4. 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.

5. 14-2 Factorial Experiments Definition

6. 14-2 Factorial Experiments Figure 14-3 Factorial Experiment, no interaction.

7. 14-2 Factorial Experiments Figure 14-4 Factorial Experiment, with interaction.

8. 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.

9. 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.

10. 14-2 Factorial Experiments Figure 14-7 Yield versus reaction time with temperature constant at 155º F.

11. 14-2 Factorial Experiments Figure 14-8 Yield versus temperature with reaction time constant at 1.7 hours.

12. 14-2 Factorial Experiments Figure 14-9 Optimization experiment using the one-factor-at-a-time method.

13. 14-3 Two-Factor Factorial Experiments

14. The observations may be described by the linear statistical model:

15. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

16. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

17. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

18. 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

19. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Definition

20. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

21. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

22. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

23. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

24. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

25. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

26. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

27. 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.

28. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1

29. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking

30. 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

31. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-12 Plot of residuals versus primer type.

32. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-13 Plot of residuals versus application method.

33. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-14 Plot of residuals versus predicted values.

34. 14-4 General Factorial Experiments Model for a three-factor factorial experiment

36. 14-4 General Factorial Experiments Example 14-2

38. 14-4 General Factorial Experiments Example 14-2

39. 14-5 2 k Factorial Designs 14-5.1 2 2 Design Figure 14-15 The 2 2 factorial design.

40. 14-5 2 k Factorial Designs 14-5.1 2 2 Design The main effect of a factor A is estimated by

41. 14-5 2 k Factorial Designs 14-5.1 2 2 Design The main effect of a factor B is estimated by

42. 14-5 2 k Factorial Designs 14-5.1 2 2 Design The AB interaction effect is estimated by

43. 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)

44. 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

45. 14-5 2 k Factorial Designs Example 14-3

46. 14-5 2 k Factorial Designs Example 14-3

47. 14-5 2 k Factorial Designs Example 14-3

48. 14-5 2 k Factorial Designs Residual Analysis Figure 14-16 Normal probability plot of residuals for the epitaxial process experiment.

49. 14-5 2 k Factorial Designs Residual Analysis Figure 14-17 Plot of residuals versus deposition time.

50. 14-5 2 k Factorial Designs Residual Analysis Figure 14-18 Plot of residuals versus arsenic flow rate.

51. 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.

52. 14-5 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors Figure 14-20 The 2 3 design.

53. 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.

54. 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

55. 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

56. 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

57. 14-5 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors

58. 14-5 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors

59. 14-5 2 k Factorial Designs 14-5.2 2 k Design for k  3 Factors Contrasts can be used to calculate several quantities:

60. 14-5 2 k Factorial Designs Example 14-4

61. 14-5 2 k Factorial Designs Example 14-4

62. 14-5 2 k Factorial Designs Example 14-4

63. 14-5 2 k Factorial Designs Example 14-4

64. 14-5 2 k Factorial Designs Example 14-4

66. 14-5 2 k Factorial Designs Residual Analysis Figure 14-22 Normal probability plot of residuals from the surface roughness experiment.

67. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

68. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

69. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

70. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

71. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

72. 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.

73. 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.

74. 14-5 2 k Factorial Designs 14-5.3 Single Replicate of the 2 k Design Example 14-5

75. 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.

76. 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.

77. 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.

78. 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:

79. 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.

80. 14-5 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

81. 14-5 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

82. 14-5 2 k Factorial Designs 14-5.4 Additional Center Points to a 2k Design Example 14-6

83. 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.

84. 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.

85. 14-6 Blocking and Confounding in the 2 k Design General method of constructing blocks employs a defining contrast

86. 14-6 Blocking and Confounding in the 2 k Design Example 14-7

87. 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. 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. 14-6 Blocking and Confounding in the 2 k Design Example 14-7

90. 14-7 Fractional Replication of the 2 k Design 14-7.1 One-Half Fraction of the 2 k Design

91. 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. 14-7 Fractional Replication of the 2 k Design Example 14-8

93. 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. 14-7 Fractional Replication of the 2 k Design Example 14-8

95. 14-7 Fractional Replication of the 2 k Design Example 14-8

96. 14-7 Fractional Replication of the 2 k Design Example 14-8

97. 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. 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. 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. 14-7 Fractional Replication of the 2 k Design Design Resolution

101. 14-7 Fractional Replication of the 2 k Design 14-7.2 Smaller Fractions: The 2 k-p Fractional Factorial

102. 14-7 Fractional Replication of the 2 k Design Example 14-9

103. Example 14-8

104. 14-7 Fractional Replication of the 2 k Design Example 14-9

105. 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. 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.

108. 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. 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. 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. 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. 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. 14-8 Response Surface Methods and Designs Figure 14-43 A contour plot of yield response surface in Figure 14-42.

114. 14-8 Response Surface Methods and Designs The first-order model The second-order model

115. 14-8 Response Surface Methods and Designs Method of Steepest Ascent

116. 14-8 Response Surface Methods and Designs Method of Steepest Ascent Figure 14-41 First-order response surface and path of steepest ascent.

117. 14-8 Response Surface Methods and Designs Example 14-11

118. 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. 14-8 Response Surface Methods and Designs Example 14-11 Figure 14-46 Steepest ascent experiment for Example 14-11.