Design and Analysis of Experiments

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

3. Outline • Introduction • The 22 Design • The 23 Design • The general 2k Design • A single Replicate of the 2k design • Additional Examples of Unreplicated 2k Designs • 2k Designs are Optimal Designs • The additional of center Point to the 2k 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 2k 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 22 Design

6. The 22 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. The 22 Design • 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

8. The 22 Design

9. The 22 Design • Standard order Yates’s order • Effects A, B, AB are orthogonal contrasts with one degree of freedom • Thus 2k designs are orthogonal designs

10. The 22 Design • ANOVA table

11. The 22 Design • Algebraic sign for calculating effects in 22 design

12. The 22 Design • Regression model • x1 and x2 are code variable in this case • Where con and catalyst are natural variables

13. The 22 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 22 Design • Regression model

15. The 22 Design • Regression model

16. The 22 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 • Regression model (without interaction) Estimated Coefficients for Yield using data in uncoded units Term Coef Constant 18.3333 Conc. 0.833333 Catalyst -5.00000

17. The 22 Design • Response surface

18. The 22 Design • Response surface (note: the axis of catalyst is reversed with the one from textbook)

19. The 23 Design • 3 factors, each at two level. • Eight combinations

20. The 23 Design • Design matrix • Or geometric notation

21. The 23 Design • Algebraic sign

22. The 23 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. The 23 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 23 Design -- example • Nitride etch process • Gap, gas flow, and RF power

25. The 23 Design -- example • Nitride etch process • Gap, gas flow, and RF power

26. The 23 Design -- example • Full model 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

27. The 23 Design -- example • Reduced model 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

28. The 23 Design – example -- Model Summary Statistics for Reduced Model • R2 and adjusted R2 • R2for prediction (based on PRESS)

29. The 23 Design -- example

30. The 23 Design -- example

31. The Regression Model

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

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

34. The General 2kFactorial Design • Statistical Analysis

35. The General 2kFactorial Design • Statistical Analysis

36. Unreplicated2kFactorial Designs • These are 2k factorial designs with one observationat each corner of the “cube” • An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k • 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. Unreplicated2kFactorial Designs 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

38. Unreplicated2kFactorial Designs • Lack of replication causes potential problemsin statistical testing • Replication admits an estimate of “pure error” (a better phrase is an internal estimateof 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 plottingof effects (Daniels, 1959)

39. Unreplicated2kFactorial Designs -- example • A 24 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

40. Unreplicated2kFactorial Designs -- example

41. Unreplicated 2kFactorial Designs -- example

42. Unreplicated 2kFactorial Designs – example –full model

43. Unreplicated 2kFactorial Designs -- example –full model

44. Unreplicated 2kFactorial Designs -- example –full model

45. Unreplicated 2kFactorial 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. Unreplicated 2kFactorial Designs -- example –reduced model

47. Unreplicated 2kFactorial Designs -- example –reduced model

48. Unreplicated 2kFactorial Designs -- example –reduced model

49. Unreplicated 2kFactorial Designs -- example –Design projection • Since factor B is negligible, the experiment can be interpreted as a 23 factorial design with factors A, C, D. • 2 replicates

50. Unreplicated 2kFactorial Designs -- example –Design projection