Chapter 4 Fractional factorial Experiments at two levels

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

God Jul Next time Chapter 6 Other design and analysis techniques for experiments at more than two levels Erik löfving Thursday, January 15th 2004 13.15 - 15.00 ff