Chapter 7 Factorial Experiments

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Chapter 7Factorial Experiments
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Agenda

• Factorial experiments
• Two-factor factorial experiments
• General factorial experiments
• 2k factorial design
• Blocking and confounding
• Fractional replication

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Factorial Experiments
Definition
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Factorial Experiments
Factorial Experiment, no interaction.
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Factorial Experiments
Factorial Experiment, with interaction.

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Factorial Experiments
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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Factorial Experiments
Three-dimensional surface plot of the data from
Table 14-1, showing main effects of the A and B
interaction.
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Factorial Experiments
Yield versus reaction time with temperature
constant at 155º F.
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Factorial Experiments
Yield versus temperature with reaction time
constant at 1.7 hours.
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Factorial Experiments
Optimization experiment using the
one-factor-at-a-time method.
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Two-Factor Factorial Experiments
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Two-Factor Factorial Experiments
The observations may be described by the linear
statistical model
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Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
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Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
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Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
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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
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Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
Definition
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Two-Factor Factorial Experiments
Statistical Analysis of the Fixed-Effects Model
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Example 1
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Example 1
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Example 1
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Example 1
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Example 1
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Example 1
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Example 1
Graph of average adhesion force versus primer
types for both application methods.
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Minitab Practice for Example 1

• Data file Example7_1 (rearrange data)
• Menu ? stat ?ANOVA ? Balanced ANOVA
• Response adhesion
• Model TypeMethod
• ?Options check Use Restricted form
• ? Results uncheck display expected mean ..

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Software Output for Example 1
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Two-Factor Factorial Experiments
Model Adequacy Checking
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Two-Factor Factorial Experiments
Model Adequacy Checking
Normal probability plot of the residuals from
Example 1
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Two-Factor Factorial Experiments
Model Adequacy Checking
Plot of residuals versus primer type.
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Two-Factor Factorial Experiments
Model Adequacy Checking
Plot of residuals versus application method.

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Two-Factor Factorial Experiments
Model Adequacy Checking
Plot of residuals versus predicted values.

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General Factorial Experiments
Model for a three-factor factorial experiment
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Example 2
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Minitab Practice for Example 2

• Data file Example7_2 (rearrange data)
• Menu ? stat ?ANOVA ? Balanced ANOVA
• Response Roughness
• Model FeedDepthAngle
• ?Options check Use Restricted form
• ? Results uncheck display expected mean ..

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Example 2
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Example 2
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2k Factorial Designs
22 Design
The 22 factorial design.
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2k Factorial Designs
22 Design
The main effect of a factor A is estimated by
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2k Factorial Designs
22 Design
The main effect of a factor B is estimated by
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2k Factorial Designs
22 Design
The AB interaction effect is estimated by
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2k Factorial Designs
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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2k Factorial Designs
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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Example 3
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Example 3
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Example 3
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2k Factorial Designs
Residual Analysis
Normal probability plot of residuals for the
epitaxial process experiment.
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2k Factorial Designs
Residual Analysis
Plot of residuals versus deposition time.

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2k Factorial Designs
Residual Analysis
Plot of residuals versus arsenic flow rate.

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2k Factorial Designs
Residual Analysis
The standard deviation of epitaxial layer
thickness at the four runs in the 22 design.
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2k Factorial Designs
2k Design for k ? 3 Factors
The 23 design.
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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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2k Factorial Designs
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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2k Factorial Designs
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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2k Factorial Designs
2k Design for k ? 3 Factors
Other two-factor interactions effects estimated by
The interaction effect of ABC is estimated by
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2k Factorial Designs
2k Design for k ? 3 Factors
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2k Factorial Designs
2k Design for k ? 3 Factors
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2k Factorial Designs
2k Design for k ? 3 Factors
Contrasts can be used to calculate several
quantities
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Example 4
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Example 4
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Example 4
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Example 4
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Example 4
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Minitab Practice for Example 4

• Data file Example7_2 (rearrange data)
• Menu ? stat ?DOE ? Factorial ?Analyze Factorial
  Design
• Factors Feed Depth Angle
• Select 2-level factorial
• Define low/high (unchange)
• Responses roughness
• Terms select all
• Results unchange

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Software output for example 4
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2k Factorial Designs
Residual Analysis
Normal probability plot of residuals from the
surface roughness experiment.
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Blocking and Confounding in the 2k Design
A 22 design in two blocks. (a) Geometric view.
(b) Assignment of the four runs to two blocks.

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Blocking and Confounding in the 2k Design
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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Blocking and Confounding in the 2k Design
General method of constructing blocks employs a
defining contrast
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Example 5
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Example 5
Block 2 Block 1
A 24 design in two blocks for Example 5. (a)
Geometric view. (b) Assignment of the 16 runs to
two blocks.
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The 24 design in two blocks
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Minitab Practice for Example 5

• Data file Example7_5.xls
• Menu ?DOE ? Factorial ?Create factorial Design ?
• Select 2-level factorial (default)
• Number of factor 4
• ?Designs
• Select full factorial 16 runs
• Number of blocks 2
• ?Open Example7_5.xls ? copy data and paste to
  worksheet 1
• ? Menu ?DOE ? Factorial ?Analyze factorial design
  
• Response distance
• Terms select all
• Results select all
• Graph select Normal

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Example 5
Normal probability plot of the effects from
software, Example 5.
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Example 5
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Fractional Replication of the 2k Design
One-Half Fraction of the 2k Design
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Fractional Replication of the 2k Design
One-Half Fraction of the 2k Design
The one-half fractions of the 23 design. (a) The
principal fraction, I ABC. (b) The alternate
fraction, I - ABC
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Example 6
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Example 6
The 24-1 design for the experiment of Example 6.

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Example 6
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Example 6
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Minitab Practice for Example 6

• Data file Example7_6.xls
• Menu ?DOE ? Factorial ?Create factorial Design ?
• Select 2-level factorial (default)
• Number of factor 4
• ?Designs
• Select 1/2 factorial 8 runs
• ?Open Example7_6.xls ? copy data and paste to
  worksheet 1
• ? Menu ?DOE ? Factorial ?Analyze factorial design
  
• Response Etch Rt
• Terms select default letters
• Results select default letters
• Graph select Normal

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Example 6
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Example 6
Normal probability plot of the effects from
software, Example 6
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Fractional Replication of the 2k Design
Projection of the 2k-1 Design
Projection of a 23-1 design into three 22
designs.
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Fractional Replication of the 2k Design
Projection of the 2k-1 Design
The 22 design obtained by dropping factors B and
C from the plasma etch experiment in Example 6.

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Fractional Replication of the 2k Design
Design Resolution
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Fractional Replication of the 2k Design
Smaller Fractions The 2k-p Fractional Factorial
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Example 7
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Example 7
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Example 7
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Example 7
Normal probability plot of effects for Example 7.

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Example 7
Plot of AB (mold temperature-screw speed)
interaction for Example 7.
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Example 7
Normal probability plot of residuals for Example
7.
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Example 7
Residuals versus holding time (C) for Example 7.

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Example 7
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Example 7
Average shrinkage and range of shrinkage in
factors A, B, and C