14-1 Introduction

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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.

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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-factor-at-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 14-13 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
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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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 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 14-41 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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