1. ENM 310 Design of Experiments and Regression Analysis Chapter 3
Ilgın ACAR
Spring 2019

2. 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. Factorial Experiments
Definition

4. Factorial Experiments
Figure 14-3 Factorial Experiment, no interaction.

5. Factorial Experiments
Figure 14-4 Factorial Experiment, with interaction.

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

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

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

11. Two-Factor Factorial Experiments

12. Two-Factor Factorial Experiments
The observations may be described by the linear statistical model:
where
μ is the overall mean effect,
τi is the effect of the ith level of factor A,
β j is the effect of the jth level of factor B,
(τβ)ij is the effect of the interaction between A and B, and
eijk is a random error component having a normal distribution with mean 0 and variance σ2.

13. Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

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

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

16. Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
To test H0: i = 0 use the ratio
To test H0: j = 0 use the ratio
To test H0: ()ij = 0 use the ratio

17. Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
Definition

18. Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

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

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

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

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

23. Two-Factor Factorial Experiments
Model Adequacy Checking

24. Two-Factor Factorial Experiments
Model Adequacy Checking
Figure Normal probability plot of the residuals from Example 14-1
This plot has tails that do not fall exactly along a straight line passing through the center of the plot, indicating some potential problems with the normality assumption, but the deviation from normality does not appear severe.

25. Two-Factor Factorial Experiments
Model Adequacy Checking
There is some indication that primer type 3 results in slightly lower variability in adhesion force than the other two primers.
Figure Plot of residuals versus primer type.

26. Two-Factor Factorial Experiments
Model Adequacy Checking
Figure Plot of residuals versus application method.

27. Two-Factor Factorial Experiments
Model Adequacy Checking
The graph of residuals versus fitted values in does not reveal any unusual or diagnostic pattern.
Figure Plot of residuals versus predicted values.

28. Example
As an example of a factorial design involving two factors, 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 purposes of a test.

29. 14-4: General Factorial Experiments
Model for a three-factor factorial experiment

31. 14-4: General Factorial Experiments
Example 14-2

32. Example 14-2

33. 14-4: General Factorial Experiments
Example 14-2

34. 14-5: 2k Factorial Designs 14-5.1 22 Design
Figure The 22 factorial design.

35. 14-5: 2k Factorial Designs 14-5.1 22 Design
The main effect of a factor A is estimated by

36. 14-5: 2k Factorial Designs 14-5.1 22 Design
The main effect of a factor B is estimated by

37. 14-5: 2k Factorial Designs 14-5.1 22 Design
The AB interaction effect is estimated by

38. 14-5: 2k Factorial Designs 14-5.1 22 Design
The quantities in brackets in Equations 14-11, 14-12, and are called contrasts. For example, the A contrast is
ContrastA = a + ab – b – (1)

39. 14-5: 2k Factorial Designs 14-5.1 22 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

40. 14-5: 2k Factorial Designs
Example 14-3

41. 14-5: 2k Factorial Designs
Example 14-3

42. 14-5: 2k Factorial Designs
Example 14-3

43. 14-5: 2k Factorial Designs Residual Analysis
Figure Normal probability plot of residuals for the epitaxial process experiment.

44. 14-5: 2k Factorial Designs Residual Analysis
Figure Plot of residuals versus deposition time.

45. 14-5: 2k Factorial Designs Residual Analysis
Figure Plot of residuals versus arsenic flow rate.

46. 14-5: 2k Factorial Designs Residual Analysis
Figure The standard deviation of epitaxial layer thickness at the four runs in the 22 design.

47. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
Figure The 23 design.

48. Figure Geometric presentation of contrasts corresponding to the main effects and interaction in the 23 design. (a) Main effects. (b) Two-factor interactions. (c) Three-factor interaction.

49. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
The main effect of A is estimated by
The main effect of B is estimated by

50. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
The main effect of C is estimated by
The interaction effect of AB is estimated by

51. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
Other two-factor interactions effects estimated by
The three-factor interaction effect, ABC, is estimated by

52. 14-5: 2k Factorial Designs
k Design for k  3 Factors

53. 14-5: 2k Factorial Designs
k Design for k  3 Factors

54. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
Contrasts can be used to calculate several quantities:

55. 14-5: 2k Factorial Designs
Example 14-4

56. 14-5: 2k Factorial Designs
Example 14-4

57. 14-5: 2k Factorial Designs
Example 14-4

58. 14-5: 2k Factorial Designs
Example 14-4

59. 14-5: 2k Factorial Designs
Example 14-4

60. Example 14-4

61. 14-5: 2k Factorial Designs Residual Analysis
Figure Normal probability plot of residuals from the surface roughness experiment.

62. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

63. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

64. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

65. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

66. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

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

68. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure Normal probability plot of effects from the plasma etch experiment.

69. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure AD (Gap-Power) interaction from the plasma etch experiment.

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

71. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure Normal probability plot of residuals from the plasma etch experiment.

72. 14-5: 2k Factorial Designs
Additional Center Points to a 2k 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 2k design will provide protection against curvature as well as allow an independent estimate of error to be obtained. Figure illustrates the situation.

73. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Figure A 22 Design with center points.

74. A single-degree-of-freedom sum of squares for curvature is given by:
14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
A single-degree-of-freedom sum of squares for curvature is given by:

75. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6
Figure The 22 Design with five center points for Example 14-6.

76. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

77. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

78. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

79. 14-6: Blocking and Confounding in the 2k Design
Figure A 22 design in two blocks. (a) Geometric view. (b) Assignment of the four runs to two blocks.

80. 14-6: Blocking and Confounding in the 2k Design
Figure A 23 design in two blocks with ABC confounded. (a) Geometric view. (b) Assignment of the eight runs to two blocks.

81. 14-6: Blocking and Confounding in the 2k Design
General method of constructing blocks employs a defining contrast

82. 14-6: Blocking and Confounding in the 2k Design
Example 14-7

83. 14-6: Blocking and Confounding in the 2k Design
Example 14-7

84. Example 14-7
Figure A 24 design in two blocks for Example (a) Geometric view. (b) Assignment of the 16 runs to two blocks.

85. 14-6: Blocking and Confounding in the 2k Design
Example 14-7
Figure Normal probability plot of the effects from Minitab, Example 14-7.

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

87. 14-7: Fractional Replication of the 2k Design
One-Half Fraction of the 2k Design

88. 14-7: Fractional Replication of the 2k Design
One-Half Fraction of the 2k Design
Figure The one-half fractions of the 23 design. (a) The principal fraction, I = +ABC. (B) The alternate fraction, I = -ABC

89. 14-7: Fractional Replication of the 2k Design
Example 14-8

90. 14-7: Fractional Replication of the 2k Design
Example 14-8
Figure The 24-1 design for the experiment of Example 14-8.

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

92. 14-7: Fractional Replication of the 2k Design
Example 14-8

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

94. 14-7: Fractional Replication of the 2k Design
Example 14-8
Figure Normal probability plot of the effects from Minitab, Example 14-8.

95. 14-7: Fractional Replication of the 2k Design
Projection of the 2k-1 Design
Figure Projection of a 23-1 design into three 22 designs.

96. 14-7: Fractional Replication of the 2k Design
Projection of the 2k-1 Design
Figure The 22 design obtained by dropping factors B and C from the plasma etch experiment in Example 14-8.

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

98. 14-7: Fractional Replication of the 2k Design
Smaller Fractions: The 2k-p Fractional Factorial

99. 14-7: Fractional Replication of the 2k Design
Example 14-9

100. Example 14-8

101. 14-7: Fractional Replication of the 2k Design
Example 14-9

102. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Normal probability plot of effects for Example 14-9.

103. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Plot of AB (mold temperature-screw speed) interaction for Example 14-9.

105. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Normal probability plot of residuals for Example 14-9.

106. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Residuals versus holding time (C) for Example 14-9.

107. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Average shrinkage and range of shrinkage in factors A, B, and C for Example 14-9.

108. 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.

109. 14-8: Response Surface Methods and Designs
Figure A three-dimensional response surface showing the expected yield as a function of temperature and feed concentration.

110. 14-8: Response Surface Methods and Designs
Figure A contour plot of yield response surface in Figure

111. 14-8: Response Surface Methods and Designs
The first-order model
The second-order model

112. 14-8: Response Surface Methods and Designs
Method of Steepest Ascent

113. 14-8: Response Surface Methods and Designs
Method of Steepest Ascent
Figure First-order response surface and path of steepest ascent.

114. 14-8: Response Surface Methods and Designs
Example 14-11

115. 14-8: Response Surface Methods and Designs
Example 14-11
Figure Response surface plots for the first-order model in the Example

116. 14-8: Response Surface Methods and Designs
Example 14-11
Figure Steepest ascent experiment for Example

117. 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) 2k factorial design Two-level factorial design
Chapter 14 Summary