24 - 1

X1

X2

X3

X4= X1•X
2• X

1

2

3

4

5

6

7

8

X1

X3

X
2

X
4

25-1

fractionis called aresolution Vdesign.Looking attheconfounding patternwe seethat11→
345and112→12+ 345.Thus maineffects are confoundedwith four-factorinteractions, and two-
tor interactions withthree-factor interactions.

eneral,a design ofresolutionR is oneinwhich nop-factor effectis confoundedwithany other effe

taining less than R-pfactors.

Theresolutionofadesignis denotedby theappropriate Roman lette

endedas asubscript.

To illustrate:

1.

A designof resolutionR = IIIdoes not confoundmain effects withone anotherbutdoes

confound maineffects with two-factorinteractions.

2.

A designof resolutionR = IV does notconfoundmaineffects andtwo-factor interactions but

does confoundtwo-factorinteractions with other two-factor interactions.

3.

A designof resolutionR = Vdoes notconfoundmaineffects andtwo-factorinteractions with

each other,but does confound two-factorinteractions withthree-factor interactions,andsoon.

eneral,theresolutionofatwo-level fractional designisthe length of the shortest word in the definin

tion.

Two-LevelFractionalDesignsFor K Variables And N Runs

(Numbers In ParenthesesRepresent Replication).

23 – 1

± 3= 1 2

23

24– 1
V

25 – 2

26 – 3

27 – 4

± 4 = 1 2
± 5 = 1 3
± 6 = 2 3

± 4 = 1 2
± 5 = 1 3
± 7 = 1 2 3

± 4 = 1 2
± 6 = 2 3

± 4= 1 23

±5 = 1 3

23

24

25 –1
V

26– 2
V

27– 3
V

28– 4
V

29 – 5

21 0 –6

21 1 –7

±
±
±
±
±

± 5 = 12 3
± 6 = 23 4
± 7 = 13 4
± 8 = 12 4

± 5 = 12 3
± 6 = 23 4
± 7 = 13 4

± 5 = 23 4
± 6 = 13 4

± 5 =12 3

± 5 = 1 2 3

± 6 =23 4

± 7 =12 3

± 8 =12 4

± 9 =1 23 4

±

± 5 =12 3 4

± 6 = 2 3 4

± 7 =13 4

± 8 =12 4

± 9 =1 23 4

±

= 1 2

±11

23

24

25

26 –1
V

27– 2
V

28– 3
V

29– 4
V

210 – 5
V

211 – 6
V

±
±
±

± 6 = 1 23 4
± 7 = 1 23 5

± 6 =2 34 5

± 6 =12 3

± 7 =1 34 5

± 8 =1 24 5

±

± 6 =1 23 4

± 7 =12 4

± 8 =1 24 5

± 9 =1 34 5

±

± 6= 1 23 4 5

± 7 =1 23 4

± 8 =2 34 5

± 9= 1 23 4 5

±

=2 34 5

±

11

23

24

25

26

27– 1
V

28– 2
V

29– 3
V

210 – 4
V

211 – 5
V

±
±
±

± 7 = 2 34 6
± 8 = 1 34 6

± 7 =1 23 4

± 7 =1 23 4

± 8 =1 35 6

± 9 =1 24 5

±

±7 = 1 2 34 5 6

± 8 =1 25 6

± 9 =3 45 6

±

=1 23 5

±

11

23

24

25

26

27

28– 1
V

29 –2
V

21 0– 3
V

21 1– 4
V

±
±

± 8 =1 23 7

± 8= 1 34 6 7

± 9 =2 34 5

±

± 8 = 1 2 3 45 6 7

± 9= 2 35 6 7

±

=1 34 6

±

11=

↑

↑

↑

↑

↑

↑
1?
? ?

2?

↑
1?
4? ?

?

↑
1?
? ?

8?

↑
1
?
16?

(1 6 )

( 8)

(4 )

( 2)

( 1 )

numbers of variables k
7

3

4

5

6

8

9

10

11