14
Design of Experiments
with Several Factors
CHAPTER OUTLINE
14-1 Introduction
14-6 Blocking & Confounding in the 2k
14-2 Factorial Experiments
design
14-3 Two-Factor Factorial Experiments 14-7 Fractional Replication of the 2k
14-3.1 Statistical analysis of the fixedDesign
effects model
14-3.2 Model adequacy checking
14-3.3 One observation per cell
14-4 General Factorial Experiments
14-5 2k Factorial Designs
14-5.1 2k design
14-5.2 2k design for k ≥3 factors
14-5.3 Single replicate of the 2k design
14-5.4 Addition of center points to a 2k
design
Chapter 14 Table of Contents
14-7.1 One-half fraction of the 2k design
14-7.2 Smaller fractions: The 2k-p fractional
factorial
14-8 Response Surface Methods and
Designs
1
Learning Objectives for Chapter 14
After careful study of this chapter, you should be able to do the
following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Design and conduct engineering experiments involving several factors
using the factorial design approach.
Know how to analyze and interpret main effects and interactions.
Understand how the ANOVA is used to analyze the data from these
experiments.
Assess model adequacy with residual plots.
Know how to use the two-level series of factorial designs.
Understand how two-level factorial designs can be run in blocks.
Design and conduct two-level fractional factorial designs.
Test for curvature in two-level factorial designs by using center points.
Use response surface methodology for process optimization experiments.
Chapter 14 Learning Objectives
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2
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.
3
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14-2: Factorial Experiments
Definition
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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.
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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.
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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.
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14-2: Factorial Experiments
Figure 14-7 Yield versus reaction time with temperature constant at 155º F.
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14-2: Factorial Experiments
Figure 14-8 Yield versus temperature with reaction time constant at 1.7 hours.
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14-2: Factorial Experiments
Figure 14-9 Optimization
experiment using the one-factorat-a-time method.
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14-3: Two-Factor Factorial Experiments
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14-3: Two-Factor Factorial Experiments
The observations may be described by the linear
statistical model:
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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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
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Definition
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Example 14-1
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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.
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14-3: Two-Factor Factorial Experiments
14-3.1 Statistical Analysis of the Fixed-Effects Model
Minitab Output for Example 14-1
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14-3: Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
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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
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14-3: Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-12 Plot of residuals versus primer type.
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14-3: Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-13 Plot of residuals versus application method.
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14-3: Two-Factor Factorial Experiments
14-3.2 Model Adequacy Checking
Figure 14-14 Plot of residuals versus predicted values.
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14-4: General Factorial Experiments
Model for a three-factor factorial experiment
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14-4: General Factorial Experiments
Example 14-2
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Example 14-2
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14-4: General Factorial Experiments
Example 14-2
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14-5: 2k Factorial Designs
14-5.1 22 Design
Figure 14-15 The 22 factorial design.
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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
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14-5: 2k Factorial Designs
14-5.1 22 Design
The AB interaction effect is estimated by
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14-5: 2k Factorial Designs
14-5.1 22 Design
The quantities in brackets in Equations 14-11, 14-12, and 1413 are called contrasts. For example, the A contrast is
ContrastA = a + ab – b – (1)
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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
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14-5: 2k Factorial Designs
Example 14-3
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14-5: 2k Factorial Designs
Example 14-3
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14-5: 2k Factorial Designs
Example 14-3
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14-5: 2k Factorial Designs
Residual Analysis
Figure 14-16 Normal
probability plot of
residuals for the epitaxial
process experiment.
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14-5: 2k Factorial Designs
Residual Analysis
Figure 14-17 Plot of
residuals versus
deposition time.
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14-5: 2k Factorial Designs
Residual Analysis
Figure 14-18 Plot of
residuals versus arsenic
flow rate.
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14-5: 2k Factorial Designs
Residual Analysis
Figure 14-19 The standard deviation of epitaxial layer thickness at the four runs
in the 22 design.
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14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
Figure 14-20 The 23 design.
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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.
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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
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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
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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
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14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
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14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
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14-5: 2k Factorial Designs
14-5.2 2k Design for k  3 Factors
Contrasts can be used to calculate several quantities:
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14-5: 2k Factorial Designs
Example 14-4
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14-5: 2k Factorial Designs
Example 14-4
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14-5: 2k Factorial Designs
Example 14-4
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14-5: 2k Factorial Designs
Example 14-4
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14-5: 2k Factorial Designs
Example 14-4
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Example 14-4
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14-5: 2k Factorial Designs
Residual Analysis
Figure 14-22 Normal
probability plot of residuals
from the surface roughness
experiment.
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure 14-23 Normal
probability plot of effects
from the plasma etch
experiment.
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure 14-24 AD (Gap-Power) interaction from the plasma etch experiment.
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
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14-5: 2k Factorial Designs
14-5.3 Single Replicate of the 2k Design
Example 14-5
Figure 14-25 Normal
probability plot of residuals
from the plasma etch
experiment.
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14-5: 2k Factorial Designs
14-5.4 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 14-26 illustrates the situation.
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14-5: 2k Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Figure 14-26 A 22 Design
with center points.
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14-5: 2k 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:
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14-5: 2k Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
Figure 14-27 The 22 Design
with five center points for
Example 14-6.
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14-5: 2k Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
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14-5: 2k Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
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14-5: 2k Factorial Designs
14-5.4 Additional Center Points to a 2k Design
Example 14-6
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14-6: Blocking and Confounding in the 2k Design
Figure 14-28 A 22 design in two blocks. (a) Geometric view. (b) Assignment of the four runs to
two blocks.
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14-6: Blocking and Confounding in the 2k Design
Figure 14-29 A 23 design in two blocks with ABC confounded. (a) Geometric view. (b) Assignment
of the eight runs to two blocks.
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14-6: Blocking and Confounding in the 2k Design
General method of constructing blocks employs a
defining contrast
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14-6: Blocking and Confounding in the 2k Design
Example 14-7
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14-6: Blocking and Confounding in the 2k Design
Example 14-7
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Example 14-7
Figure 14-30 A 24 design in two blocks for Example 14-7. (a) Geometric view. (b) Assignment of
the 16 runs to two blocks.
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14-6: Blocking and Confounding in the 2k Design
Example 14-7
Figure 14-31 Normal probability plot of the
effects from Minitab, Example 14-7.
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14-6: Blocking and Confounding in the 2k Design
Example 14-7
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14-7: Fractional Replication of the 2k Design
14-7.1 One-Half Fraction of the 2k Design
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14-7: Fractional Replication of the 2k Design
14-7.1 One-Half Fraction of the 2k Design
Figure 14-32 The one-half fractions of the 23 design. (a) The principal fraction, I = +ABC. (B)
The alternate fraction, I = -ABC
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14-7: Fractional Replication of the 2k Design
Example 14-8
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14-7: Fractional Replication of the 2k Design
Example 14-8
Figure 14-33 The 24-1 design for the experiment of Example 14-8.
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14-7: Fractional Replication of the 2k Design
Example 14-8
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14-7: Fractional Replication of the 2k Design
Example 14-8
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14-7: Fractional Replication of the 2k Design
Example 14-8
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14-7: Fractional Replication of the 2k Design
Example 14-8
Figure 14-34 Normal probability plot of the effects from Minitab, Example 14-8.
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14-7: Fractional Replication of the 2k Design
Projection of the 2k-1 Design
Figure 14-35 Projection of a 23-1 design into three 22 designs.
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14-7: Fractional Replication of the 2k Design
Projection of the 2k-1 Design
Figure 14-36 The 22 design obtained by dropping factors B and C from the plasma etch
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experiment in Example 14-8.
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14-7: Fractional Replication of the 2k Design
Design Resolution
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14-7: Fractional Replication of the 2k Design
14-7.2 Smaller Fractions: The 2k-p Fractional
Factorial
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14-7: Fractional Replication of the 2k Design
Example 14-9
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Example 14-8
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14-7: Fractional Replication of the 2k Design
Example 14-9
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14-7: Fractional Replication of the 2k Design
Example 14-9
Figure 14-37 Normal probability plot
of effects for Example 14-9.
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14-7: Fractional Replication of the 2k Design
Example 14-9
Figure 14-38 Plot of AB (mold
temperature-screw speed)
interaction for Example 14-9.
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14-7: Fractional Replication of the 2k Design
Example 14-9
Figure 14-39 Normal probability plot
of residuals for Example 14-9.
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14-7: Fractional Replication of the 2k Design
Example 14-9
Figure 14-40 Residuals versus
holding time (C) for Example 14-9.
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14-7: Fractional Replication of the 2k Design
Example 14-9
Figure 14-41 Average shrinkage and range of shrinkage in factors A, B, and C for Example
14-9.
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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.
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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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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
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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.
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14-8: Response Surface Methods and Designs
Example 14-11
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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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14-8: Response Surface Methods and Designs
Example 14-11
Figure 14-46 Steepest ascent experiment for Example 14-11.
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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
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