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

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

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.

5. Basic Definitions and Principles • Factorial Design without interaction

6. Basic Definitions and Principles • Factorial Design with interaction

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. Basic Definitions and Principles • With Interaction:

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

11. Basic Definitions and Principles • The case with interaction

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. 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. The Two-factor Factorial Design – An example • Two factors, each with three levels and four replicates • 32 factorial design

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

17. The Two-factor Factorial Design • Hypothesis • Row effects: • Column effects: • Interaction:

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

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

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

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

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

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

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

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

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

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

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

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

31. The Two-factor Factorial Design -- Statistical Analysis • Appendix Chart V • For n=4, giving D=40 on temperature, v1=2, v2=27, Φ 2 =1.28n. β =0.06

32. The Two-factor Factorial Design -- Statistical Analysis – example with no interaction 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 • Single replicate • The effect model

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

35. The Two-factor Factorial Design -- One observation per cell • 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 • 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 • 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 • ANOVA.

39. The General Factorial Design • where

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

41. The General Factorial Design --example • ANOVA

42. Fitting Response Curve and Surfaces • 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 • Battery life data • Factor temperature is quantitative

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

45. Fitting Response Curve and Surfaces -- example coding method: -1, 0, +1 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, 0, -1 0, 1, -1

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

47. Fitting Response Curve and Surfaces – example –32 factorial design • Tool life • Factors: cutting speed, total angle • Data are coded

48. Fitting Response Curve and Surfaces – example –32 factorial design

49. Fitting Response Curve and Surfaces – example –32 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. Fitting Response Curve and Surfaces – example –32 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 SeqSS 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