14-1 Introduction

4. 14-1 Introduction • An experiment is a test or series of tests. • The design of an experiment plays a major role in the eventual solution of the problem. • In a factorial experimental design, experimental trials (or runs) are performed at all combinations of the factor levels. • The analysis of variance (ANOVA) will be used as one of the primary tools for statistical data analysis.

5. 14-2 Factorial Experiments Definition

6. 14-2 Factorial Experiments Figure 14-3Factorial Experiment, no interaction.

7. 14-2 Factorial Experiments Figure 14-4Factorial Experiment, with interaction.

8. 14-2 Factorial Experiments Figure 14-5Three-dimensional surface plot of the data from Table 14-1, showing main effects of the two factors A and B.

9. 14-2 Factorial Experiments Figure 14-6Three-dimensional surface plot of the data from Table 14-2, showing main effects of the A and B interaction.

10. 14-2 Factorial Experiments Figure 14-7Yield versus reaction time with temperature constant at 155º F.

11. 14-2 Factorial Experiments Figure 14-8Yield versus temperature with reaction time constant at 1.7 hours.

12. 14-2 Factorial Experiments Figure 14-9Optimization experiment using the one-factor-at-a-time method.

13. 14-3 Two-Factor Factorial Experiments

14. 14-3 Two-Factor Factorial Experiments The observations may be described by the linear statistical model:

15. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

16. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

17. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

18. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model To test H0: i = 0 use the ratio To test H0: j = 0 use the ratio To test H0: ()ij = 0 use the ratio

19. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Definition

20. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

21. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

22. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

23. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

24. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

25. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

26. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

27. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1 Figure 14-10Graph of average adhesion force versus primer types for both application methods.

28. 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Minitab Output for Example 14-1

29. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking

30. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-11Normal probability plot of the residuals from Example 14-1

31. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-12Plot of residuals versus primer type.

32. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-13Plot of residuals versus application method.

33. 14-3 Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-14Plot of residuals versus predicted values.

34. 14-4 General Factorial Experiments Model for athree-factor factorial experiment

36. 14-4 General Factorial Experiments Example 14-2

37. Example 14-2

38. 14-4 General Factorial Experiments Example 14-2

39. 14-5 2k Factorial Designs 14-5.1 22 Design Figure 14-15The 22 factorial design.

40. 14-5 2k Factorial Designs 14-5.1 22 Design The main effect of a factor A is estimated by

41. 14-5 2k Factorial Designs 14-5.1 22 Design The main effect of a factor B is estimated by

42. 14-5 2k Factorial Designs 14-5.1 22 Design The AB interaction effect is estimated by

43. 14-5 2k Factorial Designs 14-5.1 22 Design The quantities in brackets in Equations 14-11, 14-12, and 14-13 are called contrasts. For example, the A contrast is ContrastA = a + ab – b – (1)

44. 14-5 2k Factorial Designs 14-5.1 22 Design Contrasts are used in calculating both the effect estimates and the sums of squares for A, B, and the AB interaction. The sums of squares formulas are

45. 14-5 2k Factorial Designs Example 14-3

46. 14-5 2k Factorial Designs Example 14-3

47. 14-5 2k Factorial Designs Example 14-3

48. 14-5 2k Factorial Designs Residual Analysis Figure 14-16Normal probability plot of residuals for the epitaxial process experiment.

49. 14-5 2k Factorial Designs Residual Analysis Figure 14-17Plot of residuals versus deposition time.

50. 14-5 2k Factorial Designs Residual Analysis Figure 14-18Plot of residuals versus arsenic flow rate.