1. 1 14 Design of Experiments with Several Factors 14-1 Introduction 14-2 Factorial Experiments 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical analysis of the fixed- effects model 14-3.2 Model adequacy checking 14-3.3 One observation per cell 14-4 General Factorial Experiments 14-5 2 k Factorial Designs 14-5.1 2 k design 14-5.2 2 k design for k ≥3 factors 14-5.3 Single replicate of the 2 k design 14-5.4 Addition of center points to a 2 k design 14-6 Blocking & Confounding in the 2 k design 14-7 Fractional Replication of the 2 k Design 14-7.1 One-half fraction of the 2k design 14-7.2 Smaller fractions: The 2 k-p fractional factorial 14-8 Response Surface Methods and Designs CHAPTER OUTLINE Chapter 14 Table of Contents

2. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Learning Objectives for Chapter 14 After careful study of this chapter, you should be able to do the following: 1.Design and conduct engineering experiments involving several factors using the factorial design approach. 2.Know how to analyze and interpret main effects and interactions. 3.Understand how the ANOVA is used to analyze the data from these experiments. 4.Assess model adequacy with residual plots. 5.Know how to use the two-level series of factorial designs. 6.Understand how two-level factorial designs can be run in blocks. 7.Design and conduct two-level fractional factorial designs. 8.Test for curvature in two-level factorial designs by using center points. 9.Use response surface methodology for process optimization experiments. 2Chapter 14 Learning Objectives

3. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-1: Introduction An experiment is a test or series of tests. The design of an experiment plays a major role in the eventual solution of the problem. In a factorial experimental design, experimental trials (or runs) are performed at all combinations of the factor levels. The analysis of variance (ANOVA) will be used as one of the primary tools for statistical data analysis. 3

4. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Definition 4

5. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-3 Factorial Experiment, no interaction. 5

6. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-4 Factorial Experiment, with interaction. 6

7. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-5 Three-dimensional surface plot of the data from Table 14-1, showing main effects of the two factors A and B. 7

8. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-6 Three-dimensional surface plot of the data from Table 14-2, showing main effects of the A and B interaction. 8

9. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-7 Yield versus reaction time with temperature constant at 155º F. 9

10. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-8 Yield versus temperature with reaction time constant at 1.7 hours. 10

11. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-2: Factorial Experiments Figure 14-9 Optimization experiment using the one-factor- at-a-time method. 11

12. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 12

13. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments The observations may be described by the linear statistical model: 13

14. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model 14

15. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model 15

16. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model 16

17. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments To test H 0 :  i = 0 use the ratio 14-3.1 Statistical Analysis of the Fixed-Effects Model To test H 0 :  j = 0 use the ratio To test H 0 : (  ) ij = 0 use the ratio 17

18. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Definition 18

19. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model 19

20. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 20

21. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 21

22. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 22

23. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 23

24. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 24

25. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 25

26. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 Figure 14-10 Graph of average adhesion force versus primer types for both application methods. 26

27. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1 27

28. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking 28

29. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-11 Normal probability plot of the residuals from Example 14-1 29

30. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-12 Plot of residuals versus primer type. 30

31. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-13 Plot of residuals versus application method. 31

32. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-3: Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-14 Plot of residuals versus predicted values. 32

33. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-4: General Factorial Experiments Model for a three-factor factorial experiment 33

34. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 34

35. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-4: General Factorial Experiments Example 14-2 35

36. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example 14-2 36

37. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-4: General Factorial Experiments Example 14-2 37

38. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design Figure 14-15 The 2 2 factorial design. 38

39. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design The main effect of a factor A is estimated by 39

40. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design The main effect of a factor B is estimated by 40

41. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design The AB interaction effect is estimated by 41

42. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design The quantities in brackets in Equations 14-11, 14-12, and 14- 13 are called contrasts. For example, the A contrast is Contrast A = a + ab – b – (1) 42

43. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.1 2 2 Design Contrasts are used in calculating both the effect estimates and the sums of squares for A, B, and the AB interaction. The sums of squares formulas are 43

44. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-3 44

45. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-3 45

46. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-3 46

47. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Residual Analysis Figure 14-16 Normal probability plot of residuals for the epitaxial process experiment. 47

48. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Residual Analysis Figure 14-17 Plot of residuals versus deposition time. 48

49. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Residual Analysis Figure 14-18 Plot of residuals versus arsenic flow rate. 49

50. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Residual Analysis Figure 14-19 The standard deviation of epitaxial layer thickness at the four runs in the 2 2 design. 50

51. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors Figure 14-20 The 2 3 design. 51

52. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Figure 14-21 Geometric presentation of contrasts corresponding to the main effects and interaction in the 2 3 design. (a) Main effects. (b) Two-factor interactions. (c) Three-factor interaction. 52

53. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors The main effect of A is estimated by The main effect of B is estimated by 53

54. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors The main effect of C is estimated by The interaction effect of AB is estimated by 54

55. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors Other two-factor interactions effects estimated by The three-factor interaction effect, ABC, is estimated by 55

56. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors 56

57. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors 57

58. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors Contrasts can be used to calculate several quantities: 58

59. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-4 59

60. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-4 60

61. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-4 61

62. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-4 62

63. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Example 14-4 63

64. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example 14-4 64

65. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs Residual Analysis Figure 14-22 Normal probability plot of residuals from the surface roughness experiment. 65

66. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 66

67. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 67

68. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 68

69. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 69

70. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 70

71. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 71

72. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 Figure 14-23 Normal probability plot of effects from the plasma etch experiment. 72

73. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 Figure 14-24 AD (Gap-Power) interaction from the plasma etch experiment. 73

74. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 74

75. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5 Figure 14-25 Normal probability plot of residuals from the plasma etch experiment. 75

76. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2 k Design A potential concern in the use of two-level factorial designs is the assumption of the linearity in the factor effect. Adding center points to the 2 k design will provide protection against curvature as well as allow an independent estimate of error to be obtained. Figure 14-26 illustrates the situation. 76

77. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Figure 14-26 A 2 2 Design with center points. 77

78. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design A single-degree-of-freedom sum of squares for curvature is given by: 78

79. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6 Figure 14-27 The 2 2 Design with five center points for Example 14-6. 79

80. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6 80

81. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6 81

82. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-5: 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6 82

83. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Figure 14-28 A 2 2 design in two blocks. (a) Geometric view. (b) Assignment of the four runs to two blocks. 83

84. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Figure 14-29 A 2 3 design in two blocks with ABC confounded. (a) Geometric view. (b) Assignment of the eight runs to two blocks. 84

85. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design General method of constructing blocks employs a defining contrast 85

86. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Example 14-7 86

87. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Example 14-7 87

88. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example 14-7 Figure 14-30 A 2 4 design in two blocks for Example 14-7. (a) Geometric view. (b) Assignment of the 16 runs to two blocks. 88

89. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Example 14-7 Figure 14-31 Normal probability plot of the effects from Minitab, Example 14-7. 89

90. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-6: Blocking and Confounding in the 2 k Design Example 14-7 90

91. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design 14-7.1 One-Half Fraction of the 2 k Design 91

92. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design 14-7.1 One-Half Fraction of the 2 k Design Figure 14-32 The one-half fractions of the 2 3 design. (a) The principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC 92

93. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 93

94. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 Figure 14-33 The 2 4-1 design for the experiment of Example 14-8. 94

95. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 95

96. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 96

97. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 97

98. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-8 Figure 14-34 Normal probability plot of the effects from Minitab, Example 14-8. 98

99. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Projection of the 2 k-1 Design Figure 14-35 Projection of a 2 3-1 design into three 2 2 designs. 99

100. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Projection of the 2 k-1 Design Figure 14-36 The 2 2 design obtained by dropping factors B and C from the plasma etch experiment in Example 14-8. 100

101. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Design Resolution 101

102. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design 14-7.2 Smaller Fractions: The 2 k-p Fractional Factorial 102

103. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 103

104. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example 14-8 104

105. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 105

106. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 Figure 14-37 Normal probability plot of effects for Example 14-9. 106

107. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 Figure 14-38 Plot of AB (mold temperature-screw speed) interaction for Example 14-9. 107

108. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 108

109. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 Figure 14-39 Normal probability plot of residuals for Example 14-9. 109

110. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 Figure 14-40 Residuals versus holding time (C) for Example 14-9. 110

111. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-7: Fractional Replication of the 2 k Design Example 14-9 Figure 14-41 Average shrinkage and range of shrinkage in factors A, B, and C for Example 14-9. 111

112. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Response surface methodology, or RSM, is a collection of mathematical and statistical techniques that are useful for modeling and analysis in applications where a response of interest is influenced by several variables and the objective is to optimize this response. 112

113. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Figure 14-42 A three-dimensional response surface showing the expected yield as a function of temperature and feed concentration. 113

114. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Figure 14-43 A contour plot of yield response surface in Figure 14-42. 114

115. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs The first-order model The second-order model 115

116. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Method of Steepest Ascent 116

117. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Method of Steepest Ascent Figure 14-44 First-order response surface and path of steepest ascent. 117

118. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Example 14-11 118

119. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Example 14-11 Figure 14-45 Response surface plots for the first-order model in the Example 14-11. 119

120. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14-8: Response Surface Methods and Designs Example 14-11 Figure 14-46 Steepest ascent experiment for Example 14-11. 120

121. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Important Terms & Concepts of Chapter 14 Analysis of variance (ANOVA) Blocking & nuisance factors Center points Central composite design Confounding Contrast Defining relation Design matrix Factorial experiment Fractional factorial design Generator Interaction Main effect Normal probability plot of factor effects Optimization experiment Orthogonal design Regression model Residual analysis Resolution Response surface Screening experiment Steepest ascent (or descent) 2 k factorial design Two-level factorial design 121Chapter 14 Summary