Resolution III Designs
• Designs with main effects aliased with twofactor interactions
• Used for screening (5 – 7 variables in 8
runs, 9 - 15 variables in 16 runs, for
example)
• A saturated design has k = N – 1 variables
74
2
• See Table 8-19, page 313 for a III
1
Resolution III Designs
Complete defining relation
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG
= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG
The aliases of an effect (e.g. B)
B = AD = ABCE = CF = ACG = CDE = ABCDF = BCDG
= AEF = EG = ABFG = BDEF = ABDEG = BCEFG = DFG = ACDEFG
2
Resolution III Designs
• dof = N – 1 = 7 (used to estimate 7 main effects)
• Assuming three and higher order interactions are
negligible
• Saturated resolution III design can be used to obtain
resolution III designs with fewer factors
74
6 3
2

2
e.g. by dropping one column (e.g. column G) III
III
Complete defining relation
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG
= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG
3
Resolution III Designs
• Dropping several factors in the defining relation to
form a new design. E.g. drop B, D, F and G
I = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG
= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG
27III4  23III1
• It’s possible to form a 23 design with A, B, and C
• Care is needed to form the best design
4
Resolution III Designs
• Sequential assembly of fractions to separate
aliased effects by combining designs generated by
switching certain signs – fold over
• Switching the signs in one column provides
estimates of that factor and all of its two-factor
interactions
• Switching the signs in all columns de-aliases all
main effects from their two-factor interaction alias
chains – called a full fold-over/reflection
5
Resolution III Designs
• Defining relation for a fold-over design
– Each separate fraction has L + U words as generators, and the
combined design will have L + U – 1 words as generators,
where
• L: words of like sign
• U-1: words of independent even products of the words of
unlike sign
• Example 27III4
– First fraction: I=ABD, I=ACE, I=BCF, and I=ABCG
– Second fraction: I=-ABD, I=-ACE, I=-BCF, and I=ABCG
– L=1, U=3, L+U=4
– Combined design: L+U-1=3 generators
I=ABCG, I=(ABD)(ACE)=BCDE, I=(ABD)(BCF)=ACDF
– Complete defining relation
I=ABCG=BCDE=ACDF=ADEG=BDFG=ABEF=CEFG 6

 
 
 
 


 
 
 
 
  
Principal Alternate
fraction fraction
7
Effects from the combined design
• Be careful – these rules only work for Resolution
III designs
• There are other rules for Resolution IV designs,
and other methods for adding runs to fractions to
de-alias effects of interest
8
Resolution III Designs – Example 8-7
• Response: eye focus time
• Seven factors, two levels each, a 27III4 design
• First screening, then concentrate on important factors
Contrasti
li 
N /2
9
• lA, lB, and lD are large, but the interpretation of the
data is not unique
• As ABD is a word in the defining relation, this 27III4
design projects into a replicated 23III1 design
• As the projected design is a resolution three
design, A is aliased with BD, and so on
• A second fraction is needed (with all the signs
reversed) to separate the main effects with the 2fis.
10
The fold-over design and (new) observations
11
Plackett-Burman Designs
• These are a different class of 2-level, resolution III
design
• k = N-1 variables, N runs
• The number of runs, N, needs only be a multiple
of four
N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
• The designs where N = 12, 20, 24, etc. are called
nongeometric PB designs
• PB designs have a messy alias structure – be
careful
12
• Generating a nongeometric PB design matrix
13
Plackett-Burman
Designs
Projection of the 12-run
design into
•2-factor full factorials, or
31
•A 23 + a 2 III , or
•A unbalanced 4 factor
design
14
Plackett-Burman Designs
• The alias structure is complex in the PB designs
• For example, with N = 12 and k = 11, every main
effect is aliased with every 2FI not involving itself
• Every 2FI alias chain has 45 terms
• Partial aliasing can greatly complicate
interpretation
• Interactions can be particularly disruptive
• Use very, very carefully (maybe never)
15
Resolution IV Designs: Section 8-6
• Designs with main effects not aliased with twofactor interactions, and some 2fis are aliased with
each other
k p
2
• Any IV design must contain at least 2k runs
• 2kIII p + its reflection (fold over)  2(IVk 1) p
16
Resolution IV and V Designs
• Aliased two-factor interactions can be separated
by folding over resolution IV designs
• Break the 2fis involving a specific factor
• Break the 2fis on a specific alias chain
• Break as many 2fi interaction alias chains as
possible
• One method is to run a second fraction in which
the sign is reversed on every design generator that
has an even number of letters
• Folding over a resolution IV designs will not
necessarily separate all 2fis
17
Resolution IV and V Designs
• In resolution V designs main effects and 2fis do
not alias with other main effects and 2fis –
powerful
• Standard resolution V designs require large
number of runs => irregular resolution V
fractional factorials
• A complete fold over of a resolution IV or V
design is usually unnecessary, as adding small
number of runs to the original fraction (partial fold
over) can de-alias the few aliased interactions of
interest
18
Partial fold over (semifold)
• An alternative to a complete fold over
• For a design 26IV2, only 8 runs (as opposed to 16
runs as in a complete fold over) are needed
• Procedure:
o Construct a single-factor fold over in the usual way by
changing the signs on a factor involved in a 2-fi of
interest
o Select only half of the fold-over runs by choosing those
runs where the chosen factor is either at its high or low
level. Select the level that you believe will generate the
most desirable response
19
Partial fold over (semifold)
• For a design 26IV2, the combined design has 16
(original fraction) + 16/2 (partial fold over) = 24
runs
• Defining relation and aliasing relations for the
combined (original fraction + partial fold over) are
the same as those for the combined (original
fraction + complete fold over)
20
General Method for Finding Aliasing Relations
•
•
For a 2k-p fractional factorial design: use the complete
defining relation
For more complex settings, e.g., irregular fractions and
partial fold-over designs: use a general method
• Procedure:
1. Make a polynomial or regression model in the form
y = X1b1 + e
y is an n × 1 vector of the responses, X1 is an n × p1 matrix containing
the design matrix expanded to the form of the model that the
experimenter is fitting, β1 is an p1 × 1 vector of the model parameters,
and ε is an n × 1 vector of errors.
o
The least squares estimate of β1 is
bˆ  ( X ' X ) 1 X ' y
1
1
1
1
21
• Procedure:
3. Adding another term to the model
y = X1b1 + X2b2+ e
where X2 is an n × p2 matrix containing additional variables that
are not in the fitted model and β2 is a p2× 1 vector of the
parameters associated with these variables.
4. It can be easily shown that
E ( bˆ1 )  b1  ( X 1' X 1 ) 1 X 1' X 2 b 2  b1  Ab 2
where A  ( X 1' X 1 ) 1 X 1' X 2 is called the alias matrix.
The elements of this matrix operating on β2 identify
the alias relationships for the parameters in the vector
β 1.
22
Example: A 23-1 Design
A
B
C
_
_
+
+
_
_
_
+
_
+
+
+
Consider a main effect model first:
y = bo + b1 x1 + b2 x2 + b3 x3 + e
b0 
b 
b1   1 
b 2 
 
 b3 
1  1  1 1 
1 1  1  1

X1  
1  1 1  1


1 1 1 1 
23
In order to consider the aliasing relations between main and two
factor interactions, add corresponding terms to the model:
y = bo + b1x1 + b2x2 + b3x3 + b12x1x2 + b13x1x3 + b23x2x3 + e
 b12 
b 2   b13 
 b 23 
( X 1 ' X 1 ) 1 
1
I4
4
 1  1  1
 1  1 1 

X2  
 1 1  1


1 1 1
0
0
X '1 X 2  
0

4
0 0
0 4
4 0

0 0
24
E ( bˆ1 )  b1  ( X 1' X 1 ) 1 X 1' X 2 b 2  b1  Ab 2
 bˆ0   b 0 
0
ˆ  
0
b
b
1
E 1    1   
 bˆ   b 2  4 0
 2  

ˆ
 b 3   b 3 
4
0 0
 b12 

0 4  
b13 

4 0
  b 23 
0 0
 b0 
b  b 
23 
 1
 b 2  b13 


 b 3  b12 
25