Handout #12: More on Fractional 2k Designs
A full 25 design:
Consider a ¼ fraction of this full factorial experiment: 8 of the 32 possible runs are to be
completed.
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Consider the design generators and defining relation given by Minitab:
Fractional Factorial Design
Factors:
Runs:
Blocks:
5
8
1
Base Design:
Replicates:
Center pts (total):
5, 8
1
0
Resolution:
Fraction:
III
1/4
* NOTE * Some main effects are confounded with two-way interactions.
Design Generators: D = AB, E = AC
Alias Structure
I + ABD + ACE + BCDE
A + BD + CE + ABCDE
B + AD + CDE + ABCE
C + AE + BDE + ABCD
D + AB + BCE + ACDE
E + AC + BCD + ABDE
BC + DE + ABE + ACD
BE + CD + ABC + ADE
The Design Matrix – to determine which setting of the factors are actually run:
Notice that the design matrix selects opposite corners of each cube in each of the four
quadrants.
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Example: Example 8.7 from page 323 of your text.
Example 8.7 is an example of a 27-4 resolution III design. That is, only 8 of the possible
128 runs were used. We consider this to be a highly fractionated design.
Letting Minitab do all the work…
Using the default design generators:
Fractional Factorial Design
Factors:
Runs:
Blocks:
7
8
1
Base Design:
Replicates:
Center pts (total):
7, 8
1
0
Resolution:
Fraction:
III
1/16
* NOTE * Some main effects are confounded with two-way interactions.
Design Generators: D = AB, E = AC, F = BC, G = ABC
Alias Structure
I + ABD + ACE + AFG + BCF + BEG + CDG + DEF + ABCG + ABEF + ACDF + ADEG + BCDE
+ BDFG + CEFG + ABCDEFG
A + BD + CE + FG +
ABDFG + ACEFG
B + AD + CF + EG +
ABDEG + BCEFG
C + AE + BF + DG +
ACDEG + BCDFG
D + AB + CG + EF +
ABDEF + CDEFG
E + AC + BG + DF +
ACDEF + BDEFG
F + AG + BC + DE +
ADEFG + BCDEF
G + AF + BE + CD +
ACDFG + BCDEG
BCG + BEF
+ BCDEFG
ACG + AEF
+ ACDEFG
ABG + ADF
+ ABDEFG
ACF + AEG
+ ABCEFG
ABF + ADG
+ ABCDFG
ABE + ACD
+ ABCDEG
ABC + ADE
+ ABCDEF
+ CDF + DEG + ABCF + ABEG + ACDG + ADEF + ABCDE +
+ CDE + DFG + ABCE + ABFG + BCDG + BDEF + ABCDF +
+ BDE + EFG + ABCD + ACFG + BCEG + CDEF + ABCEF +
+ BCE + BFG + ACDE + ADFG + BCDF + BDEG + ABCDG +
+ BCD + CFG + ABDE + AEFG + BCEF + CDEG + ABCEG +
+ BDG + CEG + ABDF + ACEF + BEFG + CDFG + ABCFG +
+ BDF + CEF + ABDG + ACEG + BCFG + DEFG + ABEFG +
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The design matrix:
After the experiment was run the following responses were obtained:
Using Lenth’s test:
Normal Probability Plot of the Effects
(response is response, Alpha = .05)
99
Effect Ty pe
Not Significant
Significant
95
B
90
Percent
80
D
70
60
50
40
A
30
F actor
A
B
C
D
E
F
G
N ame
A
B
C
D
E
F
G
20
10
5
1
0
10
20
Effect
30
40
Lenth's PSE = 0.675
4
What type of design have we arrived at?
Rep 1
Rep 2
What is the alias structure for this one-half fraction of the 23 design?
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In order to separate the main effects and the two-factor interactions, a second fraction is
run with all of the signs reversed (this is called a fold-over design).
Select Stat > DOE > Modify Design.
Minitab creates the design matrix, and the data from the second run are added to the
spreadsheet:
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Alias Structure (up to order 3)
I
Blocks = A*B*D + A*C*E + A*F*G + B*C*F + B*E*G + C*D*G + D*E*F
A + B*C*G + B*E*F + C*D*F + D*E*G
B + A*C*G + A*E*F + C*D*E + D*F*G
C + A*B*G + A*D*F + B*D*E + E*F*G
D + A*C*F + A*E*G + B*C*E + B*F*G
E + A*B*F + A*D*G + B*C*D + C*F*G
F + A*B*E + A*C*D + B*D*G + C*E*G
G + A*B*C + A*D*E + B*D*F + C*E*F
A*B + C*G + E*F
A*C + B*G + D*F
A*D + C*F + E*G
A*E + B*F + D*G
A*F + B*E + C*D
A*G + B*C + D*E
B*D + C*E + F*G
Estimated Effects and Coefficients for response (coded units)
Effect
1.475
38.050
-1.800
29.375
0.125
0.500
0.125
-0.500
-0.400
0.325
1.525
-2.550
-1.125
19.150
Coef
98.638
1.025
0.738
19.025
-0.900
14.687
0.062
0.250
0.063
-0.250
-0.200
0.162
0.763
-1.275
-0.563
9.575
Normal Probability Plot of the Effects
(response is response, Alpha = .05)
99
B
95
90
Effect Ty pe
Not Significant
Significant
F actor
A
B
C
D
E
F
G
D
BD
80
Percent
Term
Constant
Block
A
B
C
D
E
F
G
A*B
A*C
A*D
A*E
A*F
A*G
B*D
70
60
50
40
30
N ame
A
B
C
D
E
F
G
20
10
5
1
AF
0
10
20
Effect
30
40
Lenth's PSE = 0.75
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