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Design and Analysis of Experiments Lecture 2.2

Design and Analysis of Experiments Lecture 2.2. Review of Lecture 2.1 and Laboratory 1 Homework 2.1.2 Introducing the Design Matrix A 2 3 experiment 3 factors each at 2 levels. Minute Test: How Much. Minute Test: How Fast. Yield Loss Experiment: Blends in Randomised Blocks.

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Design and Analysis of Experiments Lecture 2.2

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  1. Design and Analysis of ExperimentsLecture 2.2 • Review of Lecture 2.1 and Laboratory 1 • Homework 2.1.2 • Introducing the Design Matrix • A 23 experiment • 3 factors each at 2 levels Diploma in Statistics Design and Analysis of Experiments

  2. Minute Test: How Much Diploma in Statistics Design and Analysis of Experiments

  3. Minute Test: How Fast Diploma in Statistics Design and Analysis of Experiments

  4. Yield Loss Experiment:Blends in Randomised Blocks General Linear Model: Loss, per cent versus Blend, Block Analysis of Variance for Loss,%, Source DF SS MS F P Blend 4 11.5560 2.8890 3.31 0.071 Block 2 1.6480 0.8240 0.94 0.429 Error 8 6.9920 0.8740 Total 14 Diploma in Statistics Design and Analysis of Experiments

  5. Decomposition of results Effect –0.5 0.7 0.0 –1.2 1.3 Effect –0.4 0.1 0.4 Diploma in Statistics Design and Analysis of Experiments

  6. Interaction between FactorsCase study: Emotional Arousal Diploma in Statistics Design and Analysis of Experiments

  7. Interaction between Factors:main effects of picturesvsgender differentiated effects Diploma in Statistics Design and Analysis of Experiments

  8. Yield loss experiment Diploma in Statistics Design and Analysis of Experiments

  9. Yield loss experiment Diploma in Statistics Design and Analysis of Experiments

  10. Yield loss experiment Diploma in Statistics Design and Analysis of Experiments

  11. Laboratory 1Soybean seed failure rates Diploma in Statistics Design and Analysis of Experiments

  12. A 22 experiment Project: optimisation of a chemical process yield Factors (with levels): operating temperature (Low, High) catalyst (C1, C2) Design: Process run at all four possible combinations of factor levels, in duplicate, in random order. Diploma in Statistics Design and Analysis of Experiments

  13. Set up Diploma in Statistics Design and Analysis of Experiments

  14. Randomisation Go to Excel Diploma in Statistics Design and Analysis of Experiments

  15. Set up:Run order NB: Reset factor levels each time Diploma in Statistics Design and Analysis of Experiments

  16. Results (run order) Diploma in Statistics Design and Analysis of Experiments

  17. Results (standard order) Diploma in Statistics Design and Analysis of Experiments

  18. Analysis (Minitab) • Main effects and Interaction plots • Pareto plot of effects • ANOVA results • with diagnostics • Calculation of t-statistic Diploma in Statistics Design and Analysis of Experiments

  19. Main Effects and Interactions Diploma in Statistics Design and Analysis of Experiments

  20. Bar height = t value (see next slide) Reference line is at critical t value (4 df) df = 7 – 3 = 4 Diploma in Statistics Design and Analysis of Experiments

  21. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 Coef = Effect / 2 SE(Effect) = SE(Coef) x 2 Diploma in Statistics Design and Analysis of Experiments

  22. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1062.50 1062.50 531.25 38.64 0.002 2-Way Interactions 1 200.00 200.00 200.00 14.55 0.019 Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75 Total 7 1317.50 Diploma in Statistics Design and Analysis of Experiments

  23. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1062.50 1062.50 531.25 38.64 0.002 2-Way Interactions 1 200.00 200.00 200.00 14.55 0.019 Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75 Total 7 1317.50 Diploma in Statistics Design and Analysis of Experiments

  24. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1062.50 1062.50 531.25 38.64 0.002 2-Way Interactions 1 200.00 200.00 200.00 14.55 0.019 Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75 Total 7 1317.50 Diploma in Statistics Design and Analysis of Experiments

  25. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1062.50 1062.50 531.25 38.64 0.002 2-Way Interactions 1 200.00 200.00 200.00 14.55 0.019 Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75 Total 7 1317.50 Diploma in Statistics Design and Analysis of Experiments

  26. Minitab DOEAnalyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant 64.2500 1.311 49.01 0.000 Temperature 23.0000 11.5000 1.311 8.77 0.001 Catalyst 1.5000 0.7500 1.311 0.57 0.598 Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019 S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 1062.50 1062.50 531.25 38.64 0.002 2-Way Interactions 1 200.00 200.00 200.00 14.55 0.019 Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75 Total 7 1317.50 Diploma in Statistics Design and Analysis of Experiments

  27. Direct Calculation Diploma in Statistics Design and Analysis of Experiments

  28. Homework 2.1.2 As part of a project to develop a GC method for analysing trace compounds in wine without the need for prior extraction of the compounds, a synthetic mixture of aroma compounds in ethanol-water was prepared. The effects of two factors, Injection volume and Solvent flow rate, on GC measured peak areas given by the mixture were assessed using a 22 factorial design with 3 replicate measurements at each design point. The results are shown in the table that follows. What conclusions can be drawn from these data? Display results numerically and graphically. Check model assumptions by using appropriate residual plots. Diploma in Statistics Design and Analysis of Experiments

  29. Measurements for GC study (EM, Exercise 5.1, pp. 199-200) Diploma in Statistics Design and Analysis of Experiments

  30. Steps in analysis • Produce main effects plots, interaction plot, • Calculate main effects and interaction effect • Calculate standard error of effects • Calculate t-tests • Produce diagnostic plots • Iterate? Diploma in Statistics Design and Analysis of Experiments

  31. Organising the data for direct analysis Diploma in Statistics Design and Analysis of Experiments

  32. Organising the data for direct analysis s2 = average(SD2) = ( 2.302 + 4.012 + 4.922 + 3.692 ) / 4 = 14.798 s = 3.85 df(s) = sum[df(SD)] = 2 + 2 + 2 + 2 = 8 Diploma in Statistics Design and Analysis of Experiments

  33. Minitab results Estimated Effects for Measurements Term Effect SE T P Flow rate -19.233 2.222 -8.66 0.000 Volume 98.233 2.222 44.21 0.000 Flow rate*Volume 8.767 2.222 3.95 0.004 S = 3.84816 Diploma in Statistics Design and Analysis of Experiments

  34. Minitab results Diploma in Statistics Design and Analysis of Experiments

  35. Minitab results Diploma in Statistics Design and Analysis of Experiments

  36. Minitab results Diploma in Statistics Design and Analysis of Experiments

  37. More Minitab results Means for Peak area Mean SE Mean Flow rate 200 88.10 1.571 400 68.87 1.571 Volume 100 29.37 1.571 200 127.60 1.571 Flow rate*Volume 200 100 43.37 2.222 400 100 15.37 2.222 200 200 132.83 2.222 400 200 122.37 2.222 Diploma in Statistics Design and Analysis of Experiments

  38. More calculations • Calculate confidence intervals for Flow Rate effects at Low and High Volumes. • Calculate confidence intervals for Volume effects at Low and High Flow Rates . Diploma in Statistics Design and Analysis of Experiments

  39. Minitab results; diagnostics Diploma in Statistics Design and Analysis of Experiments

  40. Minitab results; diagnostics Diploma in Statistics Design and Analysis of Experiments

  41. Part 4 Introducing the design matrix Organising the data for calculation Generic notation Diploma in Statistics Design and Analysis of Experiments

  42. The design matrix • The design matrix displays the range of experimental conditions under which the process is to be run. • Each row (design point) designates a set of experimental conditions. • With 2 factors each at 2 possible levels, there are 22 = 4 sets of experimental conditions, as listed. Diploma in Statistics Design and Analysis of Experiments

  43. Main effect of A: average at high A – average at low A = = Main effect of B: average at high B – average at low B = = Organising the calculations Columns of design matrix applied to column of means. Diploma in Statistics Design and Analysis of Experiments

  44. Dual role of the design matrix • Prior to the experiment, the rows designate the design points, the sets of conditions under which the process is to be run. • After the experiment, the columns designate the contrasts, the combinations of design point means which measure the main effects of the factors. • The extended design matrix facilitates the calculation of interaction effects Diploma in Statistics Design and Analysis of Experiments

  45. Calculating interaction effects AB Interaction = ½(A effect at high B – A effect at low B) = = The extended design matrix Check: AB = A × B Diploma in Statistics Design and Analysis of Experiments

  46. Part 4 A 23 experiment:3 factors each at 2 levels An experiment to investigate the effects on yield of a chemical process of changes to operating Temperature, raw material Concentration and type of Catalyst was conducted in a pilot plant set up for experimentation. Details were as follows. Factor settings and codes Diploma in Statistics Design and Analysis of Experiments

  47. Design matrix (standard order) Run order for design points (in duplicate) A three factor example Diploma in Statistics Design and Analysis of Experiments

  48. A three factor example Results, in standard order Ref: PilotPlant.xls Diploma in Statistics Design and Analysis of Experiments

  49. Minitab analysis Estimated Effects for Yield Term Effect SE T P T 23.0 1.414 16.26 0.000 C -5.0 1.414 -3.54 0.008 K 1.5 1.414 1.06 0.320 T*C 1.5 1.414 1.06 0.320 T*K 10.0 1.414 7.07 0.000 C*K 0.0 1.414 0.00 1.000 T*C*K 0.5 1.414 0.35 0.733 S = 2.82843 Diploma in Statistics Design and Analysis of Experiments

  50. Exercise 2.2.2 Calculate the T, C and K main effects Diploma in Statistics Design and Analysis of Experiments

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