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

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

3. Outline Introduction The 2 2 Design The 2 3 Design The general 2 k Design A single Replicate of the 2 k design Additional Examples of Unreplicated 2 k Designs 2 k Designs are Optimal Designs The additional of center Point to the 2 k Design

4. Introduction Special case of general factorial designs k factors each with two levels Factors maybe qualitative or quantitative A complete replicate of such design is 2 k factorial design Assumed factors are fixed, the design are completely randomized, and normality Used as factor screening experiments Response between levels is assumed linear

5. The 2 2 Design FactorTreatment Combination Replication ABIIIIIIIV --A low, B low28252780 +-A high, B low3632 100 -+A low, B high18192360 ++A high, B high31302990

6. The 2 2 Design “-” and “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different

7. Estimate factor effects Formulate model With replication, use full model With an unreplicated design, use normal probability plots Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results The 2 2 Design

9. Standard order  Yates’s order Effects(1)abab A+1+1 B +1 AB+1 +1 Effects A, B, AB are orthogonal contrasts with one degree of freedom Thus 2 k designs are orthogonal designs

10. The 2 2 Design ANOVA table

11. The 2 2 Design Algebraic sign for calculating effects in 2 2 design

12. The 2 2 Design Regression model x 1 and x 2 are code variable in this case Where con and catalyst are natural variables

13. The 2 2 Design Regression model Factorial Fit: Yield versus Conc., Catalyst Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 27.500 0.5713 48.14 0.000 Conc. 8.333 4.167 0.5713 7.29 0.000 Catalyst -5.000 -2.500 0.5713 -4.38 0.002 Conc.*Catalyst 1.667 0.833 0.5713 1.46 0.183 S = 1.97906 PRESS = 70.5 R-Sq = 90.30% R-Sq(pred) = 78.17% R-Sq(adj) = 86.66% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 283.333 283.333 141.667 36.17 0.000 2-Way Interactions 1 8.333 8.333 8.333 2.13 0.183 Residual Error 8 31.333 31.333 3.917 Pure Error 8 31.333 31.333 3.917 Total 11 323.000

14. The 2 2 Design Regression model

15. The 2 2 Design Regression model

16. The 2 2 Design Regression model Estimated Coefficients for Yield using data in uncoded units Term Coef Constant 28.3333 Conc. 0.333333 Catalyst -11.6667 Conc.*Catalyst 0.333333 Estimated Coefficients for Yield using data in uncoded units Term Coef Constant 18.3333 Conc. 0.833333 Catalyst -5.00000 Regression model (without interaction)

17. The 2 2 Design Response surface

18. The 2 2 Design Response surface (note: the axis of catalyst is reversed with the one from textbook)

19. The 2 3 Design 3 factors, each at two level. Eight combinations

20. The 2 3 Design Design matrix Or geometric notation

21. The 2 3 Design Algebraic sign

22. 22 The 2 3 Design -- Properties of the Table Except for column I, every column has an equal number of + and – signs The sum of the product of signs in any two columns is zero Multiplying any column by I leaves that column unchanged (identity element)

23. 23 The 2 3 Design -- Properties of the Table The product of any two columns yields a column in the table: Orthogonal design Orthogonality is an important property shared by all factorial designs

24. The 2 3 Design -- example Nitride etch process Gap, gas flow, and RF power

25. The 2 3 Design -- example Nitride etch process Gap, gas flow, and RF power

26. The 2 3 Design -- example Estimated Effects and Coefficients for Etch Rate (coded units) Term Effect Coef SE Coef T P Constant 776.06 11.87 65.41 0.000 Gap -101.62 -50.81 11.87 -4.28 0.003 Gas Flow 7.37 3.69 11.87 0.31 0.764 Power 306.12 153.06 11.87 12.90 0.000 Gap*Gas Flow -24.88 -12.44 11.87 -1.05 0.325 Gap*Power -153.63 -76.81 11.87 -6.47 0.000 Gas Flow*Power -2.12 -1.06 11.87 -0.09 0.931 Gap*Gas Flow*Power 5.62 2.81 11.87 0.24 0.819 S = 47.4612 PRESS = 72082 R-Sq = 96.61% R-Sq(pred) = 86.44% R-Sq(adj) = 93.64% Analysis of Variance for Etch Rate (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 416378 416378 138793 61.62 0.000 2-Way Interactions 3 96896 96896 32299 14.34 0.001 3-Way Interactions 1 127 127 127 0.06 0.819 Residual Error 8 18020 18020 2253 Pure Error 8 18021 18021 2253 Total 15 531421 Full model

27. The 2 3 Design -- example Factorial Fit: Etch Rate versus Gap, Power Estimated Effects and Coefficients for Etch Rate (coded units) Term Effect Coef SE Coef T P Constant 776.06 10.42 74.46 0.000 Gap -101.62 -50.81 10.42 -4.88 0.000 Power 306.12 153.06 10.42 14.69 0.000 Gap*Power -153.63 -76.81 10.42 -7.37 0.000 S = 41.6911 PRESS = 37080.4 R-Sq = 96.08% R-Sq(pred) = 93.02% R-Sq(adj) = 95.09% Analysis of Variance for Etch Rate (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 416161 416161 208080 119.71 0.000 2-Way Interactions 1 94403 94403 94403 54.31 0.000 Residual Error 12 20858 20858 1738 Pure Error 12 20858 20858 1738 Total 15 531421 Reduced model

28. 28 R 2 and adjusted R 2 R 2 for prediction (based on PRESS) The 2 3 Design – example -- Model Summary Statistics for Reduced Model

29. The 2 3 Design -- example

31. 31 The Regression Model

32. 32 Cube Plot of Ranges What do the large ranges when gap and power are at the high level tell you?

33. 33 The General 2 k Factorial Design There will be k main effects, and

34. 34 The General 2 k Factorial Design Statistical Analysis

35. 35 The General 2 k Factorial Design Statistical Analysis

36. 36 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the “cube” An unreplicated 2 k factorial design is also sometimes called a “single replicate” of the 2 k These designs are very widely used Risks…if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? Modeling “noise”?

37. 37 If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually best Unreplicated 2 k Factorial Designs

38. 38 Lack of replication causes potential problems in statistical testing Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels, 1959) Unreplicated 2 k Factorial Designs

39. 39 A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant Unreplicated 2 k Factorial Designs -- example

40. 40 Unreplicated 2 k Factorial Designs -- example

41. 41 Unreplicated 2 k Factorial Designs -- example

42. 42 Unreplicated 2 k Factorial Designs – example –full model

43. 43 Unreplicated 2 k Factorial Designs -- example –full model

44. 44 Unreplicated 2 k Factorial Designs -- example –full model

45. 45 Unreplicated 2 k Factorial Designs -- example –reduced model Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant 70.063 1.104 63.44 0.000 Temperature 21.625 10.812 1.104 9.79 0.000 Conc. 9.875 4.938 1.104 4.47 0.001 Stir Rate 14.625 7.312 1.104 6.62 0.000 Temperature*Conc. -18.125 -9.062 1.104 -8.21 0.000 Temperature*Stir Rate 16.625 8.313 1.104 7.53 0.000 S = 4.41730 PRESS = 499.52 R-Sq = 96.60% R-Sq(pred) = 91.28% R-Sq(adj) = 94.89% Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 3116.19 3116.19 1038.73 53.23 0.000 2-Way Interactions 2 2419.62 2419.62 1209.81 62.00 0.000 Residual Error 10 195.12 195.12 19.51 Lack of Fit 2 15.62 15.62 7.81 0.35 0.716 Pure Error 8 179.50 179.50 22.44 Total 15 5730.94

46. 46 Unreplicated 2 k Factorial Designs -- example –reduced model

47. 47 Unreplicated 2 k Factorial Designs -- example –reduced model

48. 48 Unreplicated 2 k Factorial Designs -- example –reduced model

49. 49 Unreplicated 2 k Factorial Designs -- example –Design projection Since factor B is negligible, the experiment can be interpreted as a 2 3 factorial design with factors A, C, D. 2 replicates

50. 50 Unreplicated 2 k Factorial Designs -- example –Design projection

51. 51 Unreplicated 2 k Factorial Designs -- example –Design projection Factorial Fit: Filtration versus Temperature, Conc., Stir Rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant 70.063 1.184 59.16 0.000 Temperature 21.625 10.812 1.184 9.13 0.000 Conc. 9.875 4.938 1.184 4.17 0.003 Stir Rate 14.625 7.312 1.184 6.18 0.000 Temperature*Conc. -18.125 -9.062 1.184 -7.65 0.000 Temperature*Stir Rate 16.625 8.313 1.184 7.02 0.000 Conc.*Stir Rate -1.125 -0.562 1.184 -0.48 0.647 Temperature*Conc.*Stir Rate -1.625 -0.813 1.184 -0.69 0.512 S = 4.73682 PRESS = 718 R-Sq = 96.87% R-Sq(pred) = 87.47% R-Sq(adj) = 94.13% Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 3116.19 3116.19 1038.73 46.29 0.000 2-Way Interactions 3 2424.69 2424.69 808.23 36.02 0.000 3-Way Interactions 1 10.56 10.56 10.56 0.47 0.512 Residual Error 8 179.50 179.50 22.44 Pure Error 8 179.50 179.50 22.44 Total 15 5730.94

52. 52 Dealing with Outliers Replace with an estimate Make the highest-order interaction zero In this case, estimate cd such that ABCD = 0 Analyze only the data you have Now the design isn’t orthogonal Consequences?

53. 53 Duplicate Measurements on the Response Four wafers are stacked in the furnace Four factors: temperature, time, gas flow, and pressure. Response: thickness Treated as duplicate not replicate Use average as the response

54. 54 Duplicate Measurements on the Response

55. 55 Duplicate Measurements on the Response Stat  DOE  Factorial  Pre- process Response for Analyze

56. 56 Duplicate Measurements on the Response Stat  DOE  Factorial  Analyze Factorial Design

57. 57 Duplicate Measurements on the Response Factorial Fit: average versus Temperature, Time, Pressure Estimated Effects and Coefficients for average (coded units) Term Effect Coef SE Coef T P Constant 399.188 1.049 380.48 0.000 Temperature 43.125 21.562 1.049 20.55 0.000 Time 18.125 9.062 1.049 8.64 0.000 Pressure -10.375 -5.187 1.049 -4.94 0.001 Temperature*Time 16.875 8.438 1.049 8.04 0.000 Temperature*Pressure -10.625 -5.312 1.049 -5.06 0.000 S = 4.19672 PRESS = 450.88 R-Sq = 98.39% R-Sq(pred) = 95.88% R-Sq(adj) = 97.59% Analysis of Variance for average (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 9183.7 9183.69 3061.23 173.81 0.000 2-Way Interactions 2 1590.6 1590.62 795.31 45.16 0.000 Residual Error 10 176.1 176.12 17.61 Lack of Fit 2 60.6 60.62 30.31 2.10 0.185 Pure Error 8 115.5 115.50 14.44 Total 15 10950.4

58. 58 Duplicate Measurements on the Response

59. 59 Duplicate Measurements on the Response

60. 60 The 2 k design and design optimality The model parameter estimates in a 2 k design (and the effect estimates) are least squares estimates. For example, for a 2 2 design the model is

61. 61 The four observations from a 2 2 design The 2 k design and design optimality In matrix form:

62. 62 The matrix is diagonal – consequences of an orthogonal design The regression coefficient estimates are exactly half of the ‘usual” effect estimates The “usual” contrasts The 2 k design and design optimality

63. 63 The 2 k design and design optimality The matrix X’X has interesting and useful properties: Minimum possible value for a four-run design Maximum possible value for a four-run design Notice that these results depend on both the design that you have chosen and the model

64. The 2 k design and design optimality The 2 2 design is called D-optimal design In fact, all 2 k design is D-optimal design for fitting first order model with interaction. Consider the variance of the predicted response in the 2 2 design:

65. The 2 k design and design optimality

66. The 2 2 design is called G-optimal design In fact, all 2 k design is G-optimal design for fitting first order model with interaction. Minimize the maximum prediction variance

67. The 2 k design and design optimality The 2 2 design is called I-optimal design In fact, all 2 k design is I-optimal design for fitting first order model with interaction. Smallest possible value of the average prediction variance

68. The 2 k design and design optimality The Minitab provide the function on “Select Optimal Design” when you have a full factorial design and are trying to reduce the it to a partial design or “fractional design”. It only provide the “D-optimal design” One needs to have a full factorial design first and the choose the number of data points to be allowed to use.

69. 69 These results give us some assurance that these designs are “good” designs in some general ways Factorial designs typically share some (most) of these properties There are excellent computer routines for finding optimal designs The 2 k design and design optimality

70. 70 Addition of Center Points to a 2 k Designs Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: Quadratic effects

71. 71 Addition of Center Points to a 2 k Designs When adding center points, we assume that the k factors are quantitative. Example on 2 2 design

72. 72 Addition of Center Points to a 2 k Designs Five point: (-,-),(-,+),(+,-),(+,+), and (0,0). n F =4 and n C =4 Let be the average of the four runs at the four factorial points and let be the average of n C run at the center point.

73. 73 Addition of Center Points to a 2 k Designs If the difference of is small, the center points lie on or near the plane passing through factorial points and there is no quadratic effects. The hypotheses are:

74. 74 Addition of Center Points to a 2 k Designs Test statistics: with one degree of freedom

75. 75 Addition of Center Points to a 2 k Designs -- example In example 6.2, it is a 2 4 factorial. By adding center points x1=x2=x3=x4=0, four additional responses (filtration rates) are : 73, 75, 66,69. So =70.75 and =70.06.

76. 76 Addition of Center Points to a 2 k Designs -- example Term Effect Coef SE Coef T P Constant 70.063 1.008 69.52 0.000 Temperature 21.625 10.812 1.008 10.73 0.002 Pressure 3.125 1.562 1.008 1.55 0.219 Conc. 9.875 4.937 1.008 4.90 0.016 Stir Rate 14.625 7.312 1.008 7.26 0.005 Temperature*Pressure 0.125 0.063 1.008 0.06 0.954 Temperature*Conc. -18.125 -9.063 1.008 -8.99 0.003 Temperature*Stir Rate 16.625 8.313 1.008 8.25 0.004 Pressure*Conc. 2.375 1.188 1.008 1.18 0.324 Pressure*Stir Rate -0.375 -0.187 1.008 -0.19 0.864 Conc.*Stir Rate -1.125 -0.563 1.008 -0.56 0.616 Temperature*Pressure*Conc. 1.875 0.937 1.008 0.93 0.421 Temperature*Pressure*Stir Rate 4.125 2.063 1.008 2.05 0.133 Temperature*Conc.*Stir Rate -1.625 -0.813 1.008 -0.81 0.479 Pressure*Conc.*Stir Rate -2.625 -1.312 1.008 -1.30 0.284 Temperature*Pressure*Conc.*Stir Rate 1.375 0.687 1.008 0.68 0.544 Ct Pt 0.687 2.253 0.31 0.780

77. 77 Addition of Center Points to a 2 k Designs -- example Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 4 3155.25 3155.25 788.813 48.54 0.005 2-Way Interactions 6 2447.88 2447.88 407.979 25.11 0.012 3-Way Interactions 4 120.25 120.25 30.062 1.85 0.320 4-Way Interactions 1 7.56 7.56 7.562 0.47 0.544 Curvature 1 1.51 1.51 1.512 0.09 0.780 Residual Error 3 48.75 48.75 16.250 Pure Error 3 48.75 48.75 16.250 Total 19 5781.20

78. 78 Addition of Center Points to a 2 k Designs If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model

79. 79 Addition of Center Points to a 2 k Designs

80. 80 Addition of Center Points to a 2 k Designs Use current operating conditions as the center point Check for “abnormal” conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error

81. 81 Center Points and Qualitative Factors