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

3. Outline Basic Definition and Principles The Advantages of Factorials The Two Factors Factorial Design The General Factorial Design Fitting Response Curve and Surfaces Blocking in Factorial Design 3

4. Basic Definitions and Principles Factorial Design—all of the possible combinations of factors’ level are investigated When factors are arranged in factorial design, they are said to be crossed Main effects – the effects of a factor is defined to be changed Interaction Effect – The effect that the difference in response between the levels of one factor is not the same at all levels of the other factors. 4

5. 5 Basic Definitions and Principles Factorial Design without interaction

6. 6 Basic Definitions and Principles Factorial Design with interaction

7. 7 Basic Definitions and Principles Average response – the average value at one factor’s level Average response increase – the average value change for a factor from low level to high level No Interaction:

8. 8 Basic Definitions and Principles With Interaction:

9. 9 Basic Definitions and Principles Another way to look at interaction: When factors are quantitative In the above fitted regression model, factors are coded in (-1, +1) for low and high levels This is a least square estimates

10. 10 Basic Definitions and Principles Since the interaction is small, we can ignore it. Next figure shows the response surface plot

11. 11 Basic Definitions and Principles The case with interaction

12. 12 Advantages of Factorial design Efficiency Necessary if interaction effects are presented The effects of a factor can be estimated at several levels of the other factors

13. 13 The Two-factor Factorial Design Two factors a levels of factor A, b levels of factor B n replicates In total, nab combinations or experiments

14. 14 The Two-factor Factorial Design – An example Two factors, each with three levels and four replicates 3 2 factorial design

15. 15 The Two-factor Factorial Design – An example Questions to be answered: What effects do material type and temperature have on the life the battery Is there a choice of material that would give uniformly long life regardless of temperature?

16. 16 Statistical (effects) model: means model The Two-factor Factorial Design

17. Hypothesis Row effects: Column effects: Interaction:

18. The Two-factor Factorial Design -- Statistical Analysis 18

19. The Two-factor Factorial Design -- Statistical Analysis 19 Mean square: A: B: Interaction:

20. The Two-factor Factorial Design -- Statistical Analysis 20 Mean square: Error:

21. The Two-factor Factorial Design -- Statistical Analysis 21 ANOVA table

22. The Two-factor Factorial Design -- Statistical Analysis 22 Example

23. The Two-factor Factorial Design -- Statistical Analysis 23 Example

24. The Two-factor Factorial Design -- Statistical Analysis 24 Example

25. The Two-factor Factorial Design -- Statistical Analysis 25 Example STAT  ANOVA--GLM General Linear Model: Life versus Material, Temp Factor Type Levels Values Material fixed 3 1, 2, 3 Temp fixed 3 15, 70, 125 Analysis of Variance for Life, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Material 2 10683.7 10683.7 5341.9 7.91 0.002 Temp 2 39118.7 39118.7 19559.4 28.97 0.000 Material*Temp 4 9613.8 9613.8 2403.4 3.56 0.019 Error 27 18230.7 18230.7 675.2 Total 35 77647.0 S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56% Unusual Observations for Life Obs Life Fit SE Fit Residual St Resid 2 74.000 134.750 12.992 -60.750 -2.70 R 8 180.000 134.750 12.992 45.250 2.01 R R denotes an observation with a large standardized residual.

26. The Two-factor Factorial Design -- Statistical Analysis 26 Example STAT  ANOVA--GLM

27. The Two-factor Factorial Design -- Statistical Analysis 27 Example STAT  ANOVA--GLM

28. The Two-factor Factorial Design -- Statistical Analysis 28 Estimating the model parameters

29. The Two-factor Factorial Design -- Statistical Analysis 29 Choice of sample size Row effects Column effects Interaction effects D:difference,  :standard deviation

30. The Two-factor Factorial Design -- Statistical Analysis 30

31. The Two-factor Factorial Design -- Statistical Analysis 31 Appendix Chart V For n=4, giving D=40 on temperature, v 1 =2, v 2 =27, Φ 2 =1.28n.  β =0.06 nΦ2Φ2 Φυ1υ1 υ2υ2 β 22.561.6290.45 33.841.962180.18 45.122.262270.06

32. The Two-factor Factorial Design -- Statistical Analysis – example with no interaction 32 Analysis of Variance for Life, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Material 2 10684 10684 5342 5.95 0.007 Temp 2 39119 39119 19559 21.78 0.000 Error 31 27845 27845 898 Total 35 77647 S = 29.9702 R-Sq = 64.14% R-Sq(adj) = 59.51%

33. The Two-factor Factorial Design – One observation per cell 33 Single replicate The effect model

34. The Two-factor Factorial Design – One observation per cell 34 ANOVA table

35. The Two-factor Factorial Design -- One observation per cell 35 The error variance is not estimable unless interaction effect is zero Needs Tuckey’s method to test if the interaction exists. Check page 183 for details.

36. The General Factorial Design 36 In general, there will be abc…n total observations if there are n replicates of the complete experiment. There are a levels for factor A, b levels of factor B, c levels of factor C,..so on. We must have at least two replicate (n ≧ 2) to include all the possible interactions in model.

37. The General Factorial Design 37 If all the factors are fixed, we may easily formulate and test hypotheses about the main effects and interaction effects using ANOVA. For example, the three factor analysis of variance model:

38. The General Factorial Design 38 ANOVA.

39. The General Factorial Design 39 where

40. The General Factorial Design -- example 40 Three factors: pressure, percent of carbonation, and line speed.

41. The General Factorial Design -- example 41 ANOVA

42. Fitting Response Curve and Surfaces 42 When factors are quantitative, one can fit a response curve (surface) to the levels of the factor so the experimenter can relate the response to the factors. These surface could be linear or quadratic. Linear regression model is generally used

43. Fitting Response Curve and Surfaces -- example 43 Battery life data Factor temperature is quantitative

44. 44 Example STAT  ANOVA—GLM Response  life Model  temp, material temp*temp, material*temp, material*temp*temp Covariates  temp Fitting Response Curve and Surfaces -- example

45. 45 General Linear Model: Life versus Material Factor Type Levels Values Material fixed 3 1, 2, 3 Analysis of Variance for Life, using Sequential SS for Tests Source DF Seq SS Adj SS Seq MS F P Temp 1 39042.7 1239.2 39042.7 57.82 0.000 Material 2 10683.7 1147.9 5341.9 7.91 0.002 Temp*Temp 1 76.1 76.1 76.1 0.11 0.740 Material*Temp 2 2315.1 7170.7 1157.5 1.71 0.199 Material*Temp*Temp 2 7298.7 7298.7 3649.3 5.40 0.011 Error 27 18230.8 18230.8 675.2 Total 35 77647.0 S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56% Term Coef SE Coef T P Constant 153.92 11.87 12.96 0.000 Temp -0.5906 0.4360 -1.35 0.187 Temp*Temp -0.001019 0.003037 -0.34 0.740 Temp*Material 1 -1.9108 0.6166 -3.10 0.005 2 0.4173 0.6166 0.68 0.504 Temp*Temp*Material 1 0.013871 0.004295 3.23 0.003 2 -0.004642 0.004295 -1.08 0.289 Two kinds of coding methods: 1.1, 0, -1 2.0, 1, -1 coding method: -1, 0, +1 Fitting Response Curve and Surfaces -- example

46. 46 Final regression equation: Fitting Response Curve and Surfaces -- example

47. 47 Tool life Factors: cutting speed, total angle Data are coded Fitting Response Curve and Surfaces – example –3 2 factorial design

48. 48 Fitting Response Curve and Surfaces – example –3 2 factorial design

49. 49 Fitting Response Curve and Surfaces – example –3 2 factorial design Regression Analysis: Life versus Speed, Angle,... The regression equation is Life = - 1068 + 14.5 Speed + 136 Angle - 4.08 Angle*Angle - 0.0496 Speed*Speed - 1.86 Angle*Speed + 0.00640 Angle*Speed*Speed + 0.0560 Angle*Angle*Speed - 0.000192 Angle*Angle*Speed*Speed Predictor Coef SE Coef T P Constant -1068.0 702.2 -1.52 0.163 Speed 14.480 9.503 1.52 0.162 Angle 136.30 72.61 1.88 0.093 Angle*Angle -4.080 1.810 -2.25 0.051 Speed*Speed -0.04960 0.03164 -1.57 0.151 Angle*Speed -1.8640 0.9827 -1.90 0.090 Angle*Speed*Speed 0.006400 0.003272 1.96 0.082 Angle*Angle*Speed 0.05600 0.02450 2.29 0.048 Angle*Angle*Speed*Speed -0.00019200 0.00008158 -2.35 0.043 S = 1.20185 R-Sq = 89.5% R-Sq(adj) = 80.2%

50. 50 Fitting Response Curve and Surfaces – example –3 2 factorial design Analysis of Variance Source DF SS MS F P Regression 8 111.000 13.875 9.61 0.001 Residual Error 9 13.000 1.444 Total 17 124.000 Source DF Seq SS Speed 1 21.333 Angle 1 8.333 Angle*Angle 1 16.000 Speed*Speed 1 4.000 Angle*Speed 1 8.000 Angle*Speed*Speed 1 42.667 Angle*Angle*Speed 1 2.667 Angle*Angle*Speed*Speed 1 8.000

51. 51 Fitting Response Curve and Surfaces – example –3 2 factorial design

52. 52 We may have a nuisance factor presented in a factorial design Original two factor factorial model: Blocking in a Factorial Design Two factor factorial design with a block factor model:

53. 53 Blocking in a Factorial Design

54. 54 Blocking in a Factorial Design - - example Response: intensity level Factors: Ground cutter and filter type Block factor: Operator

55. 55 Blocking in a Factorial Design - - example General Linear Model: Intensity versus Clutter, Filter, Blocks Factor Type Levels Values Clutter fixed 3 High, Low, Medium Filter fixed 2 1, 2 Blocks fixed 4 1, 2, 3, 4 Analysis of Variance for Intensity, using Sequential SS for Tests Source DF Seq SS Adj SS Seq MS F P Clutter 2 335.58 335.58 167.79 15.13 0.000 Filter 1 1066.67 1066.67 1066.67 96.19 0.000 Clutter*Filter 2 77.08 77.08 38.54 3.48 0.058 Blocks 3 402.17 402.17 134.06 12.09 0.000 Error 15 166.33 166.33 11.09 Total 23 2047.83 S = 3.33000 R-Sq = 91.88% R-Sq(adj) = 87.55%

56. 56 Blocking in a Factorial Design - - example General Linear Model: Intensity versus Clutter, Filter, Blocks Term Coef SE Coef T P Constant 94.9167 0.6797 139.64 0.000 Clutter High 4.3333 0.9613 4.51 0.000 Low -4.7917 0.9613 -4.98 0.000 Filter 1 6.6667 0.6797 9.81 0.000 Clutter*Filter High 1 2.0833 0.9613 2.17 0.047 Low 1 -2.2917 0.9613 -2.38 0.031 Blocks 1 0.417 1.177 0.35 0.728 2 1.583 1.177 1.34 0.199 3 4.583 1.177 3.89 0.001