Two-Level Factorial and
Fractional Factorial Designs in
Blocks of Size Two
• NORMAN R. DRAPER
• Journal of Quality Technology;
Jan 1997; 29, 1;pg. 71
• 報告者：謝瑋珊
Outlines
• Introduction
k
• 2 Factorial Estimates with
Paired Comparisons
• 2 k  p Factorial Fractional with
Paired Comparisons
• An Example
Introduction
• Experimental situations are necessary to
work with blocks of a given size…
– Size of two.
• Assume that…
– Interested factorial effects are estimable…
– There are no interactions of blocks with factors.
• Mirror-image(or foldover) pairs…
– Levels of the factor are changed completely…
– Are commonly used, but…
2 Factorial Estimates with Paired
Comparisons-Two Factors
k
• Six two-per-block factorial combinations:
– (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4)
– Each pairing causes a different block effect.
• Block-free comparison:
– C12=Y2-Y1 C13=Y3-Y1
– C14=Y4-Y1 C23=Y3-Y2
– C24=Y4-Y2 C34=Y4-Y3
TABLE1. 22 Factorial design
Run No.
x1
x2
Y
1
2
3
4
-1
1
-1
1
-1
-1
1
1
Y1
Y2
Y3
Y4
Main effects: L1 and L2
Two factor interaction: L12
• 2L1 = -Y1+Y2-Y3+Y4 = C12+C34 = C14-C23
• 2L2 = -Y1-Y2+Y3+Y4 = C13+C24 = C14+C23
• 2L12 = Y1-Y2-Y3+Y4 = -C12+C34 = -C13+C24
• (C12,C13,C24,C34)or(C13,C23,C14,C24)or
(C12,C34,C14,C23)
Figure1.
3
4
3
4
3
4
x2
1
(a )
2
1
(b)
2
1
(c ) 2
x1
• Combining mirror-image pairs in blocks of
size two permits only main effects to be
estimated free of blocks.
– (C14,C23)
• The set (C12,C13,C24,C34) requires changes of
only one factor within pairing.
2 Factorial Estimates with Paired
Comparisons-Three Factors
k
8 
8!
• Possible paired comparisons:   
 2  2!8  2  !  28
 
• Only 12 are needed to estimate all main effects and
interactions.
• An example: mirror-image pairs
(C18,C27,C36,C45)
• To add (C12,C13,C24,C34)
and (C56,C57,C68,C78)
• In general, putting together faces like those of
Figure1(a), 1(b), and 1(c) without creating
repeated pairs(using any pairing Cij only once)
will also work.
• One choice, for example, C12, C13, C15, C24,
C26, C34, C37, C48, C56, C57, C68, and C78,
which are the edges of the cube.
2 Factorial Estimates with Paired
Comparisons-Four or More Factors
k
• For four factors, for example, obtained by
splitting the points of the 16 into two sets where
any chosen factor is at its high or low level.
• 12+12+8=32 pairings are needed.
• In general, a full factorial two-level design in k
n
factor has n= 2 k , points with c2  n(n  1) 2 possible
pairings.
• Let n k be the number of pairings for a
2 k design. Then
n 2  1  1  2 21  2(2 21 )  4 pairings ;
n 3  4  4  231  3(231 )  12 pairings ;
n 4  12  12  2 41  4(2 41 )  32 pairings .
In general , nk  k 2 k 1 pairings are needed .
• The actual number of individual runs needed is
twice this,that is k 2 k .
k
• More by a factor of k than for the 2 design.
2 k  p Factorial Fractional with Paired
31
Comparisons-The 2    Design
• Consider a 2
31
   design,
defined by I=123.
• It is still possible to perform a 2 k  p
fractional factorial in blocks of size 2.
The conventional contrasts:
– 2L1 = -Y1+Y2-Y3+Y4 = C12+C34 = C14-C23
– 2L2 = -Y1-Y2+Y3+Y4 = C13+C24 = C14+C23
– 2L12 = Y1-Y2-Y3+Y4 = -C12+C34 = -C13+C24
Which estimate 1+23, 2+13, 3+12 effects by
(C12,C13,C24,C34) or (C13,C23,C14,C24) or (C12,C34,C14,C23)
2
k p
Factorial Fractional with Paired
4 1
Comparisons-The 2 V Design
• The 2
4 1
design defined by I=1234, and there are
C28  28 possible pairings.
• For example, the estimate of (1+234) is
(-Y1+Y2-Y3+Y4-Y5+Y6-Y7+Y8)/4;
and so on.
General 2
k p
Fractional Factorials
• Use the same pattern of requirement
developing as for 2 k factorials.
• For a fractional factorial design with
( k  p 1)
k  p runs we need n
k  p  ( k  p) 2
2
pairings.
(k  p)
• (k  p)2
runs are needed.
Illustrative Example
• Two manufacturers, U and G, each offer two types
of stockings, an economy(E) and a better(B) model.
• Possible combinations:
(1)UE, (2)GE, (3)UB, (4)GB.
• The factorial effects are
main effect U to G: (C12+C34)/2=15
main effect E to B: (C13+C24)/2=63
two-factor interaction: (-C12+C34)/2=11
• G-type stocking : 15 days longer
• Better stocking : 63 days longer
Conclusion
This method is useful to know in
situation where runs are cheap but
the response varies over time.
The end~
Thanks for your attention.