Post on 12-Jan-2022
Design of Engineering Experiments
Chapter 6 part 1
The 2k Factorial Design
The 2k Factorial Design
• Special case of the general factorial design; k factors, all at two levels
• The two levels are usually called low and high
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• Very widely used in industrial experimentation
• Form a basic “building block” for other very useful experimental designs (DNA)
• Special (short-cut) methods for analysis
The Simplest Case: The 22
The Simplest Case: The 22
The Simplest Case: The 22
Chemical Process Example
(1)(a)(b)
(ab)
A = reactant concentration, B = catalyst amount, y = recovery
The Simplest Case: The 22
“-” and “+” denote the low and high levels of a factor, respectively
• Low and high are arbitrary
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• 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
Analysis Procedure for a
Factorial Design
• Estimate factor effects
• Formulate model– With replication, use full model– With an unreplicated design, use normal probability
plots
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plots
• Statistical testing (ANOVA)• Refine the model• Analyze residuals (graphical)• Interpret results
Estimation of Factor Effects
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(1)2 2[ (1)]
A A
n
B B
A y y
ab a b
n n
ab a b
B y y
+ −
+ −
= −
+ += −
= + − −
= −
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(1)2 2[ (1)]
(1)2 2[ (1) ]
B B
n
n
ab b a
n n
ab b a
ab a bAB
n n
ab a b
+ += −
= + − −
+ += −
= + − −
Statistical Testing - ANOVA
Statistical Testing - ANOVA
Statistical Testing - ANOVA
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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?
Residuals and Diagnostic Checking
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The Response Surface
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The 23 Factorial Design
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Effects in The 23 Factorial Design
etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
+ −
+ −
+ −
= −
= −
= −
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Analysis done via computer
An Example of a 23 Factorial Design
• A 23 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the electrodes, the gas flow(C2F6 is used as the reactant gas), and the RF power(C2F6 is used as the reactant gas), and the RF powerapplied to the cathode (see Figure 3.1 for a schematic of the plasma etch tool). Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride (Å/m). The etch rate data are shown next.
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An Example of a 23 Factorial Design
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A = gap, B = Flow, C = Power, y = Etch Rate
Table of – and + Signs for the 23 Factorial Design (pg. 218)
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)1(−−−−+++= bccbabcacabaContrast A
bcacbaabcabcContrastAB −−−−+++= )1(
Properties of the Table
• Except for column I, every column has an equal number of + and –signs
• The sum of the product of signs in any two columns is zero• Multiplying any column by I leaves that column unchanged (identity
element)• The product of any two columns yields a column in the table:
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• Orthogonal design• Orthogonality is an important property shared by all factorial designs
2
A B AB
AB BC AB C AC
× =
× = =
Estimation of Factor Effects
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EffectX =1
4n(contrast X )
n
ContrastSS X
X 8
)( 2
=
ANOVA Summary – Full Model
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Model Summary Statistics for Reduced Model
• R2 and adjusted R2
52
5
5.106 100.9608
5.314 10Model
T
SSR
SS
×= = =
×
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5.314 10
/ 20857.75 /121 1 0.9509
/ 5.314 10 /15
T
E EAdj
T T
SS
SS dfR
SS df
×
= − = − =×
Model Summary Statistics
• Standard error of model coefficients (full model)
2 2252.56ˆ ˆ( ) ( ) 11.87EMSse V
σβ β= = = = =
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• Confidence interval on model coefficients
2252.56ˆ ˆ( ) ( ) 11.872 2 2(8)
E
k k
MSse V
n n
σβ β= = = = =
/ 2, / 2,ˆ ˆ ˆ ˆ( ) ( )
E Edf dft se t seα αβ β β β β− ≤ ≤ +
The Regression Model
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Model Interpretation
Cube plots are often useful visual displays of experimental results
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results
Cube Plot of Ranges
What do the
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What do the large ranges
when gap and power are at the high level tell
you?
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For what values of Gap and Power the each rate is close to 900 Ao/m?
The General 2k Factorial Design
1 )(21
XkX Contrastn
Effect =−
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2)(21
2
XkX Contrastn
SS
n
=