1. Diploma in Statistics Design and Analysis of Experiments Lecture 2.11 Design and Analysis of Experiments Lecture 2.1 1.Review of Lecture 1.2 2.Randomised Block Design and Analysis –Illustration –Explaining ANOVA –Interaction? –Effect of Blocking –Matched pairs as Randomised blocks 3.Introduction to 2-level factorial designs –A 2 2 experiment –Set up –Analysis –Application

2. Diploma in Statistics Design and Analysis of Experiments Lecture 2.12 Minute Test - How Much

3. Diploma in Statistics Design and Analysis of Experiments Lecture 2.13 Minute Test - How Fast

4. Diploma in Statistics Design and Analysis of Experiments Lecture 2.14 Was the blocking effective?

5. Diploma in Statistics Design and Analysis of Experiments Lecture 2.15 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

6. Diploma in Statistics Design and Analysis of Experiments Lecture 2.16 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

7. Diploma in Statistics Design and Analysis of Experiments Lecture 2.17 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.

8. Diploma in Statistics Design and Analysis of Experiments Lecture 2.18 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.

9. Diploma in Statistics Design and Analysis of Experiments Lecture 2.19 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

10. Diploma in Statistics Design and Analysis of Experiments Lecture 2.110 2Randomised blocks Illustration 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.

11. Diploma in Statistics Design and Analysis of Experiments Lecture 2.111 Randomised blocks Illustration 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)

12. Diploma in Statistics Design and Analysis of Experiments Lecture 2.112 Results

13. Diploma in Statistics Design and Analysis of Experiments Lecture 2.113 Exercise 2.1.1 What were the experimental units factor factor levels response blocks randomisation procedure

14. Diploma in Statistics Design and Analysis of Experiments Lecture 2.114 Minitab Analysis General 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

15. Diploma in Statistics Design and Analysis of Experiments Lecture 2.115

16. Diploma in Statistics Design and Analysis of Experiments Lecture 2.116 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

17. Diploma in Statistics Design and Analysis of Experiments Lecture 2.117 Deleted diagnostics

18. Diploma in Statistics Design and Analysis of Experiments Lecture 2.118 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

19. Diploma in Statistics Design and Analysis of Experiments Lecture 2.119 Deleted diagnostics

20. Diploma in Statistics Design and Analysis of Experiments Lecture 2.120 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

21. Diploma in Statistics Design and Analysis of Experiments Lecture 2.121 Explaining ANOVA ANOVA depends on a decompostion of "Total variation" into components: Total Variation = Blend effect + Block effect + chance variation;

22. Diploma in Statistics Design and Analysis of Experiments Lecture 2.122 Decomposition of results

23. Diploma in Statistics Design and Analysis of Experiments Lecture 2.123 Decomposition of results

24. Diploma in Statistics Design and Analysis of Experiments Lecture 2.124 Decomposition of results

25. Diploma in Statistics Design and Analysis of Experiments Lecture 2.125 Decomposition of results

26. Diploma in Statistics Design and Analysis of Experiments Lecture 2.126 Decomposition of results

27. Diploma in Statistics Design and Analysis of Experiments Lecture 2.127 Decomposition of results

28. Diploma in Statistics Design and Analysis of Experiments Lecture 2.128 Decomposition of results

29. Diploma in Statistics Design and Analysis of Experiments Lecture 2.129 Interaction? Blend x Block interaction?

30. Diploma in Statistics Design and Analysis of Experiments Lecture 2.130 Interaction? Blend x Block interaction?

31. Diploma in Statistics Design and Analysis of Experiments Lecture 2.131 Exercise 2.1.2 Calculate fitted values: Overall Mean + Blend Deviation + Block deviation 17.5 +

32. Diploma in Statistics Design and Analysis of Experiments Lecture 2.132 Exercise 2.1.2 (cont'd) Make a Block profile plot

33. Diploma in Statistics Design and Analysis of Experiments Lecture 2.133 Fitted values; NO INTERACTION

34. Diploma in Statistics Design and Analysis of Experiments Lecture 2.134 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.

35. Diploma in Statistics Design and Analysis of Experiments Lecture 2.135 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

36. Diploma in Statistics Design and Analysis of Experiments Lecture 2.136 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

37. Diploma in Statistics Design and Analysis of Experiments Lecture 2.137 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

38. Diploma in Statistics Design and Analysis of Experiments Lecture 2.138 t and F

39. Diploma in Statistics Design and Analysis of Experiments Lecture 2.139 t and F

40. Diploma in Statistics Design and Analysis of Experiments Lecture 2.140 More on t

41. Diploma in Statistics Design and Analysis of Experiments Lecture 2.141 More on F

42. Diploma in Statistics Design and Analysis of Experiments Lecture 2.142 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

43. Diploma in Statistics Design and Analysis of Experiments Lecture 2.143 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

44. Diploma in Statistics Design and Analysis of Experiments Lecture 2.144 3Introduction to 2-level factorial designs A 2 2 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.

45. Diploma in Statistics Design and Analysis of Experiments Lecture 2.145 Exercise 2.1.3 What were the experimental units factors factor levels response blocks randomisation procedure

46. Diploma in Statistics Design and Analysis of Experiments Lecture 2.146 Set up

47. Diploma in Statistics Design and Analysis of Experiments Lecture 2.147 Set up: Randomisation

48. Diploma in Statistics Design and Analysis of Experiments Lecture 2.148 Set up: Run order

49. Diploma in Statistics Design and Analysis of Experiments Lecture 2.149 Results (run order)

50. Diploma in Statistics Design and Analysis of Experiments Lecture 2.150 Results (standard order)

51. Diploma in Statistics Design and Analysis of Experiments Lecture 2.151 Analysis (Minitab) Main effects and Interaction plots Pareto plot of effects ANOVA results –with diagnostics Calculation of t-statistic

52. Diploma in Statistics Design and Analysis of Experiments Lecture 2.152 Main Effects and Interactions

53. Diploma in Statistics Design and Analysis of Experiments Lecture 2.153 Pareto plot of effects Bar height = t value (see slide 31) Reference line is at critical t value (4 df)

54. Diploma in Statistics Design and Analysis of Experiments Lecture 2.154 Minitab DOE Analyze 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

55. Diploma in Statistics Design and Analysis of Experiments Lecture 2.155

56. Diploma in Statistics Design and Analysis of Experiments Lecture 2.156 Minitab DOE Analyze 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

57. Diploma in Statistics Design and Analysis of Experiments Lecture 2.157 ANOVA results ANOVA superfluous for 2 k experiments "There is nothing to justify this complexity other than a misplaced belief in the universal value of an ANOVA table". BHH (2nd ed.), Section 5.10 "a convenient method of arranging the arithmetic" R.A. Fisher

58. Diploma in Statistics Design and Analysis of Experiments Lecture 2.158 Diagnostic Plots

59. Diploma in Statistics Design and Analysis of Experiments Lecture 2.159 Calculation of t-statistic Results (Temperature order)

60. Diploma in Statistics Design and Analysis of Experiments Lecture 2.160 Exercise 2.1.4 Calculate a confidence interval for the Temperature effect. All effects may be estimated and tested in this way. Homework 2.1.2 Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the Temperature × Catalyst interaction.

61. Diploma in Statistics Design and Analysis of Experiments Lecture 2.161 Application Finding the optimum More Minitab results Least Squares Means for Yield Mean SE Mean Temperature Low 52.75 1.854 High 75.75 1.854 Catalyst 1 63.50 1.854 2 65.00 1.854 Temperature*Catalyst Low 1 57.00 2.622 High 1 70.00 2.622 Low 2 48.50 2.622 High 2 81.50 2.622

62. Diploma in Statistics Design and Analysis of Experiments Lecture 2.162 13.0 33.0  8.5 11.5

63. Diploma in Statistics Design and Analysis of Experiments Lecture 2.163 Optimum operating conditions Highest yield achieved with Catalyst 2 at High temperature. Estimated yield: 81.5% 95% confidence interval: 81.5 ± 2.78 × 2.622, i.e., 81.5 ± 7.3, i.e., ( 74.2, 88.8 )

64. Diploma in Statistics Design and Analysis of Experiments Lecture 2.164 Homework 2.1.3 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 2 2 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.

65. Diploma in Statistics Design and Analysis of Experiments Lecture 2.165 Peak areas for GC study (EM, Exercise 5.2)

66. Diploma in Statistics Design and Analysis of Experiments Lecture 2.166 Reading EM §5.3, §7.4.2 DCM §§4-1, 5-1, 5-2, 6-1, 6-2