#### Transcript DOX - University of Tennessee

```IE440:PROCESS IMPROVEMENT THROUGH
PLANNED EXPERIMENTATION
The 2K Factorial Design
Dr. Xueping Li
Dept. of Industrial & Information Engineering
University of Tennessee, Knoxville
1
Design of Engineering Experiments
Part 4 – Introduction to Factorials

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Text reference, Chapter 6
The 22 Design
The 23 Design
The general 2K Design
The addition of center points to the 2K
design
2
Chemical Process Example
A = reactant concentration,
B = catalyst amount,
y = recovery
3
The Simplest Case: The 22
“-” and “+” denote
the low and high
levels of a factor,
respectively
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
4
22 Factorial Design
Involves two factors (A and B) and n replicates.
We are interested in:

main effect of A,


main effect of B, and
interaction between A and B.
5
Estimation of Factor Effects
A  y A  y A
ab  a b  (1)


2n
2n
 21n [ab  a  b  (1)]
B  yB  yB
ab  b a  (1)

2n
2n
 21n [ab  b  a  (1)]

See textbook, pg. 205-206 For
manual calculations
The effect estimates are:
A
= 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
ab  (1) a  b
AB 

2n
2n
 21n [ab  (1)  a  b]
6
Table 6.2 (p. 208)
Standard order/Yate’s order
Algebraic Signs for Calculating Effects in the 22 Design
7
Contrast
::=combinations (*) signs
Effects
::=2constrast/[n*2^(k)]
SS
::=constrast^2/[n*2^(k)]
Figure 6.1 (p. 204)
Treatment combinations in the 22 design.
8
Table 6.1 (p. 208)
Analysis of Variance for the Experiment in Figure 6.1
9
The Regression Model
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The regression model
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Coded variable vs. natural variable
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
X_Coded= [Var – (Var_L+Var_H)/2 ]/(Var_H –
Var_L)/2
Var = X_Coded * (Var_H – Var_L)/2 +
(Var_L+Var_H)/2
10
Figure 6.2 (p. 210)
Residual plots for the chemical process experiment.
11
Figure 6.3 (p. 211)
Response surface plot and contour plot of yield from the chemical process experiment.
12
The 23 factorial design.
Figure 6.4 (p. 211)
The 23 factorial design.
13
Figure 6.5 (p. 213)
Geometric presentation of
contrasts corresponding to the
main effects and interactions in
the 23 design.
14
Table 6.3 (p. 214)
Algebraic Signs for Calculating Effects in the 23 Design
15
Table 6.4 (p. 215)
The Plasma Etch Experiment, Example 6.1
16
Figure 6.6 (p. 216)
The 23 design for the plasma etch experiment for Example 6-1.
17
Table 6.5 (p. 217)
Effect Estimate Summary for Example 6.1
18
Table 6.6 (p. 218)
Analysis of Variance for the Plasma Etching Experiment
19
Figure 6.7 (p. 219)
Response surface and contour plot of etch rate for Example 6-1.
20
Table 6.7a (p. 220)
Design-Expert Output for Example 6.1
21
Table 6.7b (p. 221)
Continued from Previous Slide
22
Table 6.7 (p. 220)
Detail 1
23
Table 6.7 (p. 220)
Detail 2
24
Table 6.7 (p. 220)
Detail 3
25
Table 6.7 (p. 220)
Detail 4
26
Table 6.7 (p. 220)
Detail 5
27
Table 6.7 (p. 220)
Detail 6
28
Figure 6.8 (p. 224)
Ranges of etch rates for Example 6-1.
29
Table 6.8 (p. 225)
Analysis Procedure for a 2k Design
30
Table 6.9 (p. 225)
Analysis of Variance for a 2k
Design
31
To-do-list

Project
32
Figure 6.9 (p. 227)
The impact of the choice of factor levels in an unreplicated design.
33
Unreplicated 2k Factorial Designs

Lack of replication causes potential problems in
statistical testing
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Replication admits an estimate of “pure error” (a better
phrase is an internal estimate of error)
With no replication, fitting the full model results in zero
degrees of freedom for error
Potential solutions to this problem
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Pooling high-order interactions to estimate error
Normal probability plotting of effects (Daniels, 1959)
Other methods…see text, pp. 234
34
Example of an Unreplicated 2k Design
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A 24 factorial was used to investigate the
effects of four factors on the filtration rate of
a resin
The factors are A = temperature, B =
pressure, C = mole ratio, D= stirring rate
Experiment was performed in a pilot plant
35
Table 6.10 (p. 228)
Pilot Plant Filtration Rate Experiment
36
Figure 6.10 (p. 228)
Data from the pilot plant filtration rate experiment for Example 6-2.
37
Estimates of the Effects
38
Table 6.11 (p. 229)
Contrast Constants for the 24 Design
39
Table 6.12 (p. 229)
Factor Effect Estimates and Sums of Squares for the 24 Factorial in
Example 6.2
40
Figure 6.11 (p. 230)
Normal probability plot of the effects for the 24 factorial in Example 6-2.
41
Figure 6.12 (p. 230)
Main effect and interaction plots for Example 6-2.
42
Table 6.13 (p. 231)
Analysis of Variance for the Pilot Plant Filtration Rate Experiment in A, C, and D.
43
Figure 6.13 (p. 233)
Normal probability plot of residuals for Example 6-2.
44
Figure 6.14 (p. 233)
Contour plots of filtration rate, Example 6-2.
45
Figure 6.15 (p. 234)
Half-normal plot of the factor effects from Example 6-2.
46
Figure 6.17 (p. 237)
Data from the drilling experiment of Example 6-3.
47
Figure 6.18 (p. 237)
Normal probability plot of effects for Example 6-3.
48
Figure 6.19 (p. 237)
Normal probability plot of residuals for Example 6-3.
49
Figure 6.20 (p. 237)
Plot of residuals versus predicted rate for Example 6-3.
50
Figure 6.21 (p. 238)
Normal probability plot of effects for Example 6-3 following log transformation.
51
Figure 6.22 (p. 238)
Normal probability plot of residuals for Example 6-3 following log transformation.
52
Figure 6.23 (p. 238)
Plot of residuals versus predicted rate for Example 6-3 following log transformation.
53
Table 6.14 (p. 239)
Analysis of Variance for Example 6.3 Following the Log Transformation
54
Figure 6.24 (p. 239)
Data for the panel process experiment of Example 6-4.
55
Figure 6.25 (p. 239)
Normal probability plot of the factor effects for the panel process experiment of Example 6-4.
56
Figure 6.26 (p. 240)
Plots of residuals versus clamp time for Example 6-4.
57
Figure 6.27 (p. 240)
Cube plot of temperature, clamp time, and resin flow for Example 6-4.
58
Table 6.15 (p. 241)
Calculation of Dispersion Effects for Example 6.4
59
Figure 6.28 (p. 241)
Normal probability plot of the dispersion effects Fi* for Example 6.4.
60
Table 6.16 (p. 242)
The Oxide Thickness Experiment
61
Table 6.17 (p. 243)
Effect Estimates for Example
6.5, Response Variable Is
Average Oxide Thickness
62
Figure 6.29 (p. 243)
Normal probability plot of the effects for the average oxide thickness response, Example 6.5.
63
Table 6.18 (p. 244)
Analysis of Variance (from Design-Expert) for the Average Oxide Thickness Response,
Example 6.5
64
Figure 6.30 (p. 244)
Contour plots of average oxide thickness with pressure (x3) held constant.
65
Table 6.19 (p. 245)
Analysis of Variance (from Design-Expert) of the Individual Wafer Oxide Thickness Response
66
Figure 6.31 (p. 246)
Normal probability plot of the effects using In (s2) as the response, Example 6-5.
67
Figure 6.32 (p. 246)
Contour plot of s2 (within-run variability) with pressure at the low level and gas flow at the high
level.
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Figure 6.33 (p. 246)
Overlay of the average oxide thickness and s2 responses with pressure at the low level and gas
flow at the high level.
69
to a 2k Designs
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Based on the idea of replicating some of
the runs in a factorial design
Runs at the center provide an estimate of
error and allow the experimenter to
distinguish between two possible models:
k
k
k
First-order model (interaction) y   0    i xi    ij xi x j  
i 1
k
k
i 1 j i
k
k
Second-order model y   0    i xi    ij xi x j    ii xi2  
i 1
i 1 j i
i 1
70
yF  yC  no "curvature"
The hypotheses are:
k
H 0 :   ii  0
i 1
k
H1 :   ii  0
i 1
nF nC ( yF  yC )2

nF  nC
This sum of squares has a
single degree of freedom
71
Central Composite Design
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Second-order response surface model
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Figure 6.34 (p. 248)
A 22 design with center points.
73
Figure 6.35 (p. 248)
A 22 design with center points.
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If curvature is significant, augment the design with axial runs to create a
central composite design. The CCD is a very effective design for fitting a
second-order response surface model
75
Table 6.20 (p. 249)
Analysis of Variance for
Example 6.6
76
Table 6.20 (p. 249)
Detail 1
77
Table 6.20 (p. 249)
Detail 2
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Figure 6.36 (p. 250)
Central composite designs.
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Figure 6.37 (p. 251)
A 22 factorial design with one qualitative factor and center points.
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Table 6.21 (p. 252)
The Circuit Experiment
81
Table 6.22 (p. 252)
Regression Analysis for the Circuit Experiment Using Coded Variables
82
Table 6.23 (p. 253)
Regression Analysis for the Circuit Experiment Using Engineering Units
83
Table 6.24 (p. 254)
Regression Analysis for the Circuit Experiment (Interaction Term Only)
84
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