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ENM 310 Design of Experiments and Regression Analysis Chapter 3

ENM 310 Design of Experiments and Regression Analysis Chapter 3. Ilgın ACAR Spring 2019. 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.

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ENM 310 Design of Experiments and Regression Analysis Chapter 3

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  1. ENM 310Design of Experiments and Regression AnalysisChapter 3 Ilgın ACAR Spring 2019

  2. 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.

  3. Factorial Experiments Definition

  4. Factorial Experiments Figure 14-3Factorial Experiment, no interaction.

  5. Factorial Experiments Figure 14-4Factorial Experiment, with interaction.

  6. 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.

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

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

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

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

  11. Two-Factor Factorial Experiments

  12. Two-Factor Factorial Experiments The observations may be described by the linear statistical model: • where • μ is the overall mean effect, • τi is the effect of the ith level of factor A, • β j is the effect of the jth level of factor B, • (τβ)ij is the effect of the interaction between A and B, and • eijk is a random error component having a normal distribution with mean 0 and variance σ2.

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

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

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

  16. 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

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

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

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

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

  21. 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.

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

  23. Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking

  24. Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking Figure 14-11Normal probability plot of the residuals from Example 14-1 This plot has tails that do not fall exactly along a straight line passing through the center of the plot, indicating some potential problems with the normality assumption, but the deviation from normality does not appear severe.

  25. Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking There is some indication that primer type 3 results in slightly lower variability in adhesion force than the other two primers. Figure 14-12Plot of residuals versus primer type.

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

  27. Two-Factor Factorial Experiments 14-3.2 Model Adequacy Checking The graph of residuals versus fitted values in does not reveal any unusual or diagnostic pattern. Figure 14-14Plot of residuals versus predicted values.

  28. Example • As an example of a factorial design involving two factors, an engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select at this point is the plate material for the battery, and he has three possible choices. When the device is manufactured and is shipped to the field, the engineer has no control over the temperature extremes that the device will encounter, and he knows from experience that temperature will probably affect the effective battery life. However, temperature can be controlled in the product development laboratory for the purposes of a test.

  29. 14-4: General Factorial Experiments Model for athree-factor factorial experiment

  30. 14-4: General Factorial Experiments Example 14-2

  31. Example 14-2

  32. 14-4: General Factorial Experiments Example 14-2

  33. 14-5: 2k Factorial Designs 14-5.1 22 Design Figure 14-15The 22 factorial design.

  34. 14-5: 2k Factorial Designs 14-5.1 22 Design The main effect of a factor A is estimated by

  35. 14-5: 2k Factorial Designs 14-5.1 22 Design The main effect of a factor B is estimated by

  36. 14-5: 2k Factorial Designs 14-5.1 22 Design The AB interaction effect is estimated by

  37. 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)

  38. 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

  39. 14-5: 2k Factorial Designs Example 14-3

  40. 14-5: 2k Factorial Designs Example 14-3

  41. 14-5: 2k Factorial Designs Example 14-3

  42. 14-5: 2k Factorial Designs Residual Analysis Figure 14-16Normal probability plot of residuals for the epitaxial process experiment.

  43. 14-5: 2k Factorial Designs Residual Analysis Figure 14-17Plot of residuals versus deposition time.

  44. 14-5: 2k Factorial Designs Residual Analysis Figure 14-18Plot of residuals versus arsenic flow rate.

  45. 14-5: 2k Factorial Designs Residual Analysis Figure 14-19 The standard deviation of epitaxial layer thickness at the four runs in the 22 design.

  46. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Figure 14-20The 23 design.

  47. Figure 14-21Geometric presentation of contrasts corresponding to the main effects and interaction in the 23 design. (a) Main effects. (b) Two-factor interactions. (c) Three-factor interaction.

  48. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors The main effect of A is estimated by The main effect of B is estimated by

  49. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors The main effect of C is estimated by The interaction effect of AB is estimated by

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