Chapter 12
Factorial and Fractional Factorial Experiments
for Process Design and Improvement
Introduction to Statistical Quality
Control, 4th Edition
12-1. What is Experimental Design?
Objectives of Experimental Design
– Determine which variables (x’s) are most
influential on the response, y
– Determine where to set the influential x’s so
that y is near the nominal requirement
– Determine where to set the influential x’s so
that variability is small
– Determine where to set the influential x’s so
that the effects of the uncontrollable
variables z are minimized
Introduction to Statistical Quality
Control, 4th Edition
12-1. What is Experimental Design?
Results of Experimental Design (used early in
process development):
1. Improved yield
2. Reduced variability and closer conformance
to nominal
3. Reduced development time
4. Reduced overall costs
Introduction to Statistical Quality
Control, 4th Edition
12-1. What is Experimental Design?
Example 10-1 Characterizing a Process
• SPC has been applied to a soldering process.
Through u-charts and Pareto analysis, statistical
control has been established and the number of
defective solder joints has been reduced to 1%.
The average board contains over 2000 solder
joints, 1% may still be too large.
• Desired to reduce the defects level more.
Introduction to Statistical Quality
Control, 4th Edition
12-1. What is Experimental Design?
Example 10-1 Characterizing a Process
• Note: since the process is in statistical control,
not obvious what machine adjustments will be
necessary. There are several variables that may
affect the occurrence of defects:
–
•
Solder Temp, Preheat temp, Conveyor speed, Flux
type, Flux specific gravity, Conveyor angle.
A designed experiment involving these factors
could help determine which factors could help
significantly reduce defects. (Screening
experiment)
.
Introduction to Statistical Quality
Control, 4th Edition
12-2. Guidelines for Designing
Experiments
Procedure for designing an experiment
1.
2.
3.
4.
5.
6.
Recognition of and statement of the problem.
Choice of factors and levels.
Selection of the response variable.
Performing the experiment
Data analysis
Conclusions and recommendations
•
•
#1-#3 makeup pre-experimental planning
#2 and #3 often done simultaneously, or in reverse
order.
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
•
•
When there are several factors of interest
in an experiment, a factorial design
should be used.
A complete trial or replicate of the
experiment for all possible combinations
of the levels of the factors are
investigated.
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
•
•
•
Main effect is the change in response produced
by a change in the level of a primary factor.
An interaction is present among factors if a
change in the levels of one factor influences the
effect of another factor.
Consider an experiment with two factors A and
B
–
Interested in
•
•
•
Main effect of A
Main effect of B
Interaction effect of AB
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
•
•
Completely randomized design with two factors (A and
B) and n replicates.
The model is
y ijk     i   j  () ij   ijk
 = overall mean
i = effect of ith level of factor A
j = effect of jth level of factor B
()ij = effect of the interaction between A
and B.
 = random error component
where
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
1
2
Factor A

a
1
y111, y112, …,
y11n
y211, y212, …,
y21n

ya11, ya12, …,
ya1n
Factor B
2

y121, y122, …,

y12n
y221, y222, …,
y22n



ya21, ya22, …,
ya2n

Introduction to Statistical Quality
Control, 4th Edition
b
y1b1, y1b2, …,
y1bn
y2b1, y2b2, …,
y2bn

yab1, yab2, …,
yabn
12-3. Factorial Experiments
12-3.2 Statistical Analysis
•
a
Total corrected sum of squares decomposition
b
n
a
b
2
(
y

y
)

bn
(
y

y
)

an
(
y

y
)
   ijk
 i..
 . j.
...
...
...
2
i 1 j1 k 1
2
i 1
a
j1
b
 n   ( y ij.  y i..  y. j.  y... ) 2
i 1 j1
a
b
n
    ( y ijk  y ij. ) 2
i 1 j1 k 1
Introduction to Statistical Quality
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12-3. Factorial Experiments
12-3.2 Statistical Analysis
•
Total corrected sum of squares decomposition,
notation:
SST = SSA + SSB + SSAB + SSE
•
The corresponding degree of freedom
decomposition is
abn – 1 = (a – 1) + (b – 1) + (a – 1)(b – 1) + ab(n – 1)
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
12-3.2 Statistical Analysis
Source of
Variation
A
Sum of
Squares
SSA
Degrees of
Freedom
a-1
B
SSB
b-1
Interaction
SSAB
(a – 1)(b – 1)
Error
SSE
ab(n-1)
Total
SST
abn - 1
Mean Square
SS
MS A  A
a 1
SS
MS B  B
b 1
SSAB
MS AB 
(a  1)( b  1)
SSE
MS E 
ab(n  1)
Introduction to Statistical Quality
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F0
MS A
F0 
MS E
MS B
F0 
MS E
MS AB
F0 
MS E
12-3. Factorial Experiments
12-3.2 Statistical Analysis
Sum of Squares Computing Formulas
2
y
2
SST     y ijk
 ...
i 1 j1 k 1
abn
2
2
2
a y2
b
y
y
y
.
j
.
SSA   i..  ...
SSB  
 ...
i 1 bn
j1 an
abn
abn
2
2
a b y
y
ij.
SSAB   
 ...  SSA  SSB
i 1 j1 n
abn
a
Main Effects
Interaction
Error
b
n
SSE = SST – SSA - SSB - SSAB
Introduction to Statistical Quality
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12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example
2
y
2
SST     yijk
 ...
i 1 j1 k 1
abn
a
b
n
2
(
89
.
8
)
 (4.0) 2  (4.5) 2    (5.0) 2 
 10.72
18
y i2.. y...2
SSprimers  

i 1 bn
abn
(28.7) 2  (34.1) 2  (27.0) 2 (89.8) 2


 4.58
6
18
2
b y
y...2
. j.
SSmethods  

j1 an
abn
(40.2) 2  (49.6) 2 (89.8) 2


 4.91
9
18
a
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example
SSint eraction
y ij2.
y...2
 

 SSprimers  SSmethods
i 1 j1 n
abn
(12.8) 2  (15.9) 2  (11.5) 2  (15.9) 2  (18.2) 2  (15.5) 2

3
(89.8) 2

 4.58  4.91  0.24
18
a
b
SSE = 10.72 – 4.58 – 4.91 – 0.24 = 0.99
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example – Table 12-4. Analysis of Variance
Source of Variation
Primer types
Application methods
Interaction
Error
Total
Sum of
Squares
4.58
4.91
0.24
0.99
10.72
Degrees of
Freedom
2
1
2
12
17
Mean Square
2.29
4.91
0.12
0.08
Introduction to Statistical Quality
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F0
28.63
61.38
1.5
P-value
2.71 x 10-5
4.65 x 10-6
0.269
12-3. Factorial Experiments
Example 12-5
Aircraft Primer Example – Figure 12-12. Graph of average adhesion
force versus primer types
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
12-3.3 Residual Analysis
• Residuals are important in accessing model
adequacy
• The residuals from a two-factor factorial are
eijk  yijk  ŷijk
 yijk  yijk
Introduction to Statistical Quality
Control, 4th Edition
12-3. Factorial Experiments
12-3.3 Residual Analysis
Residuals Versus Primer
(response is Force)
0.4
0.3
Residual
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
1
2
Primer
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3
12-3. Factorial Experiments
12-3.3 Residual Analysis
Normal Probability Plot
.999
.99
Probability
.95
.80
.50
.20
.05
.01
.001
-0.4
-0.3
-0.2
-0.1
0.0
Residuals
0.1
Introduction to Statistical Quality
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0.2
0.3
12-4. 2k Factorial Design
12-4.1 The 22 Design
• 2k is the notation used to indicate that a certain
experimental design has k factors of interest,
each at two levels.
• 22 design: Two factors A and B, each at two
levels
A
B
Low
-1
-1
High
+1
+1
There are a total of four possible combinations.
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.1 The 22 Design
•
•
•
•
•
The simplest design involves two factors A and B and n
replicates.
Interested in the main effect of A, the main effect of B,
and the interaction between A and B.
Effects are calculated by:
Average Response at high level - Average
Response at the low level.
A large effect would indicate a significant factor (or
interaction). (How large is large?)
Contrasts can be calculated and used to estimate the
effects and then sums of squares.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.1 The 22 Design
•
•
Let the letters (1), a, b, and ab represent the
totals of all n observations taken at these design
points.
Effect estimate of A:
A  yA  yA
a  ab b  (1)

2n
2n
1
a  ab  b  (1)

2n

Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.1 The 22 Design
• Effect estimate of B:
B  y B  y B
b  ab a  (1)

2n
2n
1
b  ab  a  (1)

2n

•
Effect estimate of AB:
ab  (1) a  b
AB 

2n
2n
1
ab  (1)  a  b

2n
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.1 The 22 Design
• For the previous effects formulas, the quantities
in brackets are called contrasts.
• For example, ContrastA = a + ab – b – (1)
• The contrasts are used to calculate the sum of
squares for the factors and interaction.
(contrast ) 2
SS 
n  (contrast coefficien ts ) 2
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.1 The 22 Design
• The sum of squares for A, B, and AB are:

a  ab  b  (1)

2
SSA
4n
2

b  ab  a  (1)
SSB 
4n
2

ab  (1)  a  b
SSAB 
4n
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Example 12-6
•
The effect estimates for A, B, and AB for the router
example are:

a  ab  b  (1)
A
2n
1
96.1  161.1  59.7  64.4  16.64

2( 4)

b  ab  a  (1)
B
2n
1
59.7  161.1  96.1  64.4  7.54

2( 4)
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
Example 12-6
• The effect estimates for A, B, and AB for the
router example are:
AB 
ab  (1)  a  b
2n
1
161.1  64.4  96.1  59.7  8.71

2(4)
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Example 12-6
• The ANOVA table for the router example is then
Source of
Variation
Bit Size (A)
Speed (B)
AB
Error
Total
Sum of
Squares
1107.226
227.256
303.631
71.723
1709.836
Degrees of
Freedom
1
1
1
12
15
P-value
Mean Square
1107.226
227.256
303.631
5.977
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F0
185.25 1.17 x 10-8
38.03 4.82 x 10-5
50.80 1.20 x 10-5
12-4. 2k Factorial Design
Regression Model
•
A regression model could be fit to data from a factorial
design
y  0  1x1  2 x 2  12 x1x 2  
•
•
where 0 is the grand average of all observations and
each coefficient, j is  effect estimate.
For Example 12-6, the fitted regression model is
 16.64 
 7.54 
 8.71 
ŷ  23.83  
 x1  
x 2  
 x1x 2
 2 
 2 
 2 
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12-4. 2k Factorial Design
Residual Analysis
• Residual plots are used to access the adequacy
of the model once again.
• Residuals are calculated using the fitted
regression model.
• The residual plots versus the factor levels,
interactions, predicted values, and a normal
probability plot are all useful in determining the
adequacy of the model and satisfaction of
assumptions.
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Analysis Procedure for Factorial Designs
1. Estimate the factor effects
2. Form preliminary model
3. Test for significance of factor effects
4. Analyze residuals
5. Refine model, if necessary
6. Interpret results
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12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
• When k  2, you could have a single
replicate, but some assumptions need to
be made. (Can’t estimate all interactions).
• For k = 3, the main effects and
interactions of interest are A, B, C, AB, AC,
BC, ABC.
• The main effects are again represented by
a, b, c, ab, ac, bc, abc, and (1)
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for A:
A  yA  yA 
1
a  ab  ac  abc  b  c  bc  (1)
4n
Effect Estimate for B:
B  y B  y B 
1
b  ab  bc  abc  a  c  ac  (1)
4n
Effect Estimate for C:
C  y C  y C
1
c  ac  bc  abc  a  b  ab  (1)

4n
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for AB:
AB  y AB  y AB
1
ab  (1)  abc  c  b  a  bc  ac

4n
Effect Estimate for AC:
AC  y AC  y AC 
1
ac  (1)  abc  b  a  c  ab  bc
4n
Effect Estimate for BC:
BC  y BC   y BC 
1
bc  (1)  abc  a  b  c  ab  ac

4n
Introduction to Statistical Quality
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12-4. 2k Factorial Design
12-4.2 The 2k Design for k  3 Factors
Effect Estimate for ABC:
ABC  y ABC  y ABC 
1
abc  bc  ac  c  ab  b  a  (1)
4n
In general, the effects can be estimated using
Contrast
Effect 
n 2 k 1
The sum of squares for any effect is
2

Contrast 
SS 
n 2k
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Example 12-7
•
An experiment was performed to
investigate the surface finish of a metal
part. The experiment is a 23 factorial
design in the factors feed rate (A), depth
of cut (B), and tool angle (C), with n = 2
replicates.
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Example 12-7
Design Factors
Run
1
2
3
4
5
6
7
8
(1)
a
b
ab
c
ac
bc
abc
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
Introduction to Statistical Quality
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Surface
Finish
9, 7
10, 12
9, 11
12, 15
11, 10
10, 13
10, 8
16, 14
Totals
16
22
20
27
21
23
18
30
12-4. 2k Factorial Design
Example 12-7 – Analysis of Variance Table
Source of
Variation
A
B
C
AB
AC
BC
ABC
Error
Total
Sum of
Squares
45.5625
10.5625
3.0625
7.5625
0.0625
1.5625
5.5625
19.5000
92.9375
Degrees of
Freedom
1
1
1
1
1
1
1
8
15
P-value
Mean Square
45.5625
10.5625
3.0625
7.5625
0.0625
1.5625
5.5625
2.4375
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F0
18.69
4.33
1.26
3.10
0.03
0.64
2.08
2.54 x 10-3
0.07
0.29
0.12
0.88
0.45
0.19
12-4. 2k Factorial Design
Regression Model
•
For Example 12-7, the fitted regression model involving
only those factors found significant (A, B) and the next
significant interaction (AB) is
 3.375 
 1.625 
 1.375 
ŷ  11.0625  
 x1  
x 2  
 x1x 2
 2 
 2 
 2 
 11.0625  1.6875x1  0.8125x 2  0.6875x1x 2
•
Developing a model from a designed experiment can be
a valuable tool in determining optimal settings for the
factors.
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12-4. 2k Factorial Design
Other Methods for Judging the Significance of
Effects
• The standard error of any effect estimate in a 2k
design is
2
ˆ
s.e.(Effect ) 
n 2k 2
•
Two standard deviation limits on any estimated
effect is
Effect estimate  2[s.e.(effect)]
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Other Methods for Judging the Significance of
Effects
Effect estimate  2[s.e.(effect)]
• This interval is an approximate 95% confidence
interval on the estimated effect.
• Interpretation is simple: If zero is contained
within the 95% confidence interval, then that
effect is essentially zero and the corresponding
factor is not significant at the  = 0.05 level.
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12-4. 2k Factorial Design
12-4.3 A Single Replicate of the 2k Design
• As the number of factors in a factorial
experiment increase, the number of effects that
can be estimated also increases.
• In most situations, the sparsity of effects
principle applies.
• For a large number of factors, say k > 5, it is
common practice to run only a single replicate
of the 2k design and pool or combine the
higher-order interactions as estimate of error.
Introduction to Statistical Quality
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12-4. 2k Factorial Design
Example 12-8
• Nitride etch process on a single-wafer plasma
etcher. There are four factors of interest. The
response is etch rate for silicon nitride. A single
replicate is used.
Level
Low (-)
High (+)
Gap
A (cm)
0.80
1.20
Pressure
B (m Torr)
450
550
C2F6 Flow
C (SCCM)
125
200
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Power
D (W)
275
325
12-4. 2k Factorial Design
Example 12-8
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
(Gap)
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
(Pressure)
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
(C2F6 Flow)
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
(Power)
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
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Etch Rate
(? /min)
550
669
604
650
633
642
601
635
1037
749
1052
868
1075
860
1063
729
12-4. 2k Factorial Design
Example 12-8
• The estimated effects are found to be
A = -101.625
AD = -153.625
B = -1.625
BD = -0.625
AB = -7.875
ABD =
4.125
C=
7.375
CD = -2.125
AC = -24.875
ACD =
5.625
BC = -43.875
BCD = -25.375
ABC = -15.625 ABCD = -40.125
D = 306.125
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
Example 12-8
• Normal probability plot of effects
Normal Probability Plot of the Effects
(response is Etch, Alpha = .05)
D
Normal Score
1
0
-1
A
AD
-100
0
100
200
Effect
Introduction to Statistical Quality
Control, 4th Edition
300
A:
B:
C:
D:
A
B
C
D
12-4. 2k Factorial Design
Example 12-8
•
Normal probability plot reveals that A, D, and AD appear
to be significant.
•
To be sure that other main factors or two factor
interactions are not significant, pool the three- and fourfactor interactions to form the error mean square.
•
(NOTE: if the normal probability plot had indicated that
any of these interactions were important, they would
not be included in the error term.)
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
Example 12-8- Analysis of Variance
Source of
Variation
A
B
C
D
AB
AC
AD
BC
BD
CD
Error
Total
Sum of
Squares
41,310.563
10.563
217.563
374,850.063
248.063
2,475.063
94,402.563
7,700.063
1.563
18.063
10,186.815
531,420.936
Degrees of
Freedom
1
1
1
1
1
1
1
5
15
Mean Square
41,310.563
10.563
217.563
374,850.063
248.063
2,475.063
94,402.563
7,700.063
1.563
18.063
2,037.363
Introduction to Statistical Quality
Control, 4th Edition
F0
20.28
<1
<1
183.99
<1
1.21
48.79
3.78
<1
<1
12-4. 2k Factorial Design
Example 12-8- Analysis of Variance
• Factors A, D, and the interaction AD are
significant.
• The fitted regression model for this experiment
is
 101.625 
 306.125 
 153.625 
ŷ  776.0625  
 x1  
x 2  
 x1x 2
2 
2
2 




where x1 represents A, x2 represents D.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
• So far, the assumption of linearity in the factor
effects has been made.
• 2k works well when the linearity assumption
holds only approximately.
• 2k design will support the main effects and
interactions model providing some protection
against curvature.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
• There may be situations where a second-order
model is appropriate.
• Consider the case of k = 2, a model including
second-order effects is
y  0  1x1   2 x 2  12 x1x 2  11x12   22 x 22  
• The model cannot be fitted using a 22 design; in
order to fit a quadratic model, all factors must
be run at at least three levels.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
•
Center points can be added to the standard 2k design
•
Center points can provide not only some protection
against curvature, if the center points are replicated,
then an independent estimate of experimental error can
be obtained.
•
Center points consist of nc replicates run at the xi = 0 (i =
1, 2, …, k).
•
Addition of center points does not have an impact on
the usual effects estimates in a 2k design.
•
Assume the k factors are quantitative in order to have a
“center” or middle level of the factor.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
• Sum of Squares for pure quadratic error
n f n c y F  y c 
SSpure quadratic 
nF  nc
2
•
where nF = number of factorial design points
y F , y c = average of runs at factorial points,
and average of center points, respectively.
SSpure quadratic has a single degree of freedom.
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
• When center points are added to the design,
the model that can be estimated is
k
k
y  0    j x j   ijx i x j    jjx 2j  
i j
j1
•
j1
where jj are pure quadratic effects
The test for curvature then actually tests
k
H 0 :   jj  0
j1
vs.
k
H1 :   jj  0
Introduction to Statistical Quality
Control, 4th Edition
j1
12-4. 2k Factorial Design
Example 12-9
•
•
•
Reconsider the plasma etch experiment from example
12-8. Four center points (nc = 4) have been added to
the design with the responses given in Table 12-18.
Averages: yF  776.0625 yc  752.75
Curvature Sum of Squares
n f n c y F  y c 
SSpure quadratic 
nF  nc
2
16(4)(776.0625  752.75) 2

16  4
 1739.1
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
Example 12-9
• An estimate of experimental error can be
obtained by calculating the sample variance of
the center points:
20
ˆ 2 
2
(
y

752
.
75
)
 i
i 17
3
 3122.7
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
12-4.4 Addition of Center Points to the 2k Design
• The F-test for curvature is given by
MSCurvature
F0 
MSRe sidual
where MSresidual = SSresidual/df, with SSresidual a
combination of sum of squares for pure error
and sum of squares for lack of fit. (See Minitab
output, next slide).
Introduction to Statistical Quality
Control, 4th Edition
12-4. 2k Factorial Design
Minitab Output for Example 12-9
Estimated Effects and Coefficients for Etch (coded units)
Term
Constant
A
B
C
D
A*B
A*C
A*D
B*C
B*D
C*D
Ct Pt
Effect
-101.62
-1.63
7.37
306.12
-7.88
-24.88
-153.62
-43.87
-0.63
-2.13
Coef
776.06
-50.81
-0.81
3.69
153.06
-3.94
-12.44
-76.81
-21.94
-0.31
-1.06
-23.31
StDev Coef
10.20
10.20
10.20
10.20
10.20
10.20
10.20
10.20
10.20
10.20
10.20
22.80
T
76.11
-4.98
-0.08
0.36
15.01
-0.39
-1.22
-7.53
-2.15
-0.03
-0.10
-1.02
P
0.000
0.001
0.938
0.727
0.000
0.709
0.257
0.000
0.064
0.976
0.920
0.337
Analysis of Variance for Etch (coded units)
Source
Main Effects
2-Way Interactions
Curvature
Residual Error
Lack of Fit
Pure Error
Total
DF
4
6
1
8
5
3
19
Seq SS
416389
104845
1739
13310
10187
3123
536283
Adj SS
416389
104845
1739
13310
10187
3123
Introduction to Statistical Quality
Control, 4th Edition
Adj MS
104097
17474
1739
1664
2037
1041
F
62.57
10.50
1.05
P
0.000
0.002
0.337
1.96
0.308
12-5. Fractional Replication of the
2k Design
•
•
As the number of factors in a 2k increase, the
number of runs required increases rapidly.
If we can assume that some higher-order
interactions are negligible, then a fractional
factorial design can be used to gain
information on main effects and low-order
interactions.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
• Contains 2k-1 runs
• Often called a 2k-1 fractional factorial
design
• Consider a three-factor design: 23-1; that
is one half of a full 23
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
Table for 23 with all main effects and interactions:
Run
a
b
c
abc
ab
ac
bc
(1)
•
•
•
I
+
+
+
+
+
+
+
+
A
+
+
+
+
-
B
+
+
+
+
-
C
+
+
+
+
-
Factorial Effect
AB
AC
+
+
+
+
+
+
+
+
BC
+
+
+
+
ABC
+
+
+
+
-
Suppose runs a, b, c, and abc are chosen as the one-half fraction
(shown in the top half of the table)
Note that the runs selected yield a plus on the ABC effect.
ABC is called a generator of this particular fraction.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
•
•
•
The identity element I is also plus for the four runs,
therefore I = ABC.
I = ABC is the defining relation for the design.
The defining relation can be used to find aliases of the
main effects and interactions.
To find the aliases of a main effect or interaction,
multiply the entire defining relation by that term. (Note:
any column multiplied by itself always results in the
identity column, i.e. A•A = I)
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
With defining relation, I = ABC, some of the aliases are
–
–
–
The alias of main effect A is
A•I = A •ABC = BC
That is, A is aliased with (identical to) the BC interaction.
The alias of the main effect B is
B •I = B •ABC = AC
That is, B is aliased with the AC interaction
The alias of the AB interaction is
AB •I = AB •ABC = C
That is, the AB interaction is aliased with the main effect C.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
• Normal Probability Plots and Residuals
–
–
–
Normal probability plot can be useful in assessing the
significance of effects (especially when there are
many effects to be estimated)
Residuals can be obtained by the regression model
shown previously.
Residuals should be plotted against predicted values,
against levels of the factors, and on normal
probability paper for two reasons:
1.
2.
To assess the validity of the underlying model assumptions
Gain additional insight into the experimental situation.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
Projection of the 2k-1 Design
– If one or more factors from a one-half
fraction of a 2k can be dropped, the design
will project into a full factorial design.
– Projection is highly useful in screening
experiments.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
• Design Resolution
–
–
–
–
Design resolution is useful in categorizing designs.
The resolution of a design usually denoted by Roman
numerals.
Of particular importance are designs of Resolution III,
IV, and V.
The resolution of a design indicates the alias
relationship among the factors.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
Design Resolution
–
–
–
Resolution III designs. No main effects are aliased with
one another. Main effects are aliased with two factor
interactions.
Resolution IV designs. No main effect is aliased with any
other main effect or two-factor interaction. Two-factor
interactions are aliased with each other.
Resolution V designs. No main effects are aliased with
any other main effect or two-factor interactions. Twofactor interactions are not aliased with one another. Twofactor interactions are aliased with three-factor
interactions.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.1 The One-Half Fraction of the 2k
•
Design Resolution
–
For example, the notation 2 3III1indicates that
the design is a one-half fraction of a 23 with
resolution III. From this, we know that main
effects are not aliased with one another, but
are aliased with two factor interactions.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.2 Smaller Fractions: The 2k-p Fractional Factorial Design
•
While the half fraction is useful, it may be more
economical to use even smaller fractions of the 2k.
•
Consider a design with k = 11 factors.
–
–
•
A full 2k would require 2,048 runs
A half fraction, 2k-1 would require 1,024 runs: still an
unreasonable number.
How about a design for all 11 factors that would only
require 32 runs? Some assumptions would have to be
made, but this is a possible design.
–
11 factors in 32 runs would be a 1/6 fraction of the 211 or a 2116
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.2 Smaller Fractions: The 2k-p Fractional
Factorial Design
• Setting up a 2k-p design requires:
1. Setting up a full factorial design for k-p
factors.
2. Generate the remaining p columns by
selecting appropriate design generators.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
To illustrate, consider a 26-2 fractional factorial design.
1. Set up a full factorial design for 6 – 2 = 4 factors:
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
+
+
+
+
+
+
+
+
B
+
+
+
+
+
+
+
+
C
+
+
+
+
+
+
+
+
D
+
+
+
+
+
+
+
+
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
2. Generate the remaining 2 columns using a design generator that
involves the 4 factors used in part 1. For example, the column for factor
E
could be generated using ABC. The column for factor F could be
Run
A B C D E
F
1
generated using BCD
2
+
+
3
4
5
6
7
8
9
10
11
12
13
14
15
16
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
•
In this example there are two design generators:
–
–
•
•
•
•
E = ABC  I = ABCE
F = BCD  I = BCDF
To find the complete defining relation, multiply all pairs of
the design generators (in this case there are only two so we
would multiply (ABCE)(BCDF) = ADEF)
Therefore, the complete defining relation from which all
aliases can be found for main effects and interactions is
I = ABCE = BCDF = ADEF
By definition, the length of the smallest “word” in the
defining relation is also the resolution of the design. In this
case, the resolution is IV.
Introduction to Statistical Quality
Control, 4th Edition
12-5. Fractional Replication of the
2k Design
12-5.2 Smaller Fractions: The 2k-p Fractional
Factorial Design
• Selection of Design Generators
–
–
–
Design generators should not be chosen arbitrarily.
Choosing the incorrect design generator can result in
a design of smaller resolution than possible.
Select the design generators that will result in the
highest possible design resolution.
Introduction to Statistical Quality
Control, 4th Edition