ST 516
Experimental Statistics for Engineers II
The One-Quarter Fraction
Need two generating relations.
E.g. a 26−2 design, with generating relations I = ABCE and
I = BCDF .
Product of these is ADEF .
Complete defining relation is I = ABCE = BCDF = ADEF .
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Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
This is a resolution-IV design. Why?
There is no resolution-V 26−2 design. Why not?
To set up runs, either:
create the full 26 design with ABCE and BCDF confounded
with blocks, and choose the block with both positive;
or set up a basic design in 4 factors, then add the other 2.
For example, basic design is 24 in A, B, C , D, and defining relations
show that E = ABC and F = BCD.
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Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Basic Design
3 / 15
Run
A
1
2
3
4
5
6
7
8
9
10
11
12
..
.
+
+
+
+
+
+
..
.
16
B
C
D
E = ABC
- - - - +
+ - +
+ - - + +
- + + + + + +
- - +
- - +
+
+ - +
+
+ - +
.. .. ..
..
. . .
.
+ Two-level
+ Fractional
+ +Factorial Designs
+
F = BCD
+
+
+
+
+
+
..
.
+
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Projections
This 26−2
IV design projects into:
a single complete replicate of a 24 design in A, B, C , and D,
and any other of the 12 subsets of 4 factors that is not a word in
the defining relation;
a replicated one-half fraction of a 24 design in A, B, C , and E ,
and in the other two subsets of 4 factors that are a word in the
defining relation;
two replicates of a 23 design in any three factors;
four replicates of a 22 design in any two factors.
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Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Example with this design
Response is shrinkage in injection molding, and factors are:
A, mold temperature;
B, screw speed;
C , holding time;
D, cycle time;
E , gate size;
F , hold pressure.
5 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Data file
injection.txt:
A
+
+
+
+
+
+
+
+
B
+
+
+
+
+
+
+
+
C
+
+
+
+
+
+
+
+
D
+
+
+
+
+
+
+
+
6 / 15
E
+
+
+
+
+
+
+
+
F
+
+
+
+
+
+
+
+
Shrinkage
6
10
32
60
4
15
26
60
8
12
34
60
16
5
37
52
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
R commands
injection <- read.table("data/injection.txt", header = TRUE)
for (j in 1:(ncol(injection) - 1))
injection[ , j] <- coded(injection[ , j])
summary(lm(Shrinkage ~ A * B * C * D * E * F, injection))
Output
Call:
lm(formula = Shrinkage ~ A * B * C * D * E * F, data = injection)
Residuals:
ALL 16 residuals are 0: no residual degrees of freedom!
Coefficients: (48 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 27.3125
NA
NA
NA
A
6.9375
NA
NA
NA
B
17.8125
NA
NA
NA
C
-0.4375
NA
NA
NA
D
0.6875
NA
NA
NA
7 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Output, continued
E
F
A:B
A:C
B:C
A:D
B:D
C:D
D:E
A:B:D
A:C:D
0.1875
0.1875
5.9375
-0.8125
-0.9375
-2.6875
-0.0625
-0.0625
0.3125
0.0625
-2.4375
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
Residual standard error: NaN on 0 degrees of freedom
Multiple R-Squared:
1,
Adjusted R-squared:
F-statistic:
NaN on 15 and 0 DF, p-value: NA
NaN
Note that all 2-factor interactions are aliased with one or two other
2-factor interactions, and all but two 3-factor interactions are aliased
with main effects or other 3-factor interactions.
8 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Main effect alias chains
A = BCE = DEF = ABCDF
B = ACE = CDF = ABDEF
C = ABE = BDF = ACDEF
D = BCF = AEF = ABCDE
E = ABC = ADF = BCDEF
F = BCD = ADE = ABCEF
9 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Other alias chains
AB
AC
AD
AE
AF
BD
BF
ABD
ACD
10 / 15
= CE = ACDF = BDEF
= BE = ABDF = CDEF
= EF = BCDE = ABCF
= BC = DF = ABCDEF
= DE = ABCD = BCEF
= CF = ACDE = ABEF
= CD = ACEF = ABDE
= CDE = ACF = BEF
= BDE = ABF = CEF
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
70
●
40
30
Effects
50
60
B
●
A
●
0
10
20
A:B
●
●
0.0
●
●
●
●
●
●
0.5
●
●
●
●
1.0
1.5
Half Normal plot
11 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
The half-normal plot suggests that A and B are the important effects.
Interaction plot
with(injection, interaction.plot(A, B, Shrinkage))
Residual plots suggest that C is a dispersion effect.
Analyze absolute residuals
r <- residuals(aov(Shrinkage ~ A * B, injection))
summary(aov(abs(r) ~ A * B * C * D * E * F, injection))
12 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Output
A
B
C
D
E
F
A:B
A:C
B:C
A:D
B:D
C:D
D:E
A:B:D
A:C:D
13 / 15
Df Sum Sq Mean Sq
1
1.00
1.00
1
0.25
0.25
1 56.25
56.25
1
3.06
3.06
1
1.00
1.00
1
1.00
1.00
1
0.25
0.25
1
2.25
2.25
1
1.00
1.00
1
1.56
1.56
1
0.06
0.06
1
2.25
2.25
1
9.00
9.00
1
0.56
0.56
1
0.25
0.25
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
Montgomery suggests calculating, for each effect,
F ∗ = log
sum of squares of residuals at high level
sum of squares of residuals at low level
In R, not easy to calculate F ∗ , but we can look at the half-normal
plot for abs(r ) or r 2 :
qqnorm(aov(abs(r) ~ A * B * C * D * E * F, injection), label = TRUE)
qqnorm(aov(r^2 ~ A * B * C * D * E * F, injection), label = TRUE)
The effects shown in the second of these, for
r^2,
are essentially
sum of squares of residuals at high level
− sum of squares of residuals at low level
14 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction
ST 516
Experimental Statistics for Engineers II
In
p the spirit of R’s “Scale-Location” residual plot, we could use
|residual|:
qqnorm(aov(sqrt(abs(r)) ~ A * B * C * D * E * F, injection))
p
All three half-normal plots (r 2 , |r |, and |r |) give the same
indication as F ∗ : C appears to be a dispersion factor.
15 / 15
Two-level Fractional Factorial Designs
The One-Quarter Fraction