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. 14-3 Factorial Experiments Definition

6. 14-3 Factorial Experiments Figure 14-1 Factorial Experiment, no interaction.

7. 14-3 Factorial Experiments Figure 14-2 Factorial Experiment, with interaction.

8. 14-3 Factorial Experiments Figure 14-3 Three-dimensional surface plot of the data from Table 14-1, showing main effects of the two factors A and B.

9. 14-3 Factorial Experiments Figure 14-4 Three-dimensional surface plot of the data from Table 14-1, showing main effects of the A and B interaction.

10. 14-3 Factorial Experiments Figure 14-5 Yield versus reaction time with temperature constant at 155º F.

11. 14-3 Factorial Experiments Figure 14-6 Yield versus temperature with reaction time constant at 1.7 hours.

12. 14-3 Factorial Experiments Figure 14-7 Optimization experiment using the one-factor-at-a-time method.

13. 14-4 Two-Factor Factorial Experiments

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

15. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model

16. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model

17. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model

18. 14-4 Two-Factor Factorial Experiments To test H 0 :  i = 0 use the ratio 14-4.1 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. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Definition

20. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model

21. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

22. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

23. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

24. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

25. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

26. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

27. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 Figure 14-8 Graph of average adhesion force versus primer types for both application methods.

28. 14-4 Two-Factor Factorial Experiments 14-4.1 Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1

29. 14-4 Two-Factor Factorial Experiments 14-4.2 Model Adequacy Checking

30. 14-4 Two-Factor Factorial Experiments 14-4.2 Model Adequacy Checking Figure 14-9 Normal probability plot of the residuals from Example 14-1

31. 14-4 Two-Factor Factorial Experiments 14-4.2 Model Adequacy Checking Figure 14-10 Plot of residuals versus primer type.

32. 14-4 Two-Factor Factorial Experiments 14-4.2 Model Adequacy Checking Figure 14-11 Plot of residuals versus application method.

33. 14-4 Two-Factor Factorial Experiments 14-4.2 Model Adequacy Checking Figure 14-12 Plot of residuals versus predicted values.

34. 14-5 General Factorial Experiments Model for a three-factor factorial experiment

36. 14-5 General Factorial Experiments Example 14-2

38. 14-5 General Factorial Experiments Example 14-2

39. 14-7 2 k Factorial Designs 14-7.1 2 2 Design Figure 14-13 The 2 2 factorial design.

40. 14-7 2 k Factorial Designs 14-7.1 2 2 Design The main effect of a factor A is estimated by

41. 14-7 2 k Factorial Designs 14-7.1 2 2 Design The main effect of a factor B is estimated by

42. 14-7 2 k Factorial Designs 14-7.1 2 2 Design The AB interaction effect is estimated by

43. 14-7 2 k Factorial Designs 14-7.1 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. 14-7 2 k Factorial Designs 14-7.1 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. 14-7 2 k Factorial Designs Example 14-3

46. 14-7 2 k Factorial Designs Example 14-3

47. 14-7 2 k Factorial Designs Example 14-3

48. 14-7 2 k Factorial Designs Residual Analysis Figure 14-14 Normal probability plot of residuals for the epitaxial process experiment.

49. 14-7 2 k Factorial Designs Residual Analysis Figure 14-15 Plot of residuals versus deposition time.

50. 14-7 2 k Factorial Designs Residual Analysis Figure 14-16 Plot of residuals versus arsenic flow rate.

51. 14-7 2 k Factorial Designs Residual Analysis Figure 14-17 The standard deviation of epitaxial layer thickness at the four runs in the 2 2 design.

52. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors Figure 14-18 The 2 3 design.

53. Figure 14-19 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. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors The main effect of A is estimated by The main effect of B is estimated by

55. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors The main effect of C is estimated by The interaction effect of AB is estimated by

56. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors Other two-factor interactions effects estimated by The interaction effect of ABC is estimated by

57. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors

58. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors

59. 14-7 2 k Factorial Designs 14-7.2 2 k Design for k  3 Factors Contrasts can be used to calculate several quantities:

60. 14-7 2 k Factorial Designs Example 14-4

61. 14-7 2 k Factorial Designs Example 14-4

62. 14-7 2 k Factorial Designs Example 14-4

63. 14-7 2 k Factorial Designs Example 14-4

64. 14-7 2 k Factorial Designs Example 14-4

66. 14-7 2 k Factorial Designs Residual Analysis Figure 14-20 Normal probability plot of residuals from the surface roughness experiment.

67. 14-8 Blocking and Confounding in the 2 k Design Figure 14-24 A 2 2 design in two blocks. (a) Geometric view. (b) Assignment of the four runs to two blocks.

68. 14-8 Blocking and Confounding in the 2 k Design Figure 14-25 A 2 3 design in two blocks with ABC confounded. (a) Geometric view. (b) Assignment of the eight runs to two blocks.

69. 14-8 Blocking and Confounding in the 2 k Design General method of constructing blocks employs a defining contrast

70. 14-8 Blocking and Confounding in the 2 k Design Example 14-6

71. Figure 14-26 A 2 4 design in two blocks for Example 14-6. (a) Geometric view. (b) Assignment of the 16 runs to two blocks.

72. 14-8 Blocking and Confounding in the 2 k Design Example 14-6 Figure 14-27 Normal probability plot of the effects from Minitab, Example 14-6.

73. 14-8 Blocking and Confounding in the 2 k Design Example 14-6

74. 14-9 Fractional Replication of the 2 k Design 14-9.1 One-Half Fraction of the 2 k Design

75. 14-9 Fractional Replication of the 2 k Design 14-9.1 One-Half Fraction of the 2 k Design Figure 14-28 The one-half fractions of the 2 3 design. (a) The principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC

76. 14-9 Fractional Replication of the 2 k Design Example 14-7

77. 14-9 Fractional Replication of the 2 k Design Example 14-7 Figure 14-29 The 2 4-1 design for the experiment of Example 14-7.

78. 14-9 Fractional Replication of the 2 k Design Example 14-7

79. 14-9 Fractional Replication of the 2 k Design Example 14-7

80. 14-9 Fractional Replication of the 2 k Design Example 14-7

81. 14-9 Fractional Replication of the 2 k Design Example 14-7 Figure 14-30 Normal probability plot of the effects from Minitab, Example 14-7.

82. 14-9 Fractional Replication of the 2 k Design Projection of the 2 k-1 Design Figure 14-31 Projection of a 2 3-1 design into three 2 2 designs.

83. 14-9 Fractional Replication of the 2 k Design Projection of the 2 k-1 Design Figure 14-32 The 2 2 design obtained by dropping factors B and C from the plasma etch experiment in Example 14-7.

84. 14-9 Fractional Replication of the 2 k Design Design Resolution

85. 14-9 Fractional Replication of the 2 k Design 14-9.2 Smaller Fractions: The 2 k-p Fractional Factorial

86. 14-9 Fractional Replication of the 2 k Design Example 14-8

88. 14-9 Fractional Replication of the 2 k Design Example 14-8

89. 14-9 Fractional Replication of the 2 k Design Example 14-8 Figure 14-33 Normal probability plot of effects for Example 14-8.

90. 14-9 Fractional Replication of the 2 k Design Example 14-8 Figure 14-34 Plot of AB (mold temperature-screw speed) interaction for Example 14-8.

92. 14-9 Fractional Replication of the 2 k Design Example 14-8 Figure 14-35 Normal probability plot of residuals for Example 14-8.

93. 14-9 Fractional Replication of the 2 k Design Example 14-8 Figure 14-36 Residuals versus holding time (C) for Example 14-8.

94. 14-9 Fractional Replication of the 2 k Design Example 14-8

95. 14-9 Fractional Replication of the 2 k Design Example 14-8 Figure 14-37 Average shrinkage and range of shrinkage in factors A, B, and C for Example 14-8.