Resolution III DesignsDesigns with main effects aliased with two-factor interactionsUsed for screening (5 7 variables in 8 runs, 9 - 15 variables in 16 runs, for example)A saturated design has k = N 1 variablesSee Table 8-19, page 313 for a

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Resolution III DesignsComplete defining relationI = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFGThe aliases of an effect (e.g. B)B = AD = ABCE = CF = ACG = CDE = ABCDF = BCDG= AEF = EG = ABFG = BDEF = ABDEG = BCEFG = DFG = ACDEFG

Resolution III Designsdof = 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 factorse.g. by dropping one column (e.g. column G)Complete defining relationI = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG

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Resolution III DesignsDropping several factors in the defining relation to form a new design. E.g. drop B, D, F and GI = ABD = ACE = BCF = ABCG = BCDE = ACDF = CDG= ABEF = BEG = AFG = DEF = ADEG = CEFG = BDFG = ABCDEFG

Its possible to form a 23 design with A, B, and CCare is needed to form the best design

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Resolution III DesignsSequential assembly of fractions to separate aliased effects by combining designs generated by switching certain signs fold overSwitching the signs in one column provides estimates of that factor and all of its two-factor interactionsSwitching the signs in all columns de-aliases all main effects from their two-factor interaction alias chains called a full fold-over/reflection

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Resolution III DesignsDefining relation for a fold-over designEach separate fraction has L + U words as generators, and the combined design will have L + U 1 words as generators, whereL: words of like signU-1: words of independent even products of the words of unlike signExampleFirst fraction: I=ABD, I=ACE, I=BCF, and I=ABCGSecond fraction: I=-ABD, I=-ACE, I=-BCF, and I=ABCGL=1, U=3, L+U=4Combined design: L+U-1=3 generatorsI=ABCG, I=(ABD)(ACE)=BCDE, I=(ABD)(BCF)=ACDFComplete defining relationI=ABCG=BCDE=ACDF=ADEG=BDFG=ABEF=CEFG

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Principal fractionAlternate fraction

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Effects from the combined designBe careful these rules only work for Resolution III designsThere are other rules for Resolution IV designs, and other methods for adding runs to fractions to de-alias effects of interest

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Resolution III Designs Example 8-7Response: eye focus timeSeven factors, two levels each, a designFirst screening, then concentrate on important factors

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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 design projects into a replicated design

As the projected design is a resolution three design, A is aliased with BD, and so onA second fraction is needed (with all the signs reversed) to separate the main effects with the 2fis.

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The fold-over design and (new) observations

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Plackett-Burman DesignsThese are a different class of 2-level, resolution III designk = N-1 variables, N runsThe number of runs, N, needs only be a multiple of fourN = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, The designs where N = 12, 20, 24, etc. are called nongeometric PB designsPB designs have a messy alias structure be careful

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Generating a nongeometric PB design matrix

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Plackett-Burman DesignsProjection of the 12-run design into2-factor full factorials, or A 23 + a , or A unbalanced 4 factor design

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Plackett-Burman DesignsThe alias structure is complex in the PB designsFor example, with N = 12 and k = 11, every main effect is aliased with every 2FI not involving itselfEvery 2FI alias chain has 45 termsPartial aliasing can greatly complicate interpretation Interactions can be particularly disruptiveUse very, very carefully (maybe never)

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Resolution IV Designs: Section 8-6Designs with main effects not aliased with two-factor interactions, and some 2fis are aliased with each otherAny design must contain at least 2k runs + its reflection (fold over)

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Resolution IV and V DesignsAliased two-factor interactions can be separated by folding over resolution IV designsBreak the 2fis involving a specific factorBreak the 2fis on a specific alias chainBreak as many 2fi interaction alias chains as possibleOne 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

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Resolution IV and V DesignsIn resolution V designs main effects and 2fis do not alias with other main effects and 2fis powerfulStandard resolution V designs require large number of runs => irregular resolution V fractional factorialsA 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

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Partial fold over (semifold)An alternative to a complete fold overFor a design , only 8 runs (as opposed to 16 runs as in a complete fold over) are neededProcedure:Construct a single-factor fold over in the usual way by changing the signs on a factor involved in a 2-fi of interestSelect 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

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Partial fold over (semifold)For a design , the combined design has 16 (original fraction) + 16/2 (partial fold over) = 24 runsDefining 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)

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General Method for Finding Aliasing RelationsFor a 2k-p fractional factorial design: use the complete defining relationFor more complex settings, e.g., irregular fractions and partial fold-over designs: use a general method Procedure: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, 1 is an p1 1 vector of the model parameters, and is an n 1 vector of errors. The least squares estimate of 1 is

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Procedure: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 2 is a p2 1 vector of the parameters associated with these variables. It can be easily shown that

where is called the alias matrix. The elements of this matrix operating on 2 identify the alias relationships for the parameters in the vector 1.

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Example: A 23-1 DesignConsider a main effect model first:y = bo + b1x1 + b2x2 + b3x3 + e

ABC__++___+_+++

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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

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Different from projection, with more flexibility. One-half fraction. Main effects are aliased with 2fis.The second design was created by changing the signs of the generators with odd number of letters; can take (ABD)(ACE)=BCDE and (ACE)(BCF)=ABEF as generators as well.Can be A, B, D; A, B, AB; B, D, BD, A, D, AD. The projected design cannot separate the main effects from 2fis. All the signs are reversed because many main effects are aliased with 2fis.Table 8-22 changes the signs of factors in the basic design, and those of the generators with odd number of letters.If N is a power of 2, the design is a fractional factorial, and the analysis is identical to that shown previously. The non-factorial (a power of 2) design cannot be represented by cubes, so they are called nongeometric.Three- and four-factor projections are unbalanced designs.