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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. Alias Structures I Fractional Factorials and other Designs • Assuming that we use the following regression equation to fit the experimental results: where y is an n x 1 vector of the response X1is an n x p1 matrix β1is a p1x 1 vector • Thus the estimated of β1 via LSE is

  5. Alias Structures I Fractional Factorials and other Designs • Suppose that the true model is where X2is an n x p2 matrix with additional variables β2is a p2x 1 vector

  6. Alias Structures Fractional Factorials and other Designs • Thus the expected parameters • The matrix is called alias matrix • The elements of A operating on β2 identify the alias relationships for the parameters in the vector β1

  7. Alias Structures Fractional Factorials and other Designs • Example: 23-1 design with I=ABC

  8. Alias Structures Fractional Factorials and other Designs • Regression model • So, for the four runs

  9. Alias Structures Fractional Factorials and other Designs • Suppose the true model is • and

  10. Alias Structures Fractional Factorials and other Designs • Now try to find A

  11. Alias Structures Fractional Factorials and other Designs • And

  12. Alias Structures Fractional Factorials and other Designs • Comparison [A] A+BC [B] B+AC [C] C+AB

  13. Resolution III Designs -- Constructing • Resolution III designs are useful for screening (5 – 7 variables in 8 runs, 9 - 15 variables in 16 runs, for example) • A saturated design has k = N – 1 variables • Examples of saturated design:

  14. Resolution III Designs -- Constructing • Case of

  15. Resolution III Designs -- Constructing • Can be used to generate factors fewer than 7 • For example,

  16. Resolution III Designs – Fold over • By combining fractional factorial designs that certain signs are switched , one can systematically isolate effects of the potential interest • This type of sequential experiments is called a fold over of the original design

  17. Resolution III Designs – Fold over • For the case of • Reversing the sign in factor D  - + + - - + + -

  18. Resolution III Designs – Fold over • Reversed effects • [A]’  A-BD+CE+FG • [B]’  B-AD+CF+EG • [C]’  C+AE+BF+DG • [D]’  D-AB-CG-EF • [-D]’  -D+AB+CG+EF • [E]’  E+AC+BG-DF • [F]’  F+BC+AG-DE • [G]’  G-CD+BE+AF • Original effects

  19. Resolution III Designs – Fold over • Assuming the three-factor and higher interactions are insignificant, one can combine the two fractions • For effect of the factor D ½ [D]+1/2[D]’ D • For effects ½ [D]-1/2[D]’ AB+C+EF

  20. Resolution III Designs – Fold over • In general, if we add to a fractional design of resolution III or higher a further fraction with signs of a single factor reversed, the combined design will provide the estimates of the man effect of that factor and its two-factor interactions • This is a single-factor fold over

  21. Resolution III Designs – Fold over • If we add to a fractional design of resolution III a second fraction with signs of all the factors are reversed, the combined design break the alias link between all main effects and their two-factor interaction. • This is a full fold over

  22. Resolution III Designs – example (7—1/9) • Eye focus, Response= time • 7 factors • Screening experiment

  23. Resolution III Designs – example (7—2/9) • STAT>DOE>Create Factorial Design • 2 level fractional (default) • Number of factor  7 • Choose 1/8 fractional

  24. Resolution III Designs – example (7—3/9) • STAT>DOE>Analyze Factorial Design • Only A, B, D are significant

  25. Resolution III Designs – example (7—4/9) • Examining the alias structure • We are not sure if A or BD, B or AD, D or AB are significant!!!!! Alias Structure (up to order 3) I + A*B*D + A*C*E + A*F*G + B*C*F + B*E*G + C*D*G + D*E*F A + B*D + C*E + F*G + B*C*G + B*E*F + C*D*F + D*E*G B + A*D + C*F + E*G + A*C*G + A*E*F + C*D*E + D*F*G C + A*E + B*F + D*G + A*B*G + A*D*F + B*D*E + E*F*G D + A*B + C*G + E*F + A*C*F + A*E*G + B*C*E + B*F*G E + A*C + B*G + D*F + A*B*F + A*D*G + B*C*D + C*F*G F + A*G + B*C + D*E + A*B*E + A*C*D + B*D*G + C*E*G G + A*F + B*E + C*D + A*B*C + A*D*E + B*D*F + C*E*F

  26. Resolution III Designs – example (7—5/9) • Note that ABD is one of the word in defining relation, do not project into a full 23 factorial in ABD • It does project into two replicates of a 23-1 design. • 23-1 is a resolution III design, too • Try fold over

  27. Resolution III Designs – example (7—6/9) • 2nd fraction: • STAT>DOE>Modify Design • Specify  fold all factor OK

  28. Resolution III Designs – example (7—7/9)

  29. Resolution III Designs – example (7—8/9) • Collecting data • STAT>DOE>Analyze Factorial Design

  30. Resolution III Designs – example (7—9/9) • Though B, D, BD, and AF are significant, B and D are distinguishable • BD is aliased with CE and FG • AF is aliased with CD and BE. A + B*C*G + B*E*F + C*D*F + D*E*G B + A*C*G + A*E*F + C*D*E + D*F*G C + A*B*G + A*D*F + B*D*E + E*F*G D + A*C*F + A*E*G + B*C*E + B*F*G E + A*B*F + A*D*G + B*C*D + C*F*G F + A*B*E + A*C*D + B*D*G + C*E*G G + A*B*C + A*D*E + B*D*F + C*E*F A*B + C*G + E*F A*C + B*G + D*F A*D + C*F + E*G A*E + B*F + D*G A*F + B*E + C*D A*G + B*C + D*E B*D + C*E + F*G

  31. Resolution III Designs – Fold over • To find the defining relation for a combined design, one can assume that the first fraction has L words and the fold over fraction has U words. • Thus the combined design will have L+U-1 words used as a generators.

  32. Resolution III Designs – Fold over • For example, • Generators for the first fraction: I=ABD, I=ACE, I=BCF, I=ABCG • Generators for the second fraction: I=-ABD, I=-ACE, I=-BCF, I=ABCG • We have switched the signs on the generators with an odd number of letters

  33. Resolution III Designs – Fold over • The complete defining relations for the combined design are: I=ABCG=BCDE=ACDF=ADEG=BDFG =ABEF=CEFG

  34. Resolution III Designs – Fold over • Usually the second fraction are different from the first fraction in day, time, shift, material, methods. • This leads to the blocking situation.

  35. Resolution III Designs – Plackett-Burman Designs • For the case of k=N-1 variables in N runs, where N is a multiple of 4, one can use fold over if N is a power of 2. • However, N=12, 20, 24, 28 and 36, The Placket-Burman is of interest. • Because these design cannot be represented as cubes, called non-geometric designs. • Two ways to generate these designs, check example 8.

  36. Resolution III Designs – Plackett-Burman Designs • Upper half: for N=12, 20, 24, and 36 • Lower half: for N=28

  37. Resolution III Designs – Plackett-Burman Designs • Example for Upper half: N=12 and k=11 Turn into the first column

  38. Resolution III Designs – Plackett-Burman Designs Shift down one row! Add “-” sign

  39. Resolution III Designs – Plackett-Burman Designs • Example for Lower Half: N=28 and k=27 Y Z X

  40. Resolution III Designs – Plackett-Burman Designs • N=28 and k=27 • Arrange the design into X Y Z Z X Y Y Z X - - - - - - - - Add “-” sign to the 28th row

  41. Resolution III Designs – Plackett-Burman Designs • Alias structure • Messy and complicated • Main effects are partially aliased with every two-factor interaction not involving itself • Non-regular design • For the case of N=12 • Projected into three replicates of a full 22 design in any two of the original 11 factors • Projected into a full 23 factorial plus a 23-2III fractional factorial

  42. Resolution III Designs – Plackett-Burman Designs • The resolution II Placket-Burman design has Projectivity 3. • It will collapse into a full factorial in any subset of the three factors.

  43. Resolution III Designs – example (8—1/7) • 12 factors • If 212-8 fractional is used, all 12 main effects are aliased with four two-factor interactions. • Additional experiments could be required • Use 20 run Placket-Burman design • Two kinds of designs, one is to follow the text and Minitab • The other is to follow Example 8 in the text.

  44. Resolution III Designs – example (8—2/7) Add “+” sign Reverse “+” and “-” sign in text

  45. Resolution III Designs – example (8—3/7) Corrected Table 8.25

  46. Resolution III Designs – example (8—4/7) Alternate P-B design for N=20

  47. Resolution III Designs – example (8—5/7) • No effect is significant according to traditional analysis.

  48. Resolution III Designs – example (8—6/7) • Use stepwise regression Stepwise Regression: y versus X1, X2, ... Alpha-to-Enter: 0.1 Alpha-to-Remove: 0.15 Response is y on 19 predictors, with N = 20 Step 1 2 3 4 5 6 Constant 200.0 200.0 200.0 200.0 200.0 200.0 X2 11.8 11.8 11.8 10.0 10.0 9.9 T-Value 2.51 2.78 3.65 4.02 7.30 7.99 P-Value 0.022 0.013 0.002 0.001 0.000 0.000 X4 9.6 12.0 12.0 12.0 12.1 T-Value 2.27 3.64 4.82 8.76 9.78 P-Value 0.037 0.002 0.000 0.000 0.000 x1x2 -12.0 -12.0 -12.0 -12.5 T-Value -3.64 -4.82 -8.76 -9.91 P-Value 0.002 0.000 0.000 0.000 x1x4 9.0 9.0 9.5 T-Value 3.62 6.57 7.54 P-Value 0.003 0.000 0.000 X1 8.0 8.0 T-Value 5.96 6.60 P-Value 0.000 0.000 X5 2.6 T-Value 2.04 P-Value 0.062 S 21.0 18.9 14.4 10.9 6.00 5.42 R-Sq 25.95 43.12 68.89 83.38 95.30 96.44 R-Sq(adj) 21.83 36.43 63.05 78.94 93.63 94.80

  49. Resolution III Designs – example (8—7/7) • Fitted model:

  50. Resolution IV and V Designs -- Resolution IV Designs • A 2k-p fractional is of resolution IV if the main effects are clear of two-factor interactions and some two-factor interactions are aliased with each other. • Any 2k-pIV design must contain at least 2k runs. Resolution IV designs that contain 2k runs are called minimal designs. • Resolution IV designs maybe obtained from resolution III designs by the process of fold over.

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