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Design and Analysis of Engineering Experiments: 4. Introduction to Factorial Design

ITK-226 Statistika & Rancangan Percobaan. Design and Analysis of Engineering Experiments: 4. Introduction to Factorial Design. Dicky Dermawan www.dickydermawan.net78.net dickydermawan@gmail.com. Design of Engineering Experiments Part 1 – Introduction Chapter 1, Text.

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Design and Analysis of Engineering Experiments: 4. Introduction to Factorial Design

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  1. ITK-226 Statistika & RancanganPercobaan Design and Analysis of Engineering Experiments:4. Introduction to Factorial Design DickyDermawan www.dickydermawan.net78.net dickydermawan@gmail.com

  2. Design of Engineering ExperimentsPart 1 – IntroductionChapter 1, Text • Why is this trip necessary? Goals of the course • An abbreviated history of DOX • Some basic principles and terminology • The strategy of experimentation • Guidelines for planning, conducting and analyzing experiments DOX 6E Montgomery

  3. Strategy of Experimentation • “Best-guess” experiments • Used a lot • More successful than you might suspect, but there are disadvantages… • One-factor-at-a-time (OFAT) experiments • Sometimes associated with the “scientific” or “engineering” method • Devastated by interaction, also very inefficient • Statistically designedexperiments • Based on Fisher’s factorial concept DOX 6E Montgomery

  4. One-factor-at-a-time (OFAT) experiments

  5. Planning, Conducting & Analyzing an Experiment • Recognition of & statement of problem • Choice of factors, levels, and ranges • Selection of the response variable(s) • Choice of design • Conducting the experiment • Statistical analysis • Drawing conclusions, recommendations DOX 6E Montgomery

  6. Planning, Conducting & Analyzing an Experiment • Get statisticalthinking involved early • Your non-statistical knowledge is crucial to success • Pre-experimental planning (steps 1-3) vital • Think and experimentsequentially • See Coleman & Montgomery (1993) Technometrics paper + supplemental text material DOX 6E Montgomery

  7. Design of Engineering ExperimentsPart 4 – Introduction to Factorials • Text reference, Chapter 5-8 • General principles of factorial experiments • The two-factor factorial with fixed effects • The ANOVA for factorials • Extensions to more than two factors • Quantitative and qualitative factors – response curves and surfaces DOX 6E Montgomery

  8. Some Basic Definitions Definition of a factor effect: The change in the mean response when the factor is changed from low to high DOX 6E Montgomery

  9. The Case of Interaction: DOX 6E Montgomery

  10. Regression Model & The Associated Response Surface DOX 6E Montgomery

  11. The Effect of Interaction on the Response Surface Suppose that we add an interaction term to the model: Interaction is actually a form of curvature DOX 6E Montgomery

  12. Example 5-1 The Battery Life ExperimentText reference pg. 165 • A = Material type; B = Temperature (A quantitative variable) • What effects do material type & temperature have on life? • 2. Is there a choice of material that would give long life regardless of temperature (a robust product)? DOX 6E Montgomery

  13. The General Two-Factor Factorial Experiment a levels of factor A; b levels of factor B; n replicates This is a completely randomized design DOX 6E Montgomery

  14. Statistical (effects) model: Other models (means model, regression models) can be useful DOX 6E Montgomery

  15. Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177 DOX 6E Montgomery

  16. ANOVA Table – Fixed Effects Case Design-Expert will perform the computations Text gives details of manual computing (ugh!) – see pp. 169 & 170 DOX 6E Montgomery

  17. Design-Expert Output – Example 5-1 DOX 6E Montgomery

  18. Residual Analysis – Example 5-1 DOX 6E Montgomery

  19. Residual Analysis – Example 5-1 DOX 6E Montgomery

  20. Interaction Plot DOX 6E Montgomery

  21. Exercise

  22. The 2k Factorial Design 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 cases is that of k factors, each at only 2 levels. A complete replicate of such a design requires 2 x 2 x 2 x … x 2 = 2k observations. This extremely important class of design is particularly useful in the early stages of experimental works when there are likely to be many factors to be investigated. It provides the smallest number of runs with which k factors can be studied in a complete factorial design. Consequently, 2k factorial designs are widely used in factor screening experiments.

  23. The 22 Factorial Design Effect of Reactant Concentration and Catalyst Amount on The Conversion in A Chemical Process A = Concentration “-” = 15% “+” = 25% B = Catalyst “-” = 1 lb “+” = 2 lb

  24. The 22 Factorial Design: An Example • Anova & Effect of Factor Effect of Factors: -Main effect : effect of factor A; effect of factor B - Interaction effect : AB

  25. The 22 Factorial Design: An Example • Effect of a main factor is the change in response produce by a change in the level of that factor, averaged over the levels of the other factors. • Interaction effect AB is the average difference between the effect of A at the high level B and the effect of A at low level of B Magnitude & direction of effect of factors: -Main effect: A = {½ [(36-28)+(31-18)] + ½ [(32-25)+(30-19)] + ½ [(32-27)+(29-23)]}/3 = 8,33 B = {½ [(18-28)+(31-36)] + ½ [(19-25)+(30-32)] + ½ [(23-27)+(29-32)]}/3 = -5,00 - Interaction : AB= {½ [(31-18) - (36-28)]+ ½ [(30-19) - (32-25)] + ½ [29-23)-(32-27)] }/3 = 1.67 A = Concentration “-” = 15% “+” = 25% B = Catalyst “-” = 1 lb “+” = 2 lb

  26. Regression Model, Surface Response & Contour Plot • Suppose we conclude that interaction AB is not significant. The regression model is: • x1 is coded variable representing natural variable A, i.e. reactant concentration • x2 is coded variable representing natural variable B, i.e. catalyst amount • The fitted regression model is:

  27. Residual & Model Adequacy • Error calculation: • Model adequacy checking: • Normal Probability Plot of Residual • Residual vs Predicted Conversion

  28. Exercise

  29. Analysis Procedure for 2k Factorial Design • Estimate factor effect • Form initial model • Perform statistical testing • Refine model • Analyze residual • Interpret results

  30. The 23 Factorial Design Effect of Percentage of Carbonation, Operating Pressure and Line Speed on Uniformity of Filling Height of A Soft Drink Bottler

  31. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment • Estimate factor effect • Form initial model • Perform statistical testing • Refine model • Analyze residual • Interpret results

  32. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Contrast Constant

  33. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Effect estimate

  34. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Normal Probability Plot of Effect

  35. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Significant Effect

  36. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Error Checking

  37. The Unreplicated 24 Factorial Design Pilot Plan Filtration Rate Experiment: Interpretation – Surface Response

  38. The Unreplicated 24 Factorial DesignExercise

  39. The Unreplicated 24 Factorial DesignExercise (cont’)

  40. Fractional Factorial Design: A 25-1 Design • Motivation for fractional factorials is obvious; as the number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly • Emphasis is on factorscreening; efficiently identify the factors with large effects • There may be many variables (often because we don’t know much about the system) Construction of a One-half Fraction Basic design Defining Relation = Design Generator Principal & The Alternate Fraction of the 25-1 Confounding & Aliasing

  41. Basic Design: Full 25 Design A fractional25-1 Design is a half fraction of Full 25 factorial design

  42. Defining Relation = Design Generator Principal Fraction of the 25-1 The Alternate Fraction

  43. Confounding & Aliasing Since E = ABCD: • Effect of E & effect of ABCD are indistinguishable or • ABCDE = EE = E2 = I, thus I= ABCDE  A = AI = A2BCDE = BCDE, or also:  AB = ABI = A2B2CDE = CDE, thus

  44. Data Analysis • Estimate factor effect • Form initial model • Perform statistical testing • Refine model • Analyze residual • Interpret results

  45. Estimates of Factor Effect & Initial Model

  46. Normal Probability Plot of Effects

  47. Perform Statistical Testing: Analysis of Variance Table

  48. Refine Model Small Apperture Setting: Large Apperture Setting:

  49. Analysis of Residual:Normal Plot of Residual

  50. Analysis of Residual:Plot of Residual vs Predicted Yield

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