Factorial Exp Design

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


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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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


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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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)


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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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


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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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


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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11



df = n - 1

Factor A

Variation (SSA)df = r - 1


df = c - 1




12


( )2

1 1 1

'

∑ ∑ ∑= = =

−=r

i

c

j

n

k

ijk X X SST


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13



df = n - 1

Factor A



df = c - 1




14

Factor A Variation (SSA)

( )2

1

..'∑

=

−=r

i

i X X cnSSA


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15



df = n - 1

Factor A



df = c - 1




16

Factor B Variation (SSB)

( )2

1

..'∑

=

−=c

j

j X X rnSSB


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17



df = n - 1

Factor A



df = c - 1




18

Variation due to Interaction (SSAB)

( )2

1

.....

1

'∑ ∑= =

+−−=r

i

jiij

c

j

X X X X nSSAB


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19



df = n - 1

Factor A



df = c - 1




20

Random Error (SSE)

( )2

1

.

1 1

'

∑ ∑ ∑= = =

−=r

i

ijijk

c

j

n

k

X X SSE


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21

Mean Squares

)1(

)1)(1(

1

1

'−

=

−−=

−=

−=

nrc

SSE MSE

cr

SSAB MSAB

c

SSB MSB

r

SSA MSA


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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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


Fu F H >if Reject 0


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


Fu F H >if Reject 0


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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.


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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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!


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!

Documents

Factorial Exp Design