Statistical Analysis
Professor Lynne Stokes
Department of Statistical Science
Lecture 13
Fractional Factorials
Confounding, Aliases, Design
Resolution,
Pilot Plant Experiment
45
52
80
C2
Catalyst
83
54
68
C1
40
60
160
Concentration
72
Temperature
180
20
Pilot Plant Experiment :
Aliasing/Confounding with Operators
Complete Factorial :
1/2 Replicate for Each of 2
Operators
45
80
C2
Catalyst
52
Operator 1 Operator 2
83
54
68
C1
40
60
Concentration
72
160
180
Temperature
20
Aliasing/Confounding of Effects :
Pilot Plant Experiment
y = Constant + Main Effects + Interaction Effects
+ Operator Effect + Error
M(Temp)
= {180 Temp + Operator 2} - {160 Temp + Operator 1}
= 75.75 - 52.75
= 23.0
Does 23.0 Measure the Effect of
Temperatures, Operators, or Both ?
Aliases : Main Effect for Temperatures and Main Effect for Operators
Main Effect for Operator
Aliased with
Main Effect for Temperature
Aliasing/Confounding of Effects :
Pilot Plant Experiment
y = Constant + Main Effects + Interaction Effects
+ Operator Effect + Error
M(Temp)
M(Cat)
= {180 Temp + Operator 2} - {160 Temp + Operator 1}
= 75.75 - 52.75
= 23.0
= {Cat C2 + (Operator 1 + Operator 2)/2}
- {Cat C1 + (Operator 1 + Operator 2)/2}
= {Cat C2 – Cat C1}
= 65.0 - 63.5 = 1.5
Operator Effect Not Aliased with the
Main Effect for Catalyst
Effects Representations
Overall Average
1
8
y  = c' y , c = 18
Includes Average Influences From All Sources
Main Effect for Temperature
M(Temp) = y 2 - y1 = c T ' y , c T =
1
( -14 '
4
14 ' )'
Catalyst Effect
M(Cat) = y 2 - y 1 = c C ' y , c C =
1
(-1 + 1 - 1 + 1 - 1 + 1 - 1 + 1)'
4
Pilot Plant Experiment :
Aliased Effects
Operator Effect
M(Operator ) = yOp2 - yOp1 = cD ' y , cD =
1
( -14 ' 14 ' )'
4
Overall Average
c’cD = 0
Not Aliased
Main Effect for Temperature
cT’cD = 2
Aliased
Catalyst Effect
cC’cD = 0
Not Aliased
Aliasing / Confounding of Factor
Effects
Factor effects are Aliased or Confounded when differences in
average responses cannot uniquely be attributed to a single effect
Factor effects are Aliased or Confounded when they are estimated
by the same linear combination of response values
Factor effects are Partially Aliased or Partially Confounded when they are
estimated by nonorthogonal linear combinations of response values


Unplanned confounding can result in loss of ability to evaluate
important main effects and interactions
Planned aliasing of unimportant interactions can enable the size
of the experiment to be reduced while still enabling the estimation
of important effects
General Confounding Principle for 2k
Balanced Factoral Experiments
Effects Representations
Effect 1 = c1’y
Effect 2 = c2’y
Two Effects are Confounded or Aliased if
c1 ' c 2  0
Aliases :
c1 = const x c2
Partial Aliases : c1 ' c 2  0
Effects Representation for a
Complete 23 Factorial
Lower Level = -1
Upper level = +1
Run No.
Mean
A
B
C
AB
AC
BC
ABC
Response
1
1
-1
-1
-1
1
1
1
-1
y 111
2
1
-1
-1
1
1
-1
-1
1
y 112
3
1
-1
1
-1
-1
1
-1
1
y 121
4
1
-1
1
1
-1
-1
1
-1
y 122
5
1
1
-1
-1
-1
-1
1
1
y 211
6
1
1
-1
1
-1
1
-1
-1
y 212
7
1
1
1
-1
1
-1
-1
-1
y 221
8
Divisor
1
8
1
4
1
4
1
4
1
4
1
4
1
4
1
4
y 222
Effect = c’y / Divisor
y = Vector of Responses or Average Responses
Aliasing with Operator
Run No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Operator
1
1
1
1
-1
-1
-1
-1
Same
Alias if
All Signs
Reversed
Aliasing with Operators
Run No.
1
2
3
4
5
6
7
8
Confounded
With
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Operator
1
1
1
1
-1
-1
-1
-1
---
Operator
---
---
---
---
---
---
A
Better design for operator aliasing?
Aliasing with Operators
Run No.
1
2
3
4
5
6
7
8
Confounded
With
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Operator
-1
1
1
-1
1
-1
-1
1
---
---
---
---
---
---
---
Operator
ABC
Note: Operator effect is unconfounded with all effects except ABC;
Good choice of contrast for aliasing with operators
Summary


Some designs have one or more factors aliased with one
another
 Sums of squares measure the same effect or partially
measure the same effect
 The sums of squares are not statistically independent
Determining Aliases
 If two-level factors, multiply effect contrasts
 If nonzero, the effects are partially aliased
 If one is a multiple of another, the effects are aliased
Summary (con’t)

Accommodation
 Eliminate one of the aliased effects
 Leave all In but properly interpret analysis of variance
results
(to be discussed in subsequent classes)
Two Types of Aliasing
Fractional Factorials in
Completely Randomized Designs:
Can’t Run All Combinations
Distinguish
Randomized Incomplete Block Designs :
Insufficient Homogeneous Experimental Units
or Homogeneous Test Conditions
in Each Block – Must Include Combinations
in Two or More Blocks
Fractional Factorials

Pilot Plant Chemical Yield Study





Temperature: 160, 180 oC
Concentration: 20, 40 %
Catalysts: 1, 2
Too costly to run all 8 combinations
Must run fewer combinations
Fractional Factorial
Run No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
A
-1
-1
-1
-1
1
1
1
1
Ad-Hoc Fraction
B
-1
-1
1
1
-1
-1
1
1
C
-1
1
-1
1
-1
1
-1
1
Effect
Mean
A
C
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
Partial Aliases
A, B, AB
Mean, B, AB
AC, BC, ABC
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Half-Fraction Fractional Factorial
Run No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Half Fraction
# Possible Combinations
# Combinations in Design
 8
  = 1,680 Ways of Selecting a Half Fraction
 4
Poor Choice for a Fractional Factorial
Run No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Poor Choice for a Fractional Factorial
Run No.
Mean
A
B
C
AB
AC
BC
ABC
1
1
-1
-1
-1
1
1
1
-1
4
1
-1
1
1
-1
-1
1
-1
5
1
1
-1
-1
-1
-1
1
1
7
1
1
1
-1
1
-1
-1
-1
Confounded C,AC,
C,AC,
C,AC,
Mean,A,B, C,AC,
Mean,A, Mean,A, Mean,A,
With
BC,ABC BC,ABC BC,ABC C,AB,BC BC,ABC B,AB
B,C,AB B,AB
Good Choice for a Fractional
Factorial
Run No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
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
AB
1
1
-1
-1
-1
-1
1
1
AC
1
-1
1
-1
-1
1
-1
1
Notation
Defining Equation (Contrast)
The effect(s) aliased with the mean
I = ABC
Convention
Designate the mean by I (Identity)
BC
1
-1
-1
1
1
-1
-1
1
ABC
-1
1
1
-1
1
-1
-1
1
Confounding Pattern
Run No.
2
3
5
8
Confounded
With
Mean
1
1
1
1
A
-1
-1
1
1
B
-1
1
-1
1
C
1
-1
-1
1
AB
1
-1
-1
1
AC
-1
1
-1
1
BC
-1
-1
1
1
ABC
1
1
1
1
ABC
BC
AC
AB
C
B
A
Mean
Main effects only aliased with interactions
Defining Contrast
I = ABC
Design Resolution
Resolution R
Effects involving s factors are unconfounded
with effects involving fewer than R-s factors
Resolution III (R = 3)
Main Effects (s = 1) are unconfounded with
other main effects (R - s = 2)
Example : Half-Fraction of 23 (23-1)
Design Resolution
Resolution R
Effects involving s factors are unconfounded
with effects involving fewer than R-s factors
Resolution IV (R = 4)
Main Effects (s = 1) are unconfounded with
other main effects & two-factor interactions(R - s = 3)
Two-factor interactions (s = 2) are unconfounded with
main effects (R - s = 2); confounded with other
two-factor interactions
Confounding Pattern
Run No.
2
3
5
8
Confounded
With
Mean
1
1
1
1
A
-1
-1
1
1
B
-1
1
-1
1
C
1
-1
-1
1
AB
1
-1
-1
1
AC
-1
1
-1
1
BC
-1
-1
1
1
ABC
1
1
1
1
ABC
BC
AC
AB
C
B
A
Mean
Resolution III
Main Effects (s = 1) unaliased with other main effects (R - s = 2)
Importance of Design Resolution


Quickly identifies the overall structure of the
confounding pattern
A design of resolution R is a complete
factorial in any R-1 or fewer factors
B
A
C
C
B
B
A
C
A
Figure 7.3 Projections of a half fraction of a three-factor complete factorial
experiment (I=ABC).
Pilot Plant Experiment :
Half Fraction
45
80
C2
Catalyst
52
83
54
I = ABC
68
C1
40
60
160
Concentration
72
Temperature
180
20
Pilot Plant Experiment : RIII is a Complete
Factorial in any R-1 = 2 Factors
80
52
80
54
80
52
54
54
72
52
72
72
Catalyst
Concentration
Temperature
Importance of Fractional Factorial
Experiments
Design Efficiency
Reduce the size of the experiment
through
intentional aliasing of relatively unimportant
effects
Effects Representation for a
Complete 23 Factorial
Lower Level = -1
Upper level = +1
Run No.
Mean
A
B
C
AB
AC
BC
ABC
y
1
1
-1
-1
-1
1
1
1
-1
y 111
2
1
-1
-1
1
1
-1
-1
1
y 112
3
1
-1
1
-1
-1
1
-1
1
y 121
4
1
-1
1
1
-1
-1
1
-1
y 122
5
1
1
-1
-1
-1
-1
1
1
y 211
6
1
1
-1
1
-1
1
-1
-1
y 212
7
1
1
1
-1
1
-1
-1
-1
y 221
8
Divisor
1
8
1
4
1
4
1
4
1
4
1
4
1
4
1
4
y 222
Effect = c’y / Divisor
y = Vector of responses or average responses for the run numbers
Designing a 1/2 Fraction of a 2k
Complete Factorial
Resolution = k





Write the effects representation for the main
effects and the highest-order interaction for a
complete factorial in k factors
Randomly choose the +1 or -1 level for the
highest-order interaction (defining contrast,
defining equation)
Eliminate all rows except those of the chosen
level (+1 or -1) in the highest-order interaction
Add randomly chosen repeat tests, if possible
Randomize the test order or assignment to
experimental units
Resolution III Fractional Factorial
Combination No.
1
2
3
4
5
6
7
8
Mean
1
1
1
1
1
1
1
1
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
I = +ABC
Defining Contrast
ABC
-1
1
1
-1
1
-1
-1
1
Aliasing Pattern



Write the defining equation (contrast)
(I = Highest-order interaction)
Symbolically multiply both sides of the defining equation by
each of the other effects
Reduce the right side of the equations:
XxI=X
X x X = X2 = I (powers mod(2) )
Defining Equation:
Aliases :
I
A
B
C
=
=
=
=
ABC
AABC = BC
ABBC = AC
ABCC = AB
Resolution = III
(# factors in the
defining contrast)
Acid Plant Corrosion Rate Study
Factor
Raw-material feed rate
Levels
3000 pph 6000 pph
100 oC
200 oC
Scrubber water
5%
20%
Reactor-bed acid
20%
30%
Exit temperature
300 oC
360 oC
East
West
Gas temperature
Reactant distribution
point
64 Combinations
Cannot Test All Possible Combinations
Acid Plant Corrosion Rate Study: Half Fraction
(I = - ABCDEF)
Factor-level
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
E
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
F
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
ABCDEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
RVI
F
A
E
C
B
D
Figure 7.4 Half Fraction (RVI) of a 26 Experiment: I = -ABCDEF.
Designing Higher-Order Fractions







Total number of factor-level combinations = 2k
Experiment size desired = 2k/2p = 2k-p
Choose p defining contrasts (equations)
For each defining contrast randomly decide which
level will be included in the design
Select those combinations which simultaneously
satisfy all the selected levels
Add randomly selected repeat test runs
Randomize
Acid Plant Corrosion Rate Study: Half Fraction
(I = - ABCDEF)
Half
Fraction
26-1
Factor-level
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
E
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
F
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
ABCDEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
RVI
Acid Plant Corrosion Rate Study: Quarter Fractions
I = - ABCDEF & I = ABC
Factor-level
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
E
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
F
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
ABCDEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
ABC
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
Quarter
Fraction
26-2
Acid Plant Corrosion Rate Study: Quarter Fraction
(I = - ABCDEF = +ABC)
Factor-level
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
C
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
E
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
F
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
ABCDEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
ABC
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Quarter
Fraction
26-2
F
A
E
C
B
D
Figure 7.5 Quarter fraction (RIII) of a 26 experiment: I = -ABCDEF
= ABC (= -DEF).
Acid Plant Corrosion Rate Study: Half Fraction
(I = - ABCDEF = +ABC = -DEF)
Factor-level
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
C
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
E
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
F
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
ABCDEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
ABC
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Implicit Contrast
-ABCDEF x ABC = -AABBCCDEF = -DEF
DEF
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Design Resolution for Fractional
Factorials


Determine the p defining equations
Determine the 2p - p - 1 implicit defining equations:
symbolically multiply all of the defining equations
Resolution = Smallest ‘Word’ length in
the defining & implicit equations

Each effect has 2p aliases
26-2 Fractional Factorials :
Confounding Pattern
Build From 1/4 Fraction
I = ABCDEF = ABC = DEF
A = BCDEF = BC = ADEF
B = ACDEF = AC = BDEF
...
RIII
(I + ABCDEF)(I + ABC) = I + ABCDEF + ABC + DEF
Defining Contrasts
Implicit Contrast
26-2 Fractional Factorials :
Confounding Pattern
Build From 1/2 Fraction
I = ABCDEF = ABC = DEF
A = BCDEF = BC = ADEF
B = ACDEF = AC = BDEF
...
RIII
Optimal 1/4 Fraction
I = ABCD = CDEF
= ABEF
A = BCD = ACDEF = BEF
B = ACD = BCDEF = AEF
...
RIV