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Design and Analysis of Experiments

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

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  1. Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

  2. Two-Level Fractional Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

  3. 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

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. The One-Half Fraction of the 2kDesign – Definitions and Basic Principles • Consider a 23 factorial

  11. 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.

  12. 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.

  13. 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

  14. 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)

  15. 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

  16. 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

  17. 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.

  18. 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

  19. 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

  20. 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

  21. 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.

  22. 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

  23. 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

  24. 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

  25. The One-Half Fraction of the 2kDesign – Construction and analysis of the one-half fraction • Example: • C‧I=C‧ABC=AB

  26. 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).

  27. 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.

  28. 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.

  29. 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.

  30. 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

  31. 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

  32. The One-Half Fraction of the 2kDesign – example (1--3/7)

  33. The One-Half Fraction of the 2kDesign – example(1--4/7)

  34. 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

  35. 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

  36. 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 :

  37. The One-Half Fraction of the 2kDesign – example (2—1/8) • 5 Factors • 25-1 design • Response: Yield

  38. The One-Half Fraction of the 2kDesign – example (2--2/8)

  39. The One-Half Fraction of the 2kDesign – example (2--3/8)

  40. 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

  41. The One-Half Fraction of the 2kDesign – example (2--5/8) • Reduced to A, B, C, AB

  42. 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

  43. The One-Half Fraction of the 2kDesign – example (2--7/8)

  44. The One-Half Fraction of the 2kDesign – example (2--8/8) • Collapse into two replicate of a 23 design

  45. 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.

  46. 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.

  47. Possible Strategies for Follow-Up Experimentation Following a Fractional Factorial Design

  48. 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

  49. The One-Half Fraction of the 2kDesign – example (3—2/4)

  50. The One-Half Fraction of the 2kDesign – example (3—3/4)

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