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Design of Experiments with Several Factors

14. Design of Experiments with Several Factors. CHAPTER OUTLINE. 14-1 Introduction 14-2 Factorial Experiments 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical analysis of the fixed- effects model 14-3.2 Model adequacy checking 14-3.3 One observation per cell

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Design of Experiments with Several Factors

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  1. 14 Design of Experiments with Several Factors CHAPTER OUTLINE 14-1 Introduction 14-2 Factorial Experiments 14-3 Two-Factor Factorial Experiments 14-3.1 Statistical analysis of the fixed- effects model 14-3.2 Model adequacy checking 14-3.3 One observation per cell 14-4 General Factorial Experiments 14-5 2k Factorial Designs 14-5.1 2k design 14-5.2 2k design for k ≥3 factors 14-5.3 Single replicate of the 2k design 14-5.4 Addition of center points to a 2k design 14-6 Blocking & Confounding in the 2k design 14-7 Fractional Replication of the 2k Design 14-7.1 One-half fraction of the 2k design 14-7.2 Smaller fractions: The 2k-p fractional factorial 14-8 Response Surface Methods and Designs Chapter 14 Table of Contents

  2. 14-2: Factorial Experiments Definition

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

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

  5. 14-3: Two-Factor Factorial Experiments

  6. 14-3: Two-Factor Factorial Experiments The observations may be described by the linear statistical model:

  7. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

  8. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

  9. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

  10. 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

  11. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Definition

  12. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model

  13. 14-3: Two-Factor Factorial Experiments 14-3.1 Statistical Analysis of the Fixed-Effects Model Example 14-1

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

  15. 14-4: General Factorial Experiments Example 14-2

  16. Example 14-2

  17. 14-5: 2k Factorial Designs • Factorial designs are widely used in experiments involving several factors where it is necessary to study the joint effect of the factors on a response • The most important of these special cases is that of k factors, each at only two levels. • These levels may be • quantitative, such as two values of temperature, pressure, or time; • or they may be qualitative, such as two machines, two operators, the “high” and “low” levels of a factor, or perhaps the presence and absence of a factor. • A complete replicate of such a design requires 2 × 2 × · · · × 2 = 2k observations and is called a 𝟐k factorial design.

  18. Analysis Procedure for a Factorial 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

  19. 14-5: 2k Factorial Designs 14-5.1 22 Design Figure 14-15The 22 factorial design.

  20. Example • As an example, consider an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the conversion (yield) in a chemical process. • The objective of the experiment was to determine if adjustments to either of these two factors would increase the yield. • Let the reactant concentration be factor A and let the two levels of interest be 15 and 25 percent. • The catalyst is factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound. • The experiment is replicated three times, so there are 12 runs. The order in which the runs are made is random, so this is a completely randomized experiment.

  21. Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery

  22. The Simplest Case: The 22 “-” 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

  23. 14-5: 2k Factorial Designs 14-5.1 22 Design The main effect of a factor A is estimated by The main effect of a factor B is estimated by The AB interaction effect is estimated by

  24. Chemical Process Example • The effect of A (reactant concentration) is positive; this suggests that increasing A from the low level (15%) to the high level (25%) will increase the yield. • The effect of B (catalyst) is negative; this suggests that increasing the amount of catalyst added to the process will decrease the yield. • The interaction effect appears to be small relative to the two main effects.

  25. 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) It is often convenient to write down the treatment combinations in the order (1), a, b, ab. This is referred to as standard order (or Yates’s order, for Frank Yates

  26. 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

  27. Statistical Testing - ANOVA The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important? Design & Analysis of Experiments 7E 2009 Montgomery

  28. Example • It is important to study the effect of the concentration of the reactant and the feed rate on the viscosity of the product from a chemical process. Let the reactant concentration be factor A at levels 15 % and 25 %. Let the feed rate be factor B, with levels of 20 lb/hr and 30 lb/hr. The experiment involves two experimental runs at each of the four combinations (L= low and H= high). The viscosity readings are as follows. • Assuming a model containing two main effects and interaction, calculate the htree effects. • Sketch the interaction plot. • Do ANOVA and test for interaction. Give conclusion • Test for main effects and give final conclusion regarding the importance of all these effects.

  29. 14-5: 2k Factorial Designs

  30. 14-5: 2k Factorial Designs Example 14-3

  31. 14-5: 2k Factorial Designs Residual Analysis Figure 14-16Normal probability plot of residuals for the epitaxial process experiment.

  32. 14-5: 2k Factorial Designs Residual Analysis Figure 14-17Plot of residuals versus deposition time.

  33. 14-5: 2k Factorial Designs Residual Analysis Figure 14-18Plot of residuals versus arsenic flow rate.

  34. 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.

  35. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Figure 14-20The 23 design.

  36. 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.

  37. 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

  38. 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

  39. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Other two-factor interactions effects estimated by The three-factor interaction effect, ABC, is estimated by

  40. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors

  41. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors

  42. 14-5: 2k Factorial Designs 14-5.2 2k Design for k  3 Factors Contrasts can be used to calculate several quantities:

  43. 14-5: 2k Factorial Designs Example 14-4

  44. 14-5: 2k Factorial Designs Example 14-4

  45. 14-5: 2k Factorial Designs Example 14-4

  46. 14-5: 2k Factorial Designs Example 14-4

  47. 14-5: 2k Factorial Designs Example 14-4

  48. Example 14-4

  49. 14-5: 2k Factorial Designs Residual Analysis Figure 14-22Normal probability plot of residuals from the surface roughness experiment.

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