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Design and Analysis of Experiments. Dr. Tai- Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC. Two-Level Fractional Factorial Designs. Dr. Tai- Yue Wang Department of Industrial and Information Management
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Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC Two-Level Fractional Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC Outline
  • Introduction
  • The One-Half Fraction of the 2k factorial Design
  • The One-Quarter Fraction of the 2k factorial Design
  • The General 2k-p Fractional Factorial Design
  • Alias Structures in Fractional Factorials and Other Designs
  • Resolution III Designs
  • Resolution IV and V Designs
  • Supersaturated Designs
  • Introduction(1/5)
  • As the number of factors in 2k factorial design increases, the number of runs required for a complete replicate of the design outgrows the resources of most experimenters.
  • In 26factorial design, 64 runs for one replicate.
  • Among them, 6 df for main effects, 15 df for two-factor interaction.
  • That is, only 21 of them are majorly interested in.
  • Introduction(2/5)
  • The remaining 42 df are for three of higher interactions.
  • If the experimenter can reasonably assume that certain high-order interactions are negligible, information on the main effects and low-order interactions may be obtained by running only a fraction of the complete factorial design.
  • Introduction(3/5)
  • The Fractional Factorial Designs are among the most widely used types of designs for product and process design and process improvement.
  • A major use of fraction factorials is in screening experiments.
  • Introduction(4/5)
  • Three key ideas that fractional factorial can be used effectively:
  • The sparsity of effects principle – When there are several variables, the system or process is likely to be driven primarily by some of the main effects an lower-order interactions.
  • The projection property -- Fractional factorials can be projected into stronger designs in the subset of significant factors.
  • Introduction
  • Sequential experimentation – It is possible to combine the runs of two or more fractional factorials to assemble sequentially a larger design to estimate the factor effects and interactions interested.
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • Consider a 23 factorial design but an experimenter cannot afford to run all (8) the treatment combinations but only 4 runs.
  • This suggests a one-half fraction of a 23 design.
  • Because the design contains 23-1=4 treatment combinations, a one-half fraction of the 23 design is often called a 23-1 design.
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • Consider a 23 factorial
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • We can have tow options:
  • One is the “+” sign in column ABC and the other is the “-” sign in column ABC.
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • For the “+” in column ABC, effects a, b, c, and abc are selected.
  • For the “-” in column ABC, effects ab, ac, bc, and (1) are selected.
  • Since we use ABC to determine which half to be used, ABC is called generator.
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • We look further to see if the “+” sign half is used, the sign in column I is identical to the one we used.
  • We call
  • I=ABC
  • is the defining relation in our design.
  • Note: C=AB is factor relation.
  • C=AB I=ABC
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • In general, the defining relation for a fractional factorials will always be the set of all columns that are equal to the identity column I.
  • If one examines the main effects:
  • [A]=1/2(a-b-c+abc)
  • [B]= 1/2(-a+b-c+abc)
  • [C]= 1/2(-a-b+c+abc)
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • The two-factor interactions effects:
  • [BC]=1/2(a-b-c+abc)
  • [AC]= 1/2(-a+b-c+abc)
  • [AB]= 1/2(-a-b+c+abc)
  • Thus,
  • A = BC, B = AC, C = AB
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • So
  • [A]A+BC [B]B+AC [C]C+AB
  • The alias structure can be found by using the defining relation I=ABC.
  • AI = A(ABC) = A2BC = BC
  • BI =B(ABC) = AC
  • CI = C(ABC) = AB
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • The contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction.
  • In fact, when estimating A, we are estimating A+BC.
  • This phenomena is called aliasing and it occurs in all fractional designs.
  • Aliases can be found directly from the columns in the table of + and – signs.
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • This one-half fraction, with I=ABC, is usually called the principal fraction.
  • That is, we could choose the other half of the factorial design from Table.
  • This alternate, or complementary, one-half fraction (consisting the runs (1), ab, ac, and bc) must be chosen on purpose.
  • The defining relation of this design is
  • I=-ABC
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • So
  • [A]’A-BC [B]’B-AC [C]’C-AB
  • The alias structure can be found by using the defining relation I=-ABC.
  • AI = A(-ABC) = A2BC = -BC
  • BI =B(-ABC) = -AC
  • CI = C(-ABC) = -AB
  • The One-Half Fraction of the 2kDesign – Definitions and Basic Principles
  • In practice, it does not matter which fraction is actually used.
  • Both fractions belong to the same family.
  • Two of them form a complete 23 design.
  • The two groups of runs can be combined to form a full factorial – an example of sequential experimentation
  • The One-Half Fraction of the 2kDesign – Design Resolution
  • The 23-1 design is called a resolution III design.
  • In this design, main effects are aliased with two-factor interactions.
  • In general, a design is of resolution R if no p factor effect is aliased with another effect containing less than R-p factors.
  • For a 23-1 design, no one (p) factor effect is aliased with one (less than 3(R) – 1(p)) factor effect.
  • The One-Half Fraction of the 2kDesign – Design Resolution
  • Resolution III designs – These are designs in which no main effects are aliased with any other main effect. But main effects are aliased with two-factor interactions and some two-factor interactions maybe aliased with each other.
  • The 23-1 design is a resolution III design.
  • Noted as
  • The One-Half Fraction of the 2kDesign – Design Resolution
  • Resolution IV designs – These are designs in which no main effects are aliased with any other main effect or with any two-factor interaction. But two-factor interaction are aliased with each other.
  • The 24-1 design with I=ABCD is a resolution IV design.
  • Noted as
  • The One-Half Fraction of the 2kDesign – Design Resolution
  • Resolution V designs – These are designs in which no main effects or two-factor interaction is aliased with any other main effect or with any two-factor interaction. But two-factor interaction are aliased with three-factor interaction.
  • The 25-1 design with I=ABCDE is a resolution V design.
  • Noted as
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • Example:
  • C‧I=C‧ABC=AB
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • The one-half fraction of the 2k design of the highest resolution may be constructed by writing down a basic design consisting of the runs for a full 2k-1 factorial and then adding the kth factor by identifying its plus and minus levels with the plus and minus signs of the highest order interaction ABC..(K-1).
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • Note: Any interaction effect could be used to generate the column for the kth factor.
  • However, use any effect other than ABC…(K-1) will not product a design of the highest possible resolution.
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • Any fractional factorial design of resolution R contains complete factorial designs (possibly replicated factorials) in any subset of R-1 factors. Important and useful !!!
  • Example, if an experiment has several factors of potential interest but believes that only R-1 of them have important effects, the a fractional factorial design of resolution R is the appropriate choice of design.
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction Because the maximum possible resolution of a one-half fraction of the 2k design is R=k, every 2k-1 design will project into a full factorial in any (k-1) of the original k factors. the 2k-1 design may be projected into two replicates of a full factorial in any subset of k-2 factors., four replicates of a full factorial in any subset of k-3 factors, and so on. The One-Half Fraction of the 2kDesign – example (1--1/7)
  • Y=filtration rate
  • Fours factors: A, B, C, and D.
  • Use 24-1 with I=ABCD
  • The One-Half Fraction of the 2kDesign – example (1--2/7)
  • STAT > DOE > Factorial > Create Factorial Design
  • Number of factors  4
  • Design  ½ fraction  OK
  • Factors  Fill names for each factor
  • Fractional Factorial Design Factors: 4 Base Design: 4, 8 Resolution: IV Runs: 8 Replicates: 1 Fraction: 1/2 Blocks: 1 Center pts (total): 0 Design Generators: D = ABC Alias Structure I + ABCD A + BCD B + ACD C + ABD D + ABC AB + CD AC + BD AD + BC The One-Half Fraction of the 2kDesign – example (1--3/7) The One-Half Fraction of the 2kDesign – example(1--4/7) The One-Half Fraction of the 2kDesign – example (1--5/7)
  • After collecting data
  • STAT > DOE > Factorial > Analyze Factorial Design
  • Response  Filtration  OK
  • Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant 70.750 Temperature 19.000 9.500 Pressure 1.500 0.750 Conc. 14.000 7.000 Stir Rate 16.500 8.250 Temperature*Pressure -1.000 -0.500 Temperature*Conc. -18.500 -9.250 Temperature*Stir Rate 19.000 9.500 The One-Half Fraction of the 2kDesign – example (1--6/7)
  • Obviously, no effect is significant
  • B is less important
  • Try A, C, and D  projection 23 with A, C, D
  • The One-Half Fraction of the 2kDesign – example (1--7/7) Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant 70.750 0.7500 94.33 0.007 Temperature 19.000 9.500 0.7500 12.67 0.050 Conc. 14.000 7.000 0.7500 9.33 0.068 Stir Rate 16.500 8.250 0.7500 11.00 0.058 Temperature*Conc. -18.500 -9.250 0.7500 -12.33 0.052 Temperature*Stir Rate 19.000 9.500 0.7500 12.67 0.050 Conc.*Stir Rate -1.000 -0.500 0.7500 -0.67 0.626 S = 2.12132 PRESS = 288 R-Sq = 99.85% R-Sq(pred) = 90.62% R-Sq(adj) = 98.97%
  • Prediction equation:
  • Coded variable :
  • The One-Half Fraction of the 2kDesign – example (2—1/8)
  • 5 Factors
  • 25-1 design
  • Response: Yield
  • The One-Half Fraction of the 2kDesign – example (2--2/8) The One-Half Fraction of the 2kDesign – example (2--3/8) The One-Half Fraction of the 2kDesign – example (2--4/8) Factorial Fit: Yield versus Aperture, Exposure, Develop, Mask, Etch Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef Constant 30.3125 Aperture 11.1250 5.5625 Exposure 33.8750 16.9375 Develop 10.8750 5.4375 Mask -0.8750 -0.4375 Etch 0.6250 0.3125 Aperture*Exposure 6.8750 3.4375 Aperture*Develop 0.3750 0.1875 Aperture*Mask 1.1250 0.5625 Aperture*Etch 1.1250 0.5625 Exposure*Develop 0.6250 0.3125 Exposure*Mask -0.1250 -0.0625 Exposure*Etch -0.1250 -0.0625 Develop*Mask 0.8750 0.4375 Develop*Etch 0.3750 0.1875 Mask*Etch -1.3750 -0.6875 S = * PRESS = * Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 5 5562.8 5562.8 1112.56 * * 2-Way Interactions 10 212.6 212.6 21.26 * * Residual Error 0 * * * Total 15 5775.4 The One-Half Fraction of the 2kDesign – example (2--5/8)
  • Reduced to A, B, C, AB
  • The One-Half Fraction of the 2kDesign – example (2--6/8) Factorial Fit: Yield versus Aperture, Exposure, Develop Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 30.313 0.4002 75.74 0.000 Aperture 11.125 5.562 0.4002 13.90 0.000 Exposure 33.875 16.937 0.4002 42.32 0.000 Develop 10.875 5.437 0.4002 13.59 0.000 Aperture*Exposure 6.875 3.438 0.4002 8.59 0.000 S = 1.60078 PRESS = 59.6364 R-Sq = 99.51% R-Sq(pred) = 98.97% R-Sq(adj) = 99.33% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 5558.19 5558.19 1852.73 723.02 0.000 2-Way Interactions 1 189.06 189.06 189.06 73.78 0.000 Residual Error 11 28.19 28.19 2.56 Lack of Fit 3 9.69 9.69 3.23 1.40 0.313 Pure Error 8 18.50 18.50 2.31 Total 15 5775.44 The One-Half Fraction of the 2kDesign – example (2--7/8) The One-Half Fraction of the 2kDesign – example (2--8/8)
  • Collapse into two replicate of a 23 design
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • Using fractional factorial designs often leads to greater economy and efficiency in experimentation. Particularly if the runs can be made sequentially.
  • For example, suppose that we are investigating k=4 factors (24=16 runs). It is almost always preferable to run 24-1IV fractional design (four runs), analyze the results, and then decide on the best set of runs to perform next.
  • The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • If it is necessary to resolve ambiguities, we can always run the alternate fraction and complete the 24 design.
  • When this method is used to complete the design, both one-half fractions represent blocks of the complete design with the highest order interaction (ABCD) confounded with blocks.
  • Sequential experimentation has the result of losing only the highest order interaction.
  • Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design The One-Half Fraction of the 2kDesign – example (3—1/4)
  • From Example 1, 24-1IV design
  • Use I=-ABCD
  • STAT>DOE>Factorial>Create Factorial Design
  • Create base design first
  • 2-level factorial(specify generators)
  • Number of factors  3
  • Design  Full factorial
  • Generators D=-ABC  OK
  • The One-Half Fraction of the 2kDesign – example (3—2/4) The One-Half Fraction of the 2kDesign – example (3—3/4) The One-Half Fraction of the 2kDesign – example (3—4/4) Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant 69.375 Temperature 24.250 12.125 Pressure 4.750 2.375 Conc. 5.750 2.875 Stir Rate 12.750 6.375 Temperature*Pressure 1.250 0.625 Temperature*Conc. -17.750 -8.875 Temperature*Stir Rate 14.250 7.125 S = * PRESS = * Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 4 1612 1612 403.1 * * 2-Way Interactions 3 1039 1039 346.5 * * Residual Error 0 * * * Total 7 2652 The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction
  • Adding the alternate fraction to the principal fraction may be thought of as a type of confirmation experiment that will allow us to strengthen our initial conclusions about the two-factor interaction effects.
  • A simple confirmation experiment is to compare the results from regression and actual runs.
  • The One-Quarter Fraction of the 2kDesign
  • For a moderately large number of factors, smaller fractions of the 2kdesign are frequently useful.
  • One-quarter fraction of the 2k design
  • 2k-2 runs
  • called 2k-2 fractional factorial
  • The One-Quarter Fraction of the 2kDesign
  • Constructed by writing down a basic design consisting of runs associated with a full factorial in k-2 factors and then associating the two additional columns with appropriately chosen interactions involving the first k-2 factors.
  • Thus, two generators are needed.
  • I=P and I=Q are called generating relations for the design.
  • The One-Quarter Fraction of the 2kDesign
  • The signs of P and Q determine which one of the one-quarter fractions is produced.
  • All four fractions associated with the choice of generators ±P or ±Q are members of the same family.
  • +P and +Q are principal fraction.
  • I=P=Q=PQ
  • P, Q, and PQ are defining relation words
  • The One-Quarter Fraction of the 2kDesign
  • Example:
  • P=ABCE, Q=BCDF, PQ=ADEF
  • Thus A=BCE=ABCDF=DEF
  • When estimating A, one is really estimating
  • A+BCE+DEF+ABCDF
  • The One-Quarter Fraction of the 2kDesign
  • Complete defining relation: I = ABCE = BCDF = ADEF The One-Quarter Fraction of the 2kDesign The One-Quarter Fraction of the 2kDesign
  • Factor relations: E=ABC, F=BCD
  • I=ABCE=BCDF=ADEF
  • The One-Quarter Fraction of the 2kDesign
  • STAT>DOE>Factorial>Create factorial Design
  • Design  Full factorialOKOK
  • The One-Quarter Fraction of the 2kDesign The One-Quarter Fraction of the 2kDesign
  • Alternate fractions of 26-2 design
  • P=ABCE, -Q=-BCDF
  • -P=-ABCE, Q=BCDF
  • -P=-ABCE, -Q=-BCDF
  • [A]  A+BCE-DEF-ABCDF
  • The One-Quarter Fraction of the 2kDesign
  • A 26-2 design will project into a single replicate of a 24 design in any subset of fours factors that is not a word in the defining relation.
  • It also collapses to a replicated one-half fraction of a 24 in any subset of four factors that is a word in the defining relation.
  • The One-Quarter Fraction of the 2kDesign
  • Projectionof the design into subsets of the original six variables
  • Any subset of the original six variables that is not a word in the complete defining relation will result in a full factorial design
  • Consider ABCD (full factorial)
  • Consider ABCE (replicated half fraction)
  • Consider ABCF (full factorial)
  • The One-Quarter Fraction of the 2kDesign
  • In general, any 2k-2 fractional factorial design can be collapsed into either a full factorial or a fractional factorial in some subset of r≦k-2 of the original factors.
  • Those subset of variables that form full factorials are not words in the complete defining relation.
  • The One-Quarter Fraction of the 2kDesign— example(4—1/10)
  • Injection molding process
  • Response: Shrinkage
  • Factors: Mold temp, screw speed, holding time, cycle time, gate size, holding pressure.
  • Each at two levels
  • To run a 26-2 design, 16 runs
  • The One-Quarter Fraction of the 2kDesign— example(4—2/10) The One-Quarter Fraction of the 2kDesign— example(4—3/10)
  • Full model
  • The One-Quarter Fraction of the 2kDesign— example(4—4/10) Factorial Fit: Shrinkage versus Temperature, Screw, ... Estimated Effects and Coefficients for Shrinkage (coded units) Term Effect Coef Constant 27.313 Temperature 13.875 6.937 Screw 35.625 17.812 Hold Time -0.875 -0.437 Cycle Time 1.375 0.688 Gate 0.375 0.187 Pressure 0.375 0.187 Temperature*Screw 11.875 5.938 Temperature*Hold Time -1.625 -0.813 Temperature*Cycle Time -5.375 -2.688 Temperature*Gate -1.875 -0.937 Temperature*Pressure 0.625 0.313 Screw*Cycle Time -0.125 -0.062 Screw*Pressure -0.125 -0.063 Temperature*Screw*Cycle Time 0.125 0.062 Temperature*Hold Time*Cycle Time -4.875 -2.437 The One-Quarter Fraction of the 2kDesign— example(4—5/10)
  • Reduced model
  • The One-Quarter Fraction of the 2kDesign— example(4—6/10)
  • Reduced model
  • Factori
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