Design and Analysis of Experiments

Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Two-Level Fractional Factorial Designs

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

2/33

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 26 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 2k Design – 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.


Consider a 23 factorial


We can have tow options: One is the “+” sign in column ABC and the

other is the “-” sign in column ABC.


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.


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


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 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


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 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.


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 isI=-ABC


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


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 2k Design – 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.


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 132

III


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 142 IV


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 152 IV

The One-Half Fraction of the 2k Design – Construction and analysis of the one-half fraction

Example: C I=C ABC=AB‧ ‧

132 III


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).


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.


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.

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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 2k Design – example (1--1/7)

Y=filtration rate Fours factors: A, B, C, and D. Use 24-1 with I=ABCD


Fractional Factorial Design Factors: 4 Base Design: 4, 8 Resolution: IVRuns: 8 Replicates: 1 Fraction: 1/2Blocks: 1 Center pts (total): 0

Design Generators: D = ABC

Alias StructureI + ABCD A + BCD B + ACD C + ABD D + ABCAB + CD AC + BD AD + 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 2k Design – example(1--4/7)


Estimated Effects and Coefficients for Filtration (coded units)

Term Effect CoefConstant 70.750Temperature 19.000 9.500Pressure 1.500 0.750Conc. 14.000 7.000Stir Rate 16.500 8.250Temperature*Pressure -1.000 -0.500Temperature*Conc. -18.500 -9.250Temperature*Stir Rate 19.000 9.500

After collecting data STAT > DOE > Factorial > Analyze Factorial Design Response Filtration OK


Obviously, no effect is significant B is less important Try A, C, and D projection 23 with A, C, D


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 PConstant 70.750 0.7500 94.33 0.007Temperature 19.000 9.500 0.7500 12.67 0.050Conc. 14.000 7.000 0.7500 9.33 0.068Stir Rate 16.500 8.250 0.7500 11.00 0.058Temperature*Conc. -18.500 -9.250 0.7500 -12.33 0.052Temperature*Stir Rate 19.000 9.500 0.7500 12.67 0.050Conc.*Stir Rate -1.000 -0.500 0.7500 -0.67 0.626

S = 2.12132 PRESS = 288R-Sq = 99.85% R-Sq(pred) = 90.62% R-Sq(adj) = 98.97%

4131431 5.925.925.875.975.70 xxxxxxxy

}1,1{ ix

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The One-Half Fraction of the 2k Design – example (2—1/8)

5 Factors 25-1 design Response: Yield

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39


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The One-Half Fraction of the 2k Design – example (2--4/8)Factorial Fit: Yield versus Aperture, Exposure, Develop, Mask, Etch Estimated Effects and Coefficients for Yield (coded units)Term Effect CoefConstant 30.3125Aperture 11.1250 5.5625Exposure 33.8750 16.9375Develop 10.8750 5.4375Mask -0.8750 -0.4375Etch 0.6250 0.3125Aperture*Exposure 6.8750 3.4375Aperture*Develop 0.3750 0.1875Aperture*Mask 1.1250 0.5625Aperture*Etch 1.1250 0.5625Exposure*Develop 0.6250 0.3125Exposure*Mask -0.1250 -0.0625Exposure*Etch -0.1250 -0.0625Develop*Mask 0.8750 0.4375Develop*Etch 0.3750 0.1875Mask*Etch -1.3750 -0.6875

S = * PRESS = *Analysis of Variance for Yield (coded units)Source DF Seq SS Adj SS Adj MS F PMain 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

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Reduced to A, B, C, AB

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Factorial Fit: Yield versus Aperture, Exposure, Develop Estimated Effects and Coefficients for Yield (coded units)Term Effect Coef SE Coef T PConstant 30.313 0.4002 75.74 0.000Aperture 11.125 5.562 0.4002 13.90 0.000Exposure 33.875 16.937 0.4002 42.32 0.000Develop 10.875 5.437 0.4002 13.59 0.000Aperture*Exposure 6.875 3.438 0.4002 8.59 0.000

S = 1.60078 PRESS = 59.6364R-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 PMain Effects 3 5558.19 5558.19 1852.73 723.02 0.0002-Way Interactions 1 189.06 189.06 189.06 73.78 0.000Residual 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 2.31Total 15 5775.44

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44


Collapse into two replicate of a 23 design


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 24-1

IV fractional design (four runs), analyze the results, and then decide on the best set of runs to perform next.


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.

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Possible Strategies for

Follow-Up Experimentation

Following a Fractional

Factorial Design

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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

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50


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Estimated Effects and Coefficients for Filtration (coded units)Term Effect CoefConstant 69.375Temperature 24.250 12.125Pressure 4.750 2.375Conc. 5.750 2.875Stir Rate 12.750 6.375Temperature*Pressure 1.250 0.625Temperature*Conc. -17.750 -8.875Temperature*Stir Rate 14.250 7.125

S = * PRESS = *

Analysis of Variance for Filtration (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 4 1612 1612 403.1 * *2-Way Interactions 3 1039 1039 346.5 * *Residual Error 0 * * *Total 7 2652


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 2k Design

For a moderately large number of factors, smaller fractions of the 2k design are frequently useful.

One-quarter fraction of the 2k design 2k-2 runs called 2k-2 fractional factorial


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 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


Example: P=ABCE, Q=BCDF, PQ=ADEF Thus A=BCE=ABCDF=DEF When estimating A, one is really estimating

A+BCE+DEF+ABCDF

262 IV

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Complete defining relation: I = ABCE = BCDF = ADEF


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Factor relations: E=ABC, F=BCD

I=ABCE=BCDF=ADEF

262 IV

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STAT>DOE>Factorial>Create factorial Design

Design Full factorialOKOK

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Alternate fractions of 26-2 design P=ABCE, -Q=-BCDF

-P=-ABCE, Q=BCDF-P=-ABCE, -Q=-BCDF

[A] A+BCE-DEF-ABCDF


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.

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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)



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 2k 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 26-2 design, 16 runs


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


Factorial Fit: Shrinkage versus Temperature, Screw, ... Estimated Effects and Coefficients for Shrinkage (coded units)

Term Effect CoefConstant 27.313Temperature 13.875 6.937Screw 35.625 17.812Hold Time -0.875 -0.437Cycle Time 1.375 0.688Gate 0.375 0.187Pressure 0.375 0.187Temperature*Screw 11.875 5.938Temperature*Hold Time -1.625 -0.813Temperature*Cycle Time -5.375 -2.688Temperature*Gate -1.875 -0.937Temperature*Pressure 0.625 0.313Screw*Cycle Time -0.125 -0.062Screw*Pressure -0.125 -0.063Temperature*Screw*Cycle Time 0.125 0.062Temperature*Hold Time*Cycle Time -4.875 -2.437

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

The One-Quarter Fraction of the 2k 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 PConstant 27.313 1.138 24.00 0.000Temperature 13.875 6.937 1.138 6.09 0.000Screw 35.625 17.812 1.138 15.65 0.000Temperature*Screw 11.875 5.938 1.138 5.22 0.000


Analysis of Variance for Shrinkage (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 2 5846.6 5846.6 2923.31 141.02 0.0002-Way Interactions 1 564.1 564.1 564.06 27.21 0.000Residual Error 12 248.7 248.7 20.73 Pure Error 12 248.8 248.8 20.73Total 15 6659.4


Reduced model Normal plot


Reduced model Residuals vs

Hold time Less scatter in

low hold time than it is high


F*C is large

)()(ln 2

2*

iSiSFi


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

2k-1 = one-half fraction, 2k-2 = one-quarter fraction, 2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p 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 2p-p-1 generalized interactions.


Important to select generators so as to maximize resolution

For example, the 26-2IV design, generators:

E=ABC, F=BCD, producing IV design. maximum resolution

If E=ABC, F=ABCD is chosen, I=ABCE=ABCDF=DEF, resolution III.


Sometimes resolution alone is insufficient to distinguish between designs.

For 27-2IV 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 next table shows the suggested generators for better designs.


The General 2k-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: 27-2

IV (32 runs) and 27-3IV (16

runs)


27-3IV (16 runs)


Design Generators: F = ABCD, G = ABDE

Alias StructureI + CEFG + ABCDF + ABDEG

A + BCDF + BDEG + ACEFG B + ACDF + ADEG + BCEFG C + EFG + ABDF + ABCDEGD + ABCF + ABEG + CDEFG E + CFG + ABDG + ABCDEF F + CEG + ABCD + ABDEFGG + CEF + ABDE + ABCDFG AB + CDF + DEG + ABCEFG AC + BDF + AEFG + BCDEGAD + BCF + BEG + ACDEFG AE + BDG + ACFG + BCDEF AF + BCD + ACEG + BDEFGAG + BDE + ACEF + BCDFG BC + ADF + BEFG + ACDEG BD + ACF + AEG + BCDEFGBE + ADG + BCFG + ACDEF BF + ACD + BCEG + ADEFG BG + ADE + BCEF + ACDFGCD + ABF + DEFG + ABCEG CE + FG + ABCDG + ABDEF CF + EG + ABD + ABCDEFGCG + EF + ABCDE + ABDFG DE + ABG + CDFG + ABCEF DF + ABC + CDEG + ABEFGDG + ABE + CDEF + ABCFG ACE + AFG + BCDG + BDEF ACG + AEF + BCDE + BDFGBCE + BFG + ACDG + ADEF BCG + BEF + ACDE + ADFG CDE + DFG + ABCG + ABEFCDG + DEF + ABCE + ABFG


27-3IV (16 runs)


Design Generators: E = ABC, F = BCD, G = ACD

Alias StructureI + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG

A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFGC + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFGE + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFGG + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFGAC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFGAE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFGAG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFGABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG


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 2k-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

27-3IV design


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 24 designs.


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 2k-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.


For example, 26-2IV

According to the suggestion in Appendix X(f), ABD and its aliases to be confounded with block.

STAT>DOE>Create Factorial Design


Design Full factorial Generators:

OK



bc


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, 28-4

IV (16 runs)and 28-3IV

(32 runs) are feasible.


However, if 28-4IV (16 runs) is used, two-factor

effects will confound with blocks If EH interaction is unlikely, 28-3

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


Fractional Factorial Design

Factors: 8 Base Design: 8, 32 Resolution with blocks: IIIRuns: 32 Replicates: 1 Fraction: 1/8Blocks: 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




Analyze Design A*D + B*G + E*F*H

Inseparable If prior knowledge

implies that AD is possible, one can use reduced model


Reduced model: A, B, D, and AD


Reduced model: A, B, D, and ADFactorial Fit: ln(std_dev) versus Block, A, B, D Estimated Effects and Coefficients for ln(std_dev) (coded units)Term Effect Coef SE Coef T PConstant 1.2801 0.02310 55.41 0.000Block 1 -0.0053 0.04001 -0.13 0.896Block 2 -0.0280 0.04001 -0.70 0.491Block 3 0.0406 0.04001 1.02 0.320A 0.2903 0.1451 0.02310 6.28 0.000B -0.2005 -0.1003 0.02310 -4.34 0.000D 0.1081 0.0541 0.02310 2.34 0.028A*D -0.3741 -0.1871 0.02310 -8.10 0.000


Analysis of Variance for ln(std_dev) (coded units)

Source DF Seq SS Adj SS Adj MS F PBlocks 3 0.02014 0.02014 0.00671 0.39 0.759Main Effects 3 1.08929 1.08929 0.36310 21.26 0.0002-Way Interactions 1 1.11970 1.11970 1.11970 65.57 0.000Residual Error 24 0.40986 0.40986 0.01708Total 31 2.63899


Reduced model: A, B, D, and AD Estimated equation

41421 1871.00541.01003.01451.0 xxxxxy


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Design and Analysis of Experiments