Table) 27

28. Latin Square Design (Example) Five different formulations of a rocket propellant Five different materials , and five operators Two nuisance factors This is a 5x5 Latin square design 28

29. Latin Square Design (Example) Coding (by subtracting 25 from each observation) 29

30. Latin Square Design (Example) The sum of squares for the total, Batches (rows), (Operators) columns are computed as follows 30

31. Latin Square Design (Example) 31

32. Latin Square Design (Example) Analysis of variance for the rocket propellant There is a significant difference in the means of formulations 32

33. Graeco -Latin Square Design There is a single factor of primary interest, typically called the treatment factor, and several nuisance factors. For Latin square designs there are 2 nuisance factors, for Graeco -Latin square designs there are 3 nuisance factors a Graeco -Latin square design is a p x p tabular grid in which p is the number of levels of the treatment factor. However, it uses 3 blocking variables instead of the 2 used by the Latin square design. 33

34. Graeco -Latin Square Design A 4x4 Graeco -Latin square design 34

35. Graeco -Latin Square Design ANOVA table for Graeco -Latin square design 35

36. Graeco -Latin Square Design A 4x4 Graeco -Latin square design 36

37. Graeco -Latin Square Design (Example) Five different formulations of a rocket propellant Five different materials, five operators, and five different test assemblies The following 5x5 Graeco -Latin Square s found 37

38. Graeco -Latin Square Design (Example) The sum of squares for the total, Batches (rows), (Operators) columns are computed as follows 38

39. Graeco -Latin Square Design (Example) 39

40. Graeco -Latin Square Design (Example) 40

41. Graeco -Latin Square Design (Example) The ANOVA table 41

42. Balanced Incomplete Block Design (BIBD) A BIBD is a design in which There are a treatments There are b blocks There are k (k<a) experimental units in each block, k is block size Each treatment occurs in the same number ( r times) Each treatments occur together in the same block the same number of times, λ (each pair of treatments occur together in λ blocks) . Each block, k different treatments are randomly assigned to the experimental units 42

43. Balanced Incomplete Block Design (BIBD) N = a ( r ) = b ( k ), total number of subjects λ must be an integer Certain combination of a, r, k, b , and λ are possible Consider the following examples Each pair of treatments occurs λ=2 times In each block there are k=3 experimental units Each treatment occurs r=3 times 43 Treatment (a=4) Block(b=4) 1 2 3 4 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1

44. Balanced Incomplete Block Design (BIBD) Consider the following examples Each pair of treatments occurs λ times? In each block there are k experimental units? Each treatment occurs r times? Is it BIBD ? 44 Treatment (a=4) Block(b=4) 1 2 3 4 1 1 1 2 1 1 3 1 1 4 1 1

45. Balanced Incomplete Block Design (BIBD) 45

46. Balanced Incomplete Block Design (BIBD) 46

47. Balanced Incomplete Block Design (BIBD) The adjusted treatment sum of square is 47

48. Balanced Incomplete Block Design (BIBD) The adjusted treatment sum of square is 48

49. Balanced Incomplete Block Design (Example) Suppose a chemical engineer thinks that the time of reaction for a chemical process is a function of the type of catalyst employed. Four catalysts are being investigated. Variation in the batches of raw material may affect the performance of the catalysts, the engineer decides the use batches of raw material as a block. However each batch is only large enough to permit three catalyst to be run . 49

50. Balanced Incomplete Block Design (Example) There are a=4, b=4, k=3, r=3, λ=2, and N=12 50

51. Balanced Incomplete Block Design (Example) The adjusted treatment totals are 51

52. Balanced Incomplete Block Design (Example) The Analysis of variance table 52

53. 5. Factorial Design 53

54. Basic Definition and Principles Two factors A and B are said to be crossed if every level of A occurs with every level of factor B, and vice versa. The various combinations of the level of factors ( i.e treatments) are some times called treatment combinations Let factor A have two levels(L,H) and factor B has two levels (L, H) When the factors crossed the treatment combinations would be as follows: Treatment 1 2 3 4 LL LH HL HH 54

55. Basic Definition and Principles A factorial experiment allows investigation into the effect of two or more factors on the mean value of a response. Various combinations of factor ‘levels’ can be examined. The effect of a factor is defined to be a change in response produced by a change in the level of the factor This is also called main effect - primary interest The different categories within each factor are called levels Denote different factors by upper case letters (A, B, C, etc ) and different levels by lower case letters with subscripts 55

56. Two factor design without interaction Let two factors ( A and B) having two levels each The main effect of factor A is the difference between average response at low level of A and average response at high level of A Increasing factor A from low level to high level causes an average response increase of 21 units 56

57. Two factor design without interaction Similarly, the main effect of B is In some experiments, we may find that the difference in response between the levels of one factor is not the same at all levels of the other factor. When this occurs, there is an interaction between the factors 57

58. Two factor design with interaction Consider the two factor experiment shown below The effect of A depends on the level chosen for factor B 58

59. Two factor design with interaction This indicates existence of interaction between A and B. The magnitude of interaction effect is the average difference in these two A effects, When an interaction is large, the corresponding main effects have little practical meaning. A significant interaction will often mask the significance of main effects. 59

60. Advantages of Factorial Designs More efficient than one factor at a time experiments Necessary when interactions may be present to avoid misleading conclusions Factorial designs allow the effect of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions. 60

61. The Two Factor Factorial Design The simplest type of factorial designs involve only two factors or treatments There are a levels of factor A and b levels of factor B, and these are arranged in a factorial design, that is, each replicate of the experiment contains all ab treatment combinations. In general there are n replicates 61

62. The Two Factor Factorial Design The observations in a factorial experiment can be described by a model. The effects model 62

63. The Two Factor Factorial Design Hypothesis to be tested 63

64. The Two Factor Factorial Design Statistical Analysis The total corrected sum of squares 64

65. The Two Factor Factorial Design Simplified form of Sum of squares 65

66. The Two Factor Factorial Design (ANOVA Table) To get sum of squares of error at least two replications is required. 66

67. The Two Factor Factorial Design Expected values for mean of squares 67

68. The Two Factor Factorial Design ( Example ) The Battery Design Experiment : An engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select at this point is the plate material for the battery, and he has three possible choices. When the device is manufactured and is shipped to the field, the engineer has no control over the temperature extremes that the device will encounter, and he knows from experience that temperature will probably affect the effective battery life. However, temperature can be controlled in the product development laboratory for the purpose of the test. All three plate materials tested at three temperature levels (15, 70, 125 o F ), four batteries are tested at each combinations plate material and temperature, and all 36 tests are performed in random order. 68

69. The Two Factor Factorial Design (Example) The life (in hours) data is as follows Two questions: What effects do material type and temperature have on the life of the battery? Is there a choice of material that would give uniformly long life regardless of temperature? 69

70. The Two Factor Factorial Design (Example) 70

71. The Two Factor Factorial Design (Example) 71

72. The Two Factor Factorial Design (Example) 72

73. The Two Factor Factorial Design (Example) There is significant effect of Material Type Temperature Temperature × material type Caution on interpreting main effects 73

74. The Two Factor Factorial Design (Example) In general longer life is attained at low temperature, regardless of material type Changing from low to intermediate temperature, battery life with material type 3 actually increases, decreases for types 1 and 2. From intermediate to high temperature, battery life decreases for material type 2 and 3, and unchanged for type 1. Material type 3 seems to give the best results if engineer want less loss of effective life as temperature changes 74

75. The Two Factor Factorial Design (Example) Checking interaction for single replication Test developed by Tukey : Partitions the residual sum of squares into a single degree-of-freedom component due to non- aditivity (interaction) and a component for error with (a-1)(b-1)-1 degrees of freedom 75

76. The Two Factor Factorial Design (Example) Impurity present in a chemical product is affected by two factors-pressure and temperature. The data from a single replicate of a factorial experiment are 76

77. The Two Factor Factorial Design (Example) 77

78. The Two Factor Factorial Design (Example) 78

79. The Two Factor Factorial Design (Example) 79

80. Three factor factorial design 80

81. Three factor factorial design 81

82. Three factor factorial design ANOVA Table 82

83. Three factor factorial design (Example) Soft Drink Bottling Problem A soft drink bottler is interested in obtaining more uniform fill heights in the bottles produced by his manufacturing process. An experiment is conducted to study three factors of the process, which are the percent carbonation (A): 10, 12, 14 percent the operating pressure (B): 25, 30 psi the line speed (C): 200, 250 bpm The response is the deviation from the target fill height. Each combination of the three factors has two replicates and all 24 runs are performed in a random order. The experiment and data are shown below 83

84. Three factor factorial design (Example) Soft Drink Bottling Problem 84

85. Three factor factorial design (Example) Soft Drink Bottling Problem 85

86. Three factor factorial design (Example) Soft Drink Bottling Problem 86

87. Three factor factorial design (Example) Soft Drink Bottling Problem 87

88. Three factor factorial design (Example) Soft Drink Bottling Problem Sum of squares of error by subtraction 88

89. Three factor factorial design (Example) ANOVA Table: Soft Drink Bottling Problem 89

90. Three factor factorial design (Example) 90

91. Blocking in factorial design We have discussed factorial experiments in a completely randomized design way. We often need to eliminate the influence of extraneous factors when running an experiment. We do this by "blocking". Consider a factorial experiment with two factors (A and B) with n replicates. The linear statistical model is 91

92. Blocking in factorial design Now suppose to run this experiment a particular raw material is needed Run each replicates in a separate raw material The batches of raw materials represent a randomization restriction or a block, and a single replicate of a complete factorial experiment is run within each block 92

93. Blocking in factorial design 93

94. Blocking in factorial design(Example) 94

95. Blocking in factorial design(Example) ANOVA table 95

96. The 2 k factorial design The 2 k designs are a major set of building blocks for many experimental designs. These designs are usually referred to as screening designs. The 2 k refers to designs with k factors where each factor has just two levels. These designs are created to explore a large number of factors, with each factor having the minimal number of levels, just two. By screening we are referring to the process of screening a large number of factors that might be important in your experiment, with the goal of selecting those important for the response that you're measuring. 96

97. The 2 k factorial design The 2 2 factorial design The simplest case is 2 k where k = 2. We will define a new notation which is known as Yates notation. We will refer to our factors using the letters A, B, C, D, etc. as arbitrary labels of the factors. In the chemical process case A is the concentration of the reactant and B is the amount of catalyst , both of which are quantitative. The yield of the process is our response variable. Since there are two levels of each of two factors, 2 k equals four treatment combinations. 97

98. The 2 2 factorial design The experiment is replicated three times, so there are 12 runs. The order in which the runs are made is random , so this is a completely randomized experiment . The data obtained are as follows: 98

99. The 2 2 factorial design The four treatment combinations in this design are shown graphically in Figure below : You can see that we have 3 observations at each of 4 = 2 k combinations for k = 2. So we have n = 3 replicates. 99

100. The 2 2 factorial design The Yates notation used for denoting the factor combinations is as follows: We use "(1)" to denote that both factors are at the low level, "a" for when A is at its high level and B is at its low level, "b" for when B is at its high level and A is at its low level, and " ab " when both A and B factors are at their high level. The use of this Yates notation indicates the high level of any factor simply by using the small letter of that level factor. 100

101. The 2 2 factorial design This notation actually is used for two purposes. One is to denote the total sum of the observations at that level. In the case below b = 60 is the sum of the three observations at the level b . 101

102. The 2 2 factorial design 102

103. The 2 2 factorial design Practical interpretation? Increasing reactant concentration increases yield Catalyst effect is negative Interaction effect is relatively smaller 103

104. The 2 2 factorial design (Sum of squares) Consider the sum of square for A, B, and AB. Contrast is used to estimate effect of A, the following contrast is total effect of A; Contrast is used to estimate effect of B, the following contrast is total effect of B; Contrast is used to estimate effect of AB, the following contrast is total effect of AB; The three contrasts are orthogonal. 104

105. The 2 2 factorial design (Sum of squares) The sum of squares for any contrast can be computed (chapter 3), Which states that the contrast sum of squares is equal to the contrast squared divided by the number of observations in each total the contrast times the sum of squares of the contrast coefficients 105

106. The 2 2 factorial design (Sum of squares) The sum of squares are 106

107. The 2 2 factorial design The main effects are statistically significant and There is no interaction between these factors 107

108. The 2 3 factorial design Here is an example in three dimensions, with factors A, B and C. Below is a figure of the factors and levels as well as the table representing this experimental space. 108

109. The 2 3 factorial design 109

110. The 2 3 factorial design Consider estimating main effects The effect of A when B and C are at low level is The effect of A when B at high and C at low level is The effect of A when B at low and C at high levels is The effect of A when B and C are at high level is The average effect of A is the average of these four effects The average effect of B is the average of these four effects The average effect of C is the average of these four effects 110

111. The 2 3 factorial design The average effect of AB interaction The average effect of AC interaction is The average effect of BC interaction is The average effect of ABC interaction is 111

112. The 2 3 factorial design Sum of squares for effects are Example: refer Soft drink bottling problem (each at two levels, eliminate 15% carbonation 112

113. The 2 3 factorial design Sum of squares for effects are Example: refer Soft drink bottling problem (each at two levels, eliminate 15% carbonation 113

114. The 2 3 factorial design 114

115. The 2 3 factorial design 115

116. The 2 3 factorial design The effects estimates, sum of squares and percent contribution. The percentage contribution is often a rough but effective guide to the relative importance of each model term Main effects dominate this process, accounting for over 87 percent of the total variability 116

117. The 2 3 factorial design The sum of squares are 117

118. 6. Nested and Split Plot Design 118

119. Nested Design Factors A ( a levels)and B ( b levels) are considered crossed if – Every combinations of A and B ( ab of them) occurs; – An example: Factor B is considered nested under A ( a levels) if – Levels of B are similar for different levels of A; – But levels of B are not identical for different levels of A. – An example (two-stage nested design): 119

120. Nested Design Consider a company that purchases material from three suppliers The material comes in batches Is the purity of the material uniform? Experimental design Select four batches at random from each supplier Make three purity determinations from each batch 120

121. Nested Design (Two stage) Note: batches from each supplier are unique for that particular supplier!! Statistical Model and ANOVA Bracket notation represents nesting factor Here factor B(level j) is nested under factor A(level i) Cannot include interaction 121

122. Nested Design (Two Stage) Sum of Squares Anova Table 122

123. Nested Design-Two stage Two-Factor Nested Model with Fixed Effects Two-Factor Nested Model with Random Effects 123

124. Nested Design-Two stage Two-Factor Nested Model with Mixed Effects 124

125. Nested Design-Two stage (Example) Example 125

126. Nested Design The sum of squares are Also note that: SS E =SS T -SS A -SS B(A) 126

127. Nested Design The ANOVA table Suppliers are fixed and batches random There is no significant effect on purity due to suppliers, but The purity of batches of raw material from the same supplier does differ significantly. 127

128. Nested Design –Three Stage 128

129. Nested Design- Three Stagae 129

130. Split Plot Design Split-plot designs are needed when the levels of some treatment factors are more difficult to change during the experiment than those of others . or Useful when the nature of the experiment requires the use of large experimental units for some factors and smaller experimental units for others The designs have a nested blocking structure: split plots are nested within whole plots, which may be nested within blocks. 130

131. Split Plot Design Split-plot designs have three main characteristics: i . The levels of all the factors are not randomly determined and reset for each experimental run. Did you hold a factor at a particular setting and then run all the combinations of the other factors? ii. The size of the experimental unit is not the same for all experimental factors. Did you apply one factor to a larger unit or group of units involving combinations of the other factors? iii. There is a restriction on the random assignment of the treatment combinations to the experimental units. Is there something that prohibits assigning the treatments to the units completely randomly? 131

132. Split Plot Design Example An experiment is to compare the yield of three varieties of oats (factor A with a=3 levels) and four different levels of fertilizer (factor B with b=2 levels). Suppose 2 farmers agree to participate in the experiment and each will designate a farm field for the experiment (blocking factor with s=2 levels). Since it is easier to plant a variety of oat in a large field, the experimenter uses a split-plot design as follows : 132

133. Split Plot Design The blocks are divided into three (equal sized) large experimental units called whole plots. The three levels of factor A (oats) are randomly assigned to these whole plots (each plot is assigned a variety of oat according to a randomized block design). Each whole plot is sub divided into two smaller experimental units called sub plots (split-plots) and the two levels of fertilizer are randomly assigned to the 2 split plots. Remark: randomization in a split plot design is completed in two stages 133

134. Split Plot Design A2 A1 A3 A3 A1 A2 134 B1 B2 B2 B1 B1 B2 B2 B1 B2 B1 B1 B2 subplots Whole plots Block 1 Block 2

135. Split Plot Design (Example) The general model for a two factor split plot experiment with in RCBD With r random blocks Fixed factor A with a levels Fixed factor B with b levels 135 Whole plot error Sub plot error Blocking/replicate Main factor A effect Main factor B effect

136. Split Plot Design ( ANOVA Table) 136 Source df Sum of Square E(MS) Blocks (R) r-1 Ss block A a-1 SS A Whole plot error (RA) (r-1)(a-1) SS RA B b-1 SS B AB (a-1)(b-1) SS AB Sub plot error a(r-1)(b-1) SS Error Total rab-1 SS T

137. Split Plot Design ( ANOVA Table) 137

138. Split Plot Design ( Example) An experiment was conducted to compare the yield of two varieties of wheat. An additional factor to be considered is type of spray for weeds and three different brands were to be considered. Two farms were selected for the study and for each farm three fields were available for planting. It was deemed impractical to use different sprays in a field; there fore a split plot design was utilized. At each farm the three sprays were randomly assigned to the fields (whole plot), with the restriction that there be one spray used per filed. Each field was divided into two subplots, again with the restriction that each variety is used in exactly one subplot in each field. The response variable is the yield in quintals 138

139. Split Plot Design ( Example) The result is as follows 139 Farm 1 Variety Spray 1 2 3 1 71 64 84 2 66 56 82 2 1 83 77 97 2 79 73 88

140. Split Plot Design ( Example) The ANOVA table The mean square for Spray is: 421.08, p-value=0.0177 (look at the anova table) The whole plot error is larger than sub plot error 140 Source SS Df MS F P-value Farm 456.33 1 456.33 82.97 0.0028 Spray 842.17 2 421.03 76.56 0.027 Farm*Spray 15.17 2 7.58 1.38 0.3761 Variety 85.33 1 85.33 15.52 0.0292 Spray*Variety 1.17 2 0.58 0.11 0.9026 Error 16.5 3 5.50 Total 1416.67 11

141. Split Plot Design an experiment in which a researcher is interested in studying the effect of technicians, dosage strength and wall thickness of the capsule on absorption time of a particular type of antibiotic. There are three technicians, three dosage strengths and four capsule wall thicknesses resulting in 36 observations per replicate and the experimenter wants to perform four replicates on different days. To do so, first, technicians are randomly assigned to units of antibiotics which are the whole plots. Next, the three dosage strengths are randomly assigned to split-plots. Finally, for each dosage strength, the capsules are created with different wall thicknesses, which is the split-split factor and then tested in random order. 141

142. 7. Analysis of Covariance (ANCOVA) 142

143. Introduction A ‘classic’ ANOVA tests for differences in mean responses to categorical factor (treatment) levels. When we have heterogeneity in experimental units sometimes restrictions on the randomization (blocking) can improve the test for treatment effects. In some cases, we don’t have the opportunity to construct blocks, but can recognize and measure a continuous variable as contributing to the heterogeneity in the experimental units. 143

144. Introduction These sources of extraneous variability historically have been referred to as ‘nuisance’ or ‘concomitant’ variables. When a continuous covariate is included in an ANOVA we have the analysis of covariance (ANCOVA). Inclusion of covariates in ANCOVA models often means the difference between concluding there are or are not significant differences among treatment means using ANOVA. ANCOVA by definition is a general linear model that includes both ANOVA (categorical) predictors and Regression (continuous) predictors. 144

145. ANCOVA for CRD An appropriate statistical model is The Model assumes that: . . . . the regression coefficients for each treatment are identical . the treatment effects sum to zero 145

146. ANCOVA for CRD the concomitant variable x ij is not affected by the treatments . all treatment regression lines have identical slopes . If the treatments interact with the covariates this can result in non-identical slopes. Covariance analysis is not appropriate in these cases. Estimating and comparing different regression models is the correct approach . . . with a(n-1)- 1 df 146

147. ANCOVA for CRD If no effect of treatment then the model will be distributed 147

148. ANCOVA for CRD . 148

149. ANCOVA for CRD Adjusted mean estimates are as follows Standard error of any adjusted treatment mean is has distribution 149

150. ANCOVA for CRD 150

151. ANCOVA for CRD Cross products are computed as follows 151

152. ANCOVA for CRD Example: Three different machines produce monofilament fiber for a textile company. Strength is also affected by thickness of monofilament. The experiment is conducted and the results are displayed below? 152

153. ANCOVA for CRD Cross products 153

154. ANCOVA for CRD Cross products 154

155. ANCOVA for CRD Cross products 155

156. ANCOVA Test Statistic The estimate of the regression coefficient is 156

157. ANCOVA Therefore, there is a linear relationship between breaking strength and diameter, and the adjustment provided by the analysis of covariance was necessary . Adjusted estimated treatment means are: 157

158. ANCOVA for CRD ANOVA Table 158

159. Diagnostic Checking Diagnostic checking of the covariance model is based on residual analysis. For the covariance model, the residuals are where the fitted values are Thus The residual for the first observation from the first machine in Example is 159

160. ANCOVA A complete listing of observations, fitted values, and residuals is given in the following table : 160