Factorial Exp Design
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Transcript of Factorial Exp Design
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8/16/2019 Factorial Exp Design
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© 2001. D. A. Menascé. All Rights Reserved.
1
Design of Experiments:
Factorial Designs
Prof. Daniel A. Menasce
Dept. of Computer Science
George Mason University
© 2001. D. A. Menascé. All Rights Reserved.
2
Basic Concepts
• Factorial design: more than one factor is
studied simultaneously.
k 2
number of levels of each factor
number of factors
23 design: three factors, each with two levels. Total of 8 (23)
combinations
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© 2001. D. A. Menascé. All Rights Reserved.
3
Two-factor Design with Equal
Number of Replicates (n’)
1 2 … c
X111 X121 … X1c1
X112 X122 … X1c2
… … … …
X11n' X12n' … X1cn'
X211 X221 … X2c1
X212 X222 … X2c2
… … … …
X21n' X22n' … X2cn'
… … … … …
Xr11 Xr21 … Xrc1Xr12 Xr22 … Xrc2
… … … …
Xr1n' Xr2n' … Xrcn'
Factor A
1
2
r
Factor B
© 2001. D. A. Menascé. All Rights Reserved.
4
Notation
Bfactorof jleveland
Afactorof ilevelfornobservatioth-k :
)rcn'(nnsobservatioof numbertotal:n
celleachfornsreplicatioof number:n
Bfactorof levelsof number:c
Afactorof levelsof number:r
'
ijk X
=
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© 2001. D. A. Menascé. All Rights Reserved.
5
Means
∑
∑ ∑
∑ ∑
∑ ∑ ∑
=
= =
= =
= = =
=
=
=
=
'
'
'
1'.
'
1 1..
'
1 1..
'1 1
'
1
n
k
ijk ij
r
i
n
k
ijk
j
c
j
n
k ijk
i
r
i
c
j
n
k
ijk
n
X X
rn
X
X
cn
X
X
rcn
X
X (overall or grand mean)
(mean of i-th level of factor A)
(mean of j-th level of factor B)
(mean of cell i,j)
© 2001. D. A. Menascé. All Rights Reserved.
6
Area for Grand Mean
1 2 … c
X111 X121 … X1c1
X112 X122 … X1c2
… … … …
X11n' X12n' … X1cn'X211 X221 … X2c1
X212 X222 … X2c2
… … … …
X21n' X22n' … X2cn'
… … … … …
Xr11 Xr21 … Xrc1
Xr12 Xr22 … Xrc2
… … … …
Xr1n' Xr2n' … Xrcn'
Factor A
1
2
r
Factor B
'
1 1
'
1
rcn
X
X
r
i
c
j
n
k
ijk ∑∑∑= = =
=
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© 2001. D. A. Menascé. All Rights Reserved.
7
Area for Mean of a Level of
Factor A
1 2 … c
X111 X121 … X1c1
X112 X122 … X1c2
… … … …
X11n' X12n' … X1cn'
X211 X221 … X2c1
X212 X222 … X2c2
… … … …
X21n' X22n' … X2cn'… … … … …
Xr11 Xr21 … Xrc1
Xr12 Xr22 … Xrc2
… … … …Xr1n' Xr2n' … Xrcn'
Factor A
1
2
r
Factor B
'
1 1
..
'
cn
X
X
c
j
n
k
ijk
i
∑ ∑= =
=
i
© 2001. D. A. Menascé. All Rights Reserved.
8
Area for Mean of a Level of Factor B
1 2 … c
X111 X121 … X1c1
X112 X122 … X1c2
… … … …
X11n' X12n' … X1cn'
X211 X221 … X2c1
X212 X222 … X2c2
… … … …X21n' X22n' … X2cn'
… … … … …
Xr11 Xr21 … Xrc1
Xr12 Xr22 … Xrc2
… … … …Xr1n' Xr2n' … Xrcn'
Factor A
1
2
r
Factor B
'
1 1..
'
rn
X
X
r
i
n
k
ijk
j
∑∑= ==
j
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© 2001. D. A. Menascé. All Rights Reserved.
9
Area for Mean of a Cell
1 2 … c
X111 X121 … X1c1X112 X122 … X1c2
… … … …
X11n' X12n' … X1cn'
X211 X221 … X2c1X212 X222 … X2c2
… … … …
X21n' X22n' … X2cn'
… … … … …
Xr11 Xr21 … Xrc1Xr12 Xr22 … Xrc2
… … … …Xr1n' Xr2n' … Xrcn'
Factor A
1
2
r
Factor B∑=
=
'
1'.
n
k
ijk ij
n
X X
© 2001. D. A. Menascé. All Rights Reserved.
10
Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor AVariation (SSA)
df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
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© 2001. D. A. Menascé. All Rights Reserved.
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Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor A
Variation (SSA)df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
© 2001. D. A. Menascé. All Rights Reserved.
12
Total Variation (SST)
( )2
1 1 1
'
∑ ∑ ∑= = =
−=r
i
c
j
n
k
ijk X X SST
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© 2001. D. A. Menascé. All Rights Reserved.
13
Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor A
Variation (SSA)df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
© 2001. D. A. Menascé. All Rights Reserved.
14
Factor A Variation (SSA)
( )2
1
..'∑
=
−=r
i
i X X cnSSA
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© 2001. D. A. Menascé. All Rights Reserved.
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Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor A
Variation (SSA)df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
© 2001. D. A. Menascé. All Rights Reserved.
16
Factor B Variation (SSB)
( )2
1
..'∑
=
−=c
j
j X X rnSSB
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© 2001. D. A. Menascé. All Rights Reserved.
17
Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor A
Variation (SSA)df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
© 2001. D. A. Menascé. All Rights Reserved.
18
Variation due to Interaction (SSAB)
( )2
1
.....
1
'∑ ∑= =
+−−=r
i
jiij
c
j
X X X X nSSAB
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© 2001. D. A. Menascé. All Rights Reserved.
19
Partitioning the Variation
Total Variation (SST)
df = n - 1
Factor A
Variation (SSA)df = r - 1
Factor BVariation (SSB)
df = c - 1
Interaction (SSAB)df = (r – 1)(c-1)
RandomVariation (SSE)df = rc(n’ – 1)
© 2001. D. A. Menascé. All Rights Reserved.
20
Random Error (SSE)
( )2
1
.
1 1
'
∑ ∑ ∑= = =
−=r
i
ijijk
c
j
n
k
X X SSE
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© 2001. D. A. Menascé. All Rights Reserved.
21
Mean Squares
)1(
)1)(1(
1
1
'−
=
−−=
−=
−=
nrc
SSE MSE
cr
SSAB MSAB
c
SSB MSB
r
SSA MSA
© 2001. D. A. Menascé. All Rights Reserved.
22
Two-Factor ANOVA Model No Difference Due to Factor A
....21..0 ...: r H µµµ ===
equal.are),...,1(all Not: ..1 r i H i =µ
MSE
MSA F =F-Test statistic for Factor A:
The F-test statistic follows an F distribution with (r-1) degrees
of freedom in the numerator and rc(n’-1) in the denominator.
Fu F H >if Reject 0
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© 2001. D. A. Menascé. All Rights Reserved.
23
Two-Factor ANOVA Model
No Difference Due to Factor B...2..1.0 ...: c H µµµ ==
equal.are),...,1(all Not: ..1 c j H j =µ
MSE
MSB F =F-Test statistic for Factor B:
The F-test statistic follows an F distribution with (c-1) degrees
of freedom in the numerator and rc(n’-1) in the denominator.
Fu F H >if Reject 0
© 2001. D. A. Menascé. All Rights Reserved.
24
Two-Factor ANOVA Model No Interaction of Factors A and B
0.isBandAof ninteractiothe:0 H
0.BandAof ninteractiothe:1 ≠ H
SE
MSAB F =F-Test statistic for the interaction:
The F-test statistic follows an F distribution with (r-1)(c-1) degrees
of freedom in the numerator and rc(n’-1) in the denominator.
Fu F H >if Reject 0
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© 2001. D. A. Menascé. All Rights Reserved.
25
Example of Two Factor Design
Analysis
1 CPU 2 CPUs
1 Server 101.0 98.0
1 Server 103.0 97.5
1 Server 102.4 99.3
1 Server 104.0 100.0
2 Servers 43.0 41.0
2 Servers 46.0 44.0
2 Servers 45.0 42.0
2 Servers 49.0 46.0
Response time (in msec) of a Web Site.
© 2001. D. A. Menascé. All Rights Reserved.
26
Anova: Two-Factor With Replication
SUMMARY 1 CPU 2 CPUs Total1 Server
Count 4 4 8
Sum 410.4 394.8 805.2
Average 102.6 98.7 100.65
Variance 1.573333 1.326667 5.588571
2 Servers
Count 4 4 8
Sum 183 173 356
Average 45.75 43.25 44.5
Variance 6.25 4.916667 6.571429
Total
Count 8 8
Sum 593.4 567.8
Average 74.175 70.975
Variance 926.7593 881.1621
ANOVA
Source of Variation SS df MS F P-value F crit
Number of Servers 12611.29 1 12611.29 3586.149 3.11E-16 4.747221
Number of CPUs 40.96 1 40.96 11.64739 0.005146 4.747221
Interaction 1.96 1 1.96 0.557346 0.469706 4.747221
Within 42.2 12 3.516667
Total 12696.41 15
Reject Hypothesis that there is no difference due to number of servers.
Reject Hypothesis that there is no difference due to number of CPUs.
Accept hypothesis that there is no interaction between number of servers and number of CPUs.
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© 2001. D. A. Menascé. All Rights Reserved.
27
98.7
102.6
43.2545.75
0
20
40
60
80
100
120
1 CPU 2 CPUs
A v g .
R e s p o n s e T i m e ( m s e c
1 Server 2 Servers
56.85 55.45
Lines are almost parallel: no interaction between factors!
© 2001. D. A. Menascé. All Rights Reserved.
28
0
20
40
60
80
100
120
1 2
A
v g .
R e s p o n s e T i m e ( m s e c
1 Server 2 Servers
Lines are not parallel: factors interact!