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Design and Analysis of Experiments Lecture 2.1

Design and Analysis of Experiments Lecture 2.1. Review of Lecture 1.2 Randomised Block Design and Analysis Illustration Explaining ANOVA Interaction? Effect of Blocking Matched pairs as Randomised blocks Introduction to 2-level factorial designs A 2 2 experiment Set up Analysis

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Design and Analysis of Experiments Lecture 2.1

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  1. Design and Analysis of ExperimentsLecture 2.1 • Review of Lecture 1.2 • Randomised Block Design and Analysis • Illustration • Explaining ANOVA • Interaction? • Effect of Blocking • Matched pairs as Randomised blocks • Introduction to 2-level factorial designs • A 22 experiment • Set up • Analysis • Application 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. Was the blocking effective? Diploma in Statistics Design and Analysis of Experiments

  5. Comparing several means Membrane A: standard Membrane B: alternative using new material Membrane C: other manufacturer Membrane D: other manufacturer Burst strength (kPa) of 10 samples of each of four filter membrane types Diploma in Statistics Design and Analysis of Experiments

  6. Comparing several means Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Membrane Membrane = A subtracted from: Membrane Lower Center Upper ------+---------+---------+---------+- B -1.46 3.24 7.94 (---*----) C -12.91 -8.21 -3.51 (----*---) D -7.65 -2.95 1.75 (----*----) ------+---------+---------+---------+- -10 0 10 20 Membrane = B subtracted from: Membrane Lower Center Upper ------+---------+---------+---------+--- C -16.15 -11.45 -6.75 (----*---) D -10.89 -6.19 -1.49 (----*----) ------+---------+---------+---------+--- -10 0 10 20 Membrane = C subtracted from: Membrane Lower Center Upper ------+---------+---------+---------+--- D 0.560 5.260 9.960 (---*----) ------+---------+---------+---------+--- -10 0 10 20 Diploma in Statistics Design and Analysis of Experiments

  7. Comparing several means • Membrane B mean is significantly bigger than Membranes C and D means and close to significantly bigger than Membrane A mean. • Membrane C mean is significantly smaller than the other three means. • Membranes A and D means are not significantly different. Diploma in Statistics Design and Analysis of Experiments

  8. Comparing several means;Conclusions • Membrane C can be eliminated from our inquiries. • Membrane D shows no sign of being an improvement on the existing Membrane A and so need not be considered further. • Membrane B shows some improvement on Membrane A but not enough to recommend a change. • It may be worth while carrying out further comparisons between Membranes A and B. Diploma in Statistics Design and Analysis of Experiments

  9. Characteristics of an experiment Experimental units: entities on which observations are made Experimental Factor: controllable input variable Factor Levels / Treatments: values of the factor Response: output variable measured on the units Diploma in Statistics Design and Analysis of Experiments

  10. 2 Randomised blocksIllustration Manufacture of an organic chemical using a filtration process • Three step process: • input chemical blended from different stocks • chemical reaction results in end product suspended in an intermediate liquid product • liquid filtered to recover end product. Diploma in Statistics Design and Analysis of Experiments

  11. Randomised blocksIllustration • Problem: yield loss at filtration stage • Proposal: adjust initial blend to reduce yield loss • Plan: • prepare five different blends • use each blend in successive process runs, in random order • repeat at later times (blocks) Diploma in Statistics Design and Analysis of Experiments

  12. Results Diploma in Statistics Design and Analysis of Experiments

  13. Exercise 2.1.1 What were the experimental units factor factor levels response blocks randomisation procedure Diploma in Statistics Design and Analysis of Experiments

  14. Minitab AnalysisGeneral Linear Model ANOVA General Linear Model: Loss, per cent versus Blend, Block Analysis of Variance for Loss,%, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Blend 4 11.5560 11.5560 2.8890 3.31 0.071 Block 2 1.6480 1.6480 0.8240 0.94 0.429 Error 8 6.9920 6.9920 0.8740 Total 14 20.1960 S = 0.934880 R-Sq = 65.38% R-Sq(adj) = 39.41% Unusual Observations for Loss, per cent Loss, per Obs cent Fit SE Fit Residual St Resid 12 17.1000 18.5267 0.6386 -1.4267 -2.09 R Diploma in Statistics Design and Analysis of Experiments

  15. Diploma in Statistics Design and Analysis of Experiments

  16. Conclusions (prelim.) F(Blends) is almost statistically significant, p = 0.07 F(Blocks) is not statistically significant, p = 0.4 Prediction standard deviation: S = 0.93 Diploma in Statistics Design and Analysis of Experiments

  17. Deleted diagnostics Diploma in Statistics Design and Analysis of Experiments

  18. Iterated analysis:delete Case 12 General Linear Model: Loss versus Blend, Block Analysis of Variance for Loss Source DF Seq SS Adj SS Adj MS F P Blend 4 13.0552 14.5723 3.6431 8.03 0.009 Block 2 3.7577 3.7577 1.8788 4.14 0.065 Error 7 3.1757 3.1757 0.4537 Total 13 19.9886 S = 0.673548 Diploma in Statistics Design and Analysis of Experiments

  19. Deleted diagnostics Diploma in Statistics Design and Analysis of Experiments

  20. Conclusions (prelim.) F(Blends) is highly statistically significant, p = 0.01 F(Blocks) is not statistically significant, p = 0.65 Prediction standard deviation: S = 0.67 Diploma in Statistics Design and Analysis of Experiments

  21. Explaining ANOVA ANOVA depends on a decompostion of "Total variation" into components: Total Variation = Blend effect + Block effect + chance variation; Diploma in Statistics Design and Analysis of Experiments

  22. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  23. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  24. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  25. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  26. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  27. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  28. Decomposition of results Diploma in Statistics Design and Analysis of Experiments

  29. Interaction? Blend x Block interaction? Diploma in Statistics Design and Analysis of Experiments

  30. Interaction? Blend x Block interaction? Diploma in Statistics Design and Analysis of Experiments

  31. Exercise 2.1.2 Calculate fitted values: Overall Mean + Blend Deviation + Block deviation 17.5 + Diploma in Statistics Design and Analysis of Experiments

  32. Exercise 2.1.2 (cont'd) Make a Block profile plot Diploma in Statistics Design and Analysis of Experiments

  33. Fitted values; NO INTERACTION Diploma in Statistics Design and Analysis of Experiments

  34. Actual plot: Interaction? Blend effects (the contributions of each blend to Loss) are similar for Blocks 1 and 2 but quite different for Block 3. Diploma in Statistics Design and Analysis of Experiments

  35. Effect of Blocking Analysis of Variance for Loss (one run deleted) Source DF Seq SS Adj SS Adj MS F P Blend 4 13.0552 14.5723 3.6431 8.03 0.009 Block 2 3.7577 3.7577 1.8788 4.14 0.065 Error 7 3.1757 3.1757 0.4537 Total 13 19.9886 Analysis of Variance for Loss (one run deleted) unblocked Source DF Seq SS Adj SS Adj MS F P Blend 4 13.0552 13.0552 3.2638 4.24 0.034 Error 9 6.9333 6.9333 0.7704 Total 13 19.9886 Diploma in Statistics Design and Analysis of Experiments

  36. Matched pairs as Randomised blocks Wear of shoe soles made of two materials, A and B, worn on opposite feet by each of 10 boys Diploma in Statistics Design and Analysis of Experiments

  37. Pairing equals Blocking Paired T for Material B - Material A T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009 Two-way ANOVA: Wear versus Material, Boy Source DF SS MS F P Material 1 0.841 0.8405 11.21 0.009 Boy 9 110.491 12.2767 163.81 0.000 Error 9 0.675 0.0749 Total 19 112.006 Diploma in Statistics Design and Analysis of Experiments

  38. t and F Diploma in Statistics Design and Analysis of Experiments

  39. t and F Diploma in Statistics Design and Analysis of Experiments

  40. More on t Diploma in Statistics Design and Analysis of Experiments

  41. More on F Diploma in Statistics Design and Analysis of Experiments

  42. Paired Comparison:Effect of Pairing / Blocking Paired T for Material B - Material A T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009 Two-sample T for Material B vs Material A T-Value = 0.37 P-Value = 0.716 Diploma in Statistics Design and Analysis of Experiments

  43. Paired Comparison:Effect of Pairing / Blocking Two-way ANOVA: Wear versus Material, Boy Source DF SS MS F P Material 1 0.841 0.8405 11.21 0.009 Boy 9 110.491 12.2767 163.81 0.000 Error 9 0.675 0.0749 Total 19 112.006 One-way ANOVA: Wear versus Material Source DF SS MS F P Material 1 0.84 0.84 0.14 0.716 Error 18 111.17 6.18 Total 19 112.01 Diploma in Statistics Design and Analysis of Experiments

  44. 3 Introduction to 2-levelfactorial designs 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

  45. Exercise 2.1.3 What were the experimental units factors factor levels response blocks randomisation procedure Diploma in Statistics Design and Analysis of Experiments

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

  47. Set up:Randomisation Diploma in Statistics Design and Analysis of Experiments

  48. Set up:Run order Diploma in Statistics Design and Analysis of Experiments

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

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

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