Design of
Experiments
with
Several Factors
Chapter 14 Table of Contents
1
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.
2
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-2: Factorial Experiments
Definition
3
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14-2: Factorial Experiments
Figure 14-3 Factorial Experiment, no interaction.
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14-2: Factorial Experiments
Figure 14-4 Factorial Experiment, with interaction.
5
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14-3: Two-Factor Factorial Experiments
6
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
7
© 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
8
© 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
To test H0: i = 0 use the ratio
To test H0: j = 0 use the ratio
To test H0: ()ij = 0 use the ratio
9
© 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
10
© 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
11
© 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
12
© 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
13
© 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
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
Example 14-1
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
Example 14-1
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
Example 14-1
17
© 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.
18
© 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
19
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
20
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-4: General Factorial Experiments
Example 14-2
21
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Example 14-2
22
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-4: General Factorial Experiments
Example 14-2
23
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
24
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
25
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.1 22 Design
Figure 14-15 The 22 factorial design.
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27
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.1 22 Design
The main effect of a factor A is estimated by
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14-5: 2k Factorial Designs
14-5.1 22 Design
The main effect of a factor B is estimated by
29
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.1 22 Design
The AB interaction effect is estimated by
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31
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.1 22 Design
The quantities in brackets are called contrasts. For example,
the A contrast is:
ContrastA = a + ab – b – (1)
32
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
33
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-3
34
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-3
35
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-3
36
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Residual Analysis
Figure 14-16 Normal
probability plot of
residuals for the epitaxial
process experiment.
37
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
Figure 14-20 The 23 design.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
38
Figure 14-21 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.
39
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
40
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
41
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
42
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
43
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
44
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
Contrasts can be used to calculate several quantities:
45
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-4
46
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-4
47
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-4
48
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-5: 2k Factorial Designs
Example 14-4
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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© 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.
51
© 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.
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© 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.
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14-8: Response Surface Methods and Designs
The first-order model
The second-order model
54
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14-8: Response Surface Methods and Designs
Method of Steepest Ascent
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14-8: Response Surface Methods and Designs
Method of Steepest Ascent
Figure 14-44 First-order response
surface and path of steepest ascent.
56
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
14-8: Response Surface Methods and Designs
Example 14-11
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© 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.
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© 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.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.