1. Chapter 4 Fractional factorial Experiments at two levels

2. 2k-p factorial designs
Full factorial designs are rarely used in practice for large k ( k  7)
Fractional factorial designs are subsets of full factorial designs
The main practical motivation for choosing fractional factorial designs is run size economy
In return a price is paid
2k-p factorial designs

3. 25 Design matrix, leaf spring experiment.
Out of 31 d.f., 16 are used for estimating 3-factor or higher interaction
Is it practical to commit half of the degrees of freedom (d.f.) to estimate 3-factor or higher interaction effects ?
Lower order effects are more likely to be important than higher order effects ( 3-factor and higher intera-ctions are usually not significant )
Effects of the same order are equally likely to be important
2k-p factorial designs

4. 2k-p factorial design (general description)
2-p-th fraction (subset) of 2k full factorial design
Consists of 2k-p runs
The fraction is determined by p defining words
The group formed by the p defining words is called the defining contrast subgroup, ( consists of 2p–1 words plus the identity element I )
2k-p factorial designs

5. Generating a 2k-p design
Let 1,…,k-p denote the k-p independent columns of the +’s and –’s that generate the 2k-p runs
Generate the remaining p columns k-p+1,…,k as interactions of the first columns
Choice of these p columns determines the defining contrast subgroup
Example: a 26-2 design d ( A,B,C,D,E,F) with E = AB & F = ACD has a defining contrast subgroup I = ABE= ACDF = BCDEF
A, B, C, D are the independent columns and E, F are generated as interactions
2k-p factorial designs

6. Design matrix and free height data, leaf spring experiment.
The experimental design chosen was a half fraction of a 25 (25-1) full factorial design.
The choice was based on the relation
E=BCD
( aliasing relation ) or
I=BCDE
(defining contrast subgroup )
2k-p factorial designs

7. Clear and strongly clear effects
A main effect or 2-factor interaction is called
clear if non of its aliases are main effects or 2-factor interactions and,
strongly clear if non of its aliases are main effects, 2-factor or 3-factor interactions
BCD = E or I=BCDE
B = CDE, C= BDE, D = BCE, E = BCD Q = BCDEQ,
BQ = CDEQ, CQ = BDEQ, DQ = BCEQ EQ = BCDQ,
BC = DE, BD = CE, CD = BE, BCQ = DEQ, BDQ = CEQ, BEQ = CDQ
Clear or strongly clear effects
Pairs of aliased effects
2k-p factorial designs

8. Fractional factorial effects for the leaf spring experiment
B = CDE, C= BDE, D = BCE, E = BCD Q = BCDEQ,
BQ = CDEQ, CQ = BDEQ, DQ = BCEQ EQ = BCDQ,
BC = DE, BD = CE, CD = BE, BCQ = DEQ, BDQ = CEQ, BEQ = CDQ .
2k-p factorial designs ff

9. Design matrix and alternative choice, leaf spring experiment.
Any 2-factor interaction involving Q is strongly clear .
BCD = E or I=BCDE
All main effects are strongly clear and all 2-factor interaction effects are clear
Q = BCDE or I=BCDEQ
(25-1) Design d1
(25-1) Alternative design d2
Preferred if 2-factor interactions involving Q are more important
The Common choice
It is less desirable to choose designs which aliases lower order effects
Resolution IV
Resolution V
2k-p factorial designs

10. Resolution ( designs) For a 2k-p design, the vector W = ( A3,A4,…,AK)
Is called the wordlength pattern of the design, where Ai denote the number of defining words of length i
The resolution is defined to be the length of the shortest word in the defining contrast subgroup ( the smallest r such that Ar  1 )
Example: The 26-2 design d ( A,B,C,D,E,F) with E = AB & F = ACD that has a defining contrast subgroup
I = ABE = ACDF = BCDEF It has a wordlength pattern
W = ( 1,1,1,0)
and is a resolution III design denoted by
2k-p factorial designs ff

11. Resolution
The maximum resolution criterion chooses 2k-p design with the maximum resolution ( Box and Hunter, 1961)
Resolution R implies that no effect involving i factors is aliased with effects involving less than R-i factors
The projection of a resolution R design onto any R-1 factors is a full factorial design in the R-1 factors
If there are at most R-1 important factors out of k factors, the fractional design yields a full factorial design in R-1 factors
2k-p factorial designs ff

12. Properties for resolution IV and V
In any resolution IV design, the main effects are clear
In any resolution V design, the main effects are strongly clear and the 2-factor interactions are clear
Among the resolution IV designs with k and p, those with the largest number of clear 2-factor interactions are the best
2k-p factorial designs ff

13. Minimum aberration criterion
Consider the following designs ( k = 7, p = 2 )
which is an example where we need to apply the minimum aberration criterion to select a better design Fries & Hunter (1980)
2k-p factorial designs ff

14. Minimum aberration criterion
For any two 2k-p designs d1 and d2, d1 is said to have less aberration than d2 if
Ar(d1) < Ar(d2) where r is the smallest integer such that Ar(d1)  Ar(d2)
If there is no design with less aberration than d1 , then d1 is said to have minimum aberration
Minimum aberration automatically implies maximum resolution
Sometimes it is supplemented by the criterion of maximizing the total number of clear effects ( main effects and 2-factor interactions ) Chen, Sun, and Wu (1993)
2k-p factorial designs ff

15. Estimation of factorial effects
An effect is estimable if its alias is negligible
The analysis is the same as in the full factorial experiments except that, the observed significance of an effect should be attributed to the combination of the effect and all its aliased effects
2k-p factorial designs ff

16. Approaches for resolving the ambiguities in aliased effects
Use a prior knowledge to dismiss some of the aliased effects
Prior knowledge may suggest that some of the aliased effects are less important.
In some situations the factors of the whole system can be grouped by subsystems. In this case interactions between factors within the same subsystem may be more important than interactions between factors between subsystems
Run a follow-up experiment
provides additional information that can be used to de-alias the aliased effects
2k-p factorial designs ff

17. Follow up experiment techniques Method of adding orthogonal effects
Orthogonality provides maximum separation between the aliased effects
Chooses additional factor settings to make the aliased effects in the original experiment orthogonal
It takes additional work to find the orthogonality constraints
the settings for the remaining effects are chosen on an ad hoc basis and requires some ingenuity on the part of the experimenter
useful when a small number of effects needs to be de-aliased
It is not so generally used
2k-p factorial designs ff

18. Augmented design matrix and model matrix Method of adding orthogonal effects, leaf spring experiment
The additional runs settings
B is orthogonal to BCQ & DEQ
E=DQDEQ,
Four additional runs are added to de-alias the aliased effect BCQ=DEQ and DQ= BCEQ
A potential block effect should be accounted, since the additional runs are performed at a different time
The augmented design is no longer orthogonal
Regression analysis should be applied to the model matrix
Standard factorial effect estimates will not be valid and should not be used
2k-p factorial designs ff

19. Follow up experiment techniques Optimal design approach for follow up experiments
More flexible and economical
Apply an optimal design criterion like D-optimal criterion and Ds-optimal criterion
The working model for design optimization consists of the grand mean, a block effect, and the variables identified to be significant from the original experiment
The approach is driven by the best model identified by the original experiment as well as the optimal design criterion
It works for any model and shape of experimental region
It can be used to solve large design problem by using fast optimal design algorithm as OPTEX program in SAS/QC
2k-p factorial designs ff

20. Optimal design approach for follow up experiments
Consider N runs in the original experiments and n runs are to be added.
Assume that this model has p columns, and the settings in the n runs are defined as d
Xd = [ X1, X2 ] is (N+n)  p matrix , where X2 is (N+n)  q submatrix of Xd
The n  p submatrix of Xd is selected so that
the D-criterion or
the Ds-criterion
where:  represents a set of candidate n  p submatrices
2k-p factorial designs ff

21. Follow up experiment techniques Fold-over technique for follow up experiments
Effective for de-aliasing
all the main effects or,
one main effect and all the 2-factor interactions involving this main effect
in resolution III designs for the original experiment ( narrow objectives).
The augmented design is still orthogonal and the analysis follow the standard methods for fractional factorial design.
Requires that the follow up experiment has the same run size as in the original experiment ( not useful)
Less flexible than the optimal design approach.
2k-p factorial designs ff

22. Augmented design matrix using Fold-over techniques.
2k-p factorial designs ff

23. Blocking in full factorial designs
We need q independent variables B1, B2, … , Bq for defining 2q blocks.
Select the factorial effects v1, v2, … , vq that shall be confounded with B1, B2, … , Bq .
Define the remaining block effects by multiplying the Bi’s
The 2q-1 products of the Bi’s and the column I form the so-called block-defining contrast subgroup
2k-p factorial designs ff

24. Blocking in fractional factorial designs
Blocking is more complicated due to the presence of two defining contrast subgroups
block defining contrast subgroup ( defines the blocking scheme )
treatment defining contrast subgroup ( defines the fraction of the design )
In the context of blocking
A main effect or 2-factor interaction is said to be,
clear if non of its aliases are main effects or 2-factor interactions as well as not confounded with any block effects and,
strongly clear if in addition non of its aliases are 3-factor interactions
2k-p factorial designs ff

25. Blocking in fractional factorial designs
There is no natural extension of the minimum aberration criterion for blocked 2k-p designs
Total number of clear effects is used to compare and rank order different blocked 2k-p designs ( should not be the sole criterion for selection )
The choice can depend on what set of main effects and 2-factor interactions is believed to be more important
2k-p factorial designs ff

26. God Jul Next time Chapter 6 Other design and analysis techniques for experiments at more than two levels Erik löfving Thursday, January 15th ff