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Factorial Experiments and Quality Improvement
III
March 13, 2007
2
3
experiments: introduction
2
3
experiments have three factors, each at two levels.
Y
ĳk
`
=
μ
ĳ
`
+
±
ĳk
`
=
μ
+
α
i
+
β
j
+
γ
k
+ (
αβ
)
ĳ
+ (
αγ
)
ĳ
+ (
βγ
)
jk
+ (
αβγ
)
ĳk
+
±
ĳk
`
Main eﬀects and interactions
sum to 0 in any subscript
, with
other subscripts held ﬁxed.
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View Full Document Overall mean and main eﬀects
μ
=
μ
111
+
μ
112
+
μ
121
+
μ
122
+
μ
211
+
μ
212
+
μ
221
+
μ
222
8
α
1
=
(
μ
111
+
μ
112
+
μ
121
+
μ
122
)

(
μ
211
+
μ
212
+
μ
221
+
μ
222
)
8
β
1
=
(
μ
111
+
μ
112
+
μ
211
+
μ
212
)

(
μ
121
+
μ
122
+
μ
221
+
μ
222
)
8
and similarly for
γ
1
Twoway interactions
(
αβ
)
11
=
(
μ
111
+
μ
112
)

(
μ
121
+
μ
122
)

(
μ
211
+
μ
212
) + (
μ
221
+
μ
222
)
8
and similarly for
(
αγ
)
11
and
(
βγ
)
11
Three way interaction
(
αβγ
)
111
=
(
μ
111

μ
121

μ
121
+
μ
221
)

(
μ
112

μ
122
+
μ
122

μ
222
)
8
=
{
(
αβ
)
11
when C=1
}  {
(
αβ
)
11
when C=2
}
8
The threeway interaction is
I
the change in the (AB) interaction as C changes from 1 to 2.
I
the change in the (AC) interaction as B changes from 1 to 2.
I
the change in the (BC) interaction as A changes from 1 to 2.
Visualizing the eﬀects (Empty slide for drawing)
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View Full Document Visualizing the eﬀects (Empty slide for drawing)
Representation of eﬀects
run
A
B
C
AB
AC
BC
ABC
1
+
+
+
+
+
+
+
2

+
+


+

3
+

+

+


4
 
+
+


+
5
+
+

+



6

+


+

+
7
+
 


+
+
8
  
+
+
+

Runs would be put in a random order
2
3
experiments: example
I
Lipton carried out an experiment with 5 factors, each at
two levels.
I
there were 16 “runs”
I
As we will see later, only three factors had eﬀects
I
Ignore the factors without eﬀects
I
⇒
2
3
experiment with two replicates
Lipton
2
3
experiment: factors
Factors
in this analysis:
I
Temperature (0=water cooled, 1=ambient temperature)
I
batch weight (1500 lb, 2000 lb)
I
delay (1 day, 7 day)
Response
= performance (std. dev.)
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View Full Document R code
lipton.data = read.table(’lipton.txt’,header=TRUE)
attach(lipton.data)
lmfit = lm(performance ~ temp*wt*delay)
pdf(’lipton_boxplot_temp_delay.pdf’)
boxplot(performance~temp*delay,xlab=’temp*delay’,
ylab=’performance’)
pdf(’lipton_boxplot_wt_delay.pdf’)
boxplot(performance~wt*delay,xlab=’wt*delay’,
ylab=’performance’)
graphics.off()
detach(lipton.data)
ANOVA table
anova(lmfit)
Analysis of Variance Table
Response: performance
Df
Sum Sq Mean Sq F value
Pr(>F)
temp
1 0.03062 0.03062
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This note was uploaded on 01/09/2009 for the course ORIE 312 taught by Professor D.ruppert,p.jacks during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 D.RUPPERT,P.JACKS

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