Design of Engineering Experiments The 2k Factorial Design
28
Design of Engineering Experiments Chapter 6 part 1 The 2 k Factorial Design
Transcript of Design of Engineering Experiments The 2k Factorial Design
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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
2
• 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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
6
• 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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
7
plots
• Statistical testing (ANOVA)• Refine the model• Analyze residuals (graphical)• Interpret results
Estimation of Factor Effects
12
(1)2 2[ (1)]
A A
n
B B
A y y
ab a b
n n
ab a b
B y y
+ −
+ −
= −
+ += −
= + − −
= −
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
8
12
12
(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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
11
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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
12
The Response Surface
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
13
The 23 Factorial Design
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
14
Effects in The 23 Factorial Design
etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
+ −
+ −
+ −
= −
= −
= −
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
15
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.
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
16
An Example of a 23 Factorial Design
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
17
A = gap, B = Flow, C = Power, y = Etch Rate
Table of – and + Signs for the 23 Factorial Design (pg. 218)
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
18
)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:
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
19
• 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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
20
EffectX =1
4n(contrast X )
n
ContrastSS X
X 8
)( 2
=
ANOVA Summary – Full Model
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
21
Model Summary Statistics for Reduced Model
• R2 and adjusted R2
52
5
5.106 100.9608
5.314 10Model
T
SSR
SS
×= = =
×
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
22
25
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= = = = =
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
23
• 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
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
24
Model Interpretation
Cube plots are often useful visual displays of experimental results
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
25
results
Cube Plot of Ranges
What do the
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
26
What do the large ranges
when gap and power are at the high level tell
you?
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
27
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 =−
Chapter 6 Design & Analysis of Experiments 8E 2012 Montgomery
28
2)(21
2
XkX Contrastn
SS
n
=
