Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC 1/33

Two-Level Fractional Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC 2/33

Outline Introduction The One-Half Fraction of the 2 k factorial Design The One-Quarter Fraction of the 2 k factorial Design The General 2 k-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 2 k factorial design increases, the number of runs required for a complete replicate of the design outgrows the resources of most experimenters. In 2 6 factorial 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 2 k Design – Definitions and Basic Principles Consider a 2 3 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 2 3 design. Because the design contains =4 treatment combinations, a one-half fraction of the 2 3 design is often called a design.

The One-Half Fraction of the 2 k Design – Definitions and Basic Principles Consider a 2 3 factorial

The One-Half Fraction of the 2 k Design – 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 2 k Design – 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 2 k Design – 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 2 k Design – 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 2 k Design – 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 2 k Design – 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) = A 2 BC = BC BI =B(ABC) = AC CI = C(ABC) = AB

The One-Half Fraction of the 2 k Design – 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 2 k Design – 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 2 k Design – 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) = A 2 BC = -BC BI =B(-ABC) = -AC CI = C(-ABC) = -AB

The One-Half Fraction of the 2 k Design – 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 2 3 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 2 k Design – Design Resolution The 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 design, no one (p) factor effect is aliased with one (less than 3(R) – 1(p)) factor effect.

The One-Half Fraction of the 2 k Design – 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 design is a resolution III design. Noted as

The One-Half Fraction of the 2 k Design – 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 design with I=ABCD is a resolution IV design. Noted as

The One-Half Fraction of the 2 k Design – 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 design with I=ABCDE is a resolution V design. Noted as

The One-Half Fraction of the 2 k Design – Construction and analysis of the one-half fraction Example: C ‧ I=C ‧ ABC=AB

The One-Half Fraction of the 2 k Design – Construction and analysis of the one-half fraction The one-half fraction of the 2 k design of the highest resolution may be constructed by writing down a basic design consisting of the runs for a full 2 k-1 factorial and then adding the k th 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 2 k Design – 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 2 k Design – 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 Because the maximum possible resolution of a one-half fraction of the 2 k design is R=k, every 2 k-1 design will project into a full factorial in any (k-1) of the original k factors. the 2 k-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 2 k Design – Construction and analysis of the one-half fraction

The One-Half Fraction of the 2 k Design – example (1--1/7) Y=filtration rate Fours factors: A, B, C, and D. Use with I=ABCD

The One-Half Fraction of the 2 k Design – example (1--2/7) 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 + BCDB + ACDC + ABDD + ABC AB + CDAC + BDAD + BC STAT > DOE > Factorial > Create Factorial Design Number of factors  4 Design  ½ fraction  OK Factors  Fill names for each factor

The One-Half Fraction of the 2 k Design – example (1--3/7)

The One-Half Fraction of the 2 k Design – example(1--4/7)

The One-Half Fraction of the 2 k Design – example (1--5/7) Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant Temperature Pressure Conc Stir Rate Temperature*Pressure Temperature*Conc Temperature*Stir Rate After collecting data STAT > DOE > Factorial > Analyze Factorial Design Response  Filtration  OK

The One-Half Fraction of the 2 k Design – example (1--6/7) Obviously, no effect is significant B is less important Try A, C, and D  projection 2 3 with A, C, D

The One-Half Fraction of the 2 k Design – example (1--7/7) Prediction equation: Coded variable : Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant Temperature Conc Stir Rate Temperature*Conc Temperature*Stir Rate Conc.*Stir Rate S = PRESS = 288 R-Sq = 99.85% R-Sq(pred) = 90.62% R-Sq(adj) = 98.97%

37 The One-Half Fraction of the 2 k Design – example (2—1/8) 5 Factors design Response: Yield

38 The One-Half Fraction of the 2 k Design – example (2--2/8)

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

40 The One-Half Fraction of the 2 k Design – 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 Aperture Exposure Develop Mask Etch Aperture*Exposure Aperture*Develop Aperture*Mask Aperture*Etch Exposure*Develop Exposure*Mask Exposure*Etch Develop*Mask Develop*Etch Mask*Etch S = * PRESS = * Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects * * 2-Way Interactions * * Residual Error 0 * * * Total

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

42 The One-Half Fraction of the 2 k Design – 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 Aperture Exposure Develop Aperture*Exposure S = PRESS = 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 Way Interactions Residual Error Lack of Fit Pure Error Total

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

44 The One-Half Fraction of the 2 k Design – example (2--8/8) Collapse into two replicate of a 2 3 design

The One-Half Fraction of the 2 k Design – 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 (2 4 =16 runs). It is almost always preferable to run IV 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 2 k Design – 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 2 4 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 2 k Design – example (3—1/4) From Example 1, IV 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 2 k Design – example (3—2/4)

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

51 The One-Half Fraction of the 2 k Design – example (3—4/4) Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant Temperature Pressure Conc Stir Rate Temperature*Pressure Temperature*Conc Temperature*Stir Rate S = * PRESS = * Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects * * 2-Way Interactions * * Residual Error 0 * * * Total

The One-Half Fraction of the 2 k Design – 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 2 k Design For a moderately large number of factors, smaller fractions of the 2 k design are frequently useful. One-quarter fraction of the 2 k design 2 k-2 runs called 2 k-2 fractional factorial

The One-Quarter Fraction of the 2 k Design 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 2 k Design 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 2 k Design Example: P=ABCE, Q=BCDF, PQ=ADEF Thus A=BCE=ABCDF=DEF When estimating A, one is really estimating A+BCE+DEF+ABCDF

57 Complete defining relation: I = ABCE = BCDF = ADEF The One-Quarter Fraction of the 2 k Design

58 The One-Quarter Fraction of the 2 k Design

Factor relations: E=ABC, F=BCD I=ABCE=BCDF=ADEF

60 The One-Quarter Fraction of the 2 k Design STAT>DOE>Factorial>Create factorial Design Design  Full factorial  OK  OK

61 The One-Quarter Fraction of the 2 k Design

Alternate fractions of design P=ABCE, -Q=-BCDF -P=-ABCE, Q=BCDF -P=-ABCE, -Q=-BCDF [A]  A+BCE-DEF-ABCDF

The One-Quarter Fraction of the 2 k Design A design will project into a single replicate of a 2 4 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 2 4 in any subset of four factors that is a word in the defining relation.

64 Projection of 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 2 k Design

In general, any 2 k-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 2 k Design— 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 design, 16 runs

The One-Quarter Fraction of the 2 k Design— example(4—2/10)

The One-Quarter Fraction of the 2 k Design— example(4—3/10) Full model

The One-Quarter Fraction of the 2 k Design— example(4—4/10) Factorial Fit: Shrinkage versus Temperature, Screw,... Estimated Effects and Coefficients for Shrinkage (coded units) Term Effect Coef Constant Temperature Screw Hold Time Cycle Time Gate Pressure Temperature*Screw Temperature*Hold Time Temperature*Cycle Time Temperature*Gate Temperature*Pressure Screw*Cycle Time Screw*Pressure Temperature*Screw*Cycle Time Temperature*Hold Time*Cycle Time

The One-Quarter Fraction of the 2 k Design— example(4—5/10) Reduced model

The One-Quarter Fraction of the 2 k Design— example(4—6/10) Reduced model Factorial Fit: Shrinkage versus Temperature, Screw Estimated Effects and Coefficients for Shrinkage (coded units) Term Effect Coef SE Coef T P Constant Temperature Screw Temperature*Screw S = PRESS = R-Sq = 96.26% R-Sq(pred) = 93.36% R-Sq(adj) = 95.33% Analysis of Variance for Shrinkage (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Pure Error Total

The One-Quarter Fraction of the 2 k Design— example(4—7/10) Reduced model Normal plot

The One-Quarter Fraction of the 2 k Design — example(4—8/10) Reduced model Residuals vs Hold time Less scatter in low hold time than it is high

The One-Quarter Fraction of the 2 k Design— example(4—9/10) F * C is large

The One-Quarter Fraction of the 2 k Design— example(4—10/10)

The General 2 k-p Fractional Factorial Design – choose a design 2 k-1 = one-half fraction, 2 k-2 = one-quarter fraction, 2 k-3 = one-eighth fraction, …, 2 k-p = 1/ 2 p fraction Add p columns to the basic design; select p independent generators The defining relation for the design consists of the p generators initially chosen and their 2 p -p-1 generalized interactions.

The General 2 k-p Fractional Factorial Design – choose a design Important to select generators so as to maximize resolution For example, the IV design, generators: E=ABC, F=BCD, producing IV design.  maximum resolution If E=ABC, F=ABCD is chosen, I=ABCE=ABCDF=DEF, resolution III.

The General 2 k-p Fractional Factorial Design – choose a design Sometimes resolution alone is insufficient to distinguish between designs. For IV design, all of the design are resolution IV but with different alias structures. Design A has more extensive two-factor aliasing and design C the least.  Choose design C

The General 2 k-p Fractional Factorial Design – choose a design

The next table shows the suggested generators for better designs.

The General 2 k-p Fractional Factorial Design – choose a design

The General 2 k-p Fractional Factorial Design— example(5—1/4) 7 factors are interested. Two-factor interactions are to be explored. Resolution IV is assumed be appropriate. Two choices: IV (32 runs) and IV (16 runs)

The General 2 k-p Fractional Factorial Design— example(5—2/4) IV (16 runs) Fractional Factorial Design Factors: 7 Base Design: 7, 32 Resolution: IV Runs: 32 Replicates: 1 Fraction: 1/4 Blocks: 1 Center pts (total): 0 Design Generators: F = ABCD, G = ABDE Alias Structure I + CEFG + ABCDF + ABDEG A + BCDF + BDEG + ACEFGB + ACDF + ADEG + BCEFGC + EFG + ABDF + ABCDEG D + ABCF + ABEG + CDEFGE + CFG + ABDG + ABCDEFF + CEG + ABCD + ABDEFG G + CEF + ABDE + ABCDFGAB + CDF + DEG + ABCEFGAC + BDF + AEFG + BCDEG AD + BCF + BEG + ACDEFGAE + BDG + ACFG + BCDEFAF + BCD + ACEG + BDEFG AG + BDE + ACEF + BCDFGBC + ADF + BEFG + ACDEGBD + ACF + AEG + BCDEFG BE + ADG + BCFG + ACDEFBF + ACD + BCEG + ADEFGBG + ADE + BCEF + ACDFG CD + ABF + DEFG + ABCEGCE + FG + ABCDG + ABDEFCF + EG + ABD + ABCDEFG CG + EF + ABCDE + ABDFGDE + ABG + CDFG + ABCEFDF + ABC + CDEG + ABEFG DG + ABE + CDEF + ABCFGACE + AFG + BCDG + BDEFACG + AEF + BCDE + BDFG BCE + BFG + ACDG + ADEFBCG + BEF + ACDE + ADFGCDE + DFG + ABCG + ABEF CDG + DEF + ABCE + ABFG

The General 2 k-p Fractional Factorial Design— example(5—3/4) IV (16 runs) Fractional Factorial Design Factors: 7 Base Design: 7, 16 Resolution: IV Runs: 16 Replicates: 1 Fraction: 1/8 Blocks: 1 Center pts (total): 0 Design Generators: E = ABC, F = BCD, G = ACD Alias Structure I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFGB + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEGD + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEFF + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFGAB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEGAD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEFAF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFGBD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG

The General 2 k-p Fractional Factorial Design — example(5—4/4) Choose better design among costs, information and resolution. Appendix X provides a good reference for choosing “better” design Do not choose a design according to one single criterion unless is “order” by your boss.

The General 2 k-p Fractional Factorial Design – Analysis Use computer soft wares. Projection – a design of resolution R contains full factorials in any R – 1 of the factors IV design

The General 2 k-p Fractional Factorial Design – Analysis It will project into a full factorial in any four of the original seven factors that is not a word in the defining relation C(7, 5)=35 subsets of four factors. 7 of them (ABCE, BCDF, ACDG, ADEF, BDEG, ABFG, and CEFG) appeared in defining relations. The rest of 28 four-factor subset would form 2 4 designs.

The General 2 k-p Fractional Factorial Design – Analysis Obviously, A, B, C, D are one of them. Consider the following situation: If the 4 of 7 factors are more important than the rest of 3 factors, we would assign the more important four factors to A, B, C, D and the less important 3 factors to E, F, and G.

The General 2 k-p Fractional Factorial Design – Blocking Sometimes the runs needed in fraction factorial can not be made under homogeneous conditions. We confound the fractional factorial with blocks. Appendix X contain recommended blocking arrangements for fractional factorial designs.

The General 2 k-p Fractional Factorial Design – Blocking For example, IV According to the suggestion in Appendix X(f), ABD and its aliases to be confounded with block. STAT>DOE>Create Factorial Design

The General 2 k-p Fractional Factorial Design – Blocking Design  Full factorial Generators: OK

The General 2 k-p Fractional Factorial Design – Blocking

bc

The General 2 k-p Fractional Factorial Design— example(6—1/4) 5 axes CNC machine Response=profile deviation 8 factors are interested. Four spindles are treated as blocks Assumed tree factor and higher interactions are negligible From Appendix X, IV (16 runs)and IV (32 runs) are feasible.

The General 2 k-p Fractional Factorial Design— example(6—1/4) However, if IV (16 runs) is used, two- factor effects will confound with blocks If EH interaction is unlikely, IV (32 runs) is chosen. STAT>DOE>Create Factorial Design Choose 2 level factorial (default generators) Number of factors  8, number of blocks  4  OK

The General 2 k-p Fractional Factorial Design— example(6—1/4) Fractional Factorial Design Factors: 8 Base Design: 8, 32 Resolution with blocks: III Runs: 32 Replicates: 1 Fraction: 1/8 Blocks: 4 Center pts (total): 0 * NOTE * Blocks are confounded with two-way interactions. Design Generators: F = ABC, G = ABD, H = BCDE Block Generators: EH, ABE

The General 2 k-p Fractional Factorial Design— example(6—1/4)

Analyze Design A*D + B*G + E*F*H Inseparable If prior knowledge implies that AD is possible, one can use reduced model

The General 2 k-p Fractional Factorial Design— example(6—1/4) Reduced model: A, B, D, and AD

The General 2 k-p Fractional Factorial Design— example(6—1/4) Reduced model: A, B, D, and AD Factorial Fit: ln(std_dev) versus Block, A, B, D Estimated Effects and Coefficients for ln(std_dev) (coded units) Term EffectCoefSE CoefTP Constant Block Block Block A B D A*D S = PRESS = R-Sq = 84.47% R-Sq(pred) = 72.39% R-Sq(adj) = 79.94% Analysis of Variance for ln(std_dev) (coded units) Source DF Seq SS Adj SS Adj MS F P Blocks Main Effects Way Interactions Residual Error Total

The General 2 k-p Fractional Factorial Design— example(6—1/4) Reduced model: A, B, D, and AD Estimated equation

The General 2 k-p Fractional Factorial Design— example(6—1/4)