1. 14 Design of Experiments with Several Factors CHAPTER OUTLINE
Introduction
Blocking & Confounding in the 2k design
Factorial Experiments
Two-Factor Factorial Experiments
Fractional Replication of the 2k Design
Statistical analysis of the fixed- effects model
One-half fraction of the 2k design
Model adequacy checking
Smaller fractions: The 2k-p fractional factorial
One observation per cell
General Factorial Experiments
Response Surface Methods and Designs
k Factorial Designs
k design
k design for k ≥3 factors
Single replicate of the 2k design
Addition of center points to a 2k design
Dear Instructor: This file is an adaptation of the 4th edition slides for the 5th edition. It will be replaced as slides are developed following the style of the Chapters 1-7 slides.
Chapter 14 Table of Contents

2. 14-2: Factorial Experiments
Definition

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

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

5. 14-3: Two-Factor Factorial Experiments

6. 14-3: Two-Factor Factorial Experiments
The observations may be described by the linear statistical model:

7. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

8. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

9. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

10. 14-3: 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

11. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
Definition

12. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model

13. 14-3: Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
Example 14-1

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

16. 14-4: General Factorial Experiments
Example 14-2

17. Example 14-2

18. 14-5: 2k Factorial Designs
Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect of the factors on a response
The most important of these special cases is that of k factors, each at only two levels.
These levels may be
quantitative, such as two values of temperature, pressure, or time;
or they may be qualitative, such as two machines, two operators, the “high” and “low” levels of a factor, or perhaps the presence and absence of a factor.
A complete replicate of such a design requires 2 × 2 × · · · × 2 = 2k observations and is called a 𝟐k factorial design.

19. Analysis Procedure for a Factorial Design
Estimate factor effects
Formulate model
With replication, use full model
With an unreplicated design, use normal probability plots
Statistical testing (ANOVA)
Refine the model
Analyze residuals (graphical)
Interpret results

20. 14-5: 2k Factorial Designs 14-5.1 22 Design
Figure The 22 factorial design.

21. Example
As an example, consider an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the conversion (yield) in a chemical process.
The objective of the experiment was to determine if adjustments to either of these two factors would increase the yield.
Let the reactant concentration be factor A and let the two levels of interest be 15 and 25 percent.
The catalyst is factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound.
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.

22. Chemical Process Example
A = reactant concentration, B = catalyst amount, y = recovery

23. The Simplest Case: The 22
“-” and “+” denote the low and high levels of a factor, respectively
Low and high are arbitrary terms
Geometrically, the four runs form the corners of a square
Factors can be quantitative or qualitative, although their treatment in the final model will be different

24. 14-5: 2k Factorial Designs 14-5.1 22 Design
The main effect of a factor A is estimated by
The main effect of a factor B is estimated by
The AB interaction effect is estimated by

25. Chemical Process Example
The effect of A (reactant concentration) is positive; this suggests that increasing A from the low level (15%) to the high level (25%) will increase the yield.
The effect of B (catalyst) is negative; this suggests that increasing the amount of catalyst added to the process will decrease the yield.
The interaction effect appears to be small relative to the two main effects.

26. 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)
It is often convenient to write down the treatment combinations in the order (1), a, b, ab. This is referred to as standard order (or Yates’s order, for Frank Yates

27. 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

28. Statistical Testing - ANOVA
The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?
Design & Analysis of Experiments 7E 2009 Montgomery
Chapter 6

29. Example
It is important to study the effect of the concentration of the reactant and the feed rate on the viscosity of the product from a chemical process. Let the reactant concentration be factor A at levels 15 % and 25 %. Let the feed rate be factor B, with levels of 20 lb/hr and 30 lb/hr. The experiment involves two experimental runs at each of the four combinations (L= low and H= high). The viscosity readings are as follows.
Assuming a model containing two main effects and interaction, calculate the htree effects.
Sketch the interaction plot.
Do ANOVA and test for interaction. Give conclusion
Test for main effects and give final conclusion regarding the importance of all these effects.

30. 14-5: 2k Factorial Designs

31. 14-5: 2k Factorial Designs
Example 14-3

32. 14-5: 2k Factorial Designs Residual Analysis
Figure Normal probability plot of residuals for the epitaxial process experiment.

33. 14-5: 2k Factorial Designs Residual Analysis
Figure Plot of residuals versus deposition time.

34. 14-5: 2k Factorial Designs Residual Analysis
Figure Plot of residuals versus arsenic flow rate.

35. 14-5: 2k Factorial Designs Residual Analysis
Figure The standard deviation of epitaxial layer thickness at the four runs in the 22 design.

36. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
Figure The 23 design.

37. 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.

38. 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

39. 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

40. 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

41. 14-5: 2k Factorial Designs
k Design for k  3 Factors

42. 14-5: 2k Factorial Designs
k Design for k  3 Factors

43. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors
Contrasts can be used to calculate several quantities:

44. 14-5: 2k Factorial Designs
Example 14-4

45. 14-5: 2k Factorial Designs
Example 14-4

46. 14-5: 2k Factorial Designs
Example 14-4

47. 14-5: 2k Factorial Designs
Example 14-4

48. 14-5: 2k Factorial Designs
Example 14-4

49. Example 14-4

50. 14-5: 2k Factorial Designs Residual Analysis
Figure Normal probability plot of residuals from the surface roughness experiment.

51. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

52. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

53. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

54. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

55. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

56. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

57. 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.

58. 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.

59. 14-5: 2k Factorial Designs 14-5.3 Single Replicate of the 2k Design
Example 14-5

60. 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.

61. 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.

62. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Figure A 22 Design with center points.

63. 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:

64. 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.

65. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

66. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

67. 14-5: 2k Factorial Designs
Additional Center Points to a 2k Design
Example 14-6

68. 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.

69. 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.

70. 14-6: Blocking and Confounding in the 2k Design
General method of constructing blocks employs a defining contrast

71. 14-6: Blocking and Confounding in the 2k Design
Example 14-7

72. 14-6: Blocking and Confounding in the 2k Design
Example 14-7

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

74. 14-6: Blocking and Confounding in the 2k Design
Example 14-7
Figure Normal probability plot of the effects from Minitab, Example 14-7.

75. 14-6: Blocking and Confounding in the 2k Design
Example 14-7

76. 14-7: Fractional Replication of the 2k Design
One-Half Fraction of the 2k Design

77. 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

78. 14-7: Fractional Replication of the 2k Design
Example 14-8

79. 14-7: Fractional Replication of the 2k Design
Example 14-8
Figure The 24-1 design for the experiment of Example 14-8.

80. 14-7: Fractional Replication of the 2k Design
Example 14-8

81. 14-7: Fractional Replication of the 2k Design
Example 14-8

82. 14-7: Fractional Replication of the 2k Design
Example 14-8

83. 14-7: Fractional Replication of the 2k Design
Example 14-8
Figure Normal probability plot of the effects from Minitab, Example 14-8.

84. 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.

85. 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.

86. 14-7: Fractional Replication of the 2k Design
Design Resolution

87. 14-7: Fractional Replication of the 2k Design
Smaller Fractions: The 2k-p Fractional Factorial

88. 14-7: Fractional Replication of the 2k Design
Example 14-9

89. Example 14-8

90. 14-7: Fractional Replication of the 2k Design
Example 14-9

91. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Normal probability plot of effects for Example 14-9.

92. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Plot of AB (mold temperature-screw speed) interaction for Example 14-9.

94. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Normal probability plot of residuals for Example 14-9.

95. 14-7: Fractional Replication of the 2k Design
Example 14-9
Figure Residuals versus holding time (C) for Example 14-9.

96. 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.

97. 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.

98. 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.

99. 14-8: Response Surface Methods and Designs
Figure A contour plot of yield response surface in Figure

100. 14-8: Response Surface Methods and Designs
The first-order model
The second-order model

101. 14-8: Response Surface Methods and Designs
Method of Steepest Ascent

102. 14-8: Response Surface Methods and Designs
Method of Steepest Ascent
Figure First-order response surface and path of steepest ascent.

103. 14-8: Response Surface Methods and Designs
Example 14-11

104. 14-8: Response Surface Methods and Designs
Example 14-11
Figure Response surface plots for the first-order model in the Example

105. 14-8: Response Surface Methods and Designs
Example 14-11
Figure Steepest ascent experiment for Example

106. 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