1. Design and Analysis of Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
This is a basic course blah, blah, blah…

2. Two-Level Fractional Factorial Designs
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
This is a basic course blah, blah, blah…

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 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.

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 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.

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

11. The One-Half Fraction of the 2k 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.

12. The One-Half Fraction of the 2k 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.

13. The One-Half Fraction of the 2k 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

14. The One-Half Fraction of the 2k 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)

15. The One-Half Fraction of the 2k 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

16. The One-Half Fraction of the 2k Design – Definitions and Basic PrinciplesSo
[A] A+BC [B] B+AC [C] C+AB
The alias structure(別名) 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 2k 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.

18. The One-Half Fraction of the 2k 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

19. The One-Half Fraction of the 2k Design – Definitions and Basic PrinciplesSo
[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 2k 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 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 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=1) factor effect is aliased with one (less than (<) 2=3(R) – 1(p)) factor effect.

22. The One-Half Fraction of the 2k Design – Design Resolution
Resolution III designs – These are designs
no main effects are aliased with any other main effect
main effects are aliased with two-factor interactions
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 2k Design – Design Resolution
Resolution IV designs – These are designs
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 2k Design – Design Resolution
Resolution V designs – These are designs
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 2k Design – Construction and analysis of the one-half fraction
Example:
C‧I=C‧ABC=AB

26. The One-Half Fraction of the 2k Design – 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 2k 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.

28. The One-Half Fraction of the 2k 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. The One-Half Fraction of the 2k Design – 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 2k Design – example (1--1/9)
Y=filtration rate
Fours factors: A, B, C, and D.
Use 24-1 with I=ABCD

31. The One-Half Fraction of the 2k Design – example (1--2/9)
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: , 8 Resolution: IV
Runs: Replicates: Fraction: 1/2
Blocks: 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 2k Design – example (1--3/9)

33. The One-Half Fraction of the 2k Design – example(1--4/9)

34. The One-Half Fraction of the 2k Design – example (1--5/9)
After collecting data, example8-1.mtw
STAT > DOE > Factorial > Analyze Factorial Design
Response  Filtration  OK
Estimated Effects and Coefficients for Filtration (coded units)
Term Effect Coef
Constant
Temperature
Pressure
Conc
Stir Rate
Temperature*Pressure
Temperature*Conc
Temperature*Stir Rate

35. The One-Half Fraction of the 2k Design – example (1--6/9)

36. The One-Half Fraction of the 2k Design – example (1--7/9)
Obviously, no effect is significant
B is less important
Try A, C, and D  projection 23 with A, C, D
Assuming CD can be neglected

37. The One-Half Fraction of the 2k Design – example (1--8/9)

38. The One-Half Fraction of the 2k Design – example (1--9/9)
Factorial Fit: Filtration versus Temperature, Conc., Stir Rate
Model Summary
S R-sq R-sq(adj) R-sq(pred)
% % %
Coded Coefficients
Term Effect Coef SE Coef T-Value P-Value VIF
Constant
Temperature
Conc
Stir Rate
Temperature*Conc
Temperature*Stir Rate
Prediction equation:
Filtration = Temperature Conc Stir Rate
Temperature*Conc Temperature*Stir Rate
Coded variable :

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

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

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

42. 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 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 * * *
Total

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

44. The One-Half Fraction of the 2k 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
2-Way Interactions
Residual Error
Lack of Fit
Pure Error
Total

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

46. The One-Half Fraction of the 2k Design – example (2--8/8)
Collapse into two replicate of a 23 design

47. The One-Half Fraction of the 2k 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 (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.

48. The One-Half Fraction of the 2k 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 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.

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

50. The One-Half Fraction of the 2k Design – 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

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

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

53. The One-Half Fraction of the 2k 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 * * *
Total

54. The One-Half Fraction of the 2k 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.

55. 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

56. The One-Quarter Fraction of the 2k 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.

57. The One-Quarter Fraction of the 2k 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

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

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

60. The One-Quarter Fraction of the 2k Design

61. The One-Quarter Fraction of the 2k Design
Factor relations: E=ABC, F=BCD
I=ABCE=BCDF=ADEF

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

63. The One-Quarter Fraction of the 2k Design

64. The One-Quarter Fraction of the 2k Design
Alternate fractions of 26-2 design
P=ABCE, -Q=-BCDF
-P=-ABCE, Q=BCDF
-P=-ABCE, -Q=-BCDF
[A]  A+BCE-DEF-ABCDF

65. The One-Quarter Fraction of the 2k Design
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.

66. The One-Quarter Fraction of the 2k Design
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)

67. The One-Quarter Fraction of the 2k Design
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.

68. 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

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

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

71. The One-Quarter Fraction of the 2k 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

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

73. 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 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
2-Way Interactions
Residual Error
Pure Error
Total

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

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

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

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

78. 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.

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

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

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

82. The General 2k-p Fractional Factorial Design – choose a design
The next table shows the suggested generators for better designs.

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

84. 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: 27-2IV (32 runs) and 27-3IV (16 runs)

85. The General 2k-p Fractional Factorial Design— example(5—2/4)
27-2IV (32 runs)
Fractional Factorial Design
Factors: Base Design: , 32 Resolution: IV
Runs: Replicates: Fraction: 1/4
Blocks: Center pts (total):
Design Generators: F = ABCD, G = ABDE
Alias Structure
I + CEFG + ABCDF + ABDEG
A + BCDF + BDEG + ACEFG B + ACDF + ADEG + BCEFG C + EFG + ABDF + ABCDEG
D + ABCF + ABEG + CDEFG E + CFG + ABDG + ABCDEF F + CEG + ABCD + ABDEFG
G + CEF + ABDE + ABCDFG AB + CDF + DEG + ABCEFG AC + BDF + AEFG + BCDEG
AD + BCF + BEG + ACDEFG AE + BDG + ACFG + BCDEF AF + BCD + ACEG + BDEFG
AG + BDE + ACEF + BCDFG BC + ADF + BEFG + ACDEG BD + ACF + AEG + BCDEFG
BE + ADG + BCFG + ACDEF BF + ACD + BCEG + ADEFG BG + ADE + BCEF + ACDFG
CD + ABF + DEFG + ABCEG CE + FG + ABCDG + ABDEF CF + EG + ABD + ABCDEFG
CG + EF + ABCDE + ABDFG DE + ABG + CDFG + ABCEF DF + ABC + CDEG + ABEFG
DG + ABE + CDEF + ABCFG ACE + AFG + BCDG + BDEF ACG + AEF + BCDE + BDFG
BCE + BFG + ACDG + ADEF BCG + BEF + ACDE + ADFG CDE + DFG + ABCG + ABEF
CDG + DEF + ABCE + ABFG

86. The General 2k-p Fractional Factorial Design— example(5—3/4)
27-3IV (16 runs)
Fractional Factorial Design
Factors: Base Design: , 16 Resolution: IV
Runs: Replicates: Fraction: 1/8
Blocks: Center pts (total):
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 + ACEFG B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG

87. The General 2k-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.

88. 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

89. The General 2k-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 24 designs.

90. The General 2k-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.

91. 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.

92. The General 2k-p Fractional Factorial Design – Blocking
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

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

94. The General 2k-p Fractional Factorial Design – Blocking

95. The General 2k-p Fractional Factorial Design – Blockingbc

96. The General 2k-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 three factor and higher interactions are negligible
From Appendix X, 28-4IV (16 runs)and 28-3IV (32 runs) are feasible.

97. The General 2k-p Fractional Factorial Design— example(6—1/4)
However, if 28-4IV (16 runs) is used, two-factor effects will confound with blocks
If EH interaction is unlikely, 28-3IV (32 runs) is chosen.
STAT>DOE>Create Factorial Design
Choose 2 level factorial (default generators)
Number of factors  8, number of blocks  4  OK

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

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

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

101. The General 2k-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

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

103. The General 2k-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 Effect Coef SE Coef T P
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
2-Way Interactions
Residual Error
Total

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

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